199 20 5666 https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/2850e668c649d47f29fbf8b7351e9ae6.pdf 901bb79f6512fd9dd3cdda25a8d50fc0 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The Collegian Description An account of the resource The Collegian began as a student newspaper; became a college community literary publication in 1952 and continued until 1969; became a student newspaper again in 1969; discontinued publication from 1980 until 1989 when it again became a student literary publication. Click on <a title="The Collegian" href="http://digitalarchives.sjc.edu/items/browse?collection=26">Items in The Collegian Collection</a> to view and sort all items in the collection. Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Page numeration Number of pages in the original item. 4 pages Original Format The type of object, such as painting, sculpture, paper, photo, and additional data paper Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Identifier An unambiguous reference to the resource within a given context Collegian_1934_2_16 Title A name given to the resource St. John's Collegian, February 16, 1934 Date A point or period of time associated with an event in the lifecycle of the resource 1934-02-16 Description An account of the resource Vol. XLV, No. 8 of the St. John's Collegian. Published on February 16, 1934. Creator An entity primarily responsible for making the resource Lampee, Robert H. (Editor-in-Chief) Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Publisher An entity responsible for making the resource available St. John's College Language A language of the resource English Type The nature or genre of the resource text Rights Information about rights held in and over the resource St. John's College owns the rights to this publication. Format The file format, physical medium, or dimensions of the resource pdf Student publication https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/abf27cb9d9b765a4a727d1761a3683c8.pdf 73fe2ad5329a52e3ce5ef01e8d794364 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The Collegian Description An account of the resource The Collegian began as a student newspaper; became a college community literary publication in 1952 and continued until 1969; became a student newspaper again in 1969; discontinued publication from 1980 until 1989 when it again became a student literary publication. Click on <a title="The Collegian" href="http://digitalarchives.sjc.edu/items/browse?collection=26">Items in The Collegian Collection</a> to view and sort all items in the collection. Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Page numeration Number of pages in the original item. 4 pages Original Format The type of object, such as painting, sculpture, paper, photo, and additional data paper Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Identifier An unambiguous reference to the resource within a given context Collegian_1934_2_9 Title A name given to the resource St. John's Collegian, February 9, 1934 Date A point or period of time associated with an event in the lifecycle of the resource 1934-02-09 Description An account of the resource Vol. XLV, No. 7 of the St. John's Collegian. Published on February 9, 1934. Creator An entity primarily responsible for making the resource Lampee, Robert H. (Editor-in-Chief) Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Publisher An entity responsible for making the resource available St. John's College Language A language of the resource English Type The nature or genre of the resource text Rights Information about rights held in and over the resource St. John's College owns the rights to this publication. Format The file format, physical medium, or dimensions of the resource pdf Student publication https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/0e57b05bc2976ef09bd01b6f32337100.pdf 387372f9c7ab1652b9a794ce21d167fd Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The Collegian Description An account of the resource The Collegian began as a student newspaper; became a college community literary publication in 1952 and continued until 1969; became a student newspaper again in 1969; discontinued publication from 1980 until 1989 when it again became a student literary publication. Click on <a title="The Collegian" href="http://digitalarchives.sjc.edu/items/browse?collection=26">Items in The Collegian Collection</a> to view and sort all items in the collection. Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Page numeration Number of pages in the original item. 4 pages Original Format The type of object, such as painting, sculpture, paper, photo, and additional data paper Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Identifier An unambiguous reference to the resource within a given context Collegian_1934_2_2 Title A name given to the resource St. John's Collegian, February 2, 1934 Date A point or period of time associated with an event in the lifecycle of the resource 1934-02-02 Description An account of the resource Vol. XLV, No. 6 of the St. John's Collegian. Published on February 2, 1934. Creator An entity primarily responsible for making the resource Lampee, Robert H. (Editor-in-Chief) Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Publisher An entity responsible for making the resource available St. John's College Language A language of the resource English Type The nature or genre of the resource text Rights Information about rights held in and over the resource St. John's College owns the rights to this publication. Format The file format, physical medium, or dimensions of the resource pdf Student publication https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/3cc3b873eafe90844b96f5dc34324e59.pdf 2491ab706578401735d40b1194ac6c01 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The Collegian Description An account of the resource The Collegian began as a student newspaper; became a college community literary publication in 1952 and continued until 1969; became a student newspaper again in 1969; discontinued publication from 1980 until 1989 when it again became a student literary publication. Click on <a title="The Collegian" href="http://digitalarchives.sjc.edu/items/browse?collection=26">Items in The Collegian Collection</a> to view and sort all items in the collection. Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Page numeration Number of pages in the original item. 4 pages Original Format The type of object, such as painting, sculpture, paper, photo, and additional data paper Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Identifier An unambiguous reference to the resource within a given context Collegian_1934_1_12 Title A name given to the resource St. John's Collegian, January 12, 1934 Date A point or period of time associated with an event in the lifecycle of the resource 1934-01-12 Description An account of the resource Vol. XLV, No. 5 of the St. John's Collegian. Published on January 12,1934. Creator An entity primarily responsible for making the resource Lampee, Robert H. (Editor-in-Chief) Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Publisher An entity responsible for making the resource available St. John's College Language A language of the resource English Type The nature or genre of the resource text Rights Information about rights held in and over the resource St. John's College owns the rights to this publication. Format The file format, physical medium, or dimensions of the resource pdf Student publication https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/641808e471d8bf3f0b6848cbc476c3fa.pdf 48b6d176b99c4cfbd9b3f4b8e68aa8c5 PDF Text Text �� Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The Collegian Description An account of the resource The Collegian began as a student newspaper; became a college community literary publication in 1952 and continued until 1969; became a student newspaper again in 1969; discontinued publication from 1980 until 1989 when it again became a student literary publication. Click on <a title="The Collegian" href="http://digitalarchives.sjc.edu/items/browse?collection=26">Items in The Collegian Collection</a> to view and sort all items in the collection. Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Page numeration Number of pages in the original item. 51, 30 pages Text Any textual data included in the document text Original Format The type of object, such as painting, sculpture, paper, photo, and additional data paper Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Creator An entity primarily responsible for making the resource Lachterman, David (Editor) Title A name given to the resource The Collegian, December 1964 Date A point or period of time associated with an event in the lifecycle of the resource 1964-12 Description An account of the resource Issue of The Collegian. Published in December 1964. Identifier An unambiguous reference to the resource within a given context Collegian Dec 1964 Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Publisher An entity responsible for making the resource available St. John's College Language A language of the resource English Rights Information about rights held in and over the resource St. John's College owns the rights to this publication. Format The file format, physical medium, or dimensions of the resource pdf The Collegian https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/a2a97a0183b427003143849ed2e0694c.pdf 80d9a928a63ef27de4eed108c6f9cce4 PDF Text Text �65 68 65 is Textbook * �A QUESTION OF CERTAINTY John Merritt Honorable Mention Junior Essay Prologue Montaigne summed up the state of knowledge in 1580, with the publication of his second volume of essays; everything is un~ertain. The doubtfulness of what Aristotle wrote is only augmented by each commentator and confusion is compounded so that the hundredth commentator leaves truth more obfuscated and arcane .for the hundred-and-first. Montaigne's solution to this problem is unique ·-- perhaps untenable. Since there is in the outside world no certain knowledge, let us not sorrow or despair, but rather, like the tortoise, withdraw into ourselves and study our nature, study ourselves. But his own remarks on the inconstancy of our actions leaves him resting on a none-too-soft pillow of doubt; if there is no knowledge, let us not bother ourselves by trying ·to add to the anarchy. Let us instead take what comfort we can in this Heracliteen world, and if our kidney stones should bother us, we may pleasantly write · them. essa~a about All this may have its roots in the nature of the world, but certainly some cause for Montaigne's observation can be found in the critical spirit of _ the fourteenth-century philosophers, notably William Ockham. This Franciscan logician concerned himself mainly with epistemology -studied the very structure of cl assical metaphys i cs . Summa as Aquinas or Duns Dominican schools Scotusj~is.influence was Though he l eft no dominant~- even i~ and for the three hundred · years during which his criticism was the philosopher's principal tool, · arguments which :once seemed uibrant and now to drop still-born from the mouths viril~ _ seemed of the old masters; what was . demonstrable was. quick! y turned to -.merely probable. "In all argum~nt ," Ockham observes, "something will be : - accepted which is either doubtful or· taken qn _aith .. " f three centuries philo.sop~ers out the truth of Ockham - agreement. Now, the and theologians seemed determined to..•bear is statement, . 2) . . ~- seemed For the next for the dialectic -- and .t he dis. interminable~ . supr~me . i~ony ot~urred in 1633, with the condemnation of Galileo �2 by the Inquisition. A board of theologians, themselves in the throes of doubt (as Ockham and Montaigne had before remarked) condemned Galileo, who had founded certainty with the science of nature. from one point of view, the establishment of a consistent, orthodox system of metaphysics which would allow for the development of certain truth in the sciences, and also be acceptable to the doctors of the Sorbonne is the well-nigh insuperable task of Descartes. I In the midst of all the uncertainty of the theologians and the doubt raised by Montaigne, Descartes proceeds to discover certainty. The proper way is to begin at the beginning, where all is dimly shadowed forth -- with doubt. Although his father and his daughter had died during the writing of the Meditations, Descartes, with admirable _philosophic fortitude, announces in the first paragraph of his book that he has banished all care from his mind and is happily agitated by no passions. He is, therefore, in the perfect state to prosecute his endeavor erection of a car.ta in foundation for the sciences. the -~ To this end he observes that reason teaches him to withhold assent no less carefully from what is uncertain than from what is evidently false; the process of dialectic is not good enough for him. dialectical, premisses. He must start from demonstrative, not The reason for this is clearly seen in the first metaphor in the book -- "When the foundation is undermined," he says, "the superstructure will collapse of itself." The delicate forms of dialectic will simply never do, for they are always hanging in the air; 3 Descartes must build a building. ) Let us now see what he do~bts, and what are his reasons for so doing. The natural reply to such a bold endeavor as Descartes' is "you must be a lunatic! no one in a normal frame of mind can doubt what you doubt; some unnatural humour has plagued your brain." It is indeed amazing that anyone should have so wondered about the outcome of the book that he completed it -- that . i t was not utterly forgotten instead,. .9f~·-t'he cornerstone of a great revolt. ..• ~· .... Certainly Descartes must shdfu us cause. To begin, his sen_es ·have been known to deceive him, and since a wise man s does not entirely trust what has once -led him into error, he therefore denies all knowledge . contingent on sense expfrienca -- including the fact that he has a body. For Descartes is not a . body; he is by nature �3 a being seeking certainty~· and ". since he must acknowledge that : he has experienced noth-ing · in waking life which can· be distinguished ·from the. life experience in a dream, he is compelled to admit that· all nf his . senseknowledge, and even the experience whence it is derived, may be a dream, and has no nece~~ary r~lation t6 t~uth ·-- perhaps is radically other than truth. The only knowledge he possesses that comes not fra·m the sens·e s comes only- it seems,from the mind--mathematical knowledge. , Certainly this is not doubtful.· But, Descartes observes, there is an old idea in the mind that.thaie i~ a God, an a1l~powerfu1 being. What if this being were not b.e~evoient, but evil, and played some complicated game with Descartes, deceiving him ·at every turn, m king him think he perceived a ·a truth clearly, when in truth he saw as in a .glass dark1y4 ) __ ~aw nothing qut . ,falsity everywhere he turned, and. under the aspect of truth;? ·Indeed, it is the more probable that this gr~at Deceiver exists~ since withbut the least consideration of this omnipotent monster, Descartes has, and with reason, come to doubt even that he has a body. If ·he could be de6eived about that, surely he might be deceived in thinking th~t the - . th~orams of p~re ~a~hematics wer~ wholly and incontrovertibli tiue. So Descartes, who sat at the feet of rriontaigne (who taught doubt); i~ now -- master; he teaches with doubt. " Committed to a doubt deeper than Montai~n~'s, Descartes mtist either ~find certainty or live in a world far worse tha~ his teache~ 1 s; · ~s to d~~bt required activity in the first place, to continue on the way to knowledge requires even more. His mind is hurled into an abyss of uncertainty&) at the end of t~e first Medit~~i6n, and he propose~ to remain there ~or the remainder of the day' les·t" -the old' habit of believing his senses· should return and prevent ~is finding certainty. One need read only the first page .of the first Meditation in order lo see . that something v~ry new is about to unfold, to chall~nge inquiry ·into what is meant by the search after truth, On the second day~ ' De~cartes pro~os~s to find a firmament to cl~av~ the deep _ sea in which he finds . himself immersed. . . ever"ything which can if there be any. He has proposed to ·doubt be . doubted; 'now. he mu·s·t find the limits to doubt' The firmament which he proposes to find, ::plication) to .c reate, he likens to ~r~himedes' or (by im- point in the se.cond �4 metaphor of the meditation. Archimedes has asked for but one fixed and unmovable point in order that he be able to move the earth. Descartes no~ This point seeks. Mastery of doubt seems to follow directly from asking the question ''do I exist?" since to answer this in the negative is plainly impossible, and if not contrary to logic, certainly contrary to nature. "I do not exist" cannot be true, since that would mean that the subject of the proposition did not exist, and if there be no subject, or no term or terms for the proposition to relate, there can be no proposition. Plainly and certainly, however, it can be answered in the affirmative, for even if Descartes be utterly deceived by the great deceiver, yet can he not fail to exist, for surely someone must be deceived. Certainly the great deceiver has already lost his battle; for if we were always deceived, we would never know it, nor be allowed to think of Descartes' method. The~e is little analysis that can be performed on Descartes' Archimedean point of a strictly logical sort, though some psychological analysis is possible-in fact, necessary. Logical inquiry is hampered by the membership of that proposition in the class of immediate inferences; it simply is true. Mindful of the need of extricating himself from his wholly and unnatural position 6 uncomforta~le ~ and as quickly as possible, Descartes finds his Archimedean point in the fact of his existence; if he thinks or experiences, doubts or in any way acts, necessarily he exists. The question that must now be raised is of the meaning of "I exist",if I lack utterly a body. It is clear that the cogito expresses a truth independent of the idea of body, but we must ask what it means to exist which Descartes nowhere answers. that belong to him. a question Rather he discusses those experiences He is a substance, a substance devoid of any of the characteristics of body, and in which inhere at least the properties, or powers, of doubt, conception, assertion, denial, and will; a substance, moreover, which experiences imaginative pictures and what (for want of a better term) we call sense perception. That Descartes possesses all these characteristics he cannot deny, even if he claims that he has no body whatever. But in order to further understand his ego, Descartes directs his efforts (in what seems like definition ignotum per ignotius) to the analysis of body. Everything we know by that mode of consciousness called sense perception �5 is not known by taste, touch, smell, sight, or hearing. Nor yet is it known by the imaginative faculty; only by a purely mental, or . intel+ectual inspection is anything known. To show this, Descartes chooses as an example a piece of wax, which we are naturally inclined to think known by the five senses perhaps by them only. it shows itself chimerical fleeing. ~ot Yet, placed down by the fire, sends our weak knowledge (itself a chimera) one of the properties we recognized the wax to have by sense did it have essentially, or in nature, since each of these properties has disappeared, and the melted wax possesses an entirely new set of sensibly known properties. Was the wax then known by the imaginative power -- our capacity to mentally form pictures of extension, change, flexibility, and so on? capable of ~assing Of course not. The piece of wax is entirely through an infinity of stages that my imagination in its finitude could never comprehend. How, then, is the wax known? nature of the wax must be known only by some purely mental The inspe~tion its perception is not by sight or touch; or even imagination, nor was it ever so, though formerly Descartes took it to be so, _ when his knowledge was imperfect and confused. Now, however, he knows the nature of the wax clearly and di$tinctly. It is difficult to imagine a conversation between Descartes and one who asked if the wax were known, or one who claimed that he did not perceive extension by the sense of mental or conceptual inspection, but by the sense organs. Hard it is to know an uncertain world. Now, what light does this throw on the nature of Descartes' mind? Descartes now claims he knows himself -- he means his mind -- far more clearly and distinctly than the piece of wax. For clearly, when he judges that the wax exish, and sees this .c learly, the result which follows is that he exists, and exists as a mind, as a being who knows by inspectio. A fortiori, I suppose he would say, the mind is c1early known, since extension is clearly known, ·and the consciousness is that which knows. It is interesting indeed that the nature of extension .is mentioned before the nature of mind -- in fact, the nature of mind does not explicitly appear until .the last Meditation, in which material things are discussed. One may also note that although Descartes claims mind to be better known than body in general, yet the only property of the mind that Descartes explicitly mentions is existence~ · Betw~en the lines, lies the the fundamental criterion of the mind·; inspectio -- activity. unw~ittan, �6 Creating his universe entirely anew, Desc~rtes has first found a point in the middle of his vast sea, by virtue of which he intends to push himself to dry land, the groundwork of the third day. This earth-moving point; is the new center of consciousness_!/understood not in the passive sense of the ancients, but in an active, free, and masterful way. The way of Bacon -- to torture and force nature to yield her secrets to man, to teach man the language in which she wrote tha world, appears to be here opened. What was once the power of the gods is now man's. The ego-centric universe cannot be understood in the obvious pejorative sense, for the sense of ego is now that of ordering force, active intellect, and this is a faculty shared in by all men. To follow Descartes through the Meditations is to become one with Descartes; whatever is accidental to both writer and reader disappears beneath the power and unity, the community of intellect. This point, the existence of active intellect, . . constitutes the separation of extension and intellect, and is . more and more clearly shown by reference to the clarity with which the nature of extension is perceived. III Entering now on a quest which has the starting point (the existence of mind), Descartes next investigates, as the proper order requires, what is meant by the conscious being; he will try to grow by degrees more familiar with himself. To his opening question, "how is it so certainly known that I exist?" Descartes replies that he clearly and distinctly perceives it, and lays it down as a general rule, as one of the properties of his mind , that whatever he clearly and distinctly perceives is true . All truth, insofar as it is just that, must have some common characteristic perceivable by men. We must ask what this is. Descartes knows one thing certainly, that when he thinks, or experiences anything whatever, even though that idea be materially false, he must exist. He knows that he possesses the capacities of doubt, conception, volition , asser t ion , and denial because he exercises these powers. act. It is Descartes' nature to These actions imply a subject in which they inhere -- namely Descartes, and the nature of this implication seems to show that what 7 acts or has power exists ). ; this simply because it cannot be otherwise. No farther into the proposition can .we go. The evidence for ·the im- plication is not any formal rule, and thus stands unique in this book. �7 It is evident for one reason that it is clear and distinct. rule of clarity and distinctness is the only rule which ~an Since this be extracted from the only truth which Descartes knows, then it must be the defining characteristic of truth; all other truths, if there be any, must possess that characteristic, too -- that it admits of being very clearly and distinctly perceived. Therefore Descartes asserts the generalization of his rule. Doubt, like the man-eating tiger, must return, for Descartes had only begun his arduous journey because he found the recognized truths uncertain. many Before careful examination of his beliefs, Descartes believed thing~ he can no longer hold; they no longer appear clear and dis- tinct, though once they did. .. Why did he arr? Because he merely thought he perceived such things as extension and mathematical theorems clearly and distinctly, and was deceived by attending to his ideas with insufficient care. Perhaps, in fact, there is a great deceiver who made all those beliefs he formerly held appear true, though they w~re false. Although he thought them to be true because he had a natural inclination . . to do so, he perceived them not by the light of nature, but confusedly only. Thia supposition is sufficient to explain how Descartes was be- fore deceived. But before he can further push his knowledge, he must inquire whether an all-powerful being exists, and whether it is within the limits of that being's power to deceiv~. To this end he proceeds methodically to classify all his thoughts, beliefs, feelings, and impressions, that he may see in which of these truth and falsity do properly inhere. Obloquy must indeed follow from the failure to show that God exists, sine~ willy-nilly on that failure follows the closure of knowledge. This can be avoided if we observe the states of Descartes' consciousness, his ideas. These ideas are such as to borrow their being . fro_ Descartes' m consciousness, since an idea cannot be grasped clearly consciousness perceives it. unle~s These ideas comprise variously imaginative pictures of objects,_ volitions, emotions, and judgments.. speaking, nothing in the _ima. ination can be false, since g exists independent ~f perc~ption true that it is . imagined. . . . the wh~ther it is wholly besi9e the point; it remains Similarly for .volitions or . , .. ' Now, formally ' emot~ons. The only species of cognitions in which truth or falsity may formally inhere are thus judgements; to these Descartes must now turn and discover �8 that by virtue of which he falls into error. Unnatur~lness is the essence of falsehood, since the mechanism by which falsehood occurs is the substitution of spontaneous inclination for the light of nature. for that inclination and the light are both "natural," and therefore the mind assumes that because there is a connection of ideas, the same connection exists between the subjects of those ideas. The implication could only be valid if the ideas were clearly and distinctly grasped, and gra~ped as existing. been laid down before the next step. The necessary precaution has Among Descartes' ideas is the idea of God. The proof Descartes now offers is a proof for the existence of a benev~ olent deity from the necessity of a first efficient cause -- a proof ancient and honorable indeed. But it acquires in Descartes' hands certain changes which are essentially ramifications of certain previously discussed Cartesian notions. Descartes has already observed that he has various sorts of ideas, and that of these ideas, some represent a greater amount of reality than others~ Thus the idea of a subs~ance has a greater amount of representative reality than the idea of an attribute, and what represents most reality is the idea of God. Now, Descartes argues, the efficient cause of an idea must have at least as much reality as the effect. Only a being with the actual perfections represented by the idea could have caused such an idea, and therefore God exists. Earlier versions of this proof had depended, at least ostensibly, on the existence of material things, which everyone had naturally taken for granted. Descartes has eliminated this supposition from his version of the proof by showing that even if nothing should exist but his own mind, one of the aspects of that mind is the idea of God, and that idea must have an ePficient cause. This cause could be God, or sensible objects, or some subject of intermediate reality, perhaps an angel. That the cause cannot be material beings is clear, since the idea of God is of an infinite, omnipotent, benevolent being, possessing characteristics which manifestly cannot inhere in extended ·being. Nor could it have come from. his parents or an angel; it would have had to be in them, and since further they are not by nature necessary beings, �9 it is impossible that they should have the idea of themselves. Nor can there be an infinity of causes, and therefore the idea was ultimately caused by God, and placed ·in the consciousness to serve as a sign that man was created in the · image and likeness of his creator. Removal of the distinction between conservation and production with . . . respect to eff,icient cause destroyed what difficulties migh~ by controvei;.s y between a Scotist and a follower of Ockham. observes that the act by which God creat~d be raised Descartes the world is not conceptually distinguishable from the act by which he preserves it in being. This also means that there is no intermediate cause between man and God. Man is completely dependent on God for his very being, in the same way that the idea of God is not only . p~oduced, but conserved immediately · by God. The idea of God cannot be ·produced, by a b~ing conserved~ any more than it :can be of less reality than Him. There is nothing be- tween Descartes and God. To the objection that this idea of God could have come from himself, Descartes has merely to point out his contingency. There is nothing in him which necessitates even his existence, and his lack of perfect kn~wledge sur~ly exis~ed marks him as a in him potentially. crea~ed being -- even if all perfections Descartes considers, too, the objection that , th~ idea of God could have come from several partial causes, and re~~ct~ .it on the ground that such a conceptual grab-bag could never give rise to the clear idea of the unicity of God. In general, although by nature the certainty of Descartes• proof is questionable, it can be made satisfactory by the el~boratio~ of certain s~holastic ?istinctions, with which Descartes never bothers -- and with good reason. . '. . Out of the abyss of doubt and uncertainty, Descartes has now emerged as it seems, victorieus. him of wit~out certa~n He had need of a benevolent deity to ' assure truths the certainty of which would be questionable a proof that such a being existed. He found this proof in an old proof, manufactured anew from certain principles clear and evident to Descartes. First, in looking fer an efficient cause of the idea of God, he searches out, as it were, the efficient cause of himself, . . for the idea of God is a mode of his own c6nsciousness. there cannot be, for God, any distinction between Se~ondly, corise~vation and �10 production; not only is God the first efficient cause of the idea of God, but also the immediate efficient cause of Descartes' own mind, and even, in some sense, a formal cause. Thirdly, what seems to be a clear- cut distinction in Descartes' writings, the distinction between the actually and potentially infinite, becomes furry at the edges. The only difference between God and man is that the one is actually infinite, while the other is potentially so. But the mind is potentially infinitl3 , and in soma sense has a hold on the actual infinite; the idea of God is a mode or aspect of Descartes' own being. IV Unless he possessed knowledge of a benevolent deity to insure that he is undeceived, Descartes would have had to remain in ·doubt. But doubt, essentially an active process, cannot remain as it is (Montaigne, for Descartes, must have been the supreme enigma), and therefore must either push forward to knowledge or disintegrate totally and be forgotten. But, under Descartes' mastery, doubt has slowly been gnawing away at the nature of ·things, leading first to the existence of the self as pure intellect, and then to the existence of God, whose goodness prevents his · potency to deceive. In the fourth Meditation, Descartes reflects on what he has done, and observes that there is little truth indeed in our perceptions or corporeal objects, though much can be known about the human mind, and so much more about God, who conserves Descartes at every moment. From the finite to the infinite is manifestly the direction of our knowledge. Contemplating his experiences on the past three days, Descartes desires to move to the knowledge of other things, and to this end, he again examines the nature of judgment. To judge is a gift of God, and since God is good, He could not have given man a faculty the right use of which could lead the mind astray. But if this is true, how could man conceivably go wrong? Descartes has a very simple, and venerable, answer to this -- the .idea being. of non- Descartes is ~~apped, as it were, between being and non~being; error proceeds from non-being. Since Descartes is not . ~imsel~ God, and thus equipped with all perfections, it is very easy to see the possibility of deception; we must employ our God-given faculty to acquire truth rightly. Truth, then, comes from God, and error comes from non-being. �11 Exceptihg one problem, this seems to be · the perfect explanation of error. We can frame that problem by observing an obvious expansion of the principle behind God 1 s inability to deceive. If God is the most perfect being, then His work must be the most perfect, too, and the existence of Descartes 1 ability to err argues a lack of . perfection that should be in him. Descartes proposes two answers to this problem. First, he recognizes the infinity of God, and observes his own finite mind to be out of all proportion thereto. He then sees that there are many things about God which are mysterious indeed, and which will forever remain so, remarking incidentally that for this reason he desires consideration of final cause removed from physical inquiry. Next, since he clearly and distinctly perceives certain other beings as ideas, and also that God is capable of creating whatever he clearly and distinctly perceives, Descartes considers, as Leibniz later does, that creation as a whole may be indeed perfect, though created individuals when considered in isolation seem imperfect. The principle of sufficient reason is a dangerous principle, applied to God. Turning now wi~h Descartes to his own case (since his case is our case) we see that he separateg the cognition turn of ~nd will. phras~, since ~ources of error into two faculties Error caused by cognition is an admittedly strange th~ intellect, strictly speaking, cannot err. Nevertheless, it can be a source of error, since the knowledge of any being but God is finite, and privation of knowledge leads to error. Will, however, is fully as infinite and perfect as the will of God. The will of the soul, that is, its impetus to act, is Descartes' mark of God's authorship, as before the idea of God had been characterized. It remains, however, to discover whence error originates, for the will is perfect in kind, and so also is the understanding, finite though it be. The answer is that the will to assert or deny the truth of a proposition is forever outrunning the ability of the intellect to judge correctly. The infinite keeps drawing the finite on -~ even beyond its c'a pacity. Luckily, however, we have discovered the method of not erring. as the outcome of discovering how we err.~)_ We_ must not. allow. the will to carry the intellect away . with . it, but must curb it so that we. assert �12 only what we clearly ·and distinctly perceive as true. And that this is entirely possible, though strange and difficult, we need only examine the work of the first three days for abundant proof. The light of nature is a powerful tool, but it must be correctly used, for the chief and greatest perfection of man is this·: that he might never go wrong. Now Descartes has discovered not only the cause of falsity, but a tool for the disc~very of ·truth. v Four Meditations have now been completed, concerned with doubt, the . existence of mind, with the nature of co.n sciousness, God, and with the manner of acquiring certain truth. odd, and seems to 'pose as an four days and the climax to The heading o~ _ this in~erlude com~ ~- Meditation The fifth Meditation is quite between . the drama of the' past a calm before the storm, as it were. ~sports that ~tis about the nature of material things and the existence of God, ·though the Meditation seems at first glance to ~ the nature of material ' things for a higher goal a proof for ·the existence of God, which is ·apparently different in kind from the proof offs.red in Meditation III. of the same fact? The Meditation is Why propose two cer·tain -proofs p~imarily about the nature of material things -- not the existence of God; the proof .is not essentially different from .the previous proof, but is a simplification thereof; the con~lusitins on which Descartes mainly seizes are different from Meditation III; the general conclusions of the fifth day are necessary to the development of the following cay. Inquiring into the nature of material things in what appears to be the preliminary part of the Meditation, Descartes asserts that among his various 'ideas he finds a state which can be called imaginaticin, -.c haracterized by .the formation of mental pictures of extended beings. For example, Descartes observes that when he thinks of a triangle, besides experiencing the purely .mental inspection of that object, he is . aware of a power within himself to picture such a thing by means of the . ·:· im~gination. Imagination can call forth these picture~ at_ will. ·These . : imaginative pictures, though they be of no thing that. ever was, can . .yet ... ,.i~ no way be called. nonentities. ror Descartes clearly and dis. . tinctly .perceives that they have their own distirict ~nd unchangeable �13 natures. . hese ·natures ·are .nat .fi9ments of· the ·imagination at all, but T ·quite independent of his thinkiryg of them -- for :various properties can . be proved .or ·,them- these ideas, . ind~pendent of their ac.t ual existence. Triangles have by nature three sides, whether I ·choose to imagine them or not; I am constrained by the light. :. o.f nature to so think. Now Descartes' di~covery that certain properties necessarily inhere in certain subje~ts independe~t _ the actual existenbe of these subjects of leads him to cons~der God's existence once again. God is the subject in which the attribute of existe~ce nece~sarily . inheres;. . .th~~efo..re, since . . . . .. .... ..·.-- .. . ., . · ~e.scartes has an idea of God, _ God exist~. _ he an'?_~_ent argum~n~-- ~' ::called T_ ."ontological, 0 of passing from the possib~lity . ~o ~h~. nece~si~y of perfaction, is revived again in a new form. Implausible as Descartes• form of the argument may b~, the idea behind the argument se~~s essentially .. ample~ - ~ strong 1 ~ersi~n of .. sound. Descartes, however, chooses not to the proof in order to show not only the a~alogy to c~ncerning .the nature of ma~_er ial his first existence proof. ·-- ~he ·. ; _ proofs of _ theorems · things, but also the resemblance to In the case of both proofs, from the idea of God, Descartes passes directly to His existence. In the first proof, existence is demonstrated by conserving efficient causes; that is, the idea of God immediately implies His existence. In the second proof, existence · is ·also ·ahowri ·oy ·immediate ·reference to the · idea. · Descartes himself elsewhere admits that the first proof is an elaboration of the second; the exceeding close relationship .between the efficient cause and the formal cuase for Descartes is thus brought · into a ·strong ·· light • . . Thus, the seeond form of the proof, being essentially a simpler ·version ot ·the fir$t; has as -its · final cause a somewhat diffe~ent idea. In diecussing ·;the .outcome· of the proof given in the third Meditation, ~escartes the mere says .that on all accounts, the conclusion must be that .from fact ~ · that clearly exists. he exists and :has :an idea of the perfect being, God But in his discussion of the second -proof , ;he says that it always comes back to this -- that he is not convinced of anything excep~ ~ha~ ~~ - elear1y · ·~rid distinctiy p~rceiv~~. different objectivas· in mind in ·each case. Descartes h~s ~ery In the fifth section of the Discourse an · Method, · De~cartes declares that resting his arguments solely on the . perfection . of.- God·, .ha· tried . to prove · all the law's · nature that of · might have beeri · doubted; like the fox, .· he ·has·· more methods .. than : 6ne of achieving a single aim. �14 To see what Descartes makes of this, we turn to the last part of the Meditation (which is quite as long as the proof proper), and observe that he returns to the .nature of truth and of material things. Truth depends on two things; potency to admit of clarity and distinctness of perception, and the light of nature, which, in turn, depends on two more things -- Descartes' intellect, and his knowledge that the deceiver does rot exist. In fact, Descartes asserts that were it not for his knowledge of God, he would be uncertain of everything else. He is not arguing in a circle, for there ·is a distinct difference between what is immediately clear and evident, and what one remembers to have been clear and evident; our trouble is weakness of intellect -- we cannot be actively knowing all the time. Even the Deceiver could not make false what Descartes clearly and distinctly perceives, though he could very easily make us falsely think that we clearly perceived what we did not. Now, at last, the argument from clarity and distinctness of properties to necessary inherence in a subject will follow, to open the entire field of mathematical physics as an exercise of the intellect -- not of the senses. VI After all his difficult preparation, Descartes is prepared to consider the final cause of his efforts in the sixth and final Meditation. He has led us from the distinction -- nay, the utter rift -- between mind and body; now we approach the existence of material things, the expl.~nation of the occurrence of error in sense-perception and doubt, and an involved discussion of Nature. In part six of the Discourse on Method, he claimed that just as the last things are proved by the first, which are the causes, so also are the first things proved by the last, which are the affects. The sixth Meditation is the triumph and justifi- cation of the unbearable doubt that initiated this struggle into the opeh air, from the cave into the light of nature . Tortured long enough by the demands of doubt, Descartes plunges into the work of the last day with a detached fervour; no moredoea he fea:r uncertainty, for his·p~th is clea~ • . His pains are compensated by dis- tinguishing the mental states of intellection, imagination, and sensa. ~ion; by considering carefully the properties of these three states, �15 he may see that sensation implies a being other than himself. Having laid down the plan before it is carried out, soma of the drama seems to be gone, but -if it is, it is ·replaced by something of essentially greater interest -- explanation. Descartes' distinguishes again between in- tellect and imagination, this time .adding to the distinctness of their functions the distinctness of their dependence on himself. The power of imagination is not essential to his nature, to the nature of his mind, and therefore it is probable that it depends on some other being than himself for existence. Descartes now makes the most preposterous state - ment in the Medi.tatiOns; · he claims that if he were joined intimately to a body so that he could contemplate it instead of ideas, then he could readily understand the phenomenon of imagination in that way. Other considerations must be taken into account before a certain judgment can be reached. Since the argument from imagination is a merely probable account of the appearances, we must push our researches into another mode of consciousness -- sensation. At this point, as at a similar point in the Meditation on the existence of God, Descartes begins an enumeration. first he considers those objects he took as real before· the regime of doubt began, then he reviews his reasons for doubting the capacity of sensation to lead to truth, and finally he sets out what he may now believe. Unquestionably the state of mind of Descartes before the opening of the Meditations was largely determined by his body. The first of the so-called sensible objects he had previously held to exist was his body, the central problem which gave rise to the Meditations. This body gave him experiences of pleasure, pain, appetites, emotions, and sensations tangible, visible, and a~dible. that he distinguished and "knew" ·the world. reason. It was, in fact, thus for this he had good First, he experienced these sensations without, and often contrary to, his will. It was further his experience that the mental state arising from sensation was far more vivid, and, in its own more distinct than anything given by his power of imagination. wa~, The inference he had made was that the things in 'the world exactly resembled his ideas. What cotild be more natural? ·-Tested against that touchstone, doubt, this inference has shown its �16 base metal~ . Sense experience is indi~tin~ui~hable from what we observe as the same mental experience in sleep, though in sleep the .subject of sensation is missing. Since Descartes , did not know the Author of his being prior to . the third Meditation, he was also obliged to admit that there might be some omnipotent ·demo·n dealing failures to all his efforts · to find truth. Even the intimate expe~ienc~ of pain could not shake his doubt that he had a body, because pain should be felt ·in a limb only . if . that .limb exists. But Descarte~ had known people who had lost . limbs to complain . of pain in their lost limbs.. Certainly it is possible that h~ erred. But his stance is now different by five days from his position prior to doubting, and stronger than on any of the preceding days. What may we balieve with certainty? Confirmed in his belief that there is a God· who would not deceive him, Descartes can now analyze both sides of- this question. the argument proceeds as follows~ Roughly speaking, Because I have a clear and distinct idea of myself as a conscious being, not an extended being, and also a clear and distinct idea of extension ·1acking consciousness, it follows th~t I am really distinct from my body and sation is contra~y ~u~h co~ld exist without it. Sen- that it causes states of consciousness to arise even to my will. Therefore it cannot be that they arise from myself, but must arise from some substance containing at least a like amount of reality with the idea as the idea represents. So mY idea~ of. extension must .arise either from corporeal objects or from some higher reality. To the end that I might distinguish between these sources of reality, God has gi ven me no faculty whatever. He has given me a strong , natural , inclination to believe that they arise from actual existing material ~bjects. And because God cannot have given me any power or . inclination · the right use of whtch · might:·:_lead me astray, I am constrained to say that my sense impressions and imaginative powers are related to corporeal reality. It may be that corporeal objects are .quite unlike my ideas, nevertheless, Descartes asserts, their nature must be what I clearly and distinctly apprehend it to be -- that is, what. ver falls within the e subject-matter of 'pure mathematics! Rhetoric's needs would seem to be now satisfied, for wha·t was to be proved has been proved; yet half of this last medita.tion rem.a ins .• . Natura appsars three tim~s as riiten. in the sixth Meditation as in any �17 of the preceding ones, and provides a clue to the nature of Descartes• book. In this Meditation, toq, the use of verbs in the first person slngular or the use of the personal pronoun ego has markedly decreased, as · als~ th~ word Deus; natura has taken their place • . Out of this proof for the existence of corporeal reality, as from the three previous proofs, has come reflection on the nature of truth and By .falsity, this time cast in the contrast between nature and habit. "my nature" is meant the complex of all that God has given me; by "nature" is meant the whole Qf creation; by "the light of nature" is understood the .principle by which truth is grasped by men. Now, the light of nature shows me many things -- that I have a body, that by means of that . body certain sense impressions of corporeal objects come to my mind, that these extended beings exist, and so forth. It is by the light of nature that I know my nature as a conscious being, as an ext~nded being, and as a compound of those two natures. I err _ nature. by And yet Was . it not natural that I took extended things to be like my ideas? No. Habit of inconsiderate judgment we take a$ a substitute. for the light of nature, and it was thus that I took extended being to be similar to my ideas. This habit of ill-considered judgment is developed by the will overcoming the intellect, a mechanism explained and overcome in the fourth Meditation. In considering nature, Descartes runs up against a further problem -, .what among physical objects should I seek or shun? These things .seem to be taught by nature, because our perception of pleasure and pain gives us our clues. Vet what is to be done about the man who is sick dropsy, and has no need of water, yet experiences thirst? wi~h In his first reply to this, Descartes, in an astounding image, likens the human body to a well-made clock, which. may yet run down and tell the wrong time. Clearly the clock obeys . all the "laws of nature" whether it runs well or ill . the ~ction h~althy, So we have been allowing our will to inadvertantly impede of our intellect again; it is only with that the nature 0f the dropsical is corrupted. ·such, the sick man is , quite as ·this ~aspect ~xplanation is not ~'natural" sufficient~ to us, the Conside~ed as thehealthy man. There temains a re~l as But even fault in the nature of the sick ma.n , since . he desires what is harmful to him. . How can the divine goodness permit this corrupted nature to deceive us (for �18 corrupted or no, it is still nature)~ Repeating again the real distinction between mind and body, Descartes observes that we have two natures,mind and body. Now in the healthy state, man's ability to perceive pleasure and pain are obviously ordered for the best, because when I must take water, r feel thirst. This is caused by the intimate association of corporeality and consciousness. But nerves, as Descartes claims to have learned from physiology, operate by pulling in the area of sensation in the body to a common sensory center fn · the brain, which in some mysterious way affects the mind, causing it to experience sensation. But by a ~ metrical example, Descartes shows that that same experience can be caused quite as well by a pull on the same nerve, though higher or lower than the original position. Therefore it is possible, because of the infirmity of our nature (that is, the nature we possess as a compound of mind and body) that our nature should deceive us. We have not all the perfections of God, however perfect we may be as part of a world. And even taken from the ·point of view of the individual, the mechanism of sense perception is still better ordered than any possible alternative. Ending this almost fantastic journey, Descartes returns to the distinction between the infinity of God, the mystery of his omnipotence and omniscience, his depth and grandeur, and our poor, confused, finite, feeble understanding. Remembering the great distance the Christian must feel between himself and the power of God, he admits that practical needs do seldom allow us to examine so carefully a situation that we be not deceived, and cautions us to recognize the infirmity of our nature. In- deed the rhetoric was not finished. EPILOGUE Because of Galileo's condemnation, Descartes suppressed an early treatise perhaps containing much that is in the Meditations. It is difficult to imagine a man convinced of the truth changing his beliefs even for the sake of the Inquisition; force merely drives the heretic underground. I therefore made it a rule of thumb that the Meditations was a very subtle book. The rule was correct. Drama is the essence of the �19 Meditations, and the order in which statements come is variously shocking, amusing, baf~ling, astonishing, and at times, perfectly logical. IDhy does the simpler form of the proof for the existence of Go'cl coma·· after the more complicated version? To focus attention on 6l~rity and dis- tinctness as criteria of truth and also on the nature of material . things. But .why .ts it there at all? Preci'sely to take attention away 'from the astonishing nature of material things. Csrtainly Newton and the Sorbonne doctors would read the Meditations differently. Or why, in the second Meditation does a general criterion of truth and a proof for the existence of God follow immediately on the inquiry into the essence of ego? Examples could be multiplied almost without limit. Execution of Descartes' task, as of any scheme of grand dimension, requires discipline and action. effort is doubt. The generating force of Descartes• stupendous It is quite curious that the first collection of properties Descartes takes as belonging to his nature are doubt, conception, affirmation, denial, and volition. attribute. Descartes has not one passive Wonder is not in his vocabulary; if he knows what it means, he despises it as the milquetoast's reaction to nature. If certainty is to be achieved, doubt must be pre-supposed, accepted, and dismissed in reasonable and orderly fashion. Mind for Descartes is in constant activity, while for the ancients, the intellective faculty was passive, open to receive the truths of Nature. Thus that which was formerly thrown under the intellect for contemplation is now thrown out of the intellect for its inspection. Both the quantity and the quality of knowledge gained under the old aegis appalled De's cartes . Such · a passionate seeker after truth was he (for he had to have more and better knowledge), that instead of founding philosophy, he created philanthropy, theodicy, and anthromorphism. Descartes thus commits his own cardinal sin; before sufficient reflection, he wills that his new subject should be metaphysics. Coming to wisdom via doubt means embarking from an odd point -- a mathematical one. One of the most fascinating (and elusive) problems of the Meditations is the ego -- what is it? Perhaps one might say that it is that .active, intellective, conscious tool which organizes the vast fund of natural knowledge; it almost takes the place of God. But still we must ask in what the being of this mind consists. Descartes �20 would, I s4ppose, say that this . is a great ·mystery, just like the being of God. Ellipsis is · D scartes 1 best; method of discussing ego. .e I have already pointed · out the at_ange sequence of the third Meditation; while r its avowed purpose is study of ego, the criterion of truth and the It is almost expected. existence of God pop out as theorems. are per~al?s There three dozen places in the meditations at which the nature of mind .is alluded · to elliptically by the dramatic -power of order. Things, ideas, impressions, truth, God, order are all in soma strange way related to this strange "I" that is yo~rs The "!",. li'ke gootj sense, days' journey. and mine after our six seems· equally shared by all men. Of the nature of this self, so little can be said; it is true. ·were we formerly more capable of understanding ourselves? ·. We before understood ourselves as integrated beings. Wonderful. What more can be said? But Doubtful. physico-in~ellective One problem, or. pseudo- problem, which the earlier view avoids, Descartes also avoids (by not · addressing himself to the proble~): how are the faculties of sense perception and imagination possible? .How can two substances utterly distinct, h~ving no properties in common but existence, interact to produce these phenomena? How can we know the outside world at all? To a certain extent, we are · creat~ng How can we determine the extent; it. To how great an extent? Descartes invented (or discovered) what came to be known as the mind-body problem; its ramifications are many and varied, especially in questions about the foundations of knowledge. Though Descartes would probably not admit to many of the problems found in his work, he has no recourse but to mystery. . . moving through the same problems under a different as.p act- · another , question comes to light. What has happened to God? Why are both Descartes' proofs cast as merely plausible though they wear the mask of demonstration? The late scholastic masters could have made eoormously impressive proofs principle of sufficient door, f~om t~e ~eason same material. But the has slipped into philosophy by the back Though no_ until· Leibniz will it be operating · af its greatest t efficiency, Descar,t,es tacitly assumes it ~o. prove his- deity benevolerat. But in scholastic thought one ·principle i~· supreme.- --~ safeguard contingency. The ru~~ of .sufficient reason amounts to a denial of that �21 principle. Until deservedly kicked back into ignorance, the "philosophers" will henceforth sit on the footstool of God. Every rule proposed as universally true must be either self-evident, or given in experience with a degree of probability approaching certainty. Whence comes this puzzling rule of clarity and distinctness? It is de- rived from an examination of the cogito, but why does a criterion so dependent on an indi~idual's mind ·come out? If I perceive something very clearly and very distinctly, and you do not, how can I possibly,convince you that it is true? If there be no fo~m to a truth, about which I can speak (and can speak because it is exterior to us, or common to us 1 all), then discourse can degenerate to .verbal squabble. The controversy between Hobbes and Descartes shows ·this with exquisite clarity; Descartes will have an idea be one else. thing~ · and Hobbes must ha~e it mean something And is there a ·difference in the self-evidence of "I exist" and "God exists"? Of the unclear · or indistinct ideas in the Meditations, surely principal is the rule of clarity .and distin6tness. : Go now from this rule, which admittedly has a certain intuitive appeal, t~ the. nature of physical reality. The material world can now be known with .certainty, since the nature of .material things is strictly mathematical. Clearly a mathematical explanation of the world. has a -greater clarity and distinctness than one dependent on some notion . of matter or mass as essential to material things. With unerring foresight, Descartes neglects to mention weight once in his description of matter; the problems that classical mechanics raised qver the nature of mass amply attest to the recalcitrance of this idea to formalization. It is . form which brings forth the clarity of everything from God to writing to matter, that is the sole nature of things; efficient cause is to become indistinguishable from formal cause hereafter, and both thus lose their original sense in the blurring of the borders. On the nature of things Descartes has imposed a great problem; if my intellect is an active principle, it must trespass on Nature's own ground. Where is the line? It must be said that ultimately we can never know the wax, though if God be benevolent, we can know it _exists. We know only the nature of the wax by a mathemat.ical description of sense -- phenomena. Nature has been re-def.ined so as to admit of �22 ultimately being intelligible, and what the world really is, we no longer care; we care about ordering the contents of our minds -- no more. Perhaps the notions which proceed from this book exceed the vision of Descartes, though his masterful manner show clearly that he knew· what he was doing; writing a book about the formal description of Nature. Death blows are often dealt by slender needles; Descartes' subtlety, insufficient to escape the censure of the Church, was yet sufficient to act as the first rock in the landslide. Not once going to the trouble to refute the metaphysics then in vogue, Descartes was never forced to define his position in an accepted vocabulary; dialectic between the ancients and the moderns could, and did, stop. Taking extreme care to avoid the Inquisition, Descartes by drama, order, metaphors, and ellipsis put into his book the seeds of a new creation -- a creation lacking even the quasi-determinate nature of the old order. Systematically Descartes destroyed the distinctions between the sensibly known and the mentally known, matter and form, life and death, actual and potential, pure and applied, nature and art, intellectual activity and the pursuit of truth, man and God, and supplied us with a distinction between mind and body that perhaps cannot be maintained. Though he reversed the world, he set it in speed, and though he doubted all, he left us technology; by the kingdom of Man he replaced the kingdom of God; and if truth be the final goal of man, or our specific excellence, justice, if man must reign on earth and wisdom be his only guide, then by his very novelty, his f aith a nd his failures , he has left us our proper gift , our most necessary asset, our nobility as man -- the certainty of question. * * * * * * * * * * 1) The Opus Oxoniense of Duns Scotus constitutes what is fully the equivalent of a Summa, and is certainly the only grand challenger of Thomas Aquinas' work. 2) cf. Plato, Apology 20d - 23c 3) cf. Genesis II, 1 - 9 4) Paul, I Corinthians 13, 12 5) cf. Genesis I~ i; . also, Genesis 1, 9; 1,14; 1, 26 6) It seems to me an important question whether one can ever get something natural out of what is contrary to nature. 7) cf. Plato, Sophist 247e - 258c �23 DE LA CRUZ ESPANOLA Lois Hoffman I've staggsrGd down through Old Mad~id Drawn to this, La Plaza d~ la Cruz Ve1de. Behold the ancient square, the rusted fountain, cluttered with lgoves, and Spanish soil. Of its youth, this only do I know; That human fleah burned here And blood ran from the lion's mouth In sacrificial torrents. Torrents of ancestral blood, To drown the soul With unspeakable horror. * * * * * * * * * CHILDREN The first Child Risa, child of the beautiful smile, Nymph-like, comforts her broken dell. The Second Child I saw him first in the Ibizan moon; I did not know his kitten had drowned. "Don't cry, Lila," he said. The Third Child The third child greeted me at the gate, A kiss for either cheek; She was a Spanish rose that But being · cut, withered d~y, away~ · But the number less child, · · Born of the infertile plain, Wandered begging through Ciudad Lineal. * �24 PRIZE-WINNING SOLUTION TO THE JUNIOR-SENIOR MATHEMATIC PROBLEM, June 1964 John Hetland Theorem: If a tangent to an ellipse meets the major and minor axes extended in P and Q, the least value of the length PQ equals the sum of the major and minor semi-axes. Let there be an ellipse with center C, major axis KA, and minor semi-axis BC. At any point D on AC, erect an ordinate meeting the ellipse in T, and drop the ordinate . TE to the minor axis. At T draw a tangent meeting the major and minor axes extended in P and Q. Let the length AC be designated as a, BC as b, and CD as d. =a 2 ; 2 4 a d2 = By Apollonius I.37, CP•d Similarly, cQ·cE = b2; co2= _ _ __ b_4 CP CE 2 KC•CA TD2 -KD•DA 2 _ b2a 2 b but by Ap. I.21, therefore CQ - 2 a -d 2 a • 2 TD 2 = (a+d)(a-d) = b2__b_2__ a - 2 2 2 2 • • (1) a -d • Since • Now let d ( 3 ) .! __ = (-a ) 2 ' . where • e is any number greater than-b. (a+b+e) Thus O <. d < a, for ( ( 3 ) 1.. 3 )1 a )2 ( _a_)2 = ( a.+ -b a ) = a, ( a+b+e) ( lim and lim 8-?' Thus, by varying for all possible the points A and same things will co ( 3 h. a )2 ( a+b+e) ( = o. the quantity e within this range, we have provided positions of the tangent on the arc AB except for B, where the tangent will not meet both axes. The be true iri the other quadrants. �25 Then substituting this value for d in ( 1)' . 2 2 )1 ( 4 . a b a (a+b+e2 + )2 PQ = ( 3 3 a ) 2 ( a a a+b+e ) ( = . - 2 2 ) ( b ( a(a+b+e)+ a . 2 ~a+b+e 3 )t 3 2 ( a · +a b+a e-a · ) ( ( a~b+eHa+b+e) b+e ( 2 . b ~a+b+e) h2 ) + b+e ) . = ~ · (ab+ae+b2 }(a+b+e) ~· ! ( . b+e ) ( . = ( a 2b+2ab2 +~3+a 2e+2abe+b2e+ae2 )) ! ( b+e ( . 2 . 2 ) '2 . 2 = ( bla +2ab+b )+e(a +2ab+b )+ae ( b+e = ( ( ( ~·b-~eHa+b} 2 +ae 2 .)' ) b+e 2 ) ) ) ! .l. 2 ) .. = PQ ( ·2 , ae 2) .l. ( (a+b) + -) 2 b+e ) · ( :2 e '/ o, ae then b+e / if -b< e < o, ·then~ If if ='.o,· ·e ae 2 b+e o, and PQ > a+b; > .o, and PQ > a+b; 2 ae then· b+e ·= o, and PQ = a+b. Therefore. the least value of the length PQ is the ~.um of the major and minor semi-axes, which was to be proved • · (corallary: the t~ngent ( . a.3) J.. d ' ~((a+b )~.) ) ., , : .rl • will have its . . mini~um value a+b when �;:i 26 WHAT IS KNOWLEDGE? A Plea for the Re-entry of the Poets into the Republic David c. Dickey All men desire to know; further, they desire the assurance that they have discovered the 'truth' in whatever matter they have been investigating. This formulation immediately raises numerous questions and objections: "What is 'truth'?" "How do we recognize the truth about anything, if there is such a thing as 'truth'?" usome men are perfectly content with the thought that there is no such thing as ascertainable truth, and yet proceed with the investigation of those things about which relative, or probable knowledge can be ascertained." and, even more radically, "The search for 'truth' is a mad undertaking!" I shall answer none of these questions at, this time, and, in one sense, I shall answer none of them at all. Let us look first at the investigating and un-disillusioned man, the man who searches for the answers that fue thinks of, however erroneously, as the truth. This man will probably call himself a philosopher and pursue his quarry with a battery of logical tools, methods of observation, and perhaps finally, dialectic. None of these seems to explain too much about the process whereby a man might hope to find out anything about that which he is investigating. A man might hope to investigate either things that are without him or things that are within him. In the former instance he senses those things tha\ are without, which is to say that he measures their effect upon his body in one of five ways. He knows a little about a thing as the light wav,es from its surface alter certain sensitive parts of his retina. He knows a little more about it as the particles from its abraded surface alter the nerves in the mucuous linings of his nose and mouth. In fine, it would appear that, at last, he is examining nothing more than himself. At the very least he is finding the relationship between two things, one of them known and the other not. This is, of course, what Socrates was warning us about when he spoke of the delusions of the senses in the Phaedo, and it is perhaps the really frightening meaning behind the Delphic commandment to "know thyself." °Know thyself," �27 we are told, "for in truth you can know nothing else." This .is qui ta possibly true, for when ·man examines nature, or - he things t that _ are without his body, he must both entangle himself with and separate himself f~om, those things which he is trying to -observe. The search for 'truth,' then, must be limited tc)those things that are -within us, but here ~ ~gain we meet the same sort of objection, for it would seem to be plainly impossible for a man to separate himself, as the observed, from himself, the observer, and thus to determine how the body-soul -that is a man· works. To do this he should have, by some device, to stop the life-process, to see i t in stasis, but this woul_ not be examining life d at all. Even were it - fruit~ul, , ~uch a procedure would be· impossible, for the device of memory, whether written or organic, must be a faulty one since words are not omni~~ignificant, nor memory ·- eliable. r .' - Perhaps if two or more men were to cooperate -- but again, the view is " bleak._ man's greatest device for wo·rking with · and cooperating with his fellow man' lang'u age, is not dependable for this task' for it is con:structed upon the very thing which we wish to investigate,- namely the workings of a body-soul which assigns to a given half-understood feeling or action a given symbol and then·strives to ma~ntain a one-to-one correspondence between symbol and perception. The problem of obtaining certain truth is now resolved into two problems, the problem . of clea.r perception, and the problem of communication~ - We . b the suggestion tha_ it is impossible ·for one person t have - een:·.laft '.: with _ to see the truth, and that it is impossible for two or more in the -search for it, since comm~nicati~n is not to cooperate acc~rate. 1 Wha.t do we mean by communication? We seem to mean a device -of persuasion ) wher_by the listener is brought to have- in his mind the same thought that e w~~ in that of the speaker. If it were possible, we should desire.it to -be a form of persuasion that produced in the mind of the listener, a true thought, though we are not sure that .either mind could have any •truth' to convey. If there is an answer to this question, it . may perh~ps lie in the dia2 . lec~ic o_ Plato's Republic ), that fourth part of the divided line, f 3 that, we are told will. lead us )·"using the hypotheses not as first prin- �28 ciples, but only as hypotheses" to "the first principle of the whole" and from there descend again "without the aid of any sensible object from ideas to ideas, and in ideas she ends." UJe should like to know, with Glaucon the route of dialectic, but when he asks, for us, what are the ways itnd turnings of its path, S.ocrates tells us that he cannot tell 4 us that, though he might perhaps show us something ) 11 like reality." If the t~uth is to be ascertained from hypotheses· used only as hypotheses, ' then it must be in a fashion independent of the hypothesis tha~ is used. But it i:s plainly not some logical trick such as that attempted in the Parmenides, of showing that a conclusion follows both from an hypothesis and its , contrary• When Parmenides leads us into like difficulties both from the assumption of the .Qr!!!. and the assumption that there is no .QD.!!. we are left With the hollow taste of rhetoric and sophistry, not wi~h 5 the knowledge df truth. ) We need something more before we declare that we have found a route to the truth. I. I, We know in advance that deductive I ' ' logic, the logic of the prior analytic can deduce .!J.Q. more than its premisses contain, and we must therefore distrust a priori any logical process that claims to attain certainty, or even something like the truth, from a simple hypothesis used as an hypothesis. UJe, the un-disillusioned, demand something a little more solid on our plates, at least until we find that there is nothing better available. But perhaps, if Socrates will not tell us directly what this dialectic is, the structure of the dialogue itself may contain some clues. At the beginning we meet Kephalos, or "head," an old man who, telling us that he has lived beyond the years of his (}vf-o's ? and his £.Tit (} I Uf U;\ is now ready to study philosophy without distractions, and ideed is admirably suited [or this pursuit. But he is a man who has not' made anything in life as a money-maker, and he will not do so as a philosopher. He must leave before the real business of the dialectic begins. At a later point in the paper we shall follow him and see to what place he has gone; for now we stay within, with the men achieve something. who~' apparently, These are the men of spirit and desire, the aggravated and the angry, the greedy for the truth, and those who wish to win ho~or by refuting the great Socrates. And now we see that there is plainly a reason for this, for the two men 6) �29 of w~om Socrates thinks . the most are the utterly impossible and irritating Thrasymachus, and the honor-greedy Glaucon, a true Timocrat . w~o with the best rhetorician in Athens for the praise of . the few. contests Neither of these men desires the truth any more than poor Kephalos, empty-headed, eager ·man, y~t th~y ~e~m to be the men best able to assist Socrates in his search for · it~ . if i~~eed, that is wh~t they are dtiing. What then is this Pt.e ces.s . for finding the truth, that it demands such strange qualities on the" part of its practitioners? for it, we are told, all of man's faculties must be trained to the utmost, yet we seem to see little result from the training in gymnastic and music, grammar, rhetoric and logic, arithmetic, geometry, solid geometry, and astronomy that we are told is necessary. We know that it is, in soma way, from the origin of the word, . either 7 r£~ a "talking-through" or a .. § ~"' - ~ ;O<>S )' something that is through two ratios or two accounts, perhaps something sL: -Al: ' common to t~am. W~ are given a rather poetic image of what it can do in one part of the Republic 8 ): "It may happen that when the two contemplate each other and rub against each other, that they will forthwith s~rike a light in which justice will shine out and certainly the vision which is then revealed, we . will fix in our souls." : If this is truly dialectic of which he speaks here, then we might guess at .what he means. When, in an interesting passage, Thrasymachus .bursts .upon the scene like an onrushing buli, Socrates uses his very conviction, his · t9vr/> i~ fact, to cause him to blush, and thereafter .something strange seems to have happened to him,' and he has .truly gained more than any other person in the room in his quest for the truth · concerning jus~ice, for, as justice is later defined in.the Republic -- the ·right ordering of the parts of the soul, he seems to develop into a just u_s man; perhaps his t3ur-Js and VC1 have indeed contemplated each other and seen each other for what they are, and as they rub together, and assume their just positions we ~ea the light of that fire in his 9 r-eddened face ). Sacra.t ea has won a great .victory in making this sophistic giver ' of discourses really care whether he finds the truth, 8ur-o5 ~nd riowi with firmly in the service of his newiy-emancipated vo Cs he may proceed justly -in the search for justice. Perhaps, in .this moment of confusion and shame, brought about by the refutation o·r his hypothesis his �-----·--------------------------------------"I 30 as an hypothesis, he has been given a glimpse of \ To ov >' of the first principle of the whole, and can thenceforth keep and nurture this plant which Socrates has engendered in him, achieving, at last, a measure of truth. What a beautiful answer this would be! but, of course, even if it works and ·i s not a glib mis-interpretation, we do not know what it is that · makes it work, and we may quite fairly condemn it as "mere mysticism" if we are unable to determine upon what that mysticism is based, if indeed it is based on anything. Why, then, does not this satisfy us? We surely did not expect to find a route to truth in logic, for deductive logic is sterile, all very well for "descending from ideas· to ideas" but useless for getting us to the first principle of the whole. Inductive logic, the logic of the posterior analytic is no better10 l., for it depends upon analogy, and the notion that what has always happened will always continue to do so. If there is a route to the truth, it must lie, at least in part, without the way of logic. Let us now leave the little-room, and its entombed occupants, and follow Kephalos. He has gone without to sacrifice to Bendis, the Thracian Artemis, a terrible creature who, at a far earlier time had caused the hounds of Actaeon to turn against their master, who had been changed into a stag to be torn apart by those creatures in life whom he loved the most. She it is who uses those dark, secret chords that run through all men and animals, uniting them by passions so unspeakable, so base, that they should certainly not be the concern of Socrates and his friends within the sheltering walls. men who have no tyrannical 6}-~;5 f7/L &vi'"' t '7<.. This place is only safe for old left to warp, and who pray to have their returned to them only for Kephalos, and perhaps other old men. But even old men are not immune from the dark forces that even the gods can only use, not mold. At a later time old Teiresias, wisest of the wise, danced with sage and powerful King Kadmos a wild debauched bachanaal upon the same ground where once Actaeon's shredded corpse bore witness to the fact that men are not thinking creatures alone. : Eu~{pides �31 shows us with a fascinat~d and a horrified .genius this scene at . the start of the Bacrihai and we watch ·as it writes its way towards another . scene, strangely like' but· far more wrenching than that of older days. There is a ·new god loosed in the land, a god who once more plays the dea.d ly fascinating tunes to which all men, . will they it or not, must dance. He is Dionysos come to claim the worship of the land that ,bore him, come to take possession of the people from whose .dark hatreds he is sprung as suroly as from his double birth of Zeus. Nothing stays him, for all the women, closer perhaps to the basic sanguine tidea, must follow him. Sven Agave, mother of the doomed King Pentheus is there, sucking at the fonts of milk that nature throws out to these her children now returning in their worship to their bestial base. No woman can be blamed for what she · does in these rites, for none can resist. no! We, the watchers, feel within ourselves these same tides -- but It is not thinkable. liver of the examined life. feel their pull somewhat. We are like Penthsus, the calm, deliberating We can resist these things, though we might "But what if?" we ask ourselves, "What if?" We strain •twixt wish and duty. We blush to feel the eagerness, but uie shall not give in. Dionysos now approaches us -- Pentheus. Would he not like to see the wild women ·disport themselves on the sacred mountain, casting their anguished bodies over ground where once brave Actaeon lost his. life? ( f"or watching women's mysteries!) He, for. us., wills to resist, but we know that he cannot. · Off he goes, will he ~- nil! he, the last and perfe·c t victim of that thing which is within him as much as in the pitied, loved, and envied Bacchai. The chorus sings for his destruction, of ripping him, impio.us male, apart• UJe feel their rage, and though we know it mad, we .cannot quite resist.· But quickly, as quickly as the climax of orgasm, the rage subsides, and, gasping, we begin to see again. .We now barely remember that we had a part in that rage, the rage against the just .man, the outsider, the .dude; the fink, and we watch, in washed-out resignation, the slowing .pulse of the play. In comes Agave, ' one . swif~ gl~dly I bearing the head of that great beast which, in ' motion of nature-given strength, she truding body. ' ha~ . ripped from its in- She has won great honor and wishes to show her son the �32 lion's head. Perhpas we envy her even now her frenzy, but we see the "real world" coming back, and the little of us that would be Agave is abhorred with her as she slowly sees • • • that it is her own son's head that she bears. We look at what has happened, and what happens, and we see, if we are honest -- and the poet, by his art, has left us little choice -- that it is all ••• ~· We do not puzzle over the meaning of the word, but recognize that here it fits. des.cent from madness is true. But we see also that the Thou·gh man may have his base in Bacchic fren~y ' . he must return to the world of the AD ros . There is something in him reasonable that, however artificial it may be, re-asserts itself upon him, or within him. We see this happen to Agave, and, in measure, are reassured. This, is poetry, and it has achieved something that may have meaning then~ for us avan within the little room. But first~ . what is poetry? Let us shy away from the easy ra~te of Aristotle, and· not say simply that poetry is imitation, though we have seen that it is that. It is something else as well. composed and presented to men, it must communicating. be It is plainly not logic. First, it is assumed, with the purpose of There is left the possibility that it is rhetoric, and therefore to be exchewed, for rhetoric, a later follower of the Platonic-Aristotelian tradition will tell us, is that form of persuasion which convinces by an undue and unfair appeal to 7 ,,Ct / the · <i7Tt. .uuf"'t.,o<. , and it is possible, by a deception. He further tells us that, to him, all things in the physical world are subject to demonstration within logic, either the prior analytic or the posterior, and that, therefore, when rhetoric convinces of the false, it can , be refuted by logic though perhaps with difficulty, for men's minds are often weak. When it convinces of the true, it is the weaker ·route to something that could be demonstrated logically, and is therefore to be shunned . alternative. But perhaps we have over looked somethi.n g. Poetry may appeal to an entirely different· part .of the mind, and we have seen that it does. appeal to the There is another tJ%,cf s , £ Poetry consists in those arguments which even as logic and rhetoric appeal to the and lTL (} vJL/-< respectively; but this is, in none of these cases, a simple demarcation, for the soul is not so much divided as zoned, VD C'S �33 and the parts are forever and inextricably mixed, which is how we may ,......., speak of the having a desire. Vou5 We begin now to grasp how it is that poetry qonvinces, for each man,in examining himself, and presenting the conclusions of his introspection with the art of the poet, is able through the audience, and without the use of the ;\; cros &vf""rJ5 of his in its more narrow sense to convince another man of what he then sees to be true about himself. Poetry is the artful suggestion of the true. art convinces, but rather· the combination. Neither the truth nor the That thing which is in itself true to the reader or the watcher of the poem is borne in upon him because 8vr!S i t is true, and as it is presented to him, through his the art of the poet. compel us, without a , by Poets convince us, and with the very sensitive, ~6rd of argument, but they must miraculously lose their power, or become rhetoricians, if they essay to convince us of the false. The power of the combination is so great that Euripides was able to convince us of those things which we willed not to admit about ourselves, and which he knew from a pitiless, and perhaps mad~ examination of himself, to be true. These things are true, for we feel them, but what is it that makes it work? In short, what are the answers to the covey of questions that we flushed in the first paragraph? moment. The first seems easy enough at the What is truth? Men have certain things basic to them and comm,on among them, for they are created of the same stuff, their elements, whether we call them earth, air, fire and water, or give them a name from organic chemistry with so many syllables that it would coil four times around this page. llie might, in desperate instant, even call them love, honor, hate, lust, treachery and virtue. Whatever these things are -- for the moment it does not matter -- they are common among men. Upon these are based their instincts and their weaknesses, .and that peculiar phenomenon recognized ... by both Euclid and Plato under different names, !'<Jt_Vd,.</ \I and' vof-~fl-~Vo< /\z_ ovt£s 0 ments that all men of good will must nature. 11) -;>/ Z:.VVDLri..£. • These are the simple state- accep~, because of their vary It is to be noted that these common-.notions can apply, conceding their origin, only to 'intuitions' and agreements about men and by man. �34 Though we may well never ~now what precisely is the substratum ~pon these notions are based, we can reccggize them, and postulate it. which It is these in large part that normally produce in us that very well-founded c'o nviction that we may now call truth. These common notions have a peculiarly force negat~ve 12) , for they force men, at least men of: good will, to accept them _ ecause they cannot, try b as .they will, find an alternative ·that is not repugnant ~o ~hat t~ing within them which makes them -men, animals, and (18tural bod_ies. in this ~ sense, Indeed, Euclid is a poet • . Some of his definitions are explicit, following from sense perception; some are arbitrary explaining _ ~he . author's own creation, but a few ("A point is that w ich has no part." _ h '!An unit is that · by virtue of which each of . the t~ings that exist is ·called one."_ do not show us the thing being defined, but lead us by the poet's art to that point where we must know what the subJect is, for we cannot think otherwise, and we accept the truth that is implied, though not demonstrated. What then is this dialectic of which we made so unsatisfactory an analysis in the first part of the paper? is from poetry that it must derive it~ It is in part poetry, and it strength. It is, however, a particular form of persuasion that has one man, the questioned, as its intended audience, and is able to work with him far more immediately than a general poem to a general audience. the interrogator is this: The problem that conf~onts He wishes to instill a certain truth in ano~her person, · thoug~ he is not ~et absolutely sure if it is indeed a trtith. He k~ows that ~oetic comm~nication will ~how that it is ·a ttuth, thus helping both parties in the search for t~uth principle that he has somehow found in particular. in g~neral and this This would be all very· well, but the one t6 be questioned is not ready to listen to any poetry, so the problem is how to· induce in him a · state of susceptibility~ If ha · is a man like Glaucon and Thrasymachus, a man who · has a strong opinion in which he takes pride, in :short, · one that is involved with his ()-o~J5 , the problem may be ·salved. I All the interrogator need do J I ·is to' iiiduce ·in the question a state of tftlTOfL<:X. , . a feeling that he has be~~ refuted in that op~niri~ or theory which he prized so highly. Then he will be ashained with liis t!f vrJs 1 ~) and be ready tc;- receive the knowledge, i f it is true, for since at this moment his &rJs is �35 very briefly -- in complete charge of his soul, only poetic communication could convince him, and that could only convince him of something true. To produce this state of :> Cl( / !Tof L.ol , the interrogator might at any time be called upon to use any of the liberal arts including rhetoric and faulty inferences from analogy: for the state of '> t:l(11 I of t_,,,<7{ is only necessary that the questioned feel that he is refuted 14 , it ). Poetry in its more formal manifestations cannot work this closely with its audience and so must limit itself, it it is to be successful, to subjects to which the audience is already susceptible. When po_ ts do e not do this, they automatically limit their real audience to those who are, either by training or chance, susceptible. But no matter which poet we read, if he conveys something to us through his poetry, we will find that it is compatible with that which we have received from all other peats, for none can communicate anything but the truth. There is a certain pleasure connected with the infusion of poetic knowledge, and since this appears to be the only way in which we can achieve certain truth, we may easily see why it is that man finds learning enjoyable. Might it not be possible that a man could receive true principles from nature as from a poem? Would this imply a divine Author? This question presents us with one possible answer to the perplexing problem of how an individual mind is able, as we have seen, to do something about itself that is true. But we do not need this suggestion. Even though the devices of memory and writing are not completely reliable, a man may determine enough concerning himself from them so that he will be able to present. something that is near en.ough "like reality" so that those elements of it which are true will be heard by the audience, and they will have learned. Poetry is present in many fields of investigation, even in some, such as &L~ Lft<itSJ whibh are thought to be strictly logical; a man may divide the whole thime and time again to find something that is within him, for though he knows not where it is, he is there, and ,like the man lost in the woods, he can guide the where he is. hun~ers to himself, though he has no idea Just as the hunters in going to the lost man determine his relation to the outside world, so the process of ~ivision, guided by poetic insight, determines as it goes the position of its object in the 15 ) �36 outside world ·of thoughts. Aristotle, the man who would like to root all knowledge of the physical war ld in sense pe.r ception, is forced in the fourth book of the Physics to appeal not to t~e senses but to the common notion that all men have concernin.g a "now" -- tha:t it is the boundary between the "before" and the "after." But be that as it may, we have delayed long . he title to this paper. t eno~gh the sec~nd part of That was: A PLEA:FOR THE RE-ENTRY or THE POETS ·INTO THE REPUBLIC .. but that plea is ·aver already, the poets, we find, are in command and always were. non-existent They founded the Republic and no argument directed to their cam~l' without the walls can dislodge them from the Acropolis. Plato fears them for the dark forces that they speak of and seem to manipulate. He fears the ba6hanaal, and is made of the same stL~ff judge~ rightly that the poet and talks in the same way• He feels that these forces are a danger, and must be eliminated, but· he cannot exclude them, for it is upon them, and the firm bedrock that they represent, that all logical thought, dialectic, and the Republic itself, must be built. The poet Plato cannot expel himself. * * * * * * * * * * NOTES 1) It may seem a little violent to refer to communication as a form of _ but when one is dealing with the mind of another, trying to communicat:i f with it, the situation is indeed analogous to the sort of persuasion whereby man seems to control nature by using her laws against her. persu~~io~, 2) There is no reason to suppose that the dialectic of the Republic is inherently superior to the two other major types of Platonic dialectic, but it is the only one that promises that it can attain certain, knowable, first principles. 3) Republic 511a 4) Republic 533a 5) In the October Collegian, Mr. Klein suggested th.at the rea$Qn the dialectic of the Parmenides seems to be so sterile i~ that .the ~artici pants are not including the possibility of the indeterminate dyad. If this is the case, then .the entire dialogue, insomuch as _it is logical in nature does not really b~ar on the possibility of ariivirig ~t . th~ �37 first principle of the whole from hypotheses used as hypoth~ses~ but rather from a sort of artistic implication. This ~s consonant with things that will be _brought up later in the paper. 6) . It must · oa gr~nted that there is some anomaly, i using·tha division of the soul from the Republic in a paper that is' trying to determine whether the method used to discover truth in the Republic is valid. However, the division of the soul is accomplished by agreement about personal : experiences among the vario~s parties in the room. Though this may not . be f9ol-proof method of investigation, it is at least a "likely story." 7) Although I do not state in the paper which of these I prefer, it is of cou·r se clear that r favor the second theory of the origin of the dialectic • . This tho.e ry ·would fit in admirably with the conclusion that I draw. · 8) Republic 434e 5. This is my own translation, though the las~phrase is borrowed verbatim from the Jowett translation • . I translate r"A~ in two places to get both the. idea of possibility, as with the oytative, and of immediacy. Instead ·of combininguKOff ou VTES and Tf1-'j30vTtS in~o the one word "frictionu .as does Jowett, . have chosen to keep. them I separate. The full quotation is: -~C){~ -rrf.x .,;v' Tfd\f) c(AA~A<:A o .Korro'C'vr£.<... ~~ T\Lgov'TE5--' a ,l , ./ .... , , / W<l1T£f ;J<. lTur~ ·t..'~, £~~o(J-L1fci.L ffOL7JQC:'-fl"EV Tr;~ r 6. L """ Lvrrv in;r ' "' fh/CJol 1U1CTof1-Lt9-o1... , f>~v Z-f ~ ~ J': v o '1't. vl v ctivr v rrcr.r 11/Z"v o<vToZ.s . 1 t" 9) This proposal is, of course, not altogether serious, but is intended as an example o·f the difficulty, and perhaps the impossibility.. of trying to deal with the . dialectic of the Republic in a logical fashion rather than a poetic one. This does not mean that - th7re is not a fa~ amount of truth in it, but -I believe that the ffv~ o ~ and vov 5 rubbing against each other to find their own place6 . is altogether too pat. 10) there .is a certain difficulty in the statefuent here, when we try to apply ~~at ·has . be~n said about inductiv~ logic to what ·Aristotle hi~self says in the Posterior Analytics (I 1, 2, 3 II 19) and the Metaphysics (I 1)~ . Thar~ is implicit behind all of his arguments for building first principles from serise perception the assumption that n~tute 'is regular and pre~ictable. Vet · Aristotle himself, in dealing with the pr~blem of Tv,X_ "1 and T::i. i rc5~d,. T ov . show~· tha.t what we call "chance" or .".l uckn is o~oL.rJ.. K..(..,t,; v-uµ_,(1Ef7J t<ov. . · . • tt,e further ~ells us, : in the metaphysics that 1tliere c'an be no scie.n ce of owf4 t<o\ Ta; Cv-?-f3z./l71 ko'v. Indee~ in the wide sense in which he ~ses the concept or c:J'°'v(JL"' ~ v{,/µ.pi.f 7 f<.d v . in the sixth book of the Metaphysics, (i.e., those/things which are only "sometimes" but not "often") it very much resembles n..'2rrf~1ov • Whatever, he concedes that all .~ of these things arise out of coincidence, and that, in the chance convergence of two causal chains a new phenomenon (vis: a rainstorm in the drought season) may occur. He does not make the next step and conclude that this new phenomenon might well be the mover of other things, producing, in the end, something new. If he had realized this circumstance to be a possibility, he would have been forced to abandon the assumption that nature is predictable, for it is precisely these things that "don't happen very often" which could render invalid the "first principles" gained from ~epeated observations. �38 11) Republic 348e 9 12) That the common notions have a negative rather than a positive force is apparent to us when we attempt to deal with th~m. We do not feel immediately their truth, but rather we find it impossible to contradict them, to find an "out" from such a proposition as "equals added to equals have equal sums." We cannot see any other possibility, for if one were to be larger, which would it be? 13) cf. Republic "• •• is not a soul equally . to be deemed halt and lame which hates voluntary falsehood and is extremely indignant at herself and others when they tell lies, but is patient of involuntary falsehood, and does not mind wallowing like a swinish beast in the mire of ignorance, and has no shame at being detected? 535e (Jowett translation) 14) We now see the full force of the scene described by Socrates at the very start of the Republic (328a) where he talks of torch-race on horseback, with horsemen carrying torches and passing them one to another during the race. This is, in fact, a paradigm of dialectic. The team of man and horse: yo v.5 and ()uf'--J ~ must cooperate to pass on the flame that has long ago been discove~ed. It must not go out! 15) This is the same problem as posed in the Meno. How is it that we are able to recongize the truth of a proposition about which we know nothing, or thought that we knew nothing? The device used in the Meno is essentially a poetic one, and we need not suppose that the knowledge of geometry was planted in the mind of the slave boy by the gods. The hypothesis, or wrong opinion which is already within the slave boy is ~ enough to give the dialectician a lever with which to manipulate his 1 (J-u~o 5 and make use of the common chord of humanity which binds the two -- Socrates and slave boy. Aristotle, as we have seen, also tried to solve the Meno problem in the first books of the Posterior analytic. Being unable to accept an "un-natural" (in the modern sense) answer until all the "natural" s·o 1utions had been exhausted, he first says: "Our own doctrine is that not all knowledge is demonstrative: on the contrary, ~nowledge of the immediate premisses is independent of demonstration. (The necessity of this is obvious; for since we must know the prior premis~es from which the demonstration is drawn, and since the regress must end in immediate truths , those truths must be indemonstrable.) This is an exciting statement, and we wait to see what this immanent knowledge is. But we find (Posterior Analytics II 19) that is merely the process of induction from ~epeated observations, a process that we have already seen to be uncertain. To this extent we must say that Plato is closer to the truth than Aristotle, for he recognizes that the principle of learning is within the learner by virtue .of his humanity. ��FOREWARD The following manual on the analytic geometry is intended to present the connective links bet1rreen the conic sections as understood by Apollonius and as they are understood by the moderns. The manual itself is divided into three sections, an introduction,, parts I and II. In the Introduction we shall present certain propositions concerning the conic sections which were either known to the ancients or which could have appeared from the point of view of the ancients even though no ancient ever recorded them as far as we know. The proofs in these sec~tons were taken from: Pappus D'Alexandrie, f:. eollecti.2!!, Mathematique trantlated by Paul Ver EEcke, Desclee De Brouwer et C 8 , Paris, 1933· (B) Morton, Pierce, "On the Focus of the Conic Sections" from Ih£ Transactions of ~ Cambridge Philosophical Society, vol. 3, pp. 185-190. (C) Newton, Isaac, Mathematical Principles, tr. Florian Cajori, University of California, 1947. The second part, which is intended to be read after the first book of Descartes' Geometrie, will present solutions to various problems of a similar nature to those dealt with by I:Escartes in the second book. Most of t.he proofs have been tak9n from: (A) Monsieur Guisnee, APPLICATION DE L'ALGEBRE A LA GEOMETRIE, OU METHODE DE DEMONTRE PAR Lv .ALGEBRE, Les Theoremes de Geometric, et de resoudre et contruire tous les Problemes. Chez Quillau, Paris, 17JJ. In connection wit h these problems it is recommended that the student refer to Q!l Analytic Geometry by Fermat which appears in English tran~lation in DRvid Eugene Smitb's ! Source 122.2k in Mathematics (Dover, .19.59), -vol. ~I, · P• 389. -· - · The second part of the manual which is intended to be read a~er the readings from .the second book of D3scartes 1 Geonietrie, concerns itself with ' the reinterpretation of D3scartes from the standpoint of • the a priori coordinate system. The sources were: ( A) (B) Charles Smith, !n, Elementary Treatise 2n. Conic Sections, Macmillan and Co., London, 1904. Smith, Salkover, and Justice, Analytic Geometry, John' Wiley and Sons, Inc., London, 1954: �INTRODUCTION Prior to the study of Descartes and the modern account of the conic it will be necessary to consider certain propositions which were or could have been known to the ancients but do not appear when the subject is considered from the Apollonian point of view. For this purpose, we must see what the ancients meant by a Locus. By the word ~ or place I mean that line or surface all the points lying on which have some definite property in common. For example, given the line AB (fig. 1), I call by the name Circle that line or place which is such that if any point C be taken lying in that pmace and if the lines AC and BC be draRn, the angle ACB will be a right angle. Or again, if CD be dropped Fig. 1 perpendicular to AB (fgg. 2), I say that the locus of any point c_ C such that the perpendicular CD is equal in square to the rectangle AD, DB, is a circle. ( I '\!. A.. k~/ / __' Now we must consider the /\ \ \) .' \) following propositions from Pappus. \ ;· \. Similarly, propositions 51 -------·--· ../ ' and 52 of the third book of Apollonius may be restated as loci problems: Fig. 2 given two points A, B, and the finite line segment, c, then the place1. in · which any point Plies such that AP plus BP equal c is an ellipse. In addition to the Apolloninn problem of the locus with respect to three and four lines, Pappus was able to present those same loci in terms of a locus with respect to a line and a point. sections~ · /!\\ / - i) " The Solid Loci Concerning the Focus and the Directrix* Book VII Proposition .238 If the line RS be given in position and the point G be given not in the same line, then 'if any line QU be drawn perpendicular to RS, meeting RS at U, and if the ratio QU : Q be given, I say that the G point Q lies on one of the solid loci which is a section of a cone. Pappus proves this proposition by means of the two proceeding propositions.· Proposition 236 Let the figure be as before, and let the ratio of QU : QG be that of equality. I say that Q lies on a parabola. ~at .. GV drawn perpendicular to RS and QN perpendicular to av. Now since · NV= QG and GQt= QN2+ ID 2 * Pappus d' Alexandrie, U C6llection MMthematique, P• 793 �-2- and QU·= NV, Tfuerefore N1J2 = QN2 + NG2 Now let VG be bisected at the point A and such that NN' = NG. But since VG,_ VN' + GN2 = VN2 and QW~ + m~2 = VN2 Therefore VG, VN' = QN2 • But VN' = VG - GN' and VG = 2 AG also GN; = 2 GN Therefore VN' = 2AG - 2GN or VN' = 2AN. But QN2 = VG~ VN' Therefore QN2 = 2AN, VG or QN2 = AN, 2VG. Therefore Q lies on a parabola having AG as diameter and 2 VG as parameter. let N' be taken on NV --, ,..i rt "I u-------- -/i\ ! ---- r/ / j 1 \ 1 · 1 \ \ I l \ I \ I I \ I \~ V j-----~~-~r--~c~ .-: , i N A: Q. Fig. 3 E: _ D; It should be stated that Pappus, himself, by no means considers what we have just given to be a method for finding the parabola. On the contrary, the above proof is followed by a sy~thetic proof in which he finds the line P which is the parameter of the proposed section, and then arrives at the figure itself by an appeal to Proposition 52 of Book I of Apollonius. In later times this property of the conic becomes a defining property and there is no longer any need to return to the cone in order to discover the plac4 on which such points lie. One might say that the place becomes defined as the collection of all such points. Once the appeal to Apollonius is dropped, one is faced with the following problem. How do you know that the collection of all the points having these properties compose a continuous line? Perhaps the collection has holes or is merely a cloud. It is at this time that continuity becomes a problem. From Pappus' point of view, however, the line still has a being of its own, and hence it is irrelevant even to raise tae problem of continuity. Pappus continues as follows: The synthesis of the place may thus be accomplished i n the fol lowing manner. Let the points V and G be given and let the (given) ratio be that of equal magnitude to an equal magnitude. Let us cut the line VG into two equal parts at the point A, and let the line P be double the length of the line VG. Now since the line A&, which is terminated at the point A, is given in position, and the line _ is given in length, then about the P axis AG we may describe a parabola AH of such a nature that if one takes a point Q on it and drops a perpendicular QN to the line AG, then the rectangle formed by the lines P and AN will be equal to the square on the line QN. (Apollonius, Book I, prop. 52) I say that the line QH is a part of that parabola. Let NN' be taken off equal to N~. Now since VG is double of the line GA and GN' is double the line GN, it follows that the line VN' is also the double of the line AN. Consequently the rectangle contained by the straight lines GV and VN' is equa 1 to twice the rectangle contained by the straight . lines· VG and AN. That is to say, to the ·square on QN. �l I -JNow let us add the square on the line ~N! ~hich is equ~1 to the square on the line NG; it follows that the square , on the line ·'VN is equal to the square on the line QN plus the square 9n the ~ine NG. Consequently, the line ADH constitutes the (required) place. Proposition·237 Again let the figure be as before. I say that if the ratio· Qil : QG is greater than the ratio of equa1 ity, then Q lies on an ellipse, but if sess than equality, it lies on an hyperbola. Since GQ2 = QN 2 + GN2 QU .= NV and NV2 : QN2 + GN2 :: QU2 GQ 2 Now take N1 such that · · · NN°·2 : GN 2 : :. QU2 : GQ 2 • 2 2 Therefore NV : QN2 + GN :: NN' 2 : GN 2 :: QU2 : GQ 2 2 2 2 Therefore NV 2 - NN°·2 QN2 + GN - GN 2 : : QU : 00 or NV2 - NN'2 i!Let NK Therefore KV, N'V Therefore · . (1) Now QN2 :: QU2 GQ2. = NN' 2 NN' KV, N' V ·: QN2 : : =VN2 - Qu2 NK : NG :: QU : GQ. And since we have already considered the case in which QU equals GQ, let us now consider the other cases. Case I Let UQ be larger than GQ Since NK:NG::QU:GQ Therefore and (2) ·KG:NG::QU-GQ:GQ Now take A' on VG in such a way that VG:GA': :KG:GN. Therefore VG+GK:A'G+GN::KG:GN or VK:A'N::KG:GN. And take A on VG in such a way that VG:GA: :N'G:GN s Fig. 4 Case II be larger than QU NG:NK: :GQ :QU NG:GK: :GQ :GQ-QU KG:QN: :GQ-QU:GQ GQ VG-GK:A'~-GN::KG:GN Therefore ·vG-GN' : AG-GN: :·GN' : GN or · VN' :AN: :GN' :GN. �Hcnco �l -4a- to tho plane AQP, and that the striight lino OS which lies in tho former plane is porpondicular to tho common section AA' of tho two plane, OS is perpendicular to ·the plane AQP?. Therefore, if a sphere be described from the centre. 0 with tho radius OS or 03, it will touch the plane AQP in the point s. Join iJE, and let 13DE be the circular s.oction: passing through BE; join VP, and let it cut tho circle DDE in D, and join OD. Thon because tho triangles VOD, VOB have two sides of tho one equal to two sides of the other, each to oa~h, and tho included angles OVD, OVB equal to one another, the. bases OD, OD arc likcWiso equal, and tho angle VDO is equal to tho angle VBO, that is, to a right angle. Therefore, if a sphere be described from tho centre 0 with the radius OS or OB,~tho point D will be in tho spherical surface, and every other point of tho line VP will be without.it, and tho same may be. shown with regard to every other slant side of the conG : therG fore tho sphere.. so · de scribed will touch the surface of the cone .in tho circle BDE. Lot tho lines DE' AA'. (produced if necessary) . moot one another in tho point X, and tho planes i3DE, AQi::> in the lino RX. Then, because RX · is common section ·o r .two plane~, each of which is perpendicular to the · plane VAA'.3, RX is perpendicular to AX. From P draw PR ponpondicular to ·Rx, and therefore parallel to AA' or AX; and through V draw V.F parallel to AA' or PR to meet ED (produced if necessary) in F, and join FD, DR. Then, because FD and DR arc parts of tho same common section (Viz., tho connnon section of the plane BDE with tho plano of the parallels VF, ·PR), i'ffi is a straight line. Lastly, ~oin SP. Now, tho section of the sphere made by tho plane VPS is a circle; and tho straight lino VP touches this circle in tho point D, because it lies mn tho. same plane with it, and moots it in that point oril.y; and for the like reason PS touches it in tho point S; therefore PD is oqual to PS. Again, because tho triangles PDR, VDF arc similar, PD is to PR as VD to VF; but ~D is equal to V13, and it has boon .shmm that PD is equal · to PS; therefore PS is to PR as VlJ to VF, that is, in a constant ratio. ihorcforo, etc. Q.E.D. .. It need scarcely be added that bhe poino S thus found is tho focus . . and RX tho directrix of the conic section. If tho plane of tho conic section be not parallel to a slant side of the cone, that is, if the section be an ellipse or an hyperbola, a second sphere may be inscribed in tho lower scbmcnt of the cone (soc fig. 5a) or in the vertical cone (sec fig. ~:e), which shall touch tho plane AQP in ~ a second point S', and tho··honical surface in a second circle, with regarld to which tho same property obtains. For to this tho foregoing demonstration is equally applicable. the letters with dashes being substituted ofr others, each for each. Thus wo arrive at tho other focus and tho other directrix, and in those cases it is easy to perceive that AS is equal to A'S', and AX to A'X'. Tho next proposition will show how readily tho inscribing of those two spheres leads to tho simple properties by which tho ellipse and hyperbola arc usually defined. Proposition II If a right cone bo cut by a plane which is not parallel to a slant side; and if two spheres be inscribed, as in the last proposition, touching the plane in two points Sand S', and tho conical surface in two circles DDE and b1 D'E'; tho sum or tho difference of the distances S~ and S'P' of any point Pin the conic section from tho points Sand S', shall be always the same; tho sum in the case of tho ellipse, and the difference in that of tho hyperbola. ��l .. 7PROPOSITION :r1 Problem Given J, K, L, M, and N, no . three . f which are in the same straight , it be required to construct the conic section passing through the given points. Let it have been done. And first let JN be parallel to KM. Let A bisect JN so that JA equals A~ and . B bisect KM so that KB equals BM. Therefore the line AB is given in position. Let it cut the figure at G and E: thus EG is a diameter. Let LZ be drawn parallel to A.B meeting the figure at z. Let KZ meet JN at P; NZ meet GE at R; LZ meet JN at C; and ll~ meet JN at H. Since the points J, K, L, M, N, A, and Bare given, then H and C are also given. But ZC:CP: :ZD:DK line~ l:et:~ and LC:CH::LD:DM therefore ZC,CL:PC,CH::ZD,IlL:KD,Jlv1 and ZC,CL:JC,CN::ZD,DL:KD,IM (cf. Apoll. III:l?) But since C, H, J and N are given, P is also given. And since L, C, and Let ZN meet GE at R and JL meet GE at s. Therefore R and S are given. And by the same reasoning as above, RA,AS = EA,AG. But R, A, and S are given. Therefore RA,AS is given. Therefore;FA,AG is given, and by the same reasoning EB,BG is also given. Th~refore as will be shown, E and G are given. Therefore the diameter EG is given. But GA,AE:.AJ.~2::GE:the parameter. Therefore both the diameter and the parameter are given and so the section may be constructed. (Apoll. !:52-59) Q.E.F. K and P are given, Z is also given. Lemma I 2 It remains to be shown that if the rectangles GA, AE and GB,BE are given; and if the points A and B are given, then the points G and E are also given. Let AB,BW = GB,BE and BA,AY = GA,AE Therefore WB:BG::EB:BA and . BW:WG: :BE:EA. 1 Pappus, Book VIII, Prep. Xii!. · 2 ~·• Book VllI, Prop. XIV. �-8But BA :AE:, :GA :AY Wherefore BE:EA: :GY :YA. Thus BW:WG: :GY :YA w B G- A- BW,AY = WG,GY. Fig. 7 But since the points B,W,A, and Y are given, G is also ·given. same reasoning, E can also be found to be given. f 'v' I and= By the Q.E.F. PROPOSITION IIJ Problem Again, let the points J, K, L, M, and N no three of which are on the same straight line, be given, and let none of the lines joining them be parallel, and let it be required to construct the conic section passing through the given points. Let it have been done, and let KN and JM be joined, and let them intersect at the point T. And through the point L let LT.n1 be parallel to JM, meeting KN at U, and the figure at ~~ Now NU,UK:LU,UW::NT,TK:MT,TJ (cf. Apoll. III:l7) But the points J, K, M, and N are given, the ratio NT,TK:MT,TJ is also given. and since the lines KN and LW are point U is given. Therefore NU,UK is also given. There fore in position, the give~ LU, m~ is given. Thus the point Wis given, and by the method given in the first part of the proof, one can find the conic section passing through the points .J, W,L,M, and N and it will also go through the point K. Q.E.F. PROPOSITION III4 Problem It now remains to find the axis of the figure, given a pair of conjugate diameters. Let AB and CD be the given conjugate diameters, and let them intersect at center E. And if they are not equal, let CD be the greater. ' 12i2.• , Book VIII, .Prop. IlV, . chap. 16 4 .... ·- 12i2.. , chapter 17 �-9Therefore E bisects AB and CD. Now if AB Gilt let rp bisect Angle AEC, and OP is the axis; but if not, through .the point A·let the line GH be drawn par.aJ..lel to CD; and on the line AB, take the point ·F such that ..-~···- ·-- -- · · · · ····<, - DE2 . · ,/,. . ... · , EA AF '· And let EF be .bisected at K. · / . · -~·-.. . ·s·-And since CD is greater than· AB / c_ // :. .· ··· ---·--\--; -·--·->-~ 1 K will fall between A and F. Now / . --. --....._ \j / ~ let the line KL be erected perH '··- ~:.:_ -·· ---·---·- Of~- -- .. . ~_\_ -~ : pendicular to AF meeting GH at the \ ---..__ _.._ \ i ..•. ·• /) - -. · - ~~-- - ·-· --) P • -----I . I ? . point L. . - ·.:. .."'-._ I / ,! -- ·· Now with radiu~ LE, let there be \ , _.~~::·.-_ /,~/ ______./D _ a circle described about L as \ / 1.1 ' ---. •. ?A ··· / t\ ;,.. center, and let it cut AL at G and · / , /" _ ·,·r ,.. H. Next, let the lines EH and EG '·. .· be drawn, and from the point A let · Fi~. /9 . the perpendic~ar AM and .AN be dropped, and take··-D ·--arra P on EH such that EO = EP 2 = GE,EN. · And take the points 2 and R on E such that ER2 = ES = GE,EN. Therefore OP and RS will be the axes. (cf. Apo~l. !:37) Q.E.F. = . . •' 1 ¥ ._ _·_ _ ·- _ '\ I / t· Pappus has just shown that if five points be given it is possible to construct the conic figure passing through the five points. We, of course, have only shown the case in which that figure turns out to be an ellipse. The other proofs, however, are very much the same thirig. I also say that if five lines be given one can find the conic section which is tangent to the five lines. The proof is taken from Newton and will require several lemmas. �But : :BD:AC = DG = F is matter I say :BG. �-11- . Therefore and or PF:AP: :GQ :BG AP :BG: :PF :GQ: :FO:OG: AP :BG: :AF:BQ. LEMMA :AF:~ . Q• . E. D. rrr3 It four sides of a pai:-allelogram indefinitely produced touch any conic .section, and are ~ut by a fifth tangent : I say, that, taking 'those segments of. any two coterminous sides that. terminate in opposite ang'1i,es of the parallelogram, either segment is to the side from which ,it is cut' off as that part of the other coterminous side which is intercepted between the point of contact and the third side is to the other segment. Let the four sides~ ML, IK, KL, MI of the parallelogram MLIK touch the , conic section in A, B, C, D; and let the fifth tangent FQ cut those sides or the segments KH, MF of the sides KL, ML: I say that, that f-··-··--·· ---- ---- -· ,_ 0\ .._..:, __ ~..1:',-~:---- ·-·---.--- ·- :;L --· __ ·, ME:MI ::.: BK :KQ, I ../ ·· \. and For~ ·:v /~-· ,, KH:KL ::: AM:MF ~ by the preceding lemma, ME: EI:..: AJ.'1 I I \ .\... //~. . ' I ; .., '·'\ I \ I / I / I ,l/' !• ME:MI ~-: BK:f:Q. -~ . ...-/ /"' -. '\. c or BK :BQ, and, by addition, . (\ _// I "- "'·, . ~ ,. , ----- :~-- ---- ~--- ··/'!< -·-· ~ ',_ B / . . "\...... KH:HL:: BK or AM:AF, Also / I I Fig. l .1 H and by subtf.action, KH:KL u AM:MF. Q.E.D. COROLLARY r4 Since triangle K is similar to triangle MFE, QH QK,ME U KH,MF~ But since KH:KL: :AM:MF, Therefore QK,ME = KH,MF = KL, COROLLARY AM, which is ·~~ also given. rr5 Let E•Q• · be a sixth tangent meeting KI at Q' ,· and MI at E'. Now it was shown that ME:MI: :BK :KQ. Therefore, ME,K~ = MI,BK • . ) id. Bk. I, Lemma X:X:V. 41d. Bk. I, Lermna· x:x:v. Cor. I. : 5id. Cor. II �F··:i . ..····· ·'-·-- ·----1:-1.t. __ ····-- -~- -· -··------~-· - - ...· L . . ., ...."'. , ,{---··· / · -12- . .. ''··,, ~ .,. But Therefore and or Therefore or = ME~ KQ' KQ' = ME,KQ -/ ME', KQ 1 :KQ: :ME:ME' ·~·: EE 0 : / / / ,.>1 i '"···~ / e, I '•• / / gQ :KQ: :EE' :ME' .' \\ ;, F'."l ME' :MI: :BK :KQ'. ~·:,BK D ·.I ,-I c. ·l '\ . \ '\, L -·----n / :QK :ME 0 • ; ---q:~§ ---_/t< l-f " ,.' ~Fig. ~1.2 /H Therefore if EQ' and QE' be drawn, they will . both be parallel to the base of the triangle IMK, and the line joini~ their mid-points will bisect the line AM and the intersection will be the center of the ellipse. _ LEMMA IV6 Given a line, and a point not on the line, find a line parallel to the first line and which will be tangent to any conic section which has the given point as center, and to which the given line is tangent. Let BF be the given line, and 0 the giv~n point. Now through the point 0 let any line MON b~ drawn meeting FB at N. And let ON=OM. I say that if a line KL be drawn through Mparallel to BF, it will be tangent to any conic having 0 as center, and to which BF is tangent. r ./-------~ -a. · Let there be such a conic, and let it be tangent to BF at T and let OT meet KL at s. Now since N . / / sr--/--~ T OM= ON Therefore I SO:OT: :OM:ON and /_,. / OS = OT. I\ / !\.r ' j/' ',..,_ )., ; ' • And therefore KL is tangent to the section. -..... I / ..... -. ... _ L .. / / I - - - .. ,.. Fig. l J Q.~.D. PROBLEM? To describe a conic that is tangent to five straight lines given in position. Let the given straight lines be ABG, BCF, G- 8D, FDE, ahd EA. TAke M and N on · AF and BE such that AM= MF and : BN = NE. And let MN be drawn Therefore MN passes through the center of the section. (Lemma III, Cor. II) 6 Newton, Principia, edited and with commentary by Thomas Le Seur and Francisci Jacquier, T. T. et J. Tegg, London, MDCCCXXXIII, p. 180, note b. 7 ~ewton, op. cit., tr. Cajori, Proposition XXVII, Problem XIX. �-13- -., Fig. Again·,~. take J . ...__ 4 P ::af.a Q on- B:O!_,'a.nd GF such that BP= PD And FQ = QG, .and let PQ be drawn; and let it intersect MN at o. ·And since the center is also on PQ, 0 is the center of the section. Let the line LK be drawn parallel to BF which will be .tangent to any conic section which has 0 as center, andis tangent to BE, (Lemma IV) and let it meet EF at K and CD at L. · Now let CK and LF be joined, and let them be produced to meet in ·R. Now if 0 and R be joined and the line be pr~duced meeting BF at H, H will be on the section. · (Cf. Lemma III, Cor. II) If .in this way the other four points throughwhich the section passes and the section may be found by the method of Pappus given before. In addition to these proofs, we .s hall include Newton's .proof of the five-line locus theorem because of its elegance. THEOREM8 To find a point P from which if four right lines PQ, PR, PS, PT are drati-m to as many other right lines AB, CO, AC, BD, given by_ position, each to each at given angles, the rectangle PQ,PR, under any two of the lines drawn, shall be to the rectang~ePS,PT, under the other two, in a given ratio. Suppose the lines AB, CD, to which the two .right lines PW, PR, containing one of the rectangles, are drawn to meet two other lines, given by position, in the points A, B, C, D. From one of those, as A, draw any right line AH, in which you would find the point P. e Ne-wt.on, ~·, Lemma XIX. �-14- 7 Let this cut the opposite lines BD~. CD, in H and I; and, because all the angles of the figure are given, the ratio of PQ to PA, and PA to PS, and therefore of PQ to PS, will also be given. This ratio taken as a divisor of the given ratio of PQ, PR to PS, PT, gives the ratio of PR to PT; and multiplying the given ratios of PI to PR, and PT to PH, the ratio of PI to PH, and therefore the point P, will be given. Q.E.I. .-- - \1- ·-- -~ - F . ) Fig. , .) Corollary I Hence also a tangent may be drawn to any point D of the locus of all points P. For the chord PD, where the points P and D meet, that is, where AH is drawn through the point D, becomes a tangent. In which case the ultimate ratio of the evanescent lines IP and PH will be found as before. Therefore drawn CF parallel to AD, meeting BD in F, and cut it in E in the same ultimate ratio, then DE w~ll be the tangent; because CF and the evanescent IH are parallel, and similarly cut in E and P. Corollary II Hence also the locus of all the points may be determined. Through any of the A, B, C, D, as A, draw AE touching the locus, and through any other point B, parallel to the tangent, draw BF. ~eeting the locus in F; and find the point F by this Lemma. Bisect BF in G and, draw!bng .the indefinite line AG, this will be the A\ r-= position of the diameter to which , \ BG and FG are drdinates. Let · c ~\ this AG meet the locus in H, and AH will be its diameter or latus / " ~ transversum, to whic~ the latus · I \,« rectum will be as BG to AG,GH. ~· . If AG nowhere meets the locus, the / / c.,-\ line AH being infinite, the locus ~ . / \ will be a parabola; and· its latus /" / rectum corresponding to the diameter \ v ' \ will be BG2/AG. But if it does meet .. /4 - · - - - - - - - it anywhere, the locus will be an \ _ hyperbola, when the points A and H ' e Fig. /h are placed on the same side of the . point G; and an ellipse, if the point G fall between the A and H~ unless, perhaps, the angle AGB is a right angle, and at the same time BG equal to the rectangle GA,§H, in which case the locus will be a circle. t And so we have given in this Corollary a solution of that famous Problem of the ancients concerning four lines, begun by Euclid, and carri ed on by Apollonius; and this not an analytical calculus but a geometrical composition, such as the required. · · � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. 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JOHN'S COLLEGE SEPT. 1964 �St tine s Grace Three Poems moses maimonides Francis Bacon 964 * 65 �s these areas or di ca ider these several s be ask cities .. tine considers for sta heaven! reference can better estimate its over-all these ques come discusses a �2 God rul cities.. ied to over the in this sense, the definition cannot over vice.. and reason rul be ap , the soul rul over an obedient Because of this tine modifies that we can define a his definition of love. a common ob which is bound cities vary in of true es of of 11 Thus upon the citiness which their citizens hold in esteem. cussions of as a group of We are ci tiness tine's dis- in , Rome and Jerusalem.. one historical and the last, with three is S One What is the of each as a It is of great is a is the first which was built founded to acknowl with the confusion of indicate that the ci this established in opposition to God and his a man s tion .. sel This statement is true in agree that the scale of true failure of the first of the citiness of veneration. does not necessaril e of the the alternatel dark and It is evil in its ends, in some sense it represents a manifestation of God s Providence .. sees the vastness and power of some as the the second is small itical. It as He often refers ike the , under one central rule, considered rule t would as will later become evident .. views of the Roman state.. to ects condemn the notion to God. Rome is an historical and this and itself, as one is true tine was eviJ. in its ends as it loved were false, and not be low to reach God, a which resulted in his • At first is considered to be an is the founda- man .. ti on from which man launched his an a since it states tine to be far and to the to that �3 vastness states small towards the Roms al izes ends, supremacy It is tual Jerusalem is a God s covenant with Abraham of God was tima tel and will Jerusalem will be the one true , the will be bo an those interests a l have the these three definition true senses acknowl and it t on the is be a understands the word commonl Old Testament allegory t alone will be a state possess scale of citiness It be the as the ect es of cities the can tine, considers a ect tine s Hobbesian terms a commonweal. bas There can onl be a in the absolute sense of the word, when God is the is not a true love and service• namel citizens it is and is evil. Rome is not a true But what come to , and as such the its nature as a the supremacy of one true because it too held the Jerusalem is ts s desire for man to Is it ills man s God-crea this question creation of man tima ive this assertion? evidence do we see to next question. held since terms revela- his fall indicate that first one considers the Creation says s �4 certainl that he be a , but that and the bond of concord commended UI, 21 , in Adam all Thus in Adam all men were created, and men fell. Both alternatives: beatitude and his man's fall, indicate that he was destined to dwell within a Had man not, "disobedient use of his free will" fallen, he would: pass into the of the and obtain, without the intervention of death a blessed and endless II, 22 the indicates tha Adam and Eve would have fall taken the tined number of saints, who with them, would dwell in the heavenl immortally -- for the life of the saints is a social life. A further indication of man's civic nature is discussed in connection with the ect of peace says "How much more the laws of man's nature move him to hold and maintain peace with all men so far as in him lies, since even wicked men wage war to maintain peace with all men. XIX, 12 Peace is man's desired end this desire can be realized. earthl He lives in cities so that nature. peace is the while eternal peace is tha But in what sense is the of the a It is clear that information about the heavenly from revelation and therefore we must tion of is comes solel upon tine s to , whether literal or all discover in what sense he thinks the , is a of the of God, the It is difficult to determine whether the heavenl more than the of the tined, God and one another in God. 11 (XIX, 1 to have institutions ike those of of the social life of this and Rome? Is this tine �5 a more that the life of a wise man must be could the of God either take a attain its were not a is meant UJha this social life? We are told of describes In its state the heavenly peace faith; and faith it lives it refers to the attainment action towards God and man IX In contrast to this statement 9 s Christian of Plato, and Platonic social order but rather rest Can seem life of eternal beatitude consists and Hims el ineffable sweetness I reconcile this Platonic of within the heaven! 1 31 with the statements tine makes social over "obedient citizens? tion a and another God of this state- firs tion is clear that God, the greatest fel the e for man. tion is more the How do we unders of one another in God?" remark that tine ife It seems that it is the existence of some sort there is this social the heavenl come The second half among men. be it will come when God is there a ? tine es es His obedient citizens between the sense �6 the conditions for the contem- social life is necessary to the most life to flourish be one where the situation would unhindered life could for social invol the tine would say that within this is not so. The eternal reward is intimatel consists connected with the social life, even in the the of one of one another diction in this because God which tells of It is this God .. y There is no contra- of God, the supreme another suggests that while the 11 life is to be basicall tive in nature, it will be embodied in the form of a deemed men, 1 The ife" of man earth, because s ' be in so far as it is and heavenl to the "social i t to exist of re- the social life. , both earthl lends its el the earth There is a for ted is s and not in the immediate manner in which it will of God,, in the ultimate There is the need for men to live within cities on earth to escape from the Hobbesian "state of nature .. s ti tutions.. We cannot do this without the This "state of na ture "unnatural 11 is, in real man' s most nature he was created to be state, since in his a social animal, and dwell institutions as such.. The as to God's man will be restored to the nature, and assist him to live the social one another in God .. 11 The of his , the institutions of the will not be necessary of the ife 11 , both y and , is rooted in God's Will for man to have a social nature .. IUl social nature among the In for his nature as a "social animal. the heavenl ultimate! with its helps man whose nature has deteriorated as a result fall in peace, and its in- would say accord this, is that God wills man's "the His creation .. and mutual peace ' 11 �es G Bell Is the garmen textured time Of their , so entwined Their very were one timeless! y Breathed cleft and e Seek and Is it the falls ed come Be as dream The and years and self-abuse memories than Less ts the else �8 EM CORE Charles G. Bell Marianne: I swear, if Eve seduced Adam to eat the fruit, to what gauds have gone unrecorded Or Lilith It was not of Of song .. brow or breast or the dear hoarded ' and brea the t old UJitness second ; and when you s flute to be that song for the secret savor Your l Of lost romantic I but ' you so confound me, wives, loves, whores To lie in the falls of your If ever Off the end, it is the drowned Yours forever, Charles. Schubert, �9 Charles C Bell To climb, that l resilient, To take To air all our arms , branch over branch, the a And trail for the nurture of centuries bone of , terraces of green to be, to unfold the one Here, earth marrer, lift the dream to be life-rooted, l are: of your prayer, that tree. �-10- ON INTERCALATION by Moses Maimonides is a very strange thing for it is to say whether it is a product of nature or of custom. On the one hand, this seems to be a treatise on astronomy and yet different peoples have count from but Not Nonetheless, each is based on motion of the same the present treatise was written many years , it uses a method known to the Babylonians prior to the rise of the Greek science of __ I've in the order that the have some notion about that inherited and to is to say or not geometry role in the earlier science. On , there are no any such on suppose that a man could spend his the other hand, it is hard fe recording the motions of the and searching for patt1rns of reoccurence on the in which the moves. The we in nature but on the has · .. foundation not passages as well: for each new moon of the month of offer a passover your God, for it was on the your you from Egypt. 0 to and who One the us are not comeach month that each new moon start a time fall during the same be composed new month month wmich we are of days �a new moon .. that summer ~ime of the year. is to so the not commensurable , even though the each have s means that some and some others 1 that some will more of the new the The believed to of .l..l.IJ.\JlJ.J~U.\.J text, .I fear that one translation from a J.J.O. . of wrote A Thou of me -- may the and in which thou thee 6 exertion, this means a teacher any of the ideas for thee book alone will not be for thee since its the teacher not be needed, but rather that in a conhim I mean that in that time it taach the names of the terms of that science and the which are lead thee on the way .. will make a that thou shalt and Teach resume to make in need of stead fast in which are not from other But as for thee the cause of the but I be brief over any take place of a I have not set it as my of the arts ifll'hich same is it even to one who is ) many other ch come was not to alone.. Now �is seven. se number the Second f s the is know the number of of the week the The se are �each out for thee to be of the for thee hours since for thee on of that from the with the sun at a a time is 29 , 2 hours and is the time of a lunar When the s is the number the total the account the 7 years among the �4is that the is 9 years of the birth of each we which totals s from the and 793 is clear that the number of a lunar month is when thou the , and 876 remainder is we have made year is 13 lunar when thou takes the number of a lunar month and it 3 what totals from them be 383 days hours and 589 when thou s from this there will remain 5 days 589 The is 5' 589' Behold It becomes to thee that what remains from a lunar month after thou hast lunar month is 1 12'79J 0 • The remainder of a the of an intercalated year is the of 9 years Jhen thou take the times and the the two 7 times temain 2 6 hours and 595 what remains from the number of days of the tr acted I, write for thee should be for thee; for the art is we have shown for thee to Know that the first year of the years birth of that The \1 of ~ the hours of hours of the of to an hour thee from the from this �5 mean matter, but thou of the fourth from to the of arrivest at the birth of in to our which -- and that is the birth of sions. The for we added We birth of the lli we clear the know the birth of the this one raise ·whatever remains from the column of go back to the hours and whatever to a to the column of not fill up write on the s from whatever totals more and subtract When thou hast done accoron the the birth of the is after this -- that is come out for thee to 9 hours and 949 hour. Thus add 2t1 595 to 260, and there thee the birth of In the same way do write for thee the births of 0 to Teach arise in of a teacher. out the in the way which we have for thee table. If what came out for thee agrees in know that work is correct and if there comes out to be between the two them a of know that that is a mistake and answer agrees. the of the of each year. year of a is known in the we have to the first birth of the years the birth of for thee this The birth of the was Ue added to it hour of hours in form sions. \,l]e i'llTote it on the third the column the the 3 hours of the hour we to the 8 hours of the line and the total that, we with the and the knew that the of the was of the �6 want to the the birth after it of each �7see Form Behold! we and so that it is not made clear and time to be clear from the become clear to him in the thus from the the deed of out the birth out the birth of each month goes on same way in the same way.. Know that for the wise men the to the conjunction of the moon and the sun in motion, and thus thou shalt it with the of Name the Know that look into my want, as I year be for become after the Deferrable to comes out in any of not be on that of the the and thou must leans upon it s of any month that thou thee to know on which New of that manners that birth The rest of them There are The First Foundation: v-Jhen the birth of a comes out for thee to be 4 or 6, New Years of that year is deferred to the Do not whatever hours you have, be few or many. of what is like: if the birth comes out for thee to be on any hour of the day or night , New of that year be if be the birth comes out for thee to be on Day New of that year comes out for thee on 6 New Years of that year 5 if the it or .. The 7 Do are never as New Years) But Day 3 or 5 or The is The Foundation: vJhen birth of a year comes out for which is for -- as New Years -- and An at half the more do not fix Years 6 hours what this is If the of any was is 2 11 and of that year J: was at less than even one -- for if the birth was 2 11 1 of 2.. for the rest of the for thee to be 2 at 8 hours or J; if the is on 3 at hours or more will be 5, because deferred from 4 to 5. We have made clear that New cannot , when the birth is on Day 5 at 8 hours or more, same year be on 7, because it is deferred from and we have explained that New Years cannot be on when the birth is on at 18 hours or more, New Years of 2 since it is Day to 2. The sign of this is what the wise men of Memory ( it before noon, it is afternoon noon is as �8 way �9 ... Full or and if there or 4 3 is on Day 5 there are between them 4 2, the year is of the 5, 6 , Day 7 and the 7 and the there are between them 3 days: Day ?, Day If the of which thou dost want to know is intercalated, do this verv same ) there are between the day upon is the day upon which the there are 5 between them the year is Deficient; if if 7 , the year is Full The 5~6v D0 0'F', and is the first an intercalated year is on ? and the that year is because between the 2 days there since the on Nhich the year which comes after the year want to know· is fixed, is not in this count as we made From what we have it become lucid to that thou are not to know a year is Deficient or Full or after thou hast that year and the year comes thee all the years of this our cycle in a and the Teach f by it the form of the see Form 10 • Know that whenever DllF' year New out. 5, if the year if it The Day 3 that year is always that is that when New nor Full, but but it is not it can or Tuficient as the which D and 2' D° Know that when New is it to be Deficient, and to be The in a is go back to make st to w:ha t we have and the and the year comes out for thee to be when New Years 7 or 2 or it comes out for thee to be when New Years is 5 and the thee to be when the know that there is a mi it is found. When thou New Years and whether the year is Full or without a doubt the of each month is known. When the of each month is the appointed times are known the are kno1,m a doubt �of Of the ��-22- the month of Nisan of the year which we is~ 918 of the creation ( year) of the of the month of Nisan of year to be 22 days from the of the month of Nisan (which was Day · came out 1, since there are 22 days between the not reach the day of the as we We 1 to 22 2; we reached to Day 3; we added .~3 days to 22 the day of the which is Day Then we knew that the ........_..._..........._ ·or year which is year 4 thousand 918 of the month of By subject thou the month the of Nisan be thou to know at which '-'""'-'""" ....... L.•C:.. .i.•..i..;"'o.... ,, .-..-~~-- in case of Nisan is knovm us at 1 hour clear from when is not but of fir st hour of' of the 1 so ��I. v ��x �kn owl of the e 1 3 2 4 9 3 4 3 5 4 7 3 7 5 6 7 8 1 2 9 1 6 2 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 5 6 5 6 1 2 3 4 6 7 1 2 4 7 1 2 3 5 �1 be seen from what we �x year a in an x excess over excess excess one each year years years excess e:gcess excess years years. excess �30 20b the consequence of , there are either months of must making the this the name of have as �remembered not ris a is s treatises faceted; as seen s neither an ancient a s a ta and doctrine are all �32 about the "natural method of scienti fie invas ' of the ti on reason proper the proper end of and (to name but a few of the many new ideas) the nature which exist in the world. to the first I limit myself because it seems necessarily both to the refutations and to the introduction of the new method: can talk about Before one kn owl certain methods of ess, I think, and before one can dismiss are (or false, and before one can reveal the true end as and the method to attain that end, and of to be true about the outside of himself, one must show how the ition of know! is In order to refute the natural human reason, it; he must for so Bacon must re-evaluate e been so mistaken and unsuccess in their He must talk about the of knowl interactions exis between the soul of man and the nature of , and ultimately he must show how this "commerce" between the soul of man and the say refutation, in themselves 11 I use Bacon s word; word, for I think that, of the can be effected When re-evaluation 11 is my own his harsh and ery ---- and that ultimatel he must either fall back upon some of its conclusions, use the force of its rhetoric order to new method or As I a ~.....__~- to a coherent the a conclusion of the of the success of the he of of this "natural human reason" and its relation to the world outside of man, I shall refer often to the the differences and similarities of the two �The as is e as for he is a master at Bacon would to lead men be raised is this: The question establ between the has thrown into confusion all affairs the divorce of Preface their words ? (in the mos that we mus em before ps is not also I a and rational facul cise order it is mos the necessary emics, and pre- understand can decide as of Baconian method I the true method of means first shows the Bacon says that man's knowl false the candl of distorted men these secrets e and able to as is necessa hers el because are too weak recesses man s true vocation to into the violentl causes of "act. been wa and laws But until now this has not been the em in most cases, s causes �34 wha tness 11 of and ca us to nature, (that is, for final and formal causes, and for the answers to questions like, what is find out ) rather than to work; the first are not the nature of tell us, rather man s unders The to the and which about the nature of , is weak, confused , and prone to abstract too in icable to notions which do not therefore, are of little, if any use man s dominion over nature. Bacon wishes to restore man to his true first des ce as ruler of Nature all the false notions and habits which stand in and then the way of man a method In certain cases means of which he can discover the true causes of the former however, cannot des out that the these habits but can has certa tendencies, so tha this insidious action of the mind may be ranked and Preface, 445 • His method, then, with a way of the to abstractions; its it is a kind of non- tial (that is know! the exclusion unders ex- is tabulated and retabulated so that after of all instances of the , the can at last be allowed to consider the "affirmative instances , arrive at a true and useful further is a s can ; Francis Bacon "it to be for the interest of the and future with his that should be He sets as his task the in- unders Bk a " which "readil from the Latin ' XCVIII) means a Bacon and notions which are "false to stand, tion of the a reconstruction, to some sense stores up confused, and and even, upon the human mind and accumulates abstracted �other sa is clear Pro em is at once ristotle, Bacon conceives of a mind both and a The an seems necessary to Bacon s telian to have an upon the mind ~~~ but at the same time has to the wrong conclusions Nature, men come up the fa ct that other blame Nature it necessary to blame the mind postual te men can be active that Bacon conceives active in the it is wa t is concern is ways, doesn therefore, since ha or pursue the wrong want for he is described skill full it is another t this is a of for in a sense 1 makes colours into actual colours. unmixed in this sense of it is since J Bacon so t so that i eternal, prepare be immortal e Nature herself have power We notice that enables of r~ature the are ..i.;......_.;...;........-.~.;;;.;_~ direct into nature than the former the �36 than the restitution and due and a of nature. II He uses the of effect upon the --~~ fa cul ties of the mind, to ......_~~- to the effect which the mind, and to axioms will have upon the unders axioms can to nature and its often, to of (and, of course, if the 2 te into nature, so can the unders clear that the mind, as Bacon describes it, does It is have an innate unders , and what else can the than over nature mean but that he is more hims el those , exercis of natural means , become immortal ultimate power over Nature even if he was to conquer, seems very like this he had positive state like l am mainl Since of the interested in the of the mind as Bacon describes it, and in the rela its "activities from the and the of those which are called the Idols the crucial s its most s to be I the discuss the Tribe in his re-evaluation of It is here that the the unders in one ' Refute the natural human reason" which of which Bacon comes between instances.. is used of 1 is said The unders like a false mirror, which, distorts and discolours the nature of nature in i t .. The refutation Bk .. I, XLI the natural human reason because Idols upon a number of hiddenness Nature, 2 not ; among them are 1 ) the not from the to be discovered Truth" itsel all the Idols is the fact that any true knowl must come from Nature that there is its tion), 3 al of Nature the belief it does and 4) the belief that man If man couldn t know, or if there were ��38 sa wa remember 9 tha of sense the senses: is misled hindrance and aberration dullness Our notions • • the immediate as Hot, Cold, Black, White, do not even are sometimes confused teration of matter and of one The senses are the the sense mislead us flux and alanother cause of all our mistakes; more subtle than therefore statements about is famous nature" are the necessary resul man s certa is to attain if kn owl For all himself. senses 0 the and of its nature prone abstractions to are to resolve nature into abstractions is less to our purpose than to dissect her (LI nature a means of these arrived at almost all cases, the remote axioms relation to because from which are useless, the "middle axioms" are also useless The Baconian method ascent" in contras , "moves in an unbroken chain immediate to final axioms a of nature he has were derived, and so tha its and it never loses way up, then are ever watered. of the back into the font 11 Thus, says Bacon his method, he have �has thrown there is tra about Bacon s claim tha his method these habits and case form abstractions and Bacon s method can When the time comes to an abstraction land? is I contras into between Bacon seems as addresses is described himself the refutation" the conclusions see a I evident: cause be Bacon: on inveterate to and raise the sciences have a relation to or �, and the sense which is in turn divided into ~~··.. ~~-~tive, mind. or and elements Sense and intellect are of sensation or the interaction between the the activities of from the and and which are outside Since these its el two faculties have "no nature of their own other than that of a certain cannot make mistakes when (III, 4), are the of sense or of sensual and like are are the so-called about which I will sa little that which makes a to be what it is, I m fa of an animal or forms" which are are sometimes of in this of as the "essential that it is certa tion: the of the definition in the sense the constitutive essense is never in error nor is it the assertion of but t as while the of the can never be in error the belief seen is a man may be mistaken so too in the case of which are without matter. (III, 6 Forms are often identified, both Aristotle, I think, and Bacon, with final and formal causes is a this soul the forms of the intellective soul, and i t is clear that for the is mind is ~mµv~~ible without sensa upon are somehow derived from which for stimulation, and these are the result of sensory stimulntion of the of sensation is it.,. (III, That is, the mind upon the mind is work when sensation is not are intense stimula up in the memory and 2 the mind is , while sensory organs are �a for we can is when we are not mind is that is thus al to outside stimula can the and acted upon think without To of an we can think whenever we want to even if we don t"believe 0 whenever we want to as Error is and we can a fallacious characterized as That is white as white can never be mistaken true as is the cause our this fact the assertion .. this kind of assertion or a role assertion is false we have not the forms are of means of 11 discursive can be fallacious this this action i t is said what sense is is universal 5 these as a is sa as well as where there (II , 3 is the desire and reason has an �movement with is never found but has no effect upon the statement, and its mind. II But, the above ica tions to the mind is always , but either or wrong ( Therefore, may be an examina and tion the conclusion is reached that inasmuch as an animal is of self movement; it is not Both e and calculative tian itself both of which are the seems to be ists sources of error in the soul this in God forbid tha we should for a pattern of the world says is out a dream of our own The Plan of the Work, llie see, then, that for Aristotle the mind seems to be the one active force for stimula upon the senses i t moves an animal desires animal mus that active it too is at first in the soul, the is have an of the desires are a result of tha ti on for which will resul from an action before he can be "moved" to to obtain it. for these activities and the soul has no nature of its own other than a to be acted p. 451 sense) is therefore be contrasted was mentioned earlier Bacon certain Thus: that the soul is to the hand; for as tool of tools, so the mind is the form sense the forms of sensible III, 8 That is, the hand uses tools, and the mind uses forms, that is made actual The Forms are those them and this is means of which that is ense �a cross a a sense as for tha , while each other sake of and final causes are sometimes that forms both the form and the final cause are that an animal "strives , Bacon says Matter rather than forms should be the for forms are of the human mind those tified them tha of of wherein it is diverted from more solid are abstractions, laws are such ther than Idols called the Tribe one 7 and the mind s most its proper was That is, mind works advancement he was itself as does tend forms O, section is man s be made, as less the • • • idols of that can be done �44 insidious of the mind (The Plan of the Work" 444-5 , so that and It is of this nature that the mind must be forced ta follow the "true the pattern set for it of induction. Aristotle 82con seems to be in the marked of ties of the soul, 11 but I think that it is very clear that u he does not blame all error on the As the at the effort is to effect of this section show, his a union of the and 11 cont.err101.a tive" intellect, of the But it seems that an additional facul , if such it may be called, necessary for this union some of the soul, that is able to stand outside both the and oversee the and the Bacon of both.. no name for this that he the now that I have and mind, it remains that I and as it were in a favorable before Bk .. 1 and leveled the floor of towards what I have to in the human unders For I am the world, such as it is in fact (Bk In his 11 refuta tion used of the a cxx 1, Idols of the Tribe, Bacon has both Aristotle's theories about the soul.. while which is mind , but if it is also himself can be e calls those in the commerce between all sensual llie see outside of man the active forces and the soul of man, and says that ions, upon those outside forces, Bacon, al that we cannot know Nature without the use of the senses and that the senses are therefore when he says that we may be deceived even about True the senses are at the mercy of choose to show herself, and is the "active mind 11 must devise herself to reveal If, as says, the mind is the form of forms ��46 the in the but insists that "abstract sensible viz both states and affections of sensible are not in the sensible forms; which govern the states and affections of sensible in Nature, but "abs tract "restless motion, 11 are of the mind s and of its eventual restfulness which is the most destructive state of all. True, what actual sensation is individuals, while what is universals II, 5 But if the so-called "universals" are forms or final causes, the conclusion of this statement universals and these are in a sense (II' the is indicative of the truth, but comes no where near within the soul; are divert it from the true choke it, infect it, , and the of the unders out to desire these "universals" is one which can be and ; it cannot be rooted out. IV The outs 1 is one which s new of the underthis essay, th2t have is innate about the innate, Idols of the Tribe for it is these which are the one of the mind s favorite idols is its tions to be subtle as Bacon calls it. linked with the fact that the that all of Bacon s careful or that we are cannot be done away with can we know where we are to land at a about the motive forces in Nature? as his t that final moment us away from the lists of That is, to abstrac- But this, I think, ( and flatteries, as , cannot when it is time to on an axiom. to which and land to land, a truth How, when the mind cannot in �are sure a let and necessary e this. �48 But if anyone conceive that my Forms too are of a somewhat abstract nature, because mix and combine geneous • • • if anyone, I say, be of this assured th:' t his mind is held in gross appearances of most certain that these and alien from each other, agree in and that the power of man cannot poss and freed from the common course of nature, and exaulted to new efficients and new modes of operation, the revelation and of of this kind. (Bk .. 2, XVII) The Aristotelian: all the ' and, as it were, And the destructive (to this examination): The· more ancient of the Greeks (whose are lost) . . . . did not the less follow up their engage with Nature; (it seems) that this very question, -- viz. whether can be known, -- was to be settled not "Preface", 456) totelian 11 answer is very interes in the l of the but surel the tables are not in the way that Bacon s rhetoric is -- a moment must come when the unders the forces of persuasion, comes to its when it is clear that the mind is But in terms of s conclusions, the of extentions of Bacon's ("of this of the under- and its intercourse with nature, the last answer to the question is the most If we remember that Bacon tells us often that the true method takes a crooked s , unaided with the senses, then to to (for purposes of • of ascent, axioms, then ' as well as Truth therefore and are here the very same and works themselves are of value as pledges of than as to the comforts of life. (Bk. 1, CXXIV) y �we can see tha a very on the one hand is be can't be sure of an axioms truth until i t to the and see if we can said we could do, and if on the other hand, one of the innate and then of the mind is to first form an all other new information so it will seem that true may say then is not nature of the That is of the it is not so much that are one and the same, but that, for purposes truth and , what proves scientific .;.;.;.;;;;:~_;;_...;...-;;;._.;;;..;;;.::.~ be called a true cause, be useful since we cannot to make mistakes of the The our examinations we are able to say that we This is govern about the motions of natural bodies that if we can re- this law to natural bodies and effect a be direction in the therefore t is and util finds this facul Truth in the direction of are here the intellective then we will same which knows Truths as such, all those in soul Bacon most prone to err. and t consequence and if is possible * 1 Roman 2) c in the we are will say tha efforts succeed * * * Henceforth will not as a rule be indicate the number of the in which the appears the case of books other than the ~~of book will after the tion Preface; 433, Preface; and I Bk. CXX Bacon is interested IX, as an of Nature �50 To say that calculative mind to Aristotle s ; the tion' seams to me to be a 4) s one .. C. F. Bk .. 1, CXX, XCII, CXVI, CXXVI ( 11 ! do not sl , but govern it" .. the under- �Class: Members of the Gradua carry ' with courage and will be times when can muster, all the which you are about to new that you well and to express first, to Permit me For many of you, I in restrictive circumstances where the find chances limited and action seem you will manage to these narrows, and to discover in exercise of your powers. a sa what I should say , however f el moral ities I feel this situation evasion. some What I shall be direction of at all it is because to or is an intellectual conscience. find f a member of a and learners who live now or who have lived and written or pass over true. t see a of its revelations, as a scroll when before. I silence the intellectual and social What is If I can said of means disconnected. seems the age we years ago, t ever shone before intellect trace the rise of constitutional steadil bend a it to trial and execution see the t their rulers into a and and the men and the men and every bondman and the rocks of the mountains fire expose the miracles, the the very 'heavens him sae the restless , see �52 immense distances violence to , and in such a fire of as less salamandrine could not have endured; and he who viewed Hell without would fall to the earth before the terrific of intellect which God has scattered broadcast over this whole age. 11 What, one wonders, could Peirce add if he were to return f ram the A. Lr. C under melvin Calvin, a molecular that the next in human evolution is the and of outer upon the be undertaken, because space; that this s that it may suit man in the each future to and read the orbit of the moon; that there may be other similar at some millions of other in the universe; that our view of life and of man is of the of a trivial to that of a ma come to this view entirel inverted, from that , we cosmic influence; and that, tal and observational upon the basis science and scientific Peirce s next reads: This 's taken ery -- clever feats -but this age is that in which once more from Revelation the sun becomes black as sack-cloth of hair, and the moon becomes as blood, and the stars of heaven fall unto the earth even as a tree casteth her , when she is shaken of a wind.' reaction to all this is less bra than visceral. man as a frail creature, a reed, reed, as Pascal says. Phrases like 'the conquest of space' or And I of nature' seem that there are ourselves and our machines I tend to think of than that of more the earth .. the surface a bit of the available and I can accredit ourselves as modest ways may be able to do , and in the earth habitable toward the waste of natural resources, the wanton destruction of trees, and to persons violence and our own, and insis the called upon to deal, and barbarism; on the enforcement of civil and criminal law; is of the situations with our in the scales standards in el and t fanaticism and education; and in other ways which will occur to you or be forced on your and my attention now and later.. Some of the details are of human life and e, but the end is not. not be the The t e �53 most consistent, we must 1 f a scale the means as universe is it is drawn and the stars are as small as dimes Now so many voices of more every in such various tones of sober sensibleness, of threat and stern realism, to truth there is an dreams All will agree: there axes hones , all will will agree. is I is reason conscience and can learn it well discoveries The course is a series a process is is a process schemes, or deductive lucid for has any the not a surface, one has earn repea Some I was clear, s as say, was �54 realize with a little shock that energy cannot be localized in space assumed it could; or as one can he had somehow sonnet he had read in one way can call I and about, as does a This This there is a which of the scholar who his way in , of course, and it s love of his shrubs and the the find that a can be taken in another. re-examination and reflection can go on for years, as • and the wall. must dwell There may be But let me say a word for the faithful scholar. It is with that can be known about to know differential scholar about every- , or to have de- astronomical tablets, or to have followed the behavior of bees detail the of ritual and traced or Kant till the loci of has its , or to have studied Philo JudaeuE of archaic are all familiar. fame; and it has its satisfactions, conscience. could about matters that anyone There is a virtue in s if he Such had the the and could take the time. Please understand: unless you want to. you know. to become am not am whoever you are, You cannot know all that to know some of the there is not time. But to the veil of a is to the swindle will detect the muddle, Here direction of the the inadvertence, when or more than with closed doors, and a tion. It is of exhibits that t can be and to progress he is ect matter of hotmislead But do not be too harsh on of science, that the sake of progress, been so divided breadth of know. Amidst a And what if you do enter the to choose you will be detect it out. has It is so that the scholar can dead, before he has any make for progress or as a �ects progress, we become side-effects the seen in the can no God and not follow that But there are reliabil a scale diminish it does And that could die old and full life because life underwent had believed confronted with a advance in ideas, be tired of life, so that scholar, as I say with melville hearts man life could him death has los scholar as its sense. described him, cannot answer this tears and hie aloft to down can add wi take the scholar to • as St. John Perse the claims He antedates science. seer archaic man cosmic rel a order of cosmic events the secret be sa in esoteric of the world. attested and so there were the sacred riddle it was necessary This and Listen questions: I ask naval where is the earth I ask where is seed of the stallion ask "Whither moons and the hal the seasons -- �their desire hasten Whither in their desire hasten the waters? them and the source of riddles; you solved them or for- the ritual riddles were feited your head. wisdom to It was accounted the a riddle no one contest among the Yanaka holds a could answer. a thousand cows. his sacrifical feast, with a , has the cows driven away for himself before- certain of wise to defeat his hand, and The One of them unable to answer, from his trunk and falls into his loses his head, which Reverend cries out when all are silenced, Ya Brahmins, if any of you wishes to ask any questions let him do so, or all of of any of you, or all of if you like; or let me ask a if or kills himself out of In Greek tradition, the seer Chalcas dies of His followers attach , when bested in a riddle contest themselves to the Menander in the presence of 500 with The Venerable lonians and Greeks and 80 000 Buddhist monks, throws out each of his dilemmas Find your way of that one, your Ma with a here We are not far from the fifth Greece. is For the cosmos, from which arise. have to render For, says Anaximander, did But the sacred of words vouchsafes the divine decision word is, but in the first instance it is a gesture and a to a activ the I propose that, with bad our words, each of us makes a world which he inhabits is to tion to be nonsense, were it not for the fact that it is not reason it is not may be that we are all supposes an intention e to the ordinance of time one another and atone for the wrong Now all this love and hate in that same must is the the furies must be rule the universal process of course and and he ike the riddle contest Strife is the father of all tic attraction and discord power But the word still has aacred, who knows it, can live his life. seer and who booked such fabulous successes It is hard to say what a like all it pre- Unlike other gestures, its convention, an idea or universal �center is a the intended The central one mere same has lifeless same or if it same , from one identi does so from one context to another to another of immutable combinations mere sound is inanimate structure this becomes evident to me if the to f several times observe it in its starkness; it that I , dried-out dead The it is the for different men. nevertheless new. on the other hand, can be called of memoriGs associated Like all it is ted and it is The memory is when a memory stirs appears as bizarre and associa a storehouse of I have an which res an earlier one , or a sentence a , or consonant I live t is dissonances works that us human heart some ultimate is says 5 • is mysterious Parse time I an unwarranted far tehsad says ict make a are those mistake. olive branch before all the wand in the hands turn systems. master �58 Here all is ear This may be, a mathematician , because he has created the entire structure himself fact that, while It is a curious discuss their role in terms of what mathematicians, in and truth do, use words like But I am more concerned at the moment el and the fact that method, but its deliberation and s mathematical , can reveal every orderly demonstrations carry me How is it situated in the mathematical in certain another The domain of mathematical of which all the are each new When the of the has of other of these new sufficient had their without The are has an internal life mathematical The various elements of mathematical are indissolubl which have led to its appearance, and related to s of any one of them returns s mathematical and he may be able to define them in an autonomous way not fixed once for all. one another. The domain is unveiled accessible. opens new to the earlier theories in to the to all the processes e of from its as succeed in it from the and represent it verifiable of associations of , and mus is in which it first which borrows from that a data, The is not before or structure may lead to the ects or structures the data of the constructions are enchained in itineraries that one can somehow traverse. structures. te ~..::.;::;~..;;;;.;:;.~ in one way The evidences obtained of a in an or These constructions progress nor e world with which we can for instance, a cons how is this as such in the field -of immediate rela lated. , as on a ife of consciousness? ects are not belong to a . in an assertion , each s We can ask: fulcrum, round the law of the a ization is ever more It must shell out, from a determinate • It must in the mathematical as more structures are the �59 conclusion? t it all too briefl was autonomous tions, and contain in itsel would constitute the a which would the to all further mathematical mathematical and here be oeonT"!::'I tion elaborated in that ti on. The be , but would go and substitute for it. the limit-idea of a silent and full own proper intell We are here before exis its the characteristics of the and those of model and all discourse I means of formalization y a total is unachievable. various theorems sta it has been detail the claim that, are content of these theorems the But I that there is s more back resolute! to reflect on its role of formalization, and one can see the limitation theorems as believe this view can be the relation between the ife character the out instance a infinite and the actual infinite. is the notion of an each o constructive ; the latter is the resul be that to substitute There is tha be said about their knowl tha heard that such a system cannot be achieved which can a realized Both and mathematics construct, and , as into an exterior space, the formulae of their achievements mountains Both mathematics, and their mathematics of an incessant with the world ways a I am Both a a in widel Where are their roots? which is not a as a seems most e one conseientious conscience. to be a indeed, the The mind s its work What �is affairs the connectedness of a manifold of elements. The grasp eness is idea correctness tive links movement is Whatever is time escapes from the immediate coincidence self with self into the evanescence of the past. itself the There is no moment of all the others, a f the as well as the future and which itself dimensions of all instant involves elements tha What is sense are these conscious pas mode Consciousness the the future , is open There is an what is not consciousness the non- direction the , is moderation. s pas as ied seems so characteristic past as i t t is there can have sense it is an outside some has had constitute itsel consciousness roots , is its attunement to can raise the question its , and to pursue courage its �virtues Let gesture this liked i t et back � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The Collegian Description An account of the resource The Collegian began as a student newspaper; became a college community literary publication in 1952 and continued until 1969; became a student newspaper again in 1969; discontinued publication from 1980 until 1989 when it again became a student literary publication. Click on <a title="The Collegian" href="http://digitalarchives.sjc.edu/items/browse?collection=26">Items in The Collegian Collection</a> to view and sort all items in the collection. Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Page numeration Number of pages in the original item. 61 pages Text Any textual data included in the document text Original Format The type of object, such as painting, sculpture, paper, photo, and additional data paper Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Creator An entity primarily responsible for making the resource Lachterman, David (Editor) Title A name given to the resource The Collegian, September 1964 (Faculty Issue) Date A point or period of time associated with an event in the lifecycle of the resource 1964-09 Description An account of the resource "Faculty Issue" of The Collegian. Published in September 1964. Identifier An unambiguous reference to the resource within a given context Collegian Faculty Issue Sept 1964 Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Publisher An entity responsible for making the resource available St. John's College Language A language of the resource English Rights Information about rights held in and over the resource St. John's College owns the rights to this publication. Format The file format, physical medium, or dimensions of the resource pdf The Collegian https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/dcd9b156df4b9437de172b572781520f.pdf 10813fe5fb8b4b7e8281b50276a919f2 PDF Text Text T HE C OLLEGIAN tv.iA y _JUNE 1964 �C0 N T E N T S A Rational Extension of the Fifth Book of Euclid 0 s Elements David Stephenson, Tutor. Poems . . . . . . . . . . . . . . . . . • Charles G. The Gnomon: ..... Bell, Tutor • 7 A Note on the "Knowern. • • • • •• Eva Brann, Tutor ••• 10 Fear and War or Fear and Peace? * * ... * * • Andrea Jacobson, * * Editor • • • • • • • • • • • • • David Lachterman Business Manager • • • • • • • • David Rasmussen Editorial Assistant • • • • • Susan Roberts Faculty Advisor • • • • • • • • • Eva Brann 9 65 ••• 39 �' ·.. -r • It is my sincere hope that the Collegian has been of value to the community during the past year. My endeavor has been to present the best writings from every area of the Program along with outstanding independent work. In addition, an effort was made to encourage creative writing, specifically in the medium of the short story. ness of the response to the announced 0 The meager- Short Story Contest 0 led me to give greater credence to the oft-expressed complaint that the requirements of the Program enervate literary energies. In the hope that the summer vacation might witness a revival of those energies , Miss Brann and I have decided to ext end t he cont est unt il t he first week of the coming school year. Stories may be submitted to either of us through the college mail. The prize is still $20.00 for use in the Bookstore. DAVID LACHTERMAN �A RATIONAL EXTENSION OF T~E FIFTH ~OOK OF EUCLID 9 S ELEMENTS David H. Stephenson "! ratio is a sort of relation in respect of size between two magnitudes of the same kind. n "When of the eouimultiples, the multiple of the first magnitude exceeds the multiple of tl'-~e secc.:-id, but the multiple of the third does not' exceed the multipl8 of the fourth, then the first ( ·.~agnitu~) is said to have a gr~ater ratio to the second than the third has to the fourth. " -;-- -In comparing these two definitions, one cannot help but be struck by the ·comparative adjective "greater" ( /'-'i ({c:- Vrf.. ) applied to ratios, for it presupposes that ratios themselves have size. And if they have size, is it not possible to determine a ratio of ratios consistent with Euclid 9 s fifth book? Definition 7, quoted above, is apparently included by Euclid to giv~ a name to one class of ratios which are not the same, but also to distinguish this class from the other class of unequal, or rather nnot samen, ratios. That is, given any ratio, there are innumerable ratios other than it, and these fall into two classes: those which Euclid would call greater than it, and those than which it is greater, i.e., those less than it. These three classes, the same, greater, and less, could of course have been distinguished by adjectives without quantitative connotations, such as same, light, and dark, Therefore the use of the word, "greater;t, to describe a relationship of ratios implies a common conception of them in some way akin to the conception of magnitudes. This kinship can be explained in two ways. · First, Euclid and his con. temporaries may have felt that in the case of ratios to the same, as in propositions 7-10 of Book V, the relative size of the two antecedent magnitudes obviously determines the relationship of any two such ratios. Thus a greater magnitude should be said to have a greater ratio to the same than the less has. Since all ratios can ·be compared by finding proportionals to the same, they can be called greater, less, or same according to this reasoning. .J Secondly, the definition of ratio quoted above.implies more about the s"ze of a ratio than 1 is immediately apparent in the translation. In particular t< 1'T ~~ ff 1:I /~ l K o 17! ,-(x , which Heath prefers to .translate "in respect of sizen gives insufficient indication of what kind of relation between two magnitudes con$titutes their ratio. There are two possible ways in which the size of two unequal magnitudes can be basically compared: one can exceed the other by a certain magnitude, or one can be a multi?le of the other, Regardless of any ambige.ity in the term {t-11)L.1c..~11·/ <; , Euclid could not have meant a relation with respect to difference of size as in the former comparison. Therefore a ratio is in some way dependent on the ~u~~titpltcative relationship of two magnituaes. This is clear in the case of number3 and commensurable magnitudes, so that Euclid does not bother to define either ratio or greater ratio for numbers. Such magnitudes (or nUJ.'nbero) obviously have rati.os 'greater than, the same as' or less than each other according as th- first antecedent magnitude contains e more, as many, or fewer pnrts of its cons$quent magnitude than the other antecedent contains of its consequent. (That is if the parts taken are �- 2 - equal to the largest common measure.) Moreover the definitions of same and greater ratio for incomrnensurables in Book V hardly differ at all from the above definition for comrnensurables. Definition 5, for example, can be restated in the following way: Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when the first magnitude contains as many parts (submultiples) of the second that the third contains of the fourth for all equisubmultiples of the third and fourth. That this is equivalent to Euclid 0 s definitions is evident from the fact that according as the multiple M of one magnitude A exceeds, equals, or is less than another multiple N of a second magnitude B, the first magnitude A itself must exceed, equal or be less than a multiple p of a part of the second magnitude B, ·where the part BQ is the same submultiple of the second magnitude B that the first magnitude A is of its multiple M, and P is the same multiple of BQ that N is of B. In modern notation : if mA ~ nB then ~ nB = n ~ m M - - - ·----·--+-------. A /If----+--- B m A------ 13 - ·---T- ---r---- ({ But this means that the relationship of two ratios is determined by the relationship of numbers of parts of the consequent magnitudes contained by the antecedent magnitUdes. If the numbers are the same for all equisubmultiple parts then the two ratios are the same. If in some instance the nu.~ber of containable parts in the first ratio exceeds the number in the second ratio, should we not say that the first ratio is larger than the second? (This is equivalent to Euclid 0 s definition?.) Although the above arguments are implicit in Euclid 0 s Fifth Book and are justifications for his assumption that ratios have size, they also point to other similar asaumptions which Euclid nevertheless refused to admit. For if the fact that ratios have size is deduced from the quantitative relationships between magnitudes or numbers inherent in different ~atios, then their nrelation in respect of size" , viz. the quantuplicative relation of these ratios should also be prescribed by these numbers or magnitudes. So it should make sense to speak of a multiple of ratios or a ratio of ratios. Ratios of Ratios Consider the following ratio of ratios: (A:B):(C:B) and the ratio of magnitudes A:C (A,B. C therefore being of the same kind). Then if A equals C, it is immediately apparent that (A:B):(C:B)::A:C since A:B::C:B, therefore A:B has the same size as C:B, i.e., the magnitude A:B equals the magnitude C:B. By dint of the com..mon notions equimultiplesof equal magnitudes are equal; larger multiples of equals being larger than smaller multiples. And Definition 5 establishes the above proportion. �- 3 However if A does not equal C, (A:B):(C:B) may be greater than the same as, or less than A:C. For instance, assume (A:B):(C:B) exceeds A:C for all A greater than C. Then this is true in particular when A is itself a multiple of C. By Euclid~ s De±;inition 7 there are equimultiples -- say W;X of A:B the same multiple as K of A, and Y:Z of C:B the same multiple as L is of C, -- such that i:v:X exceeds Y:Z but K does not exceed L. Symbolically depicted: W:X::(A:B + A:B + K= ( A + A + ... + A:B) ... + A ) > (C:B + C:B + <.. . ( C + C + ... + C ) L ... + C:B): =:Y:Z Then since A is a multiple of C so is K. But K is less than or equal to L. Therefore L is a greater multiple of C than is K, or else L and K are equimul tiples. Whatever multiple K is of C let that multiple be taken of C:B, and call it P:Q. Then K and P:Q are equimultiples; so are L and Y:Z. Since L exceeds or equals K, Y:Z exceeds or equals P:Q, and 1~:X must exceed P:Q since it exceeds Y:Z. Let S:T be the same multiple of C:B that A is of C. Then since K and P:Q are equimultiples of C and C:B, they will be equimultiples of the equimultiples A and S:T. And W:X and K are also equimultiples of A:B and A, so W:X and P:Q are equimultiples of A:B and S:T. Therefore since W:X exceeds P:Q, A:B exceeds S:T. Hence a necessary condition that the ratio of two ratios (with the same consequent) exceed the ratio of their antecedents whenever the first antecedent exceeds the second is that the multiple of a ratio be always less than the ratio of the same multiple of its antecedent to its (unmultiplied) consequent. That is, if (A:B) :(C:B) > A:C for all A.greater than C then E:F > G:H whenever E is the same multiple of some magnitude K that G:H is of K:F. (Algebraically, E:F: :mK:F > m(K:F): :G:H for m > 1) It also follows from this that if A is less than C, this condition requires that (A:B):(C:B) A:C. In other words (A:B):(C:B) exceeds, is the same as, or less than A:C according as A exceeds, equals, or is less than C. Such ratios exist already in Euclid's work: the duplicate ratio or ratio of the squares on two magnitudes, for example. Thus since the duplicate ratio of A:C does exceed, equal, or become less than A:C as A exceeds, equals, or is less than C we might simply equate or define the ratio of two ratios having the same consequent, (A:B):(C:B), as the duplicate ratio of their antecedents, Duplicate (A:C). But any such definition pr esumes the necessary condition stated above concerningmultiples of ratios. < Such a condition raises the following question: when one ratio is greater than another, h£:d: ~ greater is ii? The multiple of any magnitude, including ratios, is so many times (viz. double, triple, etq.) as great as the magnitude of which it is a multiple;. if then one ratio is a multiple of another, the larger ratio, considered as a magnitude, must be so many times as large as the smaller. To answer this question is it not most consistent to continue in the direction begun by Euclid in the seventh definition of Book V? That is, as was pointed out earlier in this article, Euclid's assumption that a ratio can be greater than another is based on the fact that some magnitudes (in the limited sense, i.e., excluding ratios) numbers, or multiples of magnitudes are greater than others. In particular it probably struck Euclid as obvious that, as proposition 8 states, '~the greater (magnitude) has to �- 4the same a greater ratio than the less has". This proposition along with the preceding one therefore imply that the relative size of two ratios with ~ ~ consequents i§. determin~d 2:z. the relative size of their antececten'ts. And thus a ratio R:S, where R is a multiple of another magnitude Q, is as much larger than Q:S as R is larger than Q, i.e., R:S is the same multiple of Q:S that R is of Q. Accepting this as a logical consequence of Euclid 0 s definitions and propositions, one must also agree that (A:B):(C:B) is not greater than A:C for all A greater than C and less than A:C for all A less than C, because we have just proved that this would require that for any R a multiple of Q, R:S must be the same multiple of Q:S, and the ratio of the multiple R of the antecedent Q to the consequent -- (R:S) -- was supposed to exceed the same multiple of the ratio (also R:S), which is impossible. With multiples of ratios so defined Euclid 0 s propositions in Book V can be easily proved, and in addition the following four propositions relate ratios of ratios and ratios of simple magnitudes. - Pronosition 1 . The ratio of two ratios having the same consequent is the same as the ratio of their antecedent magnitudes. That is given any A,B,C, magnitudes of the same kind (A:B):(C:B)::A:C for if not (A:B):(X:B) is greater (or less) than A:C. Hence if greater some multiple of A:B, say P:M, exceeds another multiple of C:B, say Q:N, but the first multiple · of A, say K, does not exceed the second multiple of C, say L. But since K and P :M are equimul tiples, P :M: :K :B. Similarly Q:N: :L.:B. Hence K:B exceeds L:B and by Euclid's Proposition 10, K exceeds L, which contradicts the hypothesis. Therefore, etc. Similarly (A:B):(A:C)::C:B Q.E.D. Proposition ~ The ratio of the ratio compounded of two ratios to either component ratio is the same as the other component ratio; and if the ratio of one r at i o to a second ratio is the same as a third ratio, the first is the same as the ratio compounded of the second and the third. That is given any six magnitudes A,B,C,D,E,F, such that (A:B comp C:D)::E:F, I say that (E:F):(A:B)::C:D and (E:F):(C:D)::A:B. As Euclid demonstrates in Book VT., to any three magnitudes a fourth proportional can be found (provided the ratios of these magnitudes can be expressed as ratios of straight lines). Hence let A:B::K:L and C:D::L:M, where K.L,M are straight lines. Then (A:B comp C:D)::(K:L comp L:M)::K:M, and so K:M: :E:F. By the preceding proposition (K:M:(L:M): :K:L and (K:M) :(K:L): :L:M. Therefore (E:F) :(C:D): :A:B and (E:F) :(A:B): :'C:D. Again let (P:Q):(R:S)::T:U then I say that P:Q::(R:S comp T:U). Obviously from the above if (R :S comp T :U): :V:W then (V:W): (R :S): :T :U, hence (V:W) :(R:S): :(P:Q) :(R:S) and V:W: :P:Q. Therefore (R:S comp T:U): :P:Q. Q.E.D. • �- 5- . .The sum of two ratio$ having the same consequent is the same as the ratio of the sum of the antecedents to the consequent. That is, given any three magnitudes (of the same kina) A,B,C then (A:B + C:B): :(A+ C) :B. Since (A:B):(C:B)::A:C then by Euclid 9 s Proposition 18, Book V (which it is easy to prove for ratios as magnitudes) . (A:B + C:B):(C:B)::(A + C):C and by the preceding proposition (A:B + C:B):: ((A+ C) :C comp C:B): :(A+C) :B Q.E.D. These three propositions in effect define ratio of ratios and addition of ratios and also redefine compound ratio as . the inverse of ratio of ratios. That compounding also corresponds in some way to multiplication of numbers is evident from the next proposition. Proposition ± If four ratios are proportional, the ratio compounded of the extreme ratios is the same as the ratio compounded of the m~an two ratios. That is: given four ratios A:B, C:D, E:F, G:H such that (A:B):(C:D)::(E:F): (G:H) I say that (A:B comp G:H)::(C:D comp E:F). Since it is possible to find ratios the same as the above with either the antecedent or consequent prescribed, it suffices to prove the follo-wing: if (A:B):(K:B)::(B:L):(B:M) then (A:B comp. B:M)::(K:B comp B:L) But this is immediately apparent since (A:B):(K:B)::A:K and (B:L):(B:M):: M:L, hence A:K::M:L Alternating according to Euclid 0 s Proposition 16, A:M::K:L. Therefore since (A:B comp B:M)::A:M and (K:B comp B:L)::K:L (A:B comp B:M)::(K:B compB:L) Q .E. D. Ratios and "Real Numbersn The fourth and last of these propositions closely resembles Proposition 19 from Book VII of the Elements. Someone might well ask, how are ratio and number related? And the ·startling answer is that ratio is number and number is ratio, al though in a somewhat larger sense of 99 number" than Euclid was familiar with. That is, ratios are the same as the nposi ti ve real numbers' 7 of modern mathematical jargon. All numbers are 77 relations in respect of quantuplicity, as are ratios. "Threen means nothing other than the relation with respect to quantuplicity of three things and one thing,* it cannot exist without a unit. (Even for Euclid "a number is a multitude composed of units.") The distinguishing characteristic of numbers is that they are relations 'With respect to a * Used as an adjective, of course, the number attributes this relation to the noun it modifies. �- 6 single and non-arbitary unit. 17 An unit is that by virtue of which each of the things that exist is called one." Things are called one, moreover, by virtue of a relation, namely ~ relation 1.£ respect £.!: quantuplicity between ~ thing ~ itself.. Something is called one virtually because it is the same as itself. Therefore the unit is in fact the ratio in which antecedents and consequents are the same. Euclidean numbers, i.e., integers, are multiples* of this unique ratio, or equivalently by our previous hypothesis, the rati,o s of multiples of any magnitude to the magnitude itself. Besides the integers and the unit, the real numbers include the so-called 0 rationaP~ and "irrational" numbers. The former, as their name implies, are ratios, distinguished from the latter by the comrnensurability of antecedent with consequent. The '9irrationalsn, (in the modern sense of the term) are also ratios, but ratios with incommensurable antecedents and consequents. Although neither rational nor irrational numbers are multiples of the unit, they do relate to it quantuplicatively since the ratio of any real number (i.e., ratio) to the unit is the same as itself. For example, (A:B):(C:C)::(A:B):(B:B) since obviously C:C::B:B. And (A:B):(B:B)::A:B. Therefore, (A:B):(C:C)::A:B. FUrthermore it can be proved that ratios as real numbers, ill 121£ show an isomorphism with the points of a line, that they are continuous, and that they are distributed copiously among the integers. If Euclid had not chosen the word 0 greater" to describe one class of ratios not the same as a given ratio, and implicitly the word "less" to describe the other class, if instead of greater and less he had chosen to call them light and heavy, masculine and feminine, hot and cold ratios, these conclusions would be inconsistent ~dth Euclid 9 s thought. But he did choose that word. Therefore it is consistent. Q.E.D. * something can be multiplied without reference to number simply by adding to itself. �- 7FOR HOUGt~TAGANDIE Charles G. Bell Even the Romans in the decadence That heaped the fallen world i:Ni th monuments Of their gloom, and when the bitterness of sex surprised them, Loved it, sought it, prized it. And from fabulous shade of the Dark Ages, Out of crypts and cloisters, rise nameless voices Avowed monks on their knees inclining To the dear socket of a girl 0 s inturning. Our own forbears, sanctimonious Puritans, Warmed beds with bundling; staid Victorian Dames, under bustles of silence, veiled the same fesses, Smooth, indented, thighs spread for caressing. Body, naked, cloven, supple, swaying Here while the great cloud waits in abeyance At the sun's horizon, shall we not mate and slu.mber, Our foldings wreathed:on the leaves of Indian Summer? �- 8 - . THE NEW FALL Charles G, Bell 17 Triste, triste is the fall, the sad fall, With wind and mist and leaves that fall." (Victor Hugo, who was, alas, The principal poet produced in France.) While we, in a culminance of gold -Tulip and oak, sassafras, ash, With wine and scarlet of dogwood and gum, And the filligree of creeper and thorn -We, stripped to the skin, in rivers of sun, On .the hilltop stand, and cry the love Of these twined selves and earth that moves To the waste of winter by a turning road. W swear the rapture of this new world e Is the sun°s seizure on flesh of the fall. �- 9 .. ON THE SHORE OF BIRTH Charles G. Bell I wake from a cruel dream: Odysseus, brandishing the blade At the threshold of sleep, Drives back the spirits From the pool of blood, Seeking the prophet. I, clawing through dark To the spilled sacrament And wine of force, Tear sleep with my cry: "Odysseus, stern guard, Put back the sword; I am he; let my tongue Blossom in speech. 01 I wake, my mouth filled 1Ni th the salt reek of blood. �- 10 - ~ nrNDMQN I �- 11 - THE GNOMON: A NOTE ON THE "KNOWER" Eva Brann I. II. III. IV. V. VI. VII. Shadows as Images: the Astronomical Gnomon Images as Tools: the Geometrical Gnomon Tools as '7Knowers": the Ari thmological Gnomon THE SUNDIAL AND THE ORIGIN OF CONICS THE SUNDIAL OF VITRUVIUS Notes (Correction§ to AN ANATOMY OF ORBITS, April Collegian] The gnomon,rival in interest to the canon (Collegian, Suppl., November 1962), appears thrice in the mathematical texts of the freshman year: in the fall in Euclid (II, Def. 2) as a geometric shape, in the winter in Nicomachus (I, 9) as a numerical scheme and in the spring in Ptolemy (II, 5) as an astronomical deyice. "What do its forms have in common and why is it called Uthe knower" ( 0 '!" wv ' from 'f \.'I v ~ en< H v., to know)? wr I. Shadows ~ Images: 2 Astronomical Gnomon "Proper people usually take their shadow along when they go into the sunH. Chamisso, Peter Schlemihl It is among the wonderful facets of the visibie world that every solid body under the sun casts a shadow. To be exact, there are two sorts of bodily shadow, which Leonardo da Vinci, whQ was fascinated by . this appearance, called "primary 71 and naerivative".1; The first belongs to the body itself and is a consequence of the curious fact that bodies, in their opacity, stand in their own light, just as they hide the larger part of themselves from sight (whence the eye and the sun are often compared; in optics this leads to the notion of visual rays -- cf. Euclid, Optics, D~fs. ).• , Therefore their surface, or as the Greeks said, their ~'apparencyH ( E\fLc\'o..v £ .LCA.. )2J always has a bright and a dark side. In the ancient world, which still lived under one primary source of natural light, this was an overwhelming fact. Greek pot-painting, for instance, -- a craft humble and popular enough to permit us (as go classical text ever does) to speak of the predilections of "the Greeks a-- · testifies to this. Athenian pottery, which was produced in tremendous volume and sent to the ends of the known earth, used two basic techniques, the one belonging roughly to the "archaic~' sixth, the other to the nclassical n fifth century B. c. Archaed)logists speak of them as "black-figure~'J) and nred-figure" respectively. In the former technique, the figures of men, boys, horses, armor , girls (listed in descending order of interest) and other paraphanalia are put in rich black glaze onto the bright orange ground of the Attic clay. Therefore they appear to be seen against the light, as schemata, as flat, sharp, black silhouettes, slimmed, as they would be in nature, by the encroachment of the light around them. In the later technique the reverse is done -- the figures are "reserved" out of the black glaze covering of the pot. Therefore they appear as if the sun were full on them, and, being nourished as it were by light, are generally larger than the black figures, while the addition of black interior contours gives them a more substantial look: Leonardo aptly called such body shadows �- 12 ... "the expression of body ;~ . Their use w elaborated by Greek painters into as a technique called n shadow-painting" ( o- v.. l £A y f o.. ~ (. ~ ) , a species of trompe-1°oeil which makes the image look more natural than nature; it is often cited by Plato as the .very paradigm of the art of illusion. (e.g., Republic, 365c, 52JB, 58JB, 586B, 602D). But the two grandest and most instructive examples of all such shadows were, as Empedocles taught, the darkness of night, for 11 Earth makes night by standing under the rays . of the sun > ,,. r.... ,., /c;,., c tf) ,, /' / ( vu V:.TC\ a~ yQ.Lo.. ·nv1crw V\LO"Tyt.vll\ ~o...££0-0'1.. n4) '-~€:Atov5>), and those amazing phenomena after which the sun 9I s own circuit was named the } '\. "circle of light failure~' ( £.~A£L"tt1Lv,05 V<VK/\o~) - - the eclipses. * * * * * However, the sort of shadow relevant to the gnomon is really only the second, the cast shadow, which the painter, considering it as darkening another body , calls 0 derivativen, but which is , taken in itself, a very peculiar kind of image. As an image, it is first of all natural, like a reflection (Republic 510E), and inseparable from its body, being its g 'doppelgaenger ~': O-\kLD. b..n [oTolx "S as Euripides says, (Andromache, 745). Indeed, the ability to produce a . shadow is the very sign that a thing is a proper body among othe:r bodies; it is at once the warrant and the effect of its substantiality.5J Secondly, shadows are negative images, images by deficiency, in which that which is visible in surfaces is almost entirely absent, nfor what is seen is color°' .... 'c, .l ,,.. ( 10 yo..e OfQTov £.oTL ?( f w /-L~ - Aristotle, De Anima 4 18a29 ) • To this kind of shadow tooJthe first accounts of the cosmos are indebted, for as soon as failures of light in the heavens were recognized as shadow phenomena, the outline of the earth could be seen upon the moon, and from its phases could be inferred its character as ~1 a night-shining borrowed light, ·wandering about the earth" tr f. y~1°" v t:..>-. µ.. c v 6 v ~AA6'f'-OV ~ ~s <"' u v.., L~°'" ~') ft. ). w 1 Parmenides, Fr. 14. But the most honorable function the cast shadow has ever served was to be admitted to-t~lace shown. in this diagram of the metaphor for the road to being ·iXl the Republic ( 507ff.) : eye --+ ( thought ---) shadow body ( ( mathematical object being sun < good. The properties of shadows which fit them to be the sensible representatives of mathematical objects are just those described above and developed below: shadows are schematic consequences of the shapeliness of bodies, preserving as natural images a certain truth even under deformation: they are the lightless witnesses of all visibility, more easily consulted about its ways • �- 13 than is its dazzling source itself -- and so they are to the eye what mathematical objects are to thought: the nether aspect of being. * * * * * * The third characteristic of the cast shadow, namely that it changes in length and direction independently of the body which produces it, is that which the astronomical gnomon particularly exploits. These changes are so fixed and familiar and so much more easily observed than the course of the sun t~thts~1 (the s\4.y ·'1b<tf is names after the shadow ( o ~ L ct l its clouds cast -Partrid~e~ Origins) that the reversal of their progress becomes the very signal of God 0 s special concourse · "Behold, I will bring again the shadow : of the degrees, which is gone down in the sun dial of Ahaz, ten degrees backward. So the su.n returned ten degrees, by which degrees it was gone down" (Isaiah 38: 8, II Kings 20: 11). The gnomon, by which shadows are observed, is a thin upright, set up so as to cast a slender, neat shadow, whose changes with the hours and seasons can be closely w ~t-c\.J.td. A gnomon may be no~r· more than a stick in the ground, but it can also be the pointer of a sundial with carefully calibrated base, or even a tall obelisk-like official monument.6) Such instruments were often called "shadow-viewers 0 ( O-V.t08t'\f Q ). The first skiothera are said to have ~een set up in Sparta by Anaximander in the first half of the sixth century.71 One of them seems to have been a sun clock ( ~ f o v-v. . o\\ce..l.ov) Wf o Xoy £.L ov ) to tell the hours of the day. It seems fitting enough that the Spartans, then in the early days of their discipline with its many public occasions, should have been the first to want a public clock. Later on most public places must have had one; the Athenian market place, for instance, had a very large public water clock-sometimes also called ·> "gnomon" (Athenaeus IT, 42b, but usuallyS) 97 ~A f. i' 0 J f Cl. - - nwater stealer ) . The name gnomonics came to be applied by the Greeks and Romans to the flourishing practical art of dialling generall~, and particularly to· the construction of a great variety of sun clocks (see Vitruvius_IX, 8), among them some with inclined rather than. upright pointers, such as the one which may have occasioned the early study of conic sections, described in IV. below.(lndeed Kepler thought it a fact that gnomonics 77 begot for us the geometrical doctrine of conic sections -- Eoi tome, p. 20.S.) But the first and most ubiquitous gnomon was always a natural one -- man, the upright creature (Aristotle, ~ 2f Animals 653a31), talking away the morning as his shadows waned: These three houres that we have spent, Walking here, Two shadowe?s went Along With us, which we our selves produc'd; But, now the Sunne is just above our head, We doe . those shadowes tread; And to brave clearnesse all things are reduc'd. The theme of the human gnomon will recur. * * * * * * * �- 14 - The gnomon in its properly astronomical function depends only on the shadow at high noon.9) Its construction is, of course, identical with that of the sun clock and one fundamental diagram serves both, as iri V. below. The principle on which it works is that of a projective image. The gnomon°s vital part is therefore its shadow-casting tip, called by the Romans its ~'navel'? (umbilicus) . Because the earth is to the heavenscn~ as a point (Almagest I, 6), this tip~ be taken as the center both of the rational and of the sensible horizon. Thereupon the upper half of the small circle ·drawn about it in the northsouth plane serves as a miniature m eridian of the heavens, while the part toward earth may become a sensible quadrant, measuring the elevation of the sun. Hence in the gnomon 9 s tip coincide the center of the world and a kind of cosmic eye on whose. retina (the dial) the sun°s motions are projected in reverse (such projections through the center of a circle or sphere are r1 cw called "gnomonic " ). while the gnomon itself, by blocking the sun°s rays toward the earth, produces the shadow-index distinguished by us. The interest that the ways of such travelling markers exited is reflected in the [Aristotelian] Problems (611a14ff.): "ivby is it that although the sun travels at a regular pace the increase and decline of shadows are not the same in an equal period of the time?" (La., why, in the figure of V.} is RC less than CT?); also 912a34ff.-It is:--curious and significant thought t hat within t he Copernican system the gnomon projects the earth 9 s ow motion onto itself. n a. What motion of the sun does the noon shadow given off by the gnomon follow? Anaximander, when he furnished the Spartans with a sun clock, seems also to have installed a gnomon 11 to signal solstices and >equinoxes" (Diogenes / Laertius, ·II, 1 -- the Greek, more appropriately, says Lcr~~.JA<c...f 1..-- 0.5 -'7 equal days") .10) In our latitudes the. shadows of the culminating sun get shorter toward summer, come to a standstill (solstice) and then grow longer again, being of the same length both at spring and autu.~n equinox. These seasonal changes in the shadows of the culminating sun are perfectly familiar even to primitive peoples, who use gnomons as • �- 15 agricultural calendars.11) But if the observations are reflected upon,they yield the idea of a second solar motion, independent of that of the heavens which carry the sun through its hourly and daily path, namely a gradual ascent and descent between the tropics (turning points), taking place in a direction opposite to that of the diurnal circuit. The gnomon shows the total arc of the north-south motion to be about 11 of the meridianvs 83 parts or c. 47°40°. This is the eeliptic . . . /, t"'1 its obliquity, the great dtversifying circle of astronomy. Although Ptolemy uses two different · instruments for obtaining the obliquity which he carefully describes (Almagest I, 12), the various men who were credited with this great discovery, Thales, Anaximander and Pythagoras in the sixth and Oenopides in the fifth century, 12) were all associated with the gnomon also (see II. below), and the gnomon was certainly instrumental to it. b. But there is yet something more contained in the gnomon's shadows. If the dates at which they reach their various places are carefully kept, it is found that the time from spring equinox to summer solstice is 94t days, but from summer solsti.ce to autumn equinox 92t days (Almagest, III, 4). This discovery was ~ade by Meton and Euktemon in the latter part of the fifth century B.c.13; Such an irregularity of the sun°s motion would scarcely ever, I imagine, have been noticed, had it not been for the gnomon. For no eye. can directly follow the path of "the· heaven 11 s 2lorious sun that will not be deep-search 0 d with saucy looks" -- as it may~the planets. Thus the gnomon brings down the image of an anomaly whichJonce known, becomes a prime stimulant to the formulation of those rationalizing hypothesGs which were _ the chief preoccupation of technical astronomy in antiquity (cf. Almagest III, 3). ~~o~ c. Not only the sun's motion in the heavens but also man's place under the sun is ·indicatea by the gnomon' s shadow. Vitruvius (IX, 1) says: "It is due . to divine intelligence and a very great w onder to all who r efl ect upon it, that the shadow of a gnomon at the equinox is of one length in Athens, of another in Al-exandria, of another in Rome •• ·"· Even more impressive as a direct index of the diversifying effect of the sun's obliquity are the varying directions of the noon shadows. So remarkable has this fact always seemed that the habitable zones of the earth are characterized lby the directions· of their shadows: ~ 'These [the temperate zones] ·are alone habitaple while the regions beyond the tropics · are not, for the shadow would not fall to the north, while it is known that places become uninhabitable before the shadows cease altogether or change to the south [i.e., south of the summer tropic J ·, and the regions beneath the Bear [above the arctic circle} are uninhabitable because of the coldi1 (Aristotle, Meteorologica, 362b6ff.). So that according to Aristotle,men can live only in the temperate, 17 heteroscian ': ( f. H.- ~ o- v.. L C?S - "other-shadowed 0 ) latitudes, while Ptolemy admits 1 parallels down to the equator ( II, 6; see also also the :'amphiscian ~ Copernicus, II, 6); Kepler also was fascinated by this way of identifying o �- 16 people and quotes ~nth relish some lines by Lucan remarking that the shadows of Arabs are all sinister in direction, which he explains in the way 'bhe figure shows (Epitome, '.P• 205). In the more accurate determinations of geography nastronomical and shadowviewing instruments 99 are indispensable (Ptolemy, Geography II, 1). Ptolemy OiAi\ s'-"a- explains that while t~e heai'ens turn c>.ow1 ~o'\-.oabout us ( t\£el~o-\wv V\f'\~S ), ...;1t\.\t. pv-or< so that we easily manage to see a great I (.ft part of it, the earth cannot be Kcr~r travelled over and surveyed by one or even several men; one of the obvious consequences is that to locate themselves on earth 1-ne."' must, strange to say, take their bearings on the more accessible heavens. In fact, the grid of our terrestrial maps is still a projection of celestial circles. (The figures in V. will show ~"'"'~""-~' "\\'f how the length of the equinoctial noon shadow immediately gives the latitude above the equator.) It follows that each parallel is associated with a ratio, that of the gnomon°s height to the equinoctial noon shadow (Almagest II, 6). In order to calculate this ratio the Pythagorean theorem is necessary, and it ·is perfectly possible that the gnomon problem invited its solution -- as we will see in III.'· the gnomon ·is first and foremost a Pythagorean affair. · 1 O'ri£.e <;i<LOVVl{~n c:. S\VV\'tle...~'•t'{ i!. VV\c\-trstooJJ d. And finally the gnomon car1.1,..be used to make large-scale measurements of size. Diogenes Laertius (I, 27) reports that Thales measured the pyramids by using himself as a gnomon, "having observed0 when it [the human shadow] . is ~qual in size to us 9 ~ ( na~C1..1~ ti~c-o v1~ OTi.. q _,,u.-2v tcro_rf.y£3\ \ ).14) This story is a good one on several counts, first because the thought of Thales outdoing the Egyptians, who are supposed to have been the Greeks~ masters in practical geometry, is a nice one, secondly because it seems so characteristic that a Greek, coming from a country where buildings were built to human scale,15J should apply this scale to these Egyptian monstrosities, and last, because it is a literal exemplification of man as • ------~----' measure. o\~ o \0r'j~\" .s<:o\t>Eratosthenes S""CAc\ow o\ 0bdi::.V- =hei~\.ir. (third-second century B.C.) used the gnomon to measure the size of the earth itself. The method is as follows: S is Syene, a city lying under the summer tropic; A is Alexandria, under the same 5 meridian but north of Syene, 0 is the center of the earth. ~ I\~ l\ tttaun / / ' \ I t 0 ) Since the sun is very far awaYJ its rays strike parallel to each other. At noon of the summer solstice they �- 17 - strike the gnomon at A at angle a, equal to 70030° or one-fiftieth of four right angles. Arc SA is known to be equal to §,000 stades, angle AOS to angle a. The whole circumference is therefore equal to 50 x 5, 000 or 25,000 stades = 24,662 miles. * * * * * * * We have looked at the gnomon as producing reflections of daily time, of celestial motion and of terrestrial extent. Of these works, time-keeping is the first and greatest. At least Anaximander, the supposed discoverer16) of the gnomon,must have ascribed to the measurement of time the greatest office possible. In his one extant fragment17) he makes time the assessor of the penalty of destruction and the recompense .of return which things owe to each other for the outrage they have committed by abandoning the common pool of the boundless to come into being. But if the world 0 s changes occur n according to the command of time ;; ( Vdl 10. .,-~ v To G XJ 6 vo u TC:t t..V ) , then in keeping time men discover that order. ~ Plutarch explains why men are favorably placedfor time-keeping. In the eighth Platonic Question (Ch. III), discussing the passage in the Timaeus which relates how the demiurge sowed souls, 1fsome in the Earth, some in the moon, some in all the other tools of time" ( ·O' ~yo.. vo._ /(f 6 vov -42D), he asks w hether this means that the earth too revolves, for all the other "tools of time 11 are planets. His answer is that it is best to think of the earth as standing still, thus 'wproviding those bodies which revel ve with risings and settings by which the first measures .of time, night and day, are defined • • • For the gnomons of clocks also act as tools of time not by changing positions along with the shadows but by standing still, thereby imitating the earth's blocking ( -r o £it LIT f o ere> o Ov ) of the sun as the latter revolves about the former H (There follows the Empedocles . passage quoted in I.) Evidently Plutarch interprets the earth's role as a tool of time to be complG.mentaq to that of- those parts of the heavenly clock which 0 have the epithet 'wanderers' 9 ~ and which were made for the enunciation ( cJ Lo ~Lo-r 6v ) and keeping ( ~ v .Ao. v-. ~ v ) of the numbers of time i1 ( 38C). For tne earth, by nopposing itself to the sun '~ brings about the first sensible effect, the first reflection, the first measured measure of time. Thus because it is t he home of shadows , those movi ng images of time -- just as time itself is · a ~?moving image of the motionless •918) (J7D5) -- earth becomes a measure-taker, while the planets are measuremakers. Hence it is here on earth that images of the heavenly clockwork and i~struments for fetching its motions down to earth are made. II. Images ~ ~: ~Geometrical Gnomon 97 truste wel that alle conclusions that han be founde, or ellys possibly might be founde in so noble an instrument as is an Astrelabie ben unknowe parfitly to eny mortal man in this regioun, as I suppose." Chaucer, A Treatise on the AStrOlabe This same Anaximander, signifioantly the first writer of a book of the kind �- 18 - entitled in later references 110n Nature n ( 1\ ~fl. ~ ~ c.r ~ w 5 ), not only "invented the gnomon • • • and was the first to draw an outline of land and sea, but also to build a sphereu ( t. -i('.)~V Ji v,a~ yvwJ.Aovv.... qpw\05 •.• ' ""' '(\ \ ~ <"f \.i . ) \ \..... ,.{)/"' \ /. V,O.l 'fV'\'J V~C.l VO.ACt.<70~1 ) 1~£.f~U~:rcov 11\WIO~ t.yfl.4. ft..J c/i.l\AC>.. ~£A\: 0-\0l.Lfc'.AV V,QTf.0-V....LUV..0-(-Dio.g. Laert. , II, 1-2). These are the basic tools of' the sci enc es of heaven and earth. They all share an apparently simple, but on reflection very questionable character -- they are in some way models. By a model I do not here mean something that may be talkedabout like this: "This is only [!] a model, the real atom contains nothing of the sort" (Eddington, The Nature of the Physical hTorld, Ch. IX, speaking of Bohrg s model of the hydrogen . atoffi}.'" These instrlimental models are rather to be regarded as true images, which~ before they are put to use as instruments of observation, can serve as teaching models, as '1visual aids" for acquainting the student with the rational structure of the world. I """\ )/ All ancient astronomical instruments that I know of embody some such part of the universe~~the descendent of Anaximander 0 s sphere, the .astrolabe, whose construction PtoletnY describes in the Almagest (V, 1), is the most complete in this respect.19) It coincides basically with the Pythagorean model presented in the Timaeus (and here tt is of interest to note that Pythagoras may have been Anaximander 0 s pupil 20 J ).1t consist) of fixed "colures 11 . representing hoops of the heavenly container acting as reference meridians ~;+_ . --------·. and the band of the"other", the ~><-,. zodiacal circle in whose center ~ ~/ /l ~ ' ''\ \." \ lies the . sun° s path, the ecliptic. 21 ) ll/ -~ The astrolabe, however, while i t 0 omits the axis and the earth wound 5 I/,'/ :J~I . ~ around it" of the Tim}an model' I \~ v adds sliding P?inters - . ~ ; and movable mer1d1an and horizon 6 \\~~ ~ ~2/" circles. (Such index circles are ~.::::::::. . 0 6l"' ¥ • called •0gnomonic", Kepler, Epitome, ""'-~_!is--~;::i 7 / p. 134). The whole is therefore ~ ~ I / / ./ a skeleton of the cosmos, that ~~ ~/ aspect of the visible world which ' ~ · --_. . -. it has in the reflective view, in ' \ 'gtheory a (,. '3 ~ w f CC\.. , from _ _ -9 ~ 0.:J f o _) - ~ "spectatorH);_ ,e-/y.r';/ Q , /k J' 1 , . '. u / ~/~/~I ~r . \\ J~~"\\ r) - ~.\; calibratio~s~ Here · arises an apparently simple-minded question. What need is there of observation if the best parts of the world are already before us? This question might elicit from the astronomer a most important fact. The model which he possesses has, it will turn out, a different origin and a different standing from the astronomical theory he is working on. He has taken it over from the physical philosophers, who themselves used only the simplest t ools to obtain it -- particularly tools like the:· gnomon_, which produce natural images. The artificial model they have constructed does indeed incorporate all that is knowable of the world ~- its center ~vhich is coincident with the center of refl.ection, its roundness which is the only perfectipn possible to a moving body, and its orthagonal and oblique principles which act as sources of regularity and irregularity. However, this latter principle gives rise to appearances not construable in terms of a fixed and timeless structure, but only by complex moving diagrams. The astronomer 0 s enterprise, -l '-"'e ,-e (~c r e ; consists ~n finding such diagrams as will "save the �- 19 appearances 11 , that is, as will show their irregularities to be merely apparent by producing a geometric diagram which will account for them. But since more than one such diagram can be produced, and since none are entirely satisfactory, they remain mere hypotheses, quite different in standing from the model, which simpl~ represents what is and which may therefore serve as a guiding frame for fixing the aberrant appearances. Yet even once the usefulness of the instrument is accepted, its very nature remains a problem. The model,as has just been emphasized, is not a model of the total world -- it seems to overlook some appearances, in fact most of them, while it contains all sorts of parts no one has ever seen. It is a model only of the world 0 s rational structure, of its geometry - but it is also a sensible artifact, made of Hcircles accurately turned with four perpendicular surfaces each", and existing in time. Is it not, then, a copy of some pure model, a timeless paradigm which has been espied beyond the appearances? Is it not rather the embodiment of the model for the world than an image 2f the world? The Timaeus presents a fantasy answering the perplexities of a model-making . science such as that of the Pythagoreans. It shows that such a sciencecut5 a cros~ overlooks the world as a visible, appearing world and by going always to the geometry behind it, avoids the only road which can lead to being, that which confronts the contradictions of appearances as appearances. It shows further that its principles of account giving lead to a hopeless confounding of "before" and "after ~'. For wherever the human account is to be embodied in a mathematical, i.e., rational image, the thing to be accounted for must itself be an image incorporating a rational paradigm. But this paradigm arouses the suspicion of having been construed into or placed under the world by the human inquirer himself -- he may have taken his account of the world for its being. Hence arises the following circle, in which ever;J image turns into a model and vice versa: noetic natural man-made pure.model - rational model pure image rational image t i The Timaeus grandly stops this circle by decree: 71 • • • to which of the two models ( tt 0.. d ~ ( \( _rc...n\..) did the builder make t\,e copy alike' to that which is always the same with itself and alike, gr to that which has become? Certainly, if the world is a masterpiece ( v< o. .A 05 ), and the artisan good, it is clear that he looked to the eternal (29 A) • • • And if this is so, there is, again, every necessity for the world to be an im~ge of something. The great thing is for ever;Jthing to begin according to its natu.ral beginning . ( O.f ~ ~ v ) • Thus concerning the image and its model, we must bring out distinctly that accounts are like in kind with those very things which they expound. Accounts of what is stable and strong and belongs to thought will clearly be stable and incontrovertable • • • whil~ acpounts of that which images that model, since it is itself an image ( E- L v.. o v o_s ) , will be but likely-likenesses ( f~ v.;. 6 T"t 5 ) , being like the former only by similitude'° (29 B-C). 'What we must not forget is that Timaeus has introduced the very account which is to follow as a 70 likely myth 71 ( z:.t Kw) , r J ~ 05 29 D 2), so that i t is, by the criterion of this passage, an Ctr �- 20 - account of that world which is made in the i1uage of a sensible model -- and the Timaeus itself turns out to be the likeness of a cosmologist 0 s guandar'(. One more reflection on the principles underlying the use of instrumental models: as instruments of observation these ancient tools are of course deeply and significantly different from those which most captured people 0 s imagination at the beginning of modern times22J, namely the telescope and the microscope. For the latter are extensions of sight beyond and below the range given by nature and take the observer 1£ objects beyond his scope -for instance, in his Starry Message, which announces discoveries made through the "spy-glass~', all tending to assimilate the nature of the heavens to that of the sublunar regions, Galileo says: '°It is a very beautiful thing, and most gratifying to the sight, to behold the body of the moon, distant from us almost sixty ·earthly radii, as if it were no farther away than two such measures 97 23); the very name of the treatise, Siderius Nuncius, which has the alternative interpretation, countenanced by Galileo, of Starry Messenger, implies that the observer has been to the stars. The astrolabe, on the other hand, fetches the star to the observer, as the name b.. O-T eu >. ~ f1 o v o~yCAvov - the ~'star-fetching tool" (Ptolemy1 Geogr. II, 1) implies. Thus, for instance, in giving instructions on sighting with the instrument, Ptolemy says: "the star, as if stuck to both surfaces, is sighted on the opposite side • • • " (V, 1). The rationale of this might be said to be that the star to be sighted actually becomes part of the astrolabe, being itself, as an appearance, a mere fore-sight, serving to line up .t he instrument. What is really observed is not so much an appearance, as a reading on the instrument, the numerical stuff of a geometric diagram (at least. in theoretical astronomy). The readings will be accurate because, except in the case of the moon, the sensible and the rational horizon coincide and thereby make the center of the sphere a portable little epi-center of the world, which the observer may at once be at and look at. Kepler, who was deeply interested in the 97 sphaera materialis 11 (see Note 21), particularly in the reason why it could be used even .in a heliocentric system, explains its character in just this way. The material sphere, he says, is an Heffigy of the world, such as our sight imagines for itself; it is made in such a way that the theory (ratio) of the prime movement and of movements dependent on it can be demonstrated to the eye as with an instrumentn, and the earth is the 91home of eyes" (domicilium oculoru..rn -- Epitome, pp. 101-10 2). It is my guess that the disc.overer of the gnomon conceived the very earth itself in the image on an instrument. Anaximander held, and was probably the first to hold, that the. earth was in equipoise at the center of the world. Scholars are therefore somewhat embarrassed by his notion that the earth is cylindrical, lookin~ something .like a column drum of which we inhabit the upper flat surface2 ) -- it seems so incongruous with his geocentric spherical model. But could it not be that this cylinder is conceived precisely in the i 'mage of a gnomonic column, an 71 analemma" (see V. below), the pedestal of a gnomon? Could _ t not be that Anaximander saw the .e arth 0 s shape in the i image of that very instrument of observation which had shown him something of the sun°s circuit? * * * * * * * To say,then, that instruments are rational copies of the world must first of all mean that they incorporate geometric principles. But it means above �- 21 - all that they are constructed. Now the model-making activity is itself · in want of tools; I mean tools whose function lies inbetween the ntheoric'° tool,s discussed above arid the mechanical tools which literally do w~rk ( t f'f Cl. ) and therefore give their name to the whole class ( o ~ f o..vU-. ). These tools must be a kind of embodied geometry -- we might call them 1 demiurgic tools 1' . Of course the Timaean demi urge himself, as a noetic artisan, has no need of them, since their function is to guide the body and to aid the senses: they nruleH the human hand so that it is not distracted by the fits and starts of the body, and they assist the eye in answering the questions ;'is i t straight?i 7 ,v~is it on the level?n, ~ 1 is it upright?:'. w The geometry incarnate in such tools is nothing else than Euclidean rulerand-compass geometry, a. because it i .s timeless, eschewing those shapes · · · wh\CY. o.r~ · . w:h ose '. produc t ion is mec han1ca1 , 1.e. ,;\ prouuce d by th e mo t• ion in t• 1me o f interconnected rigid parts (for instance, it~ is reported that np1ato himself blamed those • • • who reduced the doubling:the cube to a construction by ~e 'i"'-" l IA~_) l/\.Q l- )'A'\ J( fA.. V l V\ 4-s means Of instruments and machines; 7 KC<-T~O-\Z~v~s - Diels I, 429, 9; this is of course precisely the kind of construction so fascinating to the seventeenth century, whose cultivation issued in Nev.rton°s theory of fluxions; cf., for instance, Descartes Geometry, Bk. II); b. because it is''preferential", always working from the special and more 19 perfect 97 elements, such as the right line, the right angle and the circle. Now Proclus comments on Euclid I, 12, the proposition on the1 construction f the perp~ndicular' and in particular' on its name - V\ v,o. v~ -ros 9 "'{ f CA V\ ' ~the line let fall (as a plumb line) UJ as follOWS: H Thi S problem was first investigated by Oenopides [ 5 cent. B. C.J , who thought it useful for astronomy; he, however, calls the perpendicular, in the archaic manner, [a line drawn] gnomon-wise ( V-..e>.r0.. ""'(V W/Ao V 0\. ), because the gnomon is at right angles to the horizon" (Heath I, p. 271). This Oenopides, who studied the geometric construction of the gnomon, is the same man to whom the discovery of the obliquity of the ecliptic is most authentically attributed (see I.a., above). But Proclus has much more to say about the perpendicular; in fact he almost writes hymns about it. Commenting on Euclid's Definitions X-XII of right, obtuse and acute angles he says: "But the Pythagoreans take the solution of that triple distribution back to principles and do not hesitate to define the causes of those differences among rectilinear angles. In effect, since there are among the principles on the one hand those w~ich reside in the finite, causes of limit, of sameness, of equality • ' •• and"' one. the other hand the principle which resides in the infinite ••• giving diversity to things, • • • one cause, arising from the finite, produced the right angle which alone possesses equality and similarity with every right angle, which . is determined, remains al ways the same and admits neither increase nor decrease. BuL another cause, arising from the infinite, which is inferior and of double nature, produces two angles • • • which have an unlimited tendency to more or less •••• This is apparently also the reason why the Pythagoreans refer the angles back up to the pure causes of the divine arrangements • • • , for that which is right • • • belongs to the gods .• n (Proclus, Commentaires .fil!!:. le Premier Li vre des Elements d 0 Euclide, ed. Ver Eecke, p. 120.) And again: nThe right angle is in fact the symbol of unyielding power which is united to equality, limit, or boundary. It is for this reason that Timaeus calls the circle of the other, which possesses for the divine soul the ratios of sensible things, also 0 rr �- 22 - right; for in our souls the circle is broken into fragments of every sort and undergoes the various deformations which .result from generation, while in whole souls it aintains itself intact and stable in the presence of sensible things 0 (p. 249). It is clear that the demiurge of the sensible model must gork with a carpenter0 s square, in fact, with the prototype of all carpenter 0s squares. This tool is the very symbol of the builder 0 s craft, providing by art those prime shapes which our sublunar habitat cannot produce by nature. The Greek word for the carpenter• s square was gnomon • Theognes, in the mid-sixth century B.C., mentions it along with two other such tools; he says that the envoy sent to Delphi to / ·, consult the oracle should be yg straighter than the 1 0 ~ ;v o 5 [ C<?_rnpass 1 , the ctr 0: '5 ~l V) [ plumb-line 1 and the 'fV w ~ c.,;1 v Lcarpenter 0 s square] n (Heath I, p. 371 ) • It is, however, not knovm which application, that to the shadow-casting upright or to the carpenter's squ.are, was the earlier; the Theognis passage above appears to be among the earliest uses of the word in a text. I would guess that the workman°s gnomon came first -- the builder wants to know that posts are perpendicular and corners square and it is the plumb-line and the "gnomon" that can tell him. · However that may be, the name of the tool was next transferred to the corresponding geometrical shape. For instance, Aristotle, when he wants.to illustrate that alteration is different from growth, says that "the square, when the gnomon is placed around it, grows blJt does not . · al tervr ( Categor':is 15a31 )~-') Euclid seems to have been the first to extend the definition to all parallelograms (I, Def. J): " • • • let any one whatsoever of the parallelograms about its diameter with the two complements be called a gnomo.n". And last, Heron of Alexandria (Definitions, 58) defines the geometrical gnomon as· any figure which when added to any figur·e whatever makes the whole new figure similar to that to which it is added; and he goes even further: ·' "In general, a gnomon is every addition whichj~akes the whole well-proportioned ( £. u ~ "'( e/.\.A... 0 v ) or a figure ( (T A\ n• q r (). .) The scholiast, in commenting on Euclidas definition, explains the existence and the name of the gnomon thus: "It is to be noted that the gnomon was discovered by the geometers for the sake of brevity, but the name arise.s from an incidental property. For from it the whole is known �- 23 ( '{ v c-0 ~ ~) ~\C1..t ) - - either that of the whole area or of the remainder, when the gnomon is either placed about it ( 'IT£ f LT \. i.9 v( T ~ L ) or taken away. · And in sundials it only serves to make known the actual time 97 (Euclid, ed. Heiberg, V. pp. 22.5-226). Presumably he means that from the geometric gnomon one may calculate the number of an area, while from the .upright one may tell the time. Ibes this do justice to the origin, the name and the functions of ..the nk.nower'~? III. ~ ~ 7 ~Kno'\ilrers": ~ Arithmological Gnomon Gnomon: know-man Florio The Elizabethan lexicographer 0 s jeu daesprit which is quoted at the head of this section will prove to be more apt than the scholiast 0 s exegesis, for the gnomona s name is no accident. (The once popular use of ~'gnomon 11 for nnose"26) evidently alludes tq a similarly pregnant homonym -- as everyone knows, it is often the nose that knows.) But how can the gnomon, or any tool, be said to know? a. Obviously the gnomon as throwing off a shadow pointer might be said to know what it points out, while as a carpenter's square it knows the nright" angle in the same way that a plumb-line knows how to find the perpendicular O.V'ld a ruler can command the hand. So also,...rules of right conduct are called the '°plumb7 line of l}fe 1;_ ( o- •~()/A~ (3 \.. o u ) or th~ "gnomon of moralsu ( 'J v vu/Aw v V\ ~ w v - Diog. Laert. IX, 12). 27 J b. There is in Greek, as in English, a use of the word in which it is largely synonymous with the more popular word 9 ~canonP 0 • Iv\ o.. v ~ v means · simply a rod (cf. "canev') used for keeping things in shape; for instance, the staves bracing a shield are called V, o.. v 6 v £.. 5 in the Illiad (XIII, 407). Later it comes to mean a guide line or straight edge such as is used by masons. Its assumption into the theoretical realm is best exemplified in the monochord, a calibrated sounding board displaying the geometric~l realization of ~ system of inteic,~~1s worked out by compoun?ing the simplest number ratios, namely thoseAas~the Pythagoreans had discovered, sounded as consonances when expressed in string lengths. The monochord could be used theoretically for experimenting and practically for tuning other instruments. The theory of proportions according to which the canon was "cut 71 (Sectio Canonis), or any such system of proportions, came to be itself called a 19 canonf', as for instance, the Canon of Polyciei tus, which was both the title of that sculptor 0 s book on proportions and the name of the statue which embo< !ied them: ? 'Ch11tppus holds beauty to consist in the proportion not of the elements ••• but of all th~ parts to each other, as they are set forth in the Canon of Polyclei tus. n28) Thus the word ncanon° came to designate the codified didactic exoosition of fil2Y_ theo:ry_, much as we speak of 1'a theoryn or a 1'1body of knowledge 97 as if they were objects. For instance, in the corpus of Democritus' works a logical canon ( IT'<. AD"flV....W" '..<...c..vwv - Diels II, 91, 10) is listed, just as later Aristotle's logical writings were called the Organon; 9'The Tool 71 • Therein lies, of course, the whole secret of this usage -- theory is here regarded as an instrument which, when mastered, makes a man expert. This is the use which Vitruvius has in mind when, reciting the names of the great Greek theoreti~ rt �- 24 cians, he says that "they left to posterity many things §Qncerning machines and observational instruments (organicas et gnomicas) 0 • 2 J And this is the use most agreeable to the moderns -- in the initial book of modern science, called the~ Organum, theory is explicitly treated as a mental tool: '! . • It is by instruments and helps that the work is done, which are as much wanted for the understanding as for the hand. And as the instruments of the hand either give motion or guide it, so the instruments of the mind supply either suggestions for the understanding or cautions~ (Bk. I, Aphorism II -- such books of aphorfms are precisely what the ancients called a \\gnomo1 ogy'') • ,.. c. The Tables containing the records of readings are usually called canons, for instance the various catalogues and tables in the Almagest, whence instruments of observation in generaljinsofar as they yield lists of readings, are called gnomonic or canonic. In this sense a canon is a handbook of information. In all these three applications -as index, as theory, or as information, the gnomon might be said to act as a tool of .knowledge. But how can there be such tools and what is the manner of their action? * * * * * * * The possibility of the thing depends, it seems to me, on the possibility of conceiving of reason itself as an instrument or of thoughtas being at work. The notion that our bodies are tools, either our own or someone else 0 s, seems natural enough since we can indeed feel oureel ves "pushing ourselves" or being "handled 17 • And since the '° handiest ~', the most skillful part of our body, is the hand, inasmuch as it can itself wield tools and also because i t is our hand that feeds us,all men (except perhaps young men) do think of it easily as the most : instrumental part of their bodies. But the body possesses als<j 1 other tools whose very name means •'instrument", namely the 9'organs 0 ( o p 'f CA. v t;.__ ) of sense. These seem, usually, to work effortlessly and to produce nothing new but only to make us aware -- as we suppose in practice, whatever our theory may be -- of what already is outside us. Among them the eye seems to bring us the most variegated awareness and from farthest off. (All these considerations can be found in Aristotle -- proof that they are natural ; see N ote 30). But anyone who has ever thought or "used his mind", particularly if -though the effort was great- he has found it blunted and undiscerning, finds it natural to think of it too as an norgan 77 • Usually one of the two chief .bodily organs, the eye or the hand, is chosen as the model in the image of which the reason or thought is conceived. The Stoics are the great proponents of the latter metaphor31) which is always taken more or less literally by those who choose it . Consequen 1 \ y · ""\ l things tend to be reduced tQ the tangibly · material, so that even sight may become a species of touch, 32) and knowing itself is understood under the type of 91graspingn -- Zeno, the founder of the Stoic school, is said to have been in the habit of demonstrating the meaning of ;1knowingn by showing his open hand and saying, n 'A visual appearance is of this sort; 0 next he closed his fingers a little and said, 'assent is of thi s sort', and then he pressed his fingers closely together and said that that was comprehension -- and from this analogy he gave the name, which it had not had before, to the thing itself: catalepsis [v...cx 1 ~ ..X V\ 't'S -c o\"\<e?t] • .... *' (Cicero, Acacfmica II, 47). "· · �- 25 Now insofar as gnomons are said to make figures ,,hold their shape ( (Jx._~ ,10-C{.. ) by acting as confining straight edges ( v<. et.. v o..; LS ) and directrices ( ~ .J ~ u \ ~ 1 £_5 ) -- this is in fact a second account of the name. given by the Euclid scholiast (p. 226) -- they may indeed be said to "comprehend" or "grasp ~ 1 their figure. And by reason of this grasping function they may ', in turn, be used as the tools by which .reason itself can grasp figures. Some such commonplace of exegesis ha~ presented itself to several of those who have been struck by the name.33) And yet it misses the point -- for "gnomon" does not mean "instrument for knowing" but ;'knower", and first and foremost the human knower;34)(so, for instance, certain Athenian officials, whose duty it was to inspect the countryside to see if the law against cutting down olive trees had been violated,., were called _, 'V' v ~,,,)6\J £.<.) , -"\ Hmen who . are {. .) 15 ,., , _}_ '" strict because they know" - 6L CU<.~\ IC:.L) 0..\\0 OU y vWVO..L - Etymologicum Magnum, p. 236). * *· * * * * The Pythagoreans, however, understood their gnomon, the numerical gnomon, properly -- according to its name. In a passage of the Physics (20Ja10-16) Aristotle characterizes it, albeit incidentally: ~ " Again, they [the Py)thagoreans] think that the limitless (T6 Rif EL- \ CJ V ) is the even ( -re o.. \ 1 L ov ) • For it, even when it has been enclosed ( LvO.ttoA~~iA.vcJ:t.vo~ and limited by the odd ( eno \OW t\rLf\.\TOU TT € \~\. v ~ /'Lt~ov ) ,/still brings to the things that are their ii-nclefiniteness ( 1() v ~ n- c;. l -f LCLV ) • A sign of this is what happens in numbers. For when gnomons are plac~d .around ( ~ £ f '- 1 '-Ci f. /4 e_ v o v ) , the one, the form ( -ro £.'\ cJ 05 ) is one, and without the one ( KCl~ ,, ")( w t"' lS )35), i t becomes always other ( i. .\\ o & £ l. "( { 't v f. CJ v o.c ) .. The main point of the passage is that the principles of "limit '' and ~' limit less0 reveal themselves in an arithmetical form for the Pythagoreans. For the successive odd numbers can be added gnomon-wise to the original single unit, that is to say, they can be placed around it so as to preserve -· - - - - - - - -1 the sqnare. On the other hand, if d, d ... cA. ""'1 successive even numbers ar e p l aced - ---1 I about the original two, this cannot d. oi-. d... d, t be done gnomon-wise (see Note 35)_, - - ---i t i and the proportions of the resulting cA cl..•d- 1 , cJ..l: I . rectangles change with every addition (although they approach squareness). �- 26 Three particular observations must be made about the passage: a. Aristotle does not explicitly say that the gnomon meant is the square gnomon, but this is sufficiently clear 1. from the analogy with the geometrical gnomon which is primarily square (see II. above); 2. from Simplicius' commentary, who says!nThe ·Pythagoreans call the odd numbers gnomons because when they aa:e 'p laced around square [numbers} they preserve the same shape, just as"'th.e gnomons in geometry 91 (Comm. in Arist. Graec. IX, p. 457, 1-3) and from Theonvs presentation, who generates the square number no less than four times; 3. and most important, from the "table of opposites", the list of contrary principles distinguished by the Pythagoreans, which ) and the "oblong" ( b·.ep 0)A- ( ""t) ends with 17 square~ 7 ( T~Tli 'jWvOV Metaphysics 986a25). · b. Although the square gnomon is clearly the first and most important (see c.), the arithmological, like the geometrical gnomon was later given an extended meaning. For it is possible to reproduce ,,triangular' ~ numbers as well, namely by the addition of rows of numbers increasing by one, a reproductive addition which is properly speaking a canon, a straight-edge. The triangular gnomoh may even be said to have a certain priority (Nicomachus, II, ?; Theon. p. 37, ?ff.) because it is the image of the series of natural numbers, because the triangular number is the first possible kind of plane number, and because the triangle of four rows, consisting of the beginning one and the first three gnomons, constitutes the ntetractys", the perfect ten (which contains all the numbers whose ratios are the consonances36J, which is the very "nature ·of numberu, and by which the Pythagoreans swore37~. Analogously, pentagonal,38) hexagonal and all the succeeding numbers can be generated by gnomons, provided only that the beginning is the unit , . if it is two, nothing can be done to preserve the shape. c. Aristotle speaks of the ~. the figure or form of numbers, and we have mentioned triangular, tetragonal and pentagonal numbers. What are these shaped or "figurate" numbers? "It is the way ( ~~os of the Pythagoreans 0 ')'f c-~t-(v to draw shapes ( O"'>c_'l~o..\ )''says Simplicius (p. 457, 16). Schematographia was the name later writers gave to the geometrical representation of numbers. This assimilation of numbers to geometrical figures, still evident in our word "square numbers", is familiar from the Theaetetus . In order to find one appe11ation for all commensurate numbers Theaetetus "called square and equiiateral that number which can come about by multiplying a number by its equal, considerinp it in the image of a square shapen (I~ CT)(ir~ &n£.\.K 0..e>o..v\~S ""\£yo...i-l,..\;)volf.-147E), the rest were to be called oblong ( IT('o;-,\..~v,'i:.5 Cr)(~YC\. )l'i), and the sides or .ro~ts which could square these oblongs are incommensurable with thos~e\o;~{ffie square numbers as lengths but commensurable as areas, i.e., the root of the oblong with the side numbers 2 and 1, nemely . '/"2, is incommensurable with other numbers as a side-but the area upon it, namely 2, is com.~ensurable with them. The crux of this correspondence is that it points to a class of geometric magnitudes for which there are no numerical counterparts, namely the sides which square the oblongs, the irrationals, which can be made again commensurable only by the "power:' of the square. ) 0 The Pythagoreans, however, understood their figurate numbers -- although they �- 27 ranged them under the same · two heads of square and oblong -- in a way basically different from that of Theaetetus. Pythagorean numbers are first of all shapely. This appears in the fact that the extended class of the square includes in principle all regular geometric figures, · while the Thaetetan scheme, relying primarily on the ~ nature of figures, is confined to rectangles of varying shape. The square number is the EB":ramount, numb e r-- sh a pe only because it is al't'ITClys °'gnomonized" by an odd number~ f and i_p therefore the very "sign that ·odd number is figure-producing ( £ L o o TI o Lo v ) and limitative ( -rlEf c...""\t:.,rn K bv )" -- Simplicius, p. 456, 16. For in the square the leading principle of the Pythagoreans, limit, assumes the character of . the odd41); while the odd, in turn, appears in the square both as its principie -- as the beginning one, and again in the consequences -- as the series of odd gnomonic numbers, Where it is present, a number retains its eidos through all changes of size42)j where the unlimited is at work, rect.::tneJ es of infinitely many proportions arise.43) Th•+ .: u h"' + •:- r n "' r.• .._ • ~ tu1l.....:; 1: be gai_d f-.() h!"'--V v _eido__, s f'O rlll Or' figure? It should' fir st of all t be emrJh::\_;.;j _r,;-,cJ that these numbers are int. cn1 l--~~ I · · . f.m truly prior to their figure: ~ 1 from numbers carne points; from points, lines; r:om lines' plane figures; from plane figures' solid figures: from so1 i_d . figures, sensible bodies."44) It is to this priority of rcd.ng' that the Euclid scholiast refers when he says that the gevmel,,rlv grmmo n has its name Hby analogy" ( &, ti 0 _Jv- f. \ C\ ~ <:> f 8..5 ) , al though he has already~ just .as .;;orrectly Observud +..h!l.-b. :i.-b u~ o d.Lo o . 1 V 1;-:;"1 "1 l·;f 1.}1 o<> 50<1111 Pt. CL"iJ (Hni.b crg. f'• 227, 3) • Y\ l '...-. 1 "l n. 1 HI). r 0 0 The figure arises when the units composing the numbers are disposed nonlinearly as points in a plane. This means that there must be at least three uni ts in the number and so t:the triangle • • • is the most original and elementary form of plane number" (Nicom. II, 7). One of these units is always regarded as the beginning of the number and is "potentially a triangle 11 (Nicom. II, 8) or a square,45) etc.; all the following numbers, nwhich generate ( ~Tt'"O "{l(.. vvD vTct:.5 )46) the triangular, square and polygonal numbers ~ 1 , , etc. are the gnomons. The Pythagorean mode seems convincing enough if one considers that all geometrical figures seem to include in their determination some reference to number -- as is sh0wn in the name "tri-angle 99 or "tetra-gon" -- yet the language of the commentators always tends to give figure the priority. This happens in the very expression "tetragonal number 11 , which, if the intended priority were to be observed, should rather be "tri adie figure " . A particularly blatant case is that of Philolaus (see belo r,.r) who on the one hand speaks of the decaa:ihe Hsource and leader ~' of all things, and on the other, calls geometry the 11 source and metropolis ~' of all of mathematics. ( lli.els, I 399, 26; 411 , 11). Thus the besetting Pythagorean difficulty concerning priorities (see II. above) turns up once more in the thesis of the generation of figure by number. * * * * * * * It is the gnomon°s nature, then, to preserve the~· What kind of a thing is it that can have such an office? The fragments of Philolaus the Pythagorean,47) the man with whom Cebes had studied (Phaedo, 61E), can throw light on this question. The chief of these runs as follows (Diels,I.411,14ff.) �- 28 r '' The nature ( c\) u er '-_S / ) of number is to make · things knowable ( . 'iv w ~ 1..Kc...._) and ts guide and ·to teach a_ yone n who is at a loss ( C'. f\ 0 \ ou )"- 'f.. V CJ ) about anything and does not know something. 1 !or not one among th}ng9 would ) be clear to anyone~ either-' itself ( n-o.>y .\ . ct uTo or as it is towards others .( lT f ~AA 0 ) , if number were not, or its being. But .as it is, in fitting ( ~ ejv"- 'fa ~ C,.:)...; ) them to accord With the soul ( V<. 0-T\ avv I ~ u X o.., ·v ) it makes al}- of them knowable ( -y v (A) ~,-CA ) to sens~ - awar_ ness ( CA I. o 0V\. '! 'i:. L ) and conversant e ( troT o...-yo ~c... ) with ,one an~ther according iQ. the nature of ~ 0 knower 0 ( K O.\o. '(_VW rcve>S ~ 6 crLv' -Y::48) making them into /bodies, and distingui·s hing . ( () .X: L ~ w v ) the ratios ( /\ o yo v S ) of things both unlimited and limiting -- each apart 0 • os Interpretation: nu.mber makes things knowable by its very nature. For entering into those things which have the form of first principles, the unlimited and the limiting things, it makes them bodies so that they become perceptible to sense, and it also gives them their distinctive ratios so that they can stand in numerical relation with each other. Thus it disposes things according to the nature of CJ\:. knower, for the knower has a soul composed of ratios. The passage is full of allusions to Pythagorean doctrines: 1. limit and the limitless are the principles of things and are to be regarded as their stuff (Metaph. 987a13); 2. therefore things are made of numbers, the immediate consequences of· these principles (987a19); 3. hence numbers constitute sensible bodies ( 1080b16) ~ 4. The soul is a harmony, i.e., a compound of number ratios (De Anima 407b27; Timaeus 35ff.); 5. like is known by like (Timaeus 37ff.). ~e other fragments all propose one or the other of these doctrines. Concerning the last of these (AgainsLthe Mathematicians, that the logos distinguishes since thelogosth~t arises in ( . ) £. { w ~1 \L v<. ov' ship ·with her, if it ) the following passage from Sextus Empiricus VII, 92) is of interest: ~' The Pythagoreans say [ is a K ~ L-1 ~ f Lo v] , but not generally, mathematics is, as Philolaus says, contemplative an~ seein3 the nature of the wholeJ has a certain kinnatural for like to be known by like. 0 Logos, it appears, had for Philolaus a double aspect -- it distinguishes and it relates. This is precisely · what the fragment plays on. For number, in distinguishing the ratios (logoi) of things, makes it possible for the. soul to discern them and to give accounts (logoi) of them. This is why the Pythagoreans ??who simply join number with the soul ~·~ call it the ~"discerning ' ~ n tool of the world-working god" (V,fLru<ov Ko~,ou~'lcu i:)£ou of~a..v0v Diels I 109, 29). But by making things into bodies it makes them sensible and therefore available to sense and to the soul. And since the soul is a system of number ratios, so must the bodies be "conversant" (11fco-<Ayor6'.) ~-;th. or related to each other, i.e., they must have ratios (logoi) C\ W1on~ each· · other. Then they wili form a system of ratios or a harmony .and so tney will become known to the soul as like to like. n It follows: as number makes things to accord with the soul, so i t make s things according to the gnomon, which Philo la us, 9. u~ c \< to se.e o.. l\ v, e.11\ c.ss.J �- 29 interprets as the soulas representative in the visible world. For as the gives an ~ccount of nature so does the gnomon give proportion ( o. v CL A o "{ <- C\.. ) to a shape; , as the soul is the reason for the harmony of nature; so is the gnomon the source of similarity in figures; · as the soul takes to its like in world, so does the gnomon fit its figure; as the soul embraces and binds body, so the gnomon delimits Rnd clasps number;-and so both the soul and the gnomon ~re ~~~ guardians of €?l8~ , the looks of the world.49) ·. . . smy. continued Overleaf �- 30 IV. THE SUNDIAL AND THE ORIGIN OF CONIC SECTIONS O. NeugebauGr (Proceedings 2.£ the American Philosophical Society, XCII, 1948, pp. 136-138) $hows that the peculiarly restricted way of obtaining sections in the pre-Apollonian theory of conics (attributed to Menaechmus, c. 350 B.C., a pupil of Eudoxus) is accounted for if the study of conics was first suggested through observation of the shifting shadows cast in the course of a day by the gnomon of a certain type of. sundial. In the early theory ( 1) only rio-ht circular cones, i.e., cones ·with axes perpendicular to their (circula~) bases are used, and (2) the cutting plane is always perpendiculru: to a generating line. The different sections are then obtained by varying the vertical angle (see opposite page; cf. Collegian , October 1963, PP• 36ff.). Both conditions are present in the geometry of the -- mathematically very convenient -- sundial set out on the opposite page: G: GSO: ~ d: -¥. ()\ : GSn: DIAL: SfSnSa: center of rational horizon but also shadow-casting tip of gnomon GSn; -- axial triangle of a right cone with vertex G and the circle of the sun°s daily path as base; ---angle of the sun's declination, i.e., latitude above the equator; verticle anglesc;f' opposite cones; since the maximum for ~J =23°51 9 , -}( cA. = 180°-2 cf is always obtuse; the gnomon, a generating line of the opposite cone; a cutting plane perpendicular to the gnomon; a conic section traced out by the tip of the gnomon's shadow as the sun progresses; the gnomon°s base itself forms the vertex. Since the cone in this construction is always obtuse, the section will always be a hyperbola, but the other sections are easily interpolated. V. VITRUVIUS 0 SUNDIAL A mathematical construction of a sundial is called an analemma, literally, a "pedestal'°. (Ptolemy wrote a whole treatise non the Analemma", his construction is summarized by Neugebauer, ~Exact Sciences 1!1 Antiquity, sec. 87 w ith note.) Vitruvius, the Roman architect, gives such an analemma in his Ten Books on Architecture (IX, 7). The basic (observed) ratio, from which ~rest o-rthe figure follows, is that of the height of the gnomon to the length of the equinoctial shadow (5:4 for Annapolis). The numerical latitude can then be calculated (390N; for the method see Almagest II, 5). Vitruviusv dial is marked only for the sun°s annual path. ~For construction see opposite page, overleaf.) --------- �J •• Early Conics and Sundial \ -~ ; •.., ~ · �Vitruvius 0 Sundial The parts of the analemrna,in the order of their construction, are: AB: gnomon EBIL: meridian in N-S plane BC: equinoctial shadow NAG: equinoctial noon ray GF: 15th part of meridian LAR: summer solstice ray KAT: winter solstice ray FN: equator GL: summer tropic HK: winter tropic N,K,L: culminating sun PQ: N-S axis GH: "logotomus" Circle about D: "menaeus". !/ /' I E1--~~~-->,.--~~~-"ik--~~~~~~~~--t1 Arcs GF and FH, of 24° each, represent the sun's maximum declination north and south of the equator. Lines drawn from A through the even di visions of GDH, the diameter of the '°circle of months", cut the base line in the ratio of the shadows for each month. The semi-circles about Mand 0 are perpendicular to the sun°s circuit about those same centers at summer and winter solstice respectively~ they are used in constructing the hour dials for these seasons. (For the Construction of an hour dial see Sundials, Circular No. 402 of the Bureau of Standards.) �- 33 VI. Notes General references for the gnomon: The Thirteen Books of Euclid 11 s Elements, ed. T. L. Heath, Ibver, 1925; W. K. 0:-Guthrie, ~ History 2.f Greek Philosophy, Cambridge, 1962; G. S. Kirk and J. E. Raven, The Presocratic Philosophers, Cambridge, 1963; see indices. . - t. The Notebooks 2.f Leonardo da Vinci, ed. MacCurdy, pp. 947-98.5; also R. Arnheim, Art ~ Visual Perception, pp. 2.56ff. . . . / 2. the contrary of which is the transparent ( 1ci d.Lu ~ o.v £S ) , according to Aristotle (~ Anima, 418b1ff) not itself a body but that through which bodies appear. 3. See J. D. Beazley, 4. ·H. Di.els, Fragmente ~Development ~ of Attic Black-figure, Berkeley, 1951. Vorsokratiker, 7th ed., I, pp. 331, 20 • .5. This is the theme of Chamissovs wonderful story of Peter Schlemihl who sold his shadow to the devil. The general principle that the image makes the man is held in universal respect in our world, where a person is not cori,s\c~.krcd to be 11 identified 1' , which means to say, to be the same wi. th himself, until he can produce a certain paper with his own picture on it. I have noticed a more innocent version of the same{thing among the modern Greeks, who will sometimes, at a first meeting, show you photographs of themselves, intending to prove thereby that they are people of some consequence. 6. See the picture in Ptolemy, ~ .Q.f. Kings, ed. Halma, p. xliv. 7. The texts on this matter offer certain difficulties(discussed by Kirk and Raven, pp. 99-103) the solution to which may be that· Anaximander set up two sciathera, one calibrated for astronomical purposes and another for telling the hours. 8. See The Athenian Agora, ~Guide, 1962, pp. 108-109. Besides this large piece of architecture, there has been found also an example of those portable klepsydrai which were used in Athenian law courts to limit speeches to six minutes (ibid., p. 164). These pots have been kno"Wn to inspire the envy and admiration of visiting American senators. 9. The noon shadow seems to be distinguished by primitive people too, who stay indoors then, unable to bear the uncanny feeling of going about without a shadow, cf. Arnheim, p. 257. 10. For sun caves, an early form of solstice marker> pp. 52-54. 11. see Kirk and Raven, For instance in Borneo, see! History 2f. Technology, Oxford, 1954, I, P• 117, Fig. 46. 12. See Heath, Aristarchus, Oxford 1959, pp. 21, 130ff.; Kirk and Raven, pp. 81, 101, 103. There is some question whether the earlier men actually discovered the oblique ecliptic or some partial feature like the tropics and the equinoctial circle. �- 34 13. See J. .~ S. E. Dreyer, ! History of Astronomy, Dover, pp. 93, 106. 14. On the story see Kirk and Raven, pp. 8J-84. The fact that pyramids are not thin objects actually makes the geometry of ·the situation rather complicated; it is given in T. Dantzig, ~ Bequest .Qf the Greeks, New York 1955, pp. 52ff. 15. The Parthenon, for instance, (built of course a century and a half after Thales) is a good example of a human scale applied even to a superhuman building. For a Greek temple rests on a platform of steps (crepidoma) whose risers will obviously be felt to bear a relation to the human step (cf. R. D. Martienssen, 1'.h.2, Idea of Space in~ Architecture, Johannesburg, 1958, p~ 83). The Parthenon steps are just twice as high as those of a man, and for human convenience one intermediate step must be added for each rise. Its proportions are therefore double the human. 16. a. Herodotus, insisting for some purpose of his own on the practical nature and the foreign origin of both geometry and astronomy, says that the Egypt; A.nn d~veJ_opoJ geometry for the purpose of assessing the lands swept nuay by the Nile, "iuhence it came to Greece, while the Greeks learned both the gnomon and the polos from the Babylonians=' (II, 109). This seems to conflict with the reports giving .Anaximander as the discoverer of the gnomon (cf. Kirk and Raven, pp. 101-102; Kahn, p. 91, n.J). FurthermoreJmost of Anaximander 0 s inventions have also been ascribed to Thales (Guthrie, p. 74). It seems scarcely possible to settle the matter except to point out that the gnomon, the most natural of all tools, was scarcely in need of discovery in an original sense in any case -- what the sources ought to be read to mean is that he was among the first to develop it as a theoretical tool, i.e., in a cosmological context. b. It . is not kno~m what instrument in particular the polos was; some identify it with the gnomon (Guthrie, p. 33). ·If it had anything to do with the gnomon, 'it mo.i ,ho.v.t \oee\' just~~~ ·t•~, fc;( t~tcorresponds to the geometric understanding of "pole~ 7 ; Proclus, for instance, says that if an upright gnomon is set at the center of a circle its tip will be a pole (Heath, I, p. 185 -actually the gnomon itself i.s a locus of geometric poles, a pole being any point not in the plane of a circle which is equidistant from all the points on the circu..mference). Now the common later meaning of the word is pivot, as in the (Aristotelian J treatise De Hundo: "The whole of the heavens f as I have said; but there a~e n9cess?.rily .two -.:points which are unmoved, opposj-te one another, just as in the case of a ball being· turned in the lathe ( -r- D f v lf ) ; they remain solidly fixed, holding the sphere in its place ( ()1.;vl-:><._0-V-r-ct ), and the whole burden moves in a circle around them; these points are called poles. If we think of a straight line joining these two together, which some call the axis, it ~Qll be a diameter of the cosmos, having the earth at its ce~ter, and the poles at its extremities;' (J91b). However, this meaning misses the original sense of the word, which is 11 something going round 19 , from "tTeAoyo..L , .,,go~ 1 • Thus i t is not surprising to find an earlier use in which polos means nheavens 11 because nit itself goes round and through it all things gov' (Schol. to Aristophanes, Birds, 179). It is therefore equally likely that the polos was a spherical model, perhaps Hi th movable parts. (The pivotal poles become the crucial parts of the first planetary model to embody a hypothetical account of p\0V'c.tq~1mo.t.ions, that of Eudoxus, which consisted of numerous homocentric snheres each with different ooles (see Heath, Aristarchus, Ch. XVI). L 1-. t'-"i!. who\e L.05\.'V\O!i I·,~ ~~~L.'(''1cc.\ C'\v"cA VV\OY~S · CC>V\t-'1V'\UOV.5\';, ' �- 35 17. c. The text: Kirk and Raven, pp. 117-121. On Anaximander in general see ~Origins of Greek Cosmology, Columbia 1960. H. Kahn, Anaximander and ;, > I "" 18. Translating the pun on ex - LW v -- ttun:...going" in c:A L co.v eterni tyu > " cf. 37fJ7: Lou CJ°' v 0\ '-' wv LC>v i.L-V-... o vv.-. • . The pun expresses the character of time as the source of self-contradiction. ;I / ;:. / 19. Ptolemy's astrolabe became what was later known as an '7armillary sphere", used, at least by the fifteenth century, more for demonstration than for observation. A picture of it is given in Cornford, Plato 0 s Cosmology, frontispiece. It is significant that the Neoplatonist Nicholas c;· Cusa was r one of the best known designers of such spheres (see I. Hart, the Mechanical Investigations of Leonardo da Vinci, p. 21; this book contains a brief survey of simple-astronomical instruments, pp. 19-31.) 1he astrolabe of later times, on the other hand, is a plane instrument, as was Ptolemy 0 s 11 planisphaerium11 • Neugebauer (The Exact Sciences in Antiquity pp. 18 5, 219) says that both were based on project ions of the heavens upon a plane. Such plane instruments as a class are called ntheoricn (Kepler, Epitome, p. 31). 20. See Kirk and Raven, p. 100, n. 1. 21. Archimedes says expli.citly that n 'universe 0 is the name given by most astronomers to the sphere whose cent~e is the centre of the earth and whose radius is equal to the straight line between the centre of the sun and the centre of the earth" (The i-Jorks of Archime·fos, ed. Heath, Dover, p. 221 ). This notion must have been suggested to technical astronomers by the cosmological models,forin th~se. the ecliptic is one of the containing circles. 22. See M. Nicolson, Science and Imagination, Ch. I, "The Telescope and the Imagination"; Ch. VI, "The Microscope and the English Imaginationu. Discoveries 24. cf. Kirk and Raven, p. 1J4. ~ Opinions 2.f Galileo, ed. Drake, Anchor, pp. 27ff. 23. a+ b Va2+2ab~b2 -a2 a+b ~ab+b2 gnomon~ ... (2ab+b2) 0 26. Hence in the arithmetical algorism for taking the square root the second subtrahend is still called the gnomon. The N.O.E.D. can document this usage from the time of Ben Jonson: • • Her nose, the .r!nomon of Loues diall, that tells you how the clocke of your heart goes 10 0 • to Cowper: ~'The emphatic speaker dearly loves to oppose, In contact inconvenient, nose to nose, As if the griomo.J2 in his neighbour 0 s phiz, Touched with the magnet had attracted his". �- 36 ... the word gnomon the former mostly one curious those of a of sdom cal and common of both with the which tell its Hto judge The for the Freemasons it becomes a the way we live now that· It says much bears the title books on 28 .. 957 of cf .. G.. pp .. 248-249 New Vitruvius I 6 .. men upon whom nature memory that they are able music, and other arts, pure Hence arts because many are the as tool: as 980a2J .. • The analogy hand, for the hand is a sense of ~ is like a intellect a form of forms and Ch See for instance terms of touch .. for conceived Boeckh 9 P• nos See Heath 2 has given a ' pp 542-545 are: v,o,_ ~ In the see or or ·---!-- -- l ~ ,, I - _[ J._~ The is a be there may be no two, or mnnber the gnomons wc)uld look as arranged in the It is not however, that gnomons around but a can be �- 37 when it changes the proportion of t he figure. In its latest and over-extended such an addition ·c ould be called a gnomon if i t preserved, as do the gnomons of 2., the numerical relations of the figure, i.e., n(n+1). But this is surely not a possible function of the original Pythagorean gnomon which Aristotle and Philolaos mean; even Simplicius still shows some hesitation about using the word in this way: "And although even numbers are not called gnomons in the primary sense, because they do not preserve the same form, yet they may .be added as gnoU},ons to show the unlike in figurate numbers" ( CJ~ Y\ ) -t °' Ta 1 f· O \ ~- 458, 5-7). --c\ On t _ e other hand it is evj.dently h possible to speak of numbers being "pl,.aced around" ( 1H: ~Li l..-$ E.,.,P.- t v o .4 ) as non-gnomons. Therefore V-.:G1.I.. )\<-0r~ must mean that numbers ' which are not gnomons are placed around the two, which is the first even number, and that the resulting figure is even, and different with each addition. 36. cf. Nicomachus II, 26. 37. cf. Aetius, translated in Kirk and Raven, pp. 230-231.• J8. ~i \~ There are alternative ways of ge~eratins the pentagon, as show'n; see Nicomacbus, Intrgduction to Arithmet i c (trans. n~ooge), II, 8-11, especially~· 244, n. 1. See also Euclid VII, Def. 16, and 39. Lt . ,•, ~ ' r 1 ~-~~· i. jj.. !' .S Cotvt-.,,.Ui'-._-("_ (IT J ~ p. 287.t.f.. • ) 40. Galileo, too, felt . the fascination of the tetrasonal gnomon, for he recognized in it the form of his rllle for free fall. 110.vin_g; 9i ven its geometric proof, Galileo adds ~ corollary in which he formulates the rule numerically; m .;Jhile therefore during equal intervals of time the velocities increase as the natural n~1:11bers, the increments • • • are to each other as the odd numbers beginning with n.a:nityn (Two New Sciences, Third Day, Th. II, Cor. 1). 41. which is, in t urn, one of the two elements ( as Aristotle says (Metaph. 986a18). CT x.__o .......... l )( t . i.. C\. ) of number, /"1 42. The Euclid scholiast says that "when the gnomon is added the er\v\ LA.o. grows' while the £.1' J 0) does not change in quality ( 0 ~) v, b. Ax 0 L o~ ) " where the former must mean a particular figure of particular size~(Heiberg, p. 277' 16). , 43. Aristotle 0 s explanation of the way in which the odd represents the finite and the even the in~inite has an alternative)given by Si~plicius, p. 455, 20 0 (see Kirk and Raven, pp. 244-245): as the figure ---~·shows, the even can be divided as long as it r emains even, while the one in the odd blocks divi s ion and binds the number into a finite atom. 0 • • • . . ¢> - ... Qt . 44. Alexander as quoted by Cornford, ~ ~ Parmenides, p. 8, ~-~·for a discussion of Pythagorean number. Passages relevant to this theme are collected in Kirk and Raven, pp. 242ff. 45. so the Euclid scholiast , speaking of square gnomons , calls the unit the �- 38 - first square number (Heiberg, 226, 8). ,) 46. ) ""'.' }heon, p. 37, 12; it seems as if with"· o..rro\f£- vvG.J V\€.) ••• VW C>v rt.-) V,o- ;\ 0~ V "\C-\L '1 he Were trying to COmment On the gnomOn 11 S name. r 47. On Philolaus see Guthrie, pp. 329-333; Kirk and Raven, pp. 307-313. A large part of E. Frank, ~ ~ fu sogananten . tha oreer is devoted to proving that all the fragments of Philolaus are spurious esp. pp. 291ff.). Scholarly literature on the "Philolaic problem 0' exceed reflections on the content of the fragments bv better than ten to one, roughly. There is, however, an article by Newbold. _ Arch. flir Gesch. der Phil. ( 1906) which disin cusses the gnomon fragment (pp~ 17'bff:-}"arur-;.hi~Iha:Ve not seen. 48. Diels (as rendered by K. Freeman, Ancilla to the Pre-Socratic; Philosophens, p. 75) translates: IJ But in fact Number, fitting all things into the soul through sense-perception, makes them recognizable and comparable with one another as is provided by the nature of the Gnomon •• ·"· The 11 differences are crucial to the interpretation 1 es~e~1c;,\\i ~ccov-c\•~·~1-t?. for-'\V'\to.,, 11 1 -t\.-\<. !7.ov\) v\Vlc,\ ''CA.. ~or''t~c 11 ~il\C>V'\!\Of\. • 49. Philolaus applies the word eidos not to th f figure but to the two "proper forms" ( .>l er\ Cl f(JV\ ) of number, the orld and the even! ,, and each of these has many shapes ( ;vt o f ~ C\..L ) which each thing shows by itself" (Di els I, 408, 10). VII. [Corrections to AN ANATOMY OF ORBITS, April Collegian, pp. 1-28.} p. 8, ii, 1a add: In iv, 3 this is the ~enter of ~ravity. p. 9, iii, 2, 1. 3: for HgeOllletri(; center"', read o;focll.$". p. 11, iv, J, 1. 4: for ngeOW\et:rie c.ente.r",~reaci "th~ focqs of the orbits". The figure for the ~ last two should be --;> ( ~--~H; __ '-'-...____ f~N S / '1f{ ::.----- p. 12, diagram: the sun is S p. 12, vii, 2, 1. 7: 77 eccentrici ty AB=AR 0 ; so also in the diagram, p. 13. �FEAR WA.P. OR AND There is a in Thucydides 0 it concerns one of the move men .. of fear Thucydides says that the truest cause of the war the grovrth of the fear necessitated the war of mutual fear is the both as a s for war and a means for peace. and have two us in discussion The first of what fear is.. In fear is sometimes a virtue in the case of fear of God .. fear is also often taunts another with you men often hesitate to say What is the emotion that is such various And if we succeed in with the one that is behind is whether fear, which war, can ever be the basis for be, and how is it to be opposite effects to deal second ssible cause of ir1l11at of fear from the fear which has above.. T>~.s Let us begin them and begin thereof a actions and , from the to i~~~~-.~~ Athenian fear is become aware of their from everyone are of a hurt the future, for i t is the s is in the speech of the are a match for us another too for us .. at once, and Insomuch �- 40 unless we oppose them jointly and every nation and city set to it Url~nimously, tl13y hL~.i overcome us without labor". Consideration of Athct•.:i. '.3.n action ir.. the :h·'.-,Ere leads t o the concl 1 ~sion that, if no p r evi.:H~tl~T <:.~ action is tal-rnn, tiwre wili rc,su.1-t, in tbe fllture, conditions which o.re ha.cmful. At the time when this conclusion str:.kes someone, wheth8r after thought or after the rrm.ch shortc1'" pro c .-; s (:if r..:.~1 un ~·:x) 0etsrl C"lgt .J::;:··: ve action by someone or something el ,'3e, we .:; 2. ~1 .:·: '.)·~' tha,t u. ;;e:i:~ son his Lean stru.:k by ~ .-n os , thls being th9 word 'Ihucyd::_o:.L:s uses in sr:e:::.king of Spz~:;:~+. . an fear .9S the c~mse of the w-ar. It is sorm:·t:.!:'ling that is oL:t side the person involved. W h9n he has been hit by it one of two things can happen, the persvn or city involved will follow one of two coursei>. 0 The first of these is blind panic. When, in the Peloponnesian exhortation before the battle at Naupactus, the soldiers are told that Athenian experience may be useless because ~ 0 fear confoundeth the memory p n5) the hope expressed is that the Athenians will feel so outnumbered and overwhelmed that they will consider their skill of lHtle VJ.lue. And so tl:?.3 i ·}·r.: s::'." tJ.nt factor for success will become chance, which could equ.~ily favor th(nu or the Peloponnesians. But if chance is thus so much more powerful th!ln skill, and there is no attempt made to use skill, the Athenians invo'L ;,red wi~Ll not bs r-:aking decisions. They will not be trying to und .... rstand what the pw.~~ )•)ses of the enemy are and dec-i_d-i_ng what their ot-m strategy should be. In short, they will not be using their reason. And if they feel th2t the injury they f8 3.~ is completely unavoidable, they will hope only fo:::· s011!e k 3_riJ of m~:racle ~ t,.r.ic! :_- , 1 ~-~~ :>.e!les by definition are of all things least subjact to ratio!1:tli t y. .MKi so the opinion of hurt, together with an opinion, produced by the circumstances involved, that the hurt is not to be avoided primarily by one 0 s own efforts, can lead men to panic or despair. But there is another course that fear can take, and this is the one exhibited For they are not e,q_sily sub j ect to panic, since their case is far from a hopeless one. 'I'ney are a po ~.-n :- ful city, \iip·c; 1·:-~me on land, and possessing powerful confederates. nWherefore, men of L3.cedaemon, decree the war, as becometh the dignity of Sparta.n6) Whether or not they want this war, they are not panicked by it. This leaves the way open for rationality. by the Lacedaemonians. The first occasion for reasoning is the decision that Athens can be stopped only by war. Sthenelaidas may have been an irrational demagogue; but even wise and temperate Archidamn\ls thought a war might be necessary -- vgand prepare withal for the war~ 1 7) he tells the Lacedaemonians. And the conduct of and preparation for a war are intensely involved with reason. The actions of the war are in the future, and the future can be known only by conjecture based on reasoning analysis of the past and its analogies to the present. The Spartans will be concerned with causal �- 41 - relationships -- what will result from an imn1ediate invasion of Attica, is it better to wait and attack somewhere else, what should be done to train a navy? The whole problem of confederates will also require much thinking. The . Spartans must decide how best to arouse and hold their own, and also how to persuade Athenian confederates to revolt. The awareness of separation that is involved in jd 05 will make tl.1.crn pay all the mo1·e attention to this aspect of the war. ff And so fear can become rational, and as such is no longer something outside the person who fears. It is more what is meant by the Greek word 6i ~ and it is, as we have seen very much concerned with the future. .J' Thus we have arrived at our second question. Can such a thing as fear be the basis for a league or for peace? Clearly panic, with its heedless thoughtless .action and its passivity, can not. But what of rational fear, what of ~:~) ? The first difficulty is that anything that is to be the basis of peace must be very strong. Hobbes says, "In the nature of man we fing)three principle causes for quarrel -- competition, diffidence, and glory. 17 And these causes are powerful, for within their categories can be grouped all the passions, desires, and prejudices that move men to war upon each other. What makes us think any fear can control them, that it can override desire for gain and glory? Do not people often act d8~ r ite their fears to obtain a desired end? But there are reasons for believing that fear, if great enough, will be even more powerful than these desires. For while gain and glory are apparent goods and certainly sought after by men, they are goods that are to be enjoyed in the future. If a man has a dec1.sion to make, and one choice' .may lead him to gain or glory, these two goods will often come to him some time in the future. The profits on his business will begin to accumulate at the end of the week and by the next week he will be enjoying greater luxury; or perhaps, in the case of glory, he will be honored for his generalship when he returns to Athens the next year. Fear, however, is usually more immediate and sure. Men tend not to have the foresight to fear things far in the future, precisely because fear is concerned with undesirable objects and it is natural not to want to think of such. The result is that when fear does arrive, the potential hurt may be very near in time. The choice beh.reen a fairly immediate evil and a good in the future will often be decided in favor of the former. And, more importantly, gain can be enjoyed only by living men, and even glory seems better if one is alive at the same time that he is glorious. In other words, life is a pre-requisite to either of these goods, and a fear that contained an opinion of hurt to one 0 s life would have necessary priority over the desire for some gain that, without life, would be worthless. Self-preservation is not only an old instinct, but it may also be of the first importance in our actions. So fear of the right kind and in great enough quantities may be strong enough �,. · - 42 to overrule the other emotions. But what is the right kind of fear? The Mytelinaeans have spoken of mutual fear. Let us say that I am afraid of you because you are strong, but you are also afraid of me, for I too am strong. We agree not to hurt each other, because neither is sure of winning, neither wants the risk of losing, and neither wants to waste energy and life in wars. Also, if the two of us combine our strength, we will be that much more sure against the world. Therefore -- let us have peace. But there is a great problem in all of this, one which makes this theory always eventually fail, and this is the factor of changes in power. My C' rr: ·~J C may fail one year, my slaves may revolt, or my city may suffci' sect· .::. _• You, observing this, will judge me eeaker and perha"9s consi r_1 c.=;r :-'. ·!T ~ 2 :-,. - ..,-; 1;. :-·rth the ,. risk, because the odds have changed. Mutual fe?..r i.s c 1 - ;» :..~ .-•._-.,.._-,_-,L~.:: ·.-. :- .:iu no longer have a predominant opinion of hurt. And so tGo i~ the peace gone. 1 : ·. ' .. Mutual fear is not sufficient for the basis of a league. Neither is a fear that cannot be understood by all. For c:i.n example of th.is let us consider the Melian dialogue. We col:ld ~; '""'·=' t llA.t w2 b.:-,-l9 the po s·-,·j)_~ i~.i ty h c :"c t.h : .:t t he Melians are so str1-.:. c.1~ 1._:J ; -:::,c~_._-'. .::i .. . ~ : : •J ~>>~: :'. :t·-;:ss ab ) r (; t ::-~s :~ r/i tuu. ~_, jc·J u~ at they are no longer bf;iqg ~· ~· .::: .L ~ . y r· : ; ·:-. ·~L•:J,~1 "':.. o rt--i t.:. .~ .'--3 r~ ~·-- C' ~-J!J..:~-i.] f..}~:?: c~' ? .::~ (~ :, -i.- 1 J'. i· henians .JP, hoped to have on the m2ss 0s i f E1 ·?-: :~.:-_:_ j L,~ ..-_..~1 c::.~l-j_.,r,,-,:_,:1 t n ~...: l.'. -: ;~:i_-~ '_:;c.J~ ~'./ ~ o t hem. Let us assume, however, that t k _a · 1. ~: b t, t .n.l8 of ~:. :.~ ~; V;:<.ian 1.e2ders, and that the fighting power of the Athe1 :i ::! ·1.::; }·; · i c) no-C c .~-i1r :.J·] c.-tcly p ani cked · t h~ra. There is s ti l.J fc: -sr i.nv0lved here, c--:r1d it i~ fo ar o f ·ct1.-:' ;_: ":'1. equ eV'l s :; ·:; of .Athenian s r u.1 ,s- . S· :o "::-J•~ !\ .:: c-~ei '.1 .s-:-.·si::: s nc :.:, or!'_:.y 1..~·1.~--: [-1 '._:;-,o r u.~:.1 0, bl:t c-·~ ~'.°; po 1 ·i.lu 1;:s ~ f o.r V ·.'.0y h e:. . ·;;~ nc- t>'.:.~· :-~'.-(:.:-: ·c: -~~ , ::~~. /. ~ !1 ~·;:~:.:, n ~1c:~ tl:w ma .:.·cer , r.;::- ·1_::i .J ~ ~"."· t r t:-! ;.; -r,:i..th se vc~ ri ty. An~! 5 f L.'.:·~~ i>:.::i·,'"-~ ·:. :.: v1 j-, :~~~- 8 ~.~ .• -:,_ ,_·; _d l;a co:ro .:n.:. r.; r- r;::: si·; . ; i :1 -c.:•·:. ,') c c._:_ ~-3 c .f :.hi.s ~ ong . : : -. a n·J ·~ Y·;.J ~·:1': ~ '.'0 v:·~-:. 1.~ ~ t, :1 °:.: J:·: :;.J. iC:t l~ l .:~ . ~dt:r:; v:on ~ _ d h ·'.iVG di. f fi c·L•lt:..- - .:1 l e ad'itig a : rE: ,-:> i_ o i' -1.1: 0 c~,_ -;·,y ~ :y::. L they had earli -, i: v:-JJ.untci.rily su.r ~ · c tic:er ed. ·t. But t li .~tt ~- 2 -!-.·'1~ :~ ::1 ~ f :·:;:::. -~.~l e_:. T<·:;-~. :~:.:_;::~\ .•e J ~ i l-:~· 1 :.~-.i.el~ i ~JJ .~:.: r:;· ,; :~. .... ,-:_,.'.-·. .' : o·i..l::·:·;_~ f . :~ .~ ~· : 1a~ i ,.· ·. 1 . 1 1i"'J:-.: •C'."t.0r:d? r..,;:;1;.!. ,.!. c ~-.Ytc, i nl. y t'.;ey know i: ·:> ~er..:: f'r ee. i;.1 !:, they uf l. l·..;:; .:. r rule, n.ot in ;~:_1:·..::::-i·f.::.'." do t l ~:-~-. ~t~e ~,J i'.:.,'! ~:..i':-r·:': . :. :l·, tt; i'::;I.i.o. ~~. s vv1:_::_J b e , ~: . _".'<:.:..'...J. term.:-J o f t.r-j ·o :lt\-"~ t:nt r..ight oe pr8sent Ly l ( "' :. :-::1:; b -u.t in t e.r·~JJ.S of pl2-cing :: one 17 ~ s ·::: U ' j 11 c} 1::: r) ower of people like t h e AU1enians. The Athenici.11 ambassadors s ~1 J, 19 You ~; > ::::. :.:. t.:,:J.-.J3 YC':'Y ab.s t; rd counsel in deciding to res i..st u ~.lr:~ ?· :~-!~ ::e-: t~-~ 1}..:i.-J. ,-!~ 1 :-:: ., ;..-01J r11c•.'>:e ~:!" ~ ) nisst ycurseT v es some more di scr c·~ t concl uJ ... ~ '~~ ·,.-•; T'" :J At t;. 2; ~~.rt~!'."'. i".:.l.:·-.=-; i . t r i::· cc~L!.:,J-~ \..;-::.:;n 3. ~"1 f' (.::.k er , s ul::: j i.:· ,u~ .J.ted city , r ar . ~ i ;;::._t\.:::'. t ..'.~ r- 7 r.i ~ .~ 1 :· t, 0 SC 1.~·1 c":.'Y1. :: '~ :~·._;-,:-::.- ;~1 : ·:::: 1·1.~-· ;. C•:'.'' _i_v;,.~~:;;-Lt!. OUS, they may E Ot re ·::i.1i za a~-:ct l16 l" na:~. icn could lhCl..-J. t he:m in t l1is W~{y ~ 8 . · And S8 pe1·hnp s t hroughout the dialogue the conversation is blocked by this Melian fe ar that the Athenians do not understand. And so it is clear that in any attempt at peace, if fear is used as a means, it must be understa rd able to all. Thus our analysis suggests that fear is a powerful enough passion to control men in the actions that lead to war. As we ha ve seen there is a problem of determining the proper object of such fear, for it must be fear of something enduring and must be a fear visible to all. There seem to me to be two possibilities. One of them is relevci.;it to Tb ._icydides 11 time and later is : developed by Hobbes. This is fear ci' the s·t a te. of war. If war plac es men in a state of nature -- 11 solitary, poor, nasty, brutish, and short''-- · man should want to avoid the evils that such a state entails. • �43 - In the past, however, war has not involved a state of nature for many individuals. The men who govern countries are often not those who fight in foxholes (which might be a good rr.odern equivctlent cf what Hobbes had in mind). Large parts of the population continue to live in civilized society, although the society may be changed in form by a w ar. But now the possibility has appeared, through atomic weapons, of actually destroying all civilization and perhaps the earth as a suitable place for life. And this fear seems to fulfill the requirements for a fear that can lead to peace. If it sounds too optimistic to place so much hope in fear -- and well it might -- we must remember that ~Jhere is a large qualification underlying this thesis. This qualification is sc-.l·;ation and resultant applied rationality. Even Hobbes says that the laws of na_j_.-..ire are arrived at by reason. If man knew them without reason, there wolli j never be any state of nature, or at least not one of the kind that Hobbes de scribes. So a man must be taught, either by experience or by other m~ ::. ~ t hat war is the greatest of dangers. But there is something else he m ust know, that may be even more difficult for him to learn. This is the fact that all men have some basic needs in common, and that all men desire many of t he kinds of things he does. If he knows this -- and on the assumption that other men ~nll be rational -- he can conclude that others have the same dread of war he does. And so he may be free to follow his aversion to war without the risk of domination by others that at first seemPd apparent. I will freely grant that this qualification is very large, and I do not kn-::>·,1 whether it is possible to educate men in such a way. But this theory may be a road -- a way applicable to both men and nations that avoids the cruelty of Machiavelli and the need for divine intervention. * * * * * * NOTES (All fo~tnotes are references to History Qi Pelooenassian otherwise noted.) 1) I: 23 2) III: 11 3) Hobbes 0 Leviathan, Part I, Chapter 6, p. 4) I : 22 5) II: 87 6) I: 86 7) I: 85 8) Leviathan, Part I, Chapter 13 , p. 106 9) v: 111 55 ~' unless � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The Collegian Description An account of the resource The Collegian began as a student newspaper; became a college community literary publication in 1952 and continued until 1969; became a student newspaper again in 1969; discontinued publication from 1980 until 1989 when it again became a student literary publication. Click on <a title="The Collegian" href="http://digitalarchives.sjc.edu/items/browse?collection=26">Items in The Collegian Collection</a> to view and sort all items in the collection. Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Page numeration Number of pages in the original item. 43 pages Text Any textual data included in the document text Original Format The type of object, such as painting, sculpture, paper, photo, and additional data paper Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Creator An entity primarily responsible for making the resource Lachterman, David (Editor) Title A name given to the resource The Collegian, May-June 1964 Date A point or period of time associated with an event in the lifecycle of the resource 1964-05 Description An account of the resource Issue of The Collegian. Published in May/June 1964. Identifier An unambiguous reference to the resource within a given context Collegian May-June 1964 Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Publisher An entity responsible for making the resource available St. John's College Language A language of the resource English Rights Information about rights held in and over the resource St. John's College owns the rights to this publication. Format The file format, physical medium, or dimensions of the resource pdf The Collegian https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/f480baa8b9c13cb85c0dcd87a6b5befc.pdf 59eca23be1459e1f36bac0077cd317b8 PDF Text Text THE '-......._ ~ c-OLLEG . IAN A PR I L 1964 .______..--_.-/" �C 0 N T E N T S AN ANATOMY OF PLANETARY ORBITS ..... . HERACLITUS: THE THAUMATURGY OF LOGOS • THE TRAGEDY OF KING LFAR • • • * * David Lachterrnan, '65 • ... * .. 1 • 29 Jessica Hoffmann,'65. • • • 50 A SELECTION FROM BACON 1 S SYLVA SYLVARlJ11 * Eva Brann, Tutor • • • • * • • • Introduced by • • • • • 65 Laurence Berns, Tutor * * * David Lachterman • • • • • • • • • • Editor Eva Brann • • • • • • • • Faculty Advisor David Rasmussen • • • Business Manager Susan Roberts • • • • • Editorial Assistant �AN ANATOMY OF PLANETARY ORBITS* Eva Brann In reading astronomical texts it is ver1;1 profitable to stop now and then to fix the attention on the mathematical diagrams themselves, not insofar as they assist the course of demonstration but insofar as they are the incarnation of conceptions perhaps never made quite explicit in the text -- to inspect them, as is sometimes said, "quali tati vely v '. No manner of lookinCj is more useful in the effort of gathering that budget of insights into the ways of science and the being of its objects which forms the capital of reflection.** Ptolemy under·wri tes this point of view when he explicitly speak. of applying the consequences of his s conceptions by demonstrations '9using the diagrammatic approach" a classical distinction between a diagrammatic and a numerical, tabular method, cf. Theon, Exposition 2.£ Mathematical Matters Useful f£!:. Reading ~. ed. Hiller, p. 177). In general, two basic kinds of diagrams may be distinguished in our astronomical texts . The first pictures a mathematical hypothesis devised to "save the appearances ~ 1 under the constraint of the axiom which requires heavenly bodies to move in circles and uniformly; the diagrams, in other words, are constructed to rationalize or correct the apparent anomalies of their motion. Such diagrams are drawn by Ptolemy _ and by Co~ernicus. kind pictures the analysis or generation of the heavens by real bodies. ~ The second orbite traced in Such diagrams are drawn by Kepler and *The relation or· such orbits to cosmologies is surveyed in the POSTSCRIPT, pp. 1&ff. . **Some consequences will be drawn in. a note -on astronomical~ thesis projected for a future Collegian. �- 2 - Newton; they meet the 99 great need of imagining the true figures in which the routes of the planets are arranged". 1 ) The characteristic figure of the first kind of diagram is the circle, of the second, the ellipse.*** I. CIRCULAR h7POTHETICAL DIAGRAMS A thorough study of these ;,yould begin with a characterization of the planetary diagrams in Ptolemy 0 s Almagest and would then tTace their transformation in each corresponding chapter of Copernicus' ~ Revolutionibus Orbium Coelestium, that is, it would detail the mathematical Copernican revolution. But that is too great a work to attempt here, especially since Copernicus' circular devices, falling betwixt and between the old and the '°New Astronomy 0 ~ as: are they properly to be conceived as mathematical hypotheses are very hard to interpret, raising questions such or as approximations to orbits? has .his ·attempt to restore the axiom of circular motion in all its purity a "physical" motive? can his diagrams really be fitted into his cosmological model? (See POSTSCRIPT.) I shall therefore confine myself in this section ·to a brief review of a typical Ptolemaic figure such as precedes all determinate adjustments to the · appearances. This inventory of charac- teristics will certainly yield nothing unfamiliar, but it will serve to illustrate what I mean by an nanatomyn of diagrams and it will provide a contrast to the ·Keplerian and the Newtonian orbital figures: 1. The paper plane in which the dtagram ·is drawn represents an ideal geometric m, or what might best be called "Euclidean ***All three-dimensional elements, i.e., those inclinations of diagrammatic planes to one another which account for lat~tudinal variations of planets, will be omitted here. �- 3. . .. un-spacett , for Ptolemy's figures, al though they have a direction (see b. below), represent neither a particular place nor a locus of positions in the world.2) This is a consequence of the pri~ marily mathematical, that is, non~physical nature of the figures they certainly do not image the real motion in the heavens. 2. Each diagram is a separate and isolated device, for there is implied neither a common scale in accordance with which the determinate ratios of the different parts of each diagram are calaulated, nor a planetary system into which they fit as wholes. 3. The diagrammatic ~ of the basic regular planetary ou~c.c- motion (r;,hich varies -in speed for each planet), is evidently ob" tained from those spherical drawings, projections of the heavens . unto the earth, in which the ecliptic lies horizontally, the vernal equinox is front and center, and the sphere is tipped forward to bring the European latitudes int.o prominence (II, 5 and 6). If the tipping is continued until the paper plane · coincides with the ecl ip- · tic as seen_from the north, the basic planetary motion, which is from west to east (see POSTSCRIPT), will be counterclockwise (contrast motion 2.!l ~auxiliary ~ Caelo, 285 b 15). The sense of the circles i§_. !!Qi unitary but is either the same or opposite to the basic motion-- (for instance, in the equ:fi;/alent planetary hypotheses of the epipycle and the moving eccentric, the former has to move in the same, the latter in the opposite sense �- 4of its deferent, XII, 1). 4. The parts of the diagrams differ . :-.somewhat for different planets and somewhat more for the planets above and below the sun, but the tyPical elements are alike. headings, a. circles They are enumerated under three 2f movement, b. lines of direction; c. special points: a. a circle of appearances (ecliptic) having the earth as a center and therefore "homocentric with the cosmos" (III, 3); of indefinite periphery since the e·arth is to it a point; about its center are measured the angles of observation. as a circle off-center with respext to the earth (eccentric deferent)~~of indifferent size, which carries the ~planet in its geometric path; it corrects an anomaly with respect to the ecliptic; its 120th part is the common measure of the diagram. a smaller circle (epicycle)' borne on the deferent and found in terms of its radius, the geometric path of~ apparent planet; it corrects an ) anomaly with respect to the sun.3 a center of regular motion (equant) situated on the deferent's diameter; it 0 equalizesn a discrepancy between the zodiacal (ecliptic) appearances of the mean planet and its correct motion in the diagram so far constructed (IX, 2). Its periphery serves as no geometric path -- it is solely the , point about which alone the planet moves uniformly, the clock-work of the diagram; this is expressed in its na.n;e: Hthe ceayer which does the revolving" ( TO ITE..fL"-yov KEVTfOV ). . b. the radii of the above circles, forming, as rays of angles, the measure of distance traversed, which is never a linear but always an angular quantity in these diagrams. The radii of regular motion are 19 thought of as revol ving 17 the planet ( n.C v o o 0rE v a..\,,. 1\~ r ~ a...-yt.Lv f..08£'l~L III, 3). The line from the earth to the apparent planet is the line of observation. �- 5the line containing the centers of the ecliptic. the eccentric and the equant (line of apsides) which gives the diagram a direction with relation to the ecliptic; anomalies are shown to occur 'S'Yilmietrically ab'ol'i'.tit; its very slow movement (IX, 6) indicates a shift of the plane of the whole _ ypothetical diagram against the h circle of observation. · an (implicit) line showing that the radius from the mean sun to the earth is always parallel either to the radius of the equant (inner planets) or of the epicycle (outer -planets); this means that each diagram is coordinated with the sun.5} c. there are a number of special p(ints such as the farther and cl.o ser poles of the line of apsides apogee and perigee), corresponding points on the epicycle, ·p oints of ™passage (I, J), etc. the most interesting of these are the station points which bound the planet's retrogradations and returns, found when. the line of observation is cut by the periphery of the epicycle in a ratio dependent on the ratio of mean to epicyclic speed (XII, 1~ since this ratio and this point have to fall out of the diagram after the numerical determination of the-rat"iO-Of the epicycle's radius, they represent a kind of corroberation of the hypothesis. If such a diagram is set in motion an epicycloidal orbit of fairly. fantastic shape is described. Ptolemy, of course, never attends to these traced paths at all within the · Almagest -- the parts Of the diagram are only an imaginarjr ( passim) machinery of motion (see POSTSCRIPT). that Ke-pler Ch •. T.) ~ v oO r (; f.V O .v' > . (It is significant M"-q' . trace"' nperplexissimae spiraen , Astronomia llirY:!J Indicative of that is his willingness to admit the equi- valence of the epicyclical hypothesis described above to an alternate hypothesis, that of the moving eccentric, for although both save the 6 same appearances, they sometimes generate qui te different orbits. ) While both the Ptolemaic and the Copernican diagrams fail either to produce real routes or to fit into ·a true system, they are nevertheless fundamentally bound to a cosmological model~ In fact the bloodless part of the ·cop,ernican revolution is ·nothing but the �6 from the center of the center s see WO'' both orien- tation but IL causes of the This is s causis motuum p s concern new The 20 external of mere appearances with cal consequences of the nature of in the of His to us in the in these terms is most Newton's a much more certain ways of the basic features not about a books are combines its may have as the thus actual orbits are as to be in the ways from the The of the is for an the ancient a �- 7true viewer's perch on the world's top. Thus not only is a common measure found for the planetary intervals, but their absolute magnitudes are also determined; the mean speeds of the planets can furthermore be related to these. 2. The diagrams are no longer constructed to Hsave" apparently irregular motions, but rather to explicate ~ and lawful motions; consequently there are no circular, i.e., closed auxiliary motions either same or opposite, but only continuously varying components. From the point of view of the fi.gure accomplished, the sense of the motion in such diagrams is indifferent and the time of the diagram therefore reversible. Since, furthermore, the common planetary direction is not, as in Ptolemaic diagrams, determined from the basic spherical model, its unitary character calls for a special cosmogonic -- account. 3. Since the mathematical explanation is no longer a mere hypothesis, the diagrams now contain parts representing and measuring physical quantities, such as lines of force, centers of force, inertial velocities, etc. 4. Since the center of the orbital diagram now represents a source, or at least a center, of force working across the orbital plane at the periphery, the mediating ~of the figure will, contrary to that of the older figures, turn out to. have a function. 5. While in the old ·diagrams all changes of speed ( accelera- tions) were only relative to the observer and all measuresof speed r,rere angular, the orbital diagrams incorporate absolute accelera.tions along the actual linear 6. ~ of the planet. While the hypothetical diagrams exist only as total figures, �- 8 having certain special places such as the apogee, perigee and points of mean motion, the elliptical diagrams can be analyzed anywhere on the periphery, being everywhere generated according to 7. ~ .fil!2. the hlhile analyses of motions are not absent from circular astronomy, they yield, in accordance with 6., whole figures such as the w~ic'-" ecliptic,~is a component of the sun°s spiral (the only impor- tant actual path in Ptolemy) after the daily revolutions have been analyzed out; only in the new, physically determined paths are motions literally generated in terms of small components; these differ according to the conception of the causes involved in the generation. A comparative analysis of Keplerian and Newtonian elliptical orbits is therefore subjoined: NEWTON KEPLER (References to the Principia, (References to the Epitome, ed. Cajori) E.B. ed.) SINGLE ORBITS i. ~ orbital figure defined Orbits are "properly speaking, that line which the planet describes around the rn" ' (p.999) ii. center ~dynamic • 2. The real, bodily 895, 91o;:- ~ (pp. Orbits are described by any body revolving about a center of force (p. 40.) of~~ 1. A mathematical point acting as a theoretical center of force in mathematical demonstrations ( p. 40). 2. The sun, in the physical interpretation-or Bk. III (p. 420). �- 9 - KEPLER NEWTON iii. ~ place 2.f ~ ceriter 1. As a mathematical point it is fixed in geometric space at the focus of the ellipse. 2. As a body it is on a (tiny) ellipse, relative and similar to the planet, with its geometric center at the center of gravity (p. 164). J. At the absolute and exact 3. In absolute space, near the center of the finite, spherical world and at the focus of all orbital ellipses (pp. 857, center of the world (p:-419). 987). iv. ~bodies in planetary diagrams 1. The sun, formost in dignity (p. 8571.fixed at the world 0 s center (p. 897) , · endot'10d with a motor ~ (p. 896),. in rota- _ tion, and both possessing and giving all light, heat, life and motion (p. 897). 2. The planets, having these 2. All bodies have the same desproperties: cription (p. 398): a. volume (p. 1). a. volume (moles), which varies according to observation(!), as the orbital radius (p. 879). b. amount of matter (copia), compounded of volume and density, and varying as the square root of the radius (c r Y~ ) both for arc he. typal reasons (p. 881) arid to suit the Third Law (p. 880). °"' . . b. mass, the measure of matter, compounded of density and volume and found to vary directly as weight (pp. 1, 411). I c. powerlessness (6.dvvo..rLo.) or natural inertia, an inability to move (p. 894). c. primary properties among which are impenetrability, hardness, etc. ( p • 399 ) • v. The forces assumed Active 1. Active force ac~~ untlate- 1 • Active force is reciprocal or �sun is source immaterial outflow which conveys mo....,.tor motrix) effecting a turning arising in of the sun 895ff. its effect in 2 .. change induces mass of the Law II, p center of force the mass not a factor: f o< a .. 5 • Motor virtue is a .; ;. . ; ;. .;. .; .; .; . ;.:. . sweeping vectoria which the planet attempts to carry it in a circular orbit .. 899 .. another. the geometry of 2, the of force ac)ts as a center ;;;;...;...;,...........,.............. 6 .. Planets possess certain a. a stubborn resistance to the.;;....;....;............ terial inertia 2c; 894) , by which it s the grasp of the s motor virtue and makes delays in p .. p .. 135. �·- 11 - KEPLER NEwrON its orbit. b. a passive magnetic ~ (p. 936) with a ~ "friendly" direction in the plane of the orbit. The more nearly this pole points toward the sun the more effect does the sun's (active) attraction have to cause "libration~, i.e., tho~e 0 small apprp.che s towa.rd the sun which deform the circular orbit (cf. v, 1; p. 965), (The permanent posture of the magnetic thread at right angles to the line of apsides is slightly deflected and restored in one revolution . (p. 941). 'This motion is nothing but a calculatory device, p. 9L~2.) b. Magnetic force differs from the power of gravity (p. 414). ~ iv. ~ shapes £! ~ ol"bits . ·--- Depending on the bodies and the . precise ~ law. involved .the . ellipses are differently generated : 1. In purely mathematical diagrams with a poinf center the orbit is a stationary ellipse (p. 56). 2. If the force law is slightly altered from the inverse square law (perhaps by a distant body's interference) , the planet moves as OV\ a prototype ellipse (whose line of apsides;~major axis J changes direction in the sky as the ellipse rotates about the focus)meanwhile tracing out a more complex (looped) path (Ne~"rton H anual, pp. 127, · 148). 3. If the center of force is itself another (reacting) body, it will itself describe a similar (small) ellipse; the geometric center of both bodies is their common center of gravity (p. 165). 4. The true orbit is a nearly 4. If there are three or more bodies exact ellipse about the sun as involved~the true orbit cannot be found but approximates _fill ellipse �2 the sun at focus focus vii. .e. circular are be a measure, for the oband hence , with the i;d th which the solar ray strikes the 969 -<~ measure can noi,r, beof the deflection of is Let there be circle PD the of P is the apogee is Let there be a center of force the an inAB in the first in- �·\., , - 13 KEPLER NEwTON D . I ." \ \ '' "' \ \ \ \ ·B \ \ I I ·A \ F I l IxM·p CtY1T~r Then 2AJ3;PB is the total libration since R is the closest approach of the planet to the sun. r Divide the circle into least equal arcs PK, KG, GD, • 7-:-:-i.e., continue the 'division to infinity (p. 973). The comparative libration in the arcs is given by the sines LX, HF, EB, • • • (vii , 1b) , and it is shown (p. 974) that these are in the ratios of the sagittae PX, XF, FB. In particular, as the whole sagitta PR of the whole circle is to the total libration PB, so is each sagitta PX, XF ,. FB, • • . ·: to='' ' the · corresponding increments 2.£ libration, which are thus founO. and laid off on the diameter as PM, MI, IF, ••• s A Now let the force act all at once to draw the body toward-Sby the amount BF. In an equal second time interval the body's uniform inertial motion would have carried it through Bc=AB. Instead the two motions BF and Be are addedVectorially (p. 14) to give the resultant BC. The actual path is \the polygon whose sides are BC, CD, etc., a series of resultants obtained by . composing the inertial motions Be, Cd with the measures £! attraction BF, CG, etc. If the time taken is least (infinitesimal), the orbit becomes curvilinear; if the force acts according to the inverse square law, · it is an ellipse with illi:, center £! force & illi:, focus. Using the intervals AM, AI, AF, • • • as radii, mark off points L, H, E ••• on the sines. These µoints can then _ shown to be on an ellipse be (p. 9 7 5 ) . - - - viii. ~ 1. The orbit's µlane is comµosed of infinitesimal ~ torial areas of shrinking radius determined by the equal elements 1. The triangles described about the center of force in the polygonal orbit are equal for equal times (p. 4o). �NE1rVTON For = BC since ~SAB bases = £::,,. same and heights Therefore ::::: The area law to • 983) so as to of the of time the a sum ix. composed v b etc �- 15 KEPLER in the direction of the radii, measuring the effects of the attractive p~~r of the sun (Cf• V, 5; Vl1 • NElrJTON petal effects of the impulse atB(p.41). 2. But the curvilinear orbit can also be analyzed in terms of (Newton Manual, p. 83): o... B S a. an inertial or centrifugal component, tangent to the curve. b. a centripetal component, the subtense of the tangent and the curve, drawn parallel to the line of action·and showing how the planet is drawn back on orbit. 3. But it can be shown (Newton Manual, p. 85) that the diagonal (1b) ultimately equals twice the subtense (2b). This is a sign of the fact that the limits of polygonal orbits are not coinctdent with curvi:.. linear-orbits, sin·c e their .measure of fore~ is not identical. In order to identify them one · must make the special. assumption that "ultimately" the impulses of 1. merge into the continuous force of 2., i.e., that the 'fO'r'Ces are identical in kind. 4. It should be observed that the Kepler's divisions are actually infinite and his integrals are sums of real infinitesimals (p. 973). 4. All components are understood to be evanescent quantities in ultimate ratios as the time becomes an .instant (p. 39). W'.nether this means that they themselves are limits or infinitesimals is not clear (Principia,Appendix, P• 653). · . (In the limit all these displacement vectors are proportional to force vectors.) �choice of the creator: in our xi time as the the inverse l .... c c. the motor ) are system �- 17 · - KEPLER NE WI' ON Now c. and d. cancel each other since as each higher . planet receives a weaker virtue, it absorbs, by reason of its greater volume, a larger amount of it (cf. iv, 2a). And: 1 cl.. c o/. l•c R (circumferences are as radii) RJ:'~ (iv• 2b) R~; ~ But l • C: c:A T, since distance and material resistance compound to decrease the speed and increase the time. : • T xii. oot. ~ R~ • numbers, ~ ~ altitudes All these have 0 archetypaln, mathematical causes (see POSTSCRIPT). xiii. ~medium 2f. ~planets These are not mathematically but theologically accounted for (pp. 544, 566). of ~ orbits 1. The planets move in a limpid ether, scarcely or not at all resistant (p. 857). 1. The planets move in a rare but resistant medium which slows them down imperceptibly (p. 419). 2. The motor virtue traverses this ether instantaneously, i . e ., i t is t r ansmitted nonmaterially (pp. 901, 902). 2. This ether may perhaps be the medium for the mechanical transm ission of force (p. 547) . III. ANALYTICAL VECTOR DIAGRAMS As a contrast to the figurative view of orbits taken above, compare the analytic treatments in the Junior Mechanics Manual (IV B, ii) and the Appendix to the Newton Manual. Attending here merely to the analytic description of the orbit, two sets of terms are to be distinguished: �- 18 - 1. a large,polar coordinate system whose origin is the center of force about which a radius of variable length (r) sweeps out uniformly increasing angles ( !::. ~ )• 2. a smaller, roving system of orthogonal vectors, one of which always lies along the radius. These vectors measure small displacements ( b,.. s) of the planet (which can also be given in terms of the x and y axes). The accelera- tions of such a path are given in a differential equation. In this context the point is merely that such planetary routes are studied mostly in terms of~ vectorial components: the orbital figure is no longer the center of attention. POSTSCRIPT ON THE RELATION OF PLANETARY ORBITS TO COSMOLOGICAL MODELS This paper has so far noticed only tha.t part of astronomy called the "doctrine . f schemata" ( Ik>ctrina Theorica) which deals with plane o diagrams rationalizing the "secondary'', that is, planetary motions. Since, however, such diagrams are obviously determined from the outset by models of the whole world, there should be at least a cursory comment on usphaerics 99 , the doctrine of the cosmological container,as well. (On the size of the sphere of the fixed stars in each model see Note 8.) �- 19 PTOLEMY: In the basic cosmic model two main features, the first physical and the other mathematical., very ill adjusted to · one another, can be distinguished: 1. the traditional geocentric, moving-sphere cosmology of the Timaeus and ~ ~ (I, 1-8) and 2. the professional astronomer 0 s planetary hypotheses, which do, indeed, take their departure from cosmological and physical considerations (for such first gave rise to the axioms of regularity and circularity, III, J; IX, 2; cf. Aristotle Physics VIII, 8), but which have, as the ancients themselves remark (Eudemus and Geminus, in G. de Santillana, The Origins£[ Scientific Thought, pp. 253ff.), long since became divorced from these. The main connection between these two is that the practical, observing astronomer coincides with the philosopher'in having a naturally geocentric point of view (I, 6). The physical part of the model is presupposed (I, 9) in the technical exposition. In this model (VIII, 3) the earth is fixed at · the center while two spheres of prime movement turn. about it, the starry outer sphere with a regula.r daily mo:ti~n westward about the main (north-south) axis, and the inner sphere, whose axes are oblique to the first, in the opposite direction -- these are the Timaeus' motions of the "~", the source of regu.lari ty, and of the '~~", the source of diversity. On a great circle perpendicular to the axis of the second (ecliptic), the sun has its yearly path, 1v-hile the planets' route is in the zodiacal band about the ecliptic circle. The sphere of fixed stars imparts the ultimate motion to the world and serves as its astronomical clock (I, 3). The planetary diagrams fit, or rather, fail to fit into this model as follows: although their center of reference is the earth, they are not conceived according to a common scalejso that their sizes in relation to each other are indeterminate. Nor is there a harmonic rule which would allow them to be fitted into a system of proportions (at least not in the Almagest; as for Ptolemy's Harmonice, Kepler devotes· an appendix of his own Harmonice Mundi to showing that it was a failure). Even the very order of the planets from the center is somewhat arbitarily settled: the inner planets shall be those whose elongations from the sun are limited (IX, i). As far as the construction of a moving · model of the complete planetary system is concerned, it is, even when a common scale for all orbits is invented, not feasible within the ancierit framework of spherical physics -- as ' Ptolemy himself admits in ·a work in which he attempts to construct physical models of the hypotheses of single planets.9) On the other hand, the feasibility and simplicity of such co·nstructions in the new astronomy (with the orbits taken as circular or almost so) is its great point of pride: e.g., Galileo, Chief Systems (Title page); Kepler, Mysterium Cosmographicum (Tab. I, IV). The difference which this fact renders concrete is that between motions ·which have mathematical complexity ( a u/. vrr/\ ov..~), :which are a mixture ( ~~~t5 ) and whose actual simplicity is inaccessible to human understanding (XIII, J), and motions which are visibly and really simple and decomposed only in the mathematical analysis. COPERNICUS: The most immediately remarkable aspect of his system is, of course, the subversion of appearances in ·the case of the primary, �- 20 - vv.-H. I the ·.day-and-night ( \/ u )\~ fl v "S ) circuit of the heavens which is now attributed to the rotation of the earth (De Revolutionibus, I, 11) ~ motion 2f ~ Hsame~ 9 is located ill~ ~· +h.,e, As far as the planetary orbits are concerne~~first, most significant consequence of this revival. of the npythagoreanu (I, 5) heliocentric model is that the earth is among the "wanderersn and these no longer move in a plane oblique to the horizon of the model: rather it is the earth's tilt which is now the cause of our oblique view of things-the constituents of the motion ._... the nether" are located in the earth, of ....,_ ......... _. ._.. which is consequently the source' of all appearances, while the true center of motion is the sun, which-, "resting on a kingly throne, governs the family of stars which wheel around" (I, 10). Next, both the ~ and the of'bital radius of the planets is now given through the "first law •••· that the magnitude of the orbital circles should be measured by the magnitude of time" (I, 10), so that they have in their motions"a sure bond of harmonyn. Now also the single orbits recover their pristine circularity and are either "truly circular or composed of many circular movements" (i.e., the equant is banished) -- but this not done only to purify the mathematical hypotheses but to eliminate a "physical" incongruency: for any deviatfon from truly uniform circular motion must occur 'gon account of the inconstancy of the I planet's] motor virtue • • • " (I, 4), and such inconstancy is unthinkable. Even the planetary paths themselves, although still composed of epicyclical and eccentric movements, do now approximate smooth circular orbits without loops; this is effected by keeping the epicycle small and giving it a speed equal to that of the eccentric. And finally, Copernicus 0 planetary diagrams contain an important new i tern -- the circulus maximus, the orbit of the planet Earth whose center is the origin of all planetary diagrams (see below) and whose radius is their common measure. --- - ·~ So far Copernicus seems to be a modern, an "orbital" astronomer. But several difficulties arise on this score: First, the geometric centers of the system of planetary orbits are not in the sun but ·form a cloud of points near it. (See .t he diagrams in Koyre, La revolution astronomique, P• 65) -- the system is neither "physically" heliocentric nor is1 it really a tight system. Secondly, Copernicus sometim&s slips into the language of solid spheres as bearers of planets -- in fact in the very title of his book "On the Revolutions of the Heavenly,..-.S pheres 91 (see J. Copernican Treatises, ed. Rosen, Dover, pp. 12ff). Third, the devices needed to save the single planets' appearances, Le., the technical parts of the work, are more complicated than Ptolemy's (Koyre, pp. 47-48) and might hence be regarded as even more "hypothetical 70 • (This does not justify Osiander 0 s unauthorized contention that the whole system is a mere hypothesis see ne· Revolutionbus, "To the Reader Concerning the Hypotheses of this Work". It is, incidentally, significant that it was Kepler who first exposed the imposture, cf. Astronomia Nova, back of title page). Finally, for Copernicus the center of each single planetary diagram. is not . · referred to the sun, b:vt to the center of the earth's orbit. (Cf. De Revolutionbus V, this is the reason for the disjointed aspect of the system mentioned above, for this center, which is in fact the ~sun, lies differently with respect to the true sun for every �---, - 21 - planet.) The result is that the sun holds the true center in the speculative, cosmological (I, 10) but not in the technical parts of the book. Thus again the hypothetical diagram diverges from the real picture. The true character of Copernican orbits with respect to the criteria of this paper can therefore scarcely be fixed · and their discussion has been omitted above. The works of the three astronomers that follow may be regarded as justificatio~ and demonstr·a tions of the Copernican system, as the following quotations show: 1. Galileo, Dialogue Concerning ~ Two Chief World Systems, "To the Discerning Reader": ". • • the celestial phenomena will be examined, strengthening the Copernican hypothesis, until it might seem as if this might triumph absolutely." (It must be remembered that the book is written under the -- transparent -- fiction of an impartial comparison of the Ptolemaic and the Copernican systems.) 2. Kepler, Astronomia Nova, Introduction: 0 • • • from which consideration [that of celestial physics) arise certain clear argwnents by which on~y Copernicus' opinion on the world (with a few changes) is shown to be true, and the other two are convicted as falsen. 3. Newton dedicated a manuscript of the Principia to the royal society in which, it was recorded, "· •• he gives a mathemati.cal demonstration of the Copernican hypothesis as proposed by Kepler • • • " (Isaac Nevrton°s Papers~ Letters 2.!1 Natural Philosophy, ed. Cohen, p~ 489 : this is not to say, however, that for Newton himself the basic facts of the heliocentric system were any longer in need of··.justification; for him, "demonstrating the hypothesis" means, strictly speaking, to deduce from the Keplerian period and area laws a force law, . and hence . -the Keplerian or bi ts). It follows that the basic diagrams of all three are heliocentrically determined from the very beginning. GALILEO: In the Two Chief Systems (ed. Drake) the discussion centers largely on thatnewly ordained planet, the earth. However, enough is said about the other planets to make it clear that all actual planetary orbits are considered to be perfectly circular (pp. 28ff., 321ff.), not, I think, merely to make the exposition to l aymen ea· ier , but because the circle is Galileo's line _Q! inertial s motion. Inertia is for Galileo a purely kinematic notion, quite distinct from the resistance of material bodies which was for Kepler one of the causes of the deformation of circular orbits; Galileo 0 s orbits remain circular precisely because his bodies have no brutely .material, unreckonable, 11 conturnaciousH properties (ibid., p. 207). (The. suggestion that Galileo's· orbits are circular because he had a classicizing esthetic preference for circularity, . proposed by Panowsky in Galileo i l ~Critic Qi~ Arts~ Hague ·1954, pp. 20ff., is, accordingly, insufficient.) Galileo derives his notion of inertia from the (idealized) �- 22 terrestrial phenomenon that a body neither gains nor loses velocity along the horizontal plane (considered as a small arc of a large circle about the earth), because there i t is everywhere equidistant from the center of gravity (Chief Systems, pp. 27-28ff). This insight he transfers to the heavens, not completely realizing that this implies that the sun is, in fa.ct, the center of gravity for the planets -- though he suspects that the causes of motion are the same in the heavens as on earth (ibid., p. 234). This same inertial motion:-taken simply as a straight line, enters into Galileo's analysis of terrestrial projectile orbits (Two New Sciences, J:bver, Fourth tay,pp. 244ff.). His treatment there marks the beginning of the analysis of orbits into two rectilinear co~po nents(one representing displacement under inertia and the other under some acceleration toward the center) and of the notion that such orbits are conics, in this case, parabolas. KEPLER: His is the last and most enthusiastic attempt to fit the planetary orbits into a closed cosmological model. He finds three patterns: . The first (and his earliest discovery: Mvsterium Cosmographicu..m) is the cosmological or '0archetypaln model proper, a solid geom.e tric construction accounting for the number of planets (6) and their intervals in terms of thick spheres containing the planets' (eccentric) orbits which are circumscribed about and inscribed in the five regular Platonic solids nested within each other (Epitome, IV, p. I, 3). Kepler cherished this Timaean artifiact, the purely mathematical, invisible structure· of the world, throughout his life, finding a rationale for every discrepancy ~nth the facts (Harmonice Mundi, V, 3 -the ruinous seventh planet, Uranus, was not discovered until 1781). The second, properly systematic, bond fits the planets into a harmony, a "system" in the Pythagorean sense of 0 scale". In this pattern the velocities at the perihelia and aphelia of the .elliptical orbits are taken to correspond to vibrations producing (inaudible) consonances. It has a 99 manifQld kinship" with the solid model · (Harmonice Mundi, V, 2ff.).10J A third rule is the°'Third Law" (cf. II, ix, 2) by which the periods of all planets are to each other as the three - halves .power of thei r mean radii. This is the only one of the three that has survived, since it is implied in Newtonian dynamics (Principia, Bk. I, Prop. IV, Cor. 6; Note ?). Within this system of planets there is for Kepler, the try.e. Qop~:rnican . one orbit of prime importance -~ the earth's (for now the ' ernbhasis is ' oh >· the "geoperipheral" along with the heliocentric aspect' of the system). among his earliest labors was that of establishing its correct ~ eccentricity as a preliminary to finding Mars 0 true orbit (Astonomia ~' Ch. XXVIIff.). Henceforth the earth's orbit becomes more than a circle about the diagrammatic center -- it becomes a moving observation post, occupying a middle station between the inner ·and the outer planets, so. that its inhabitant, the us-Peculative creature" (Epitome,IV, Pt. I, 3, 4) can, knowing its own route, rise above it, as it were, to plot that of the other planets with .respect to the true center, the sun. r �., - 23 .. NEWTON: Through two books of the Principia no particular reference to the heavens (except in the scholia and, tacitly, in the lettering of the diagrams: S (o11 , T [erra1 , . L (una}) is made. Furthermore, when the celestial application is finally made in the third book, we find that Copernicus' and Kepler 0 s much-embattled theories may now be presented as -- appearances: the. Copernican system and the Keplerian period and area laws have become "Phaenomena" III, IV and V respectively~ (This vantage ·point, permitting him to observe as phenomena·· the theories behind the common sense appearances, must have been in _ Newton's mind when he said that he stood on the shoulders of giants.) · .These two circumstances signify that Galileo's demand for a physical theory, uniform for heaven and earth,has been once and for all met; there are no special proofs for or patterns in the heavens, and guided by mathematical theory, it is easy to 0 see 0 the planets obey the laws of motion. In fact, the whole vexed question of the place of the human observer is, for the technical parts of the book, once and for all superseded in Nelvton' s notion of the relativity of "apparent" space: "the dimensions of the orbits are the same, whether the sun revolves about the earth or the earth about the sun'1 (Phae~·~"TV). The plan, then, of the first two books of the Principia (Preface, p. xvii) is ~ 7 from the phenomena of motion to investigate the forces of nature, and then from these forces to demonstrate the other phenomena'°. This requires that the causes admitted must be simple and universal (p. 398); Newton speaks of "universal bodies" (corpora uni versa), ' 7uni versal gravitationn (p. 400). Therefore the third book, "De Systemate Mundi", which is the Principia's "example" of mathematical deduction applied to phenomena of motion (p. xviii), has to do not with a cosmic edifice (mundus) bonded by a harmony of ratios (system), but with a "universe''-- the homogeneous locale of motions performed by bodies all, indifferently, subject to the laws of motion: a world of physical quantities ruled by mechanical principles. Against this background and within the context of the anatomy of orbits certain queries concerning the nature of the Newtonian enterprise arise in very particular terms: Al though Newton explicitly says that •0lines are descr ibed, and thereby generated, not by the opposition of parts, but by the continued motion of points; ••• These geneses really ~place in the nature of things, and are daily seen in the motion of bodiesu (!!::r2. Treatises on the Quadrature of Curves, trans. Stewart, London 1745), his mathe;;-ti'C'al analyses are· not, in fact, conformable with this view. For .while curvilinear orbits are said to coincide ultimately with the rectilinear elements (e.g., Principia, pp. 30-47) of which the mathematical analysis requires that they · be composed, nevertheless : 1. this very mathematical requiremant shows that his passing attempt to make the process formally accessible to the understanding by regarding the ultimate orbits as "limits" (p. 39) is vain, for it is . haracteristic of limits that they cannot •0really take place'° or be c "daily seen", i.e., they cannot become actual at all; . 2. the ultimately diverse measures of force for polygonal and curvilinear orbits brought out in II, ix, 3 above (cf. p. 192; P. Frost, Newton, Principia, Ii£.2i ~' para. 145) show that such orbits are ultimately different in generation and kind (not to speak of yet �- 24 - another kind of force tacitly introduced in the case of the't>rototype" orbit, II, vi, 2, which rotates about the center of force) 3 3. although the Laws of Motion, which require that the ultimate simple motions, inertial (Law I) or under acceleration (Law II), shall be rectilinear, and their first corollary (really a 11 Fourth Law", A. Sommerfeld, .Mechanics , p. 6) which does the same for their resultants, are said to be induced from the phenomena of nature (pp. xvii, 4oO), such ultimate motions are certainly not :r eally to be 0 daily seen'1 , i.e., are not among the phenomena; they are rather abstractions from or the assumed constituents of real motions, as the explicatory· paragraphs under each law (p. 13) bring out-~ consequently there is a gap between motions as experienced and as understood; 4. Newtong s frequent lapses into the langu· ge of infinitesimal a indivisibles (e.g., p. 45, para. 4: "least arcs"), although excused beforehand by a cautionary statement (p. 38), nevertheless bring out what is inevitably the actual vimv of the calculator. In the face of this ambiguity it is legitimate to press the inquiry into the true original nature of the motions whose determination i~ the very end of the Principia (p. 12). One possible means for discovering the ultimate character of Newtonian motion~is to conjecture about them from Newtongs cosmogony, whose stages may be assembled as follows: 1. the 0 universal mechanics" which teaches that "description of right lines and circles, upon which geometry is founded" (p. xvii) is presumably given prior to the world; 2. then are made the universal bodies (Rule III, p. 398) with their universal properties (Opticks, p. 400); 3. the future planets are then put in a sufficiently isolated place, (p. 544) at distances appropriate to the design (pp. 566) about a preponderant body, the sun (Isaac Newton~s Papers and Letters .Q.ll Natural Philosophy, ed. Cohen, pp. 281 ff.), where they remain at first at rest (ibid. P• 296); 4. thei1the "cause well skilled in Mechanicks and Geometry" (~., p. 287) imparts such initial velocities as are calculated to allow the planets to describe closed, nearly circular orbits about the sun (ibid, pp. 286, 297), at the very same instance infusing into them a certain active principle, the power of gravity working according to the inverse square law (ibid., p~ 296; Ooticks, pp. 297, 401; Princinia, pp. 4oO), selected out of an infinity of mathematically possible measures (p. 46, Cor. VII and Scholium), Now the ~'causen that knows . at genesis the future courses of the planets, comprehends, in whatever manner, the whole of the temporal consequence (p. 545; Ooticks, p. 40J), that is, the. true motions of ·a bsolute space (p. 12). I conjecture that the ultimate nature of such true motions is that which we apprehend as ~hole geometric configurations. For such objects underlie the reasoning of the Principia as beginnings and ends, much as they do the cosmogony; this notion is supported by the following items: 1. the description of such figures is regarded as the basic mathematical science (p. xvii); �- 2.5 2. the mathematic~l calculus of the generation of motions has "no regard to time formally considered", i.e., true time (Method of Fluxions, p. 20) -- .presumably because in true space this generation is somehow qualitatively different; 3. after the establishment of the area law (Bk. I, Props. I-II) and of a general formulaic relation between the derived physical quantities (velocity and acceleration) and the radii of orbits (Prop. IV) nthe determination of centripetal forces" is undertaken for special, selected geometric figures; in particular central and focal orbital ellipses are investigated (Props. X, XI). This mirrors the, indeed obvious, fact that forces became determinate only within a geometric context -- given an ellipse, if the focus is the center of force the inverse square law can be shov.111 to obtain; within the Principia the converse is not proved independently, and the generation of such an orbit (Prop. XVII) proceeds analytically (the conic section is taken as given and the initial velocities which will produce an ellipse, ·a circle, and a hyperbola respectively are then established in terms of the latus rectum); 4. the expression of force derived in this section (Prop. VI) is in terms of the "curvature" of the orbit (p. 61, Cor. I, see Courant, Differential ~Integral Calculus, II, p. 307), which is a purely geometric formulation of the shapeliness of the orbital curves; in these propositions the center of force is merely a point to which the force is directed and rectilinear generating elements are not involved; · 5. the shape of the heavenly orbits is found at the ~ of the .c ourse of inquiry (pp. xviii, Bk. III, Prop. XIII); planets are consequently said to"pursue their revolutions in orbits given in !s1!1£ (specie) and position" (General Scholium, p. 543); 6. a last speculation, indeed not to be found in the Principia but rather suggested by future developments, is based on t1·JO features of · N~wtonian force: a. that it is the only criterium for distinguishing true motions (p. 10), its effects being-:ui'e only appearances knor'l7!1 to originate in true space; b. that its nature is quite unknown (p. 547), its effects alone being measurable, and that, contrary to Leibnitz' ~~,it supports no essential methaphysical role. In view of this, might not Newton himself have been amenable to the suggestion that it is the geometry of orbits which underlies as a (formal) . principle the dynamic phenomena, i.e., the appearances of force? Tt...ro final items give rise to more radical questions related to the guiding distinction of this paper, that between mathematical hypotheses and real orbits: first, the ninth section of Book I presents the peculiar concept of the "moving orbit", a hybrid orbit composed of a geometric archetypal orbit which itself, as a whole, rotates about the center of force while the planet moving , on the archetype traces out a true path (cf. above II, vi, 2); second, it follows from the eleventh section (Prop. LXVI) that the real orbits in systems of more than two bodies are indeterminate, so that, although they are very nearly the ellipses they would be "if the sun were at rest, and the other planets did not act upon one another" (p. 421), no accurate path becomes known for any planet. (See the Junior Mechanics Manual, Ch. IV, E on the insolubility of the three-body .problem.) The first .item implies the collapse of the above distinction,forthe "moving" orbit is similar �- 26 to Ptolemy's epicyclic hypotheses: in both the planet, moving on a geometric figure, traces out an actual orbit. The second goes even further to suggest that the very nature of the objects analyzed may prevent accurate knowledge of real orbits and true motions. If so, "true 19 and nmathematical 11 are not -synonymous (Principia, p. 6) -Timaean 6... v 0..7' K vi and not Newtonian lawfulness ultimately characterize bodies and · a thor~hly mathematicizable reality may itself be a hypothesis. NOTES 1. Kepler, Epitome Astronomiae Copernicae, V, Pt. I, beginning. 2. Although Ptole~y begins by asserting that astronomical science deals with motions and seeks "place, time and similar things" (I, 1), in the face of proliferating hypotheses he admits that no humanly construable motions are thus generated --" for it is not proper to compare human matters to the gods" (XIII, 2); see aiso Note 9. 3. to which the large moving eccentric borne on a small circle is the equivalent alternate hypothesis (Ptolemy Manual, IV, 1, p. 36). The equivalence of these hypotheses is not proved in general, but only for the particular case of retrogradation (XII, 1). I . is to t be noted that in contrast to the two hypotheses of the sun (III, 3) which are genuine equivalent alternatives, the two planetary hypotheses are in principle identical, both being epicyclic. 4. This is, of course, a name transferred to the equant from the radii (see b.). The device of the equant is generally taken to be Ptolemy 0 s own innovation; the inappropriAteness of the name he gave it; which consists in the fact that a suggestion of physical agency is attached to the merest geometric point, expresses the equantPs illegitimacy -- much urged by Copernicus (V, 2). 5. This pnrallelism is one of the brutejunrationalizable facts in Ptolemy (Ptolemy Manual,' II, 2 (3), p. 17). It' points up the suspiciously special role which the sun plays in every planetary diagram of the Almagest. 6. Cf. Note 3. A sample of the paths of a planetary retrogradation under either hypothesis is reprinted in Claggett, Greek Science in Antiquitv, Collier, p. 122. 7. Kepler 0 s inquiries concerning the physical causes of motion are thoroughly intert~nned with his researches concerning orbits. Roughly speaking, one may say that the former played both a prior, heuristic and a posterior, rationalizing role with respect to his orbital diagrams. Conceiving early(Mysterium Cosmographicum,1596) that. since a planet's speed depends on its nearness to the sun, the sun must act as a center of force, he imagined that the sum of rays �- 27 .. from the sun to the planet included in any angle about the sun and interpreted as an integral area must be an (inverse) measure of the velocity and therefore a(direc~- measure of the time (so-called "Second Law0 , Astronomia ~' 1609). - Only then, having tried many figures, did he find that · for Mars .at _least an ellipse with the sun at the focus ("First Law") both saved the appearances- and obeyed the area law: "0 me ridiculum, ••• I had omitted no figures among planets in orbit except the perfect ellipse" ( Astronomia Nova IV, Ch. 58. I have r :.ead somewhere that the term focus -- "hearth" -- for the corresponding point in the ellipse was firstli'sed by Kepler; the choice of this term is an acknowledgement that one feature of his great discover-y goes back to the ancients: certain Pythagoreans believed that the planets turned about a fiery center of power which they called Hestia-Hearth, e.g., Diels, Fragmente der Vorsokratiker, I, 403, 11. 14, 19, 28). He then went on to think out causes for the compound of simple motions needed to generate the ellipse (Astronomia ~; Epitome, 1621 ) • His "Third Law" (periodic time2 o<. radi us3 for all planets) was found entirely empirically and the physical explanation is purely after the fact (Epitome IV, ii, 4). (Kepler himself called his rules not "laws", but "axiomata astronomica'', see Harmonice Mundi, V, iiiJ . . -- ' Newton began by supposing terrestrial gravity to extend to the orbit of the moon and then 11• • • from Kepler 0 s rule (Third Law] • • • deduced that the forces which keep the planets in their orbs must be reciprocally as the squares of their distances [ inverse square law] • • • " (Newton on the discoveries of the plague year of 1665-1666, Hall, ~Galileo 12, Newton, pp. 278ff. and p. 360, notes 4, 5). The next step was to demonstrate geometrically that planets in fact move in Keplerian ellipses; Newton had done this by 1684 (i!2i9,., p. 293). The Principia appeared in 1687. 8. The size assigned to the starry heavens is not of immediate importance in planetary diagrams; nevertheless here is a summary of opinions: Ptolemy (Almagest I, 6): the earth is, for purposes of observation, as a point in the center of the heavens (except in the case of the moon which is too ne.a r, VI, 1 ) • Copernicus (De Revolutionibus, I, 6): the heavens are immense at leas't in contrast to Ptolemy 0 s~ it is not the earth that is infinitesimal but the heavens that are indefinitely great. This is a necessary presupposition for a system in which the earth moves if no sidereal parallax is observable, as Aristarchus, Copernicus' ancient precursor,had already acknowledged; see Archimedesv comments on this in his Sand Reckoner (~, ed. Heath, Dover, p. 222). Kepler (Epitome, IV, end): the dimensions of the starry sphere are exactly known -- its diameter is 60 million earth diameters and its thickness(!) 2000 German miles. Newton (Principia, p. 6): from its nature we may infer that mathematical (true) space is boundless (General Scholium, p. 544; First Letter to Bentley, Ne~rton's Papers, p. 281); the material universe in this space is likely to be infinite and to contain other planetary systems . ��- 29 - HERACLITUS: THE THAUMATURGY OF LOGOS David Lachterman Ou1< tJ of ffA-<,a ct~~~ lo~ Ao 10J ~.t..oJef,1.VTo(S or~o~oyE'ZV I{;, C.J I ~ Cl oV £ () r L V 'i.. V 7l d. \/ T °" t <.. v' cU.. . 1 F r c:· j (\11~ Yl + .5 o "Listening not to me but to the logos., it is wise to speak in correspondence with the logos: 'all things are one'." "All things a:re one" -- in this dramatic utterance three words clamour for our attention: hfill, panta, einai. The utterance as a whole seems more like a solution than the setting out of a problem; nonetheless, I hope to show that the dia-~tic of the ~ and the panta adumbrated in this fragment is the fundamental problem with which Heraclitus is concerned. Furthermore, out of a consideration of this dia-~tic of the ~ and the panta emerges the possibility of cosmos and · it is of this possibility viewed from many sides that most of the Heraclitean fragments are speaking. Finally, this dia-lectic determines the form of human utterance itself, that is, o"f'human knowledge; man is the primary implicate of the logos in the sense that he, his words and his thinking are woven ~ the. fabric of the whole of things in the most responsible , that is, most accountable, way. Three phases (statements/manifestations) of this dialectic may be conveniently distinguished: its immanence in the visible, 'material' world (pvr·, tropai PY.!£.[, phvsis); its invisible presence as the structuring principle of ordered (and hence; intelligible, -ent) reality (harmonia, 'Dike/ polemos, 1£ sophon); and its re-presentation in human discourse (nous, onomata). The interwovenness of these three phases· through the logos brings 'being' as 'well-regulated, measured strife' (Dike/ polemos) to light. · In order to confront the ~-panta dialectic which is the basis of Heraclitus' thought, it is first necessary to clear away several Aristotelian distinctions and 'laws of thought' (Denkgesetz.e) which obscure our apprehension of the fragments. (Cf. esp., .Physics A, ii 185 b6 ff.; Metaphysics I, iii 1005 b23 ff.; Ibid 1012 a 24 ff.) Heraclitus presumably listened to . 0 what is' speak to ·him through the logos, rather than presuming to speak about 'what is 0 through 'logic 0 l); consequently, Aristotle's discrimination of substance 1 ) M. Heidegger, Was ist das -- die Philosonhie, p. 52 Herakli t und Parmenides weren grQ's;en in deID Sinne, dass sie noch im Einklang standen mit dem A~ros 'd.h. dem t: IJ:., rrlvTa<. �- 30 - and its accidents2), or, more generally, the division of reality into categories, the concept of unity as identity in defining form ( A cf,t~ )3), and the inviolability of the principle of contradiction as applied to a substance and the pairs of contradictory accidents potentially predicable of it4) must all be overlooked if we are to come to terms with the force of the hen in hen panta einai and, through it, 'With the 'dynamics' of the log~ PHASE I Immanence of the hen-panta dialectic in the material, visible world Aristotle (Met. A 984 a8) classes Heraclitus with the other preSocratic 'physiologoi' (986 b14) who sought a 'material cause' out of which all things come to be and into which they are ultimately resolved -- the substratum which persists through all change (983 b8-19). Heraclitus is said to have chosen fire as his 'material cause•. It was the shortcoming of these physiologoi that they distinguished only this aspect or factor in the constitution of the real, neglecting the other three causes which were left for Aristotle to discover (993 a10-15). The insufficiency of Aristotle's treatment of Heraclitus will become clear, if it is not already so, as soon as we survey the extant descriptions of the role of fire in the whole of things. This insufficiency is only another instance of the inadequacy of Aristotelian distinctions (e.g., the~ causes) for the study of his predecessors. Fragment fifty speaks of panta -- 'all things'; otger fragments also 0eak of panta or ta panta: rr °" viwv N 1.. v 1J t.v £ v w v 7 ( Fr. 1 ; ,Ti~ v ,.-rA. .,.~Lo(. ~ v I ;Ai v £I<. v ~ t, pv ') o.. w at I Fr. 41 ) ; T d.. rr ell\/ Tot 0 L ~ Kl r£1,.. K f \> °' I) v 8 s (Fr. 64) ' and so forth. What meaning m~st, we giv,e jo _this word? Frag!J.lent seven provides the answer: ~l iTi:Av'Tdt. TrA. 0V1ol 1~irvJ<, 0 £"vCiL-ro ••• ; ta panta are ta onta -- the exisfent things, the things that are. -and it is of these that the logos says: "they are one." ( I ' r - Heraclitus tells us more than this; he speaks of the manifestation of this oneness of ta onta in the world accessible to the senses, the 'natural'world, as we should say, This world is for Aristotle the world of hvle; for. Heraclitus, it is the world o·f pvr, or, more specifically, of the tropai pyros -- the transformations of fire (Fr. 31). The foundation of the oneness of this 'natural' world, the world in which all things come to be (Fr. 1) is said to be fire: 2) Physics, 185 a24. 3) Physics 185 b8-9. 4) Metaphysics 1005 b19-26 �- 31 - I u All ' things are exchanged for fire and fire for all things, just as goods are exchanged for gold and gold, for goods." (Fr. 90) How can such an exchange take place? More fundamentally, in what sense does Heraclitus understand the relationship between fire and ta pa~ta, that is, ta onta? If of of us we were to accept Aristotle's judgment, fire would be the matter all existent things, the persistent substratum of the manifold things rivers, mountains, clouds, plants, and so forth. Let listen to Heraclitus: first of the transformations of fire is sea; and of sea, one-half is earth, one-half, fiery storm-cloud." (Fr. 31) 19 The Three terms are given in this fragment: sea, earth and fire (the fiery storm-cloud). Their relations may be schematized as follows (following Kirk, The Cosmic Fragments, p. 332): Fire r le ~H ~ r-\,<f u I lff VJuff] f Sea r Earth The right-hand column is based on the second half of fragment thirty one: "Earth (following Kranz) is melted into sea (or, as sea)." One additional fragment must be taken into consideration here -with what seems to me, following the persuasive arguments of Kirk, (op. cit., p. 343 ff.), a necessary emendation: �- 32 - nv r-1s &:v:.•oV' 1<~ F1[!' .,~5 ·1 IJ\/ TfvfO~ ff~vci1.To-;J u~wf )·yi Tb\/ c:H fOS rr v ro s &J....v~To ,., ) a~ , o " os . 55 n-Cf \ I"' \ <_I ( I) cJ. .-.. T "Fire lives the death of earth and (air lives the death of fire), water lives the death of (air) fire, earth (lives ~death) of water." (Fr. 76, in the version of Ma.ximus Tyrtaeus. Cp. Fr. 36 in which air is not mentioned.) Deleting the mention of air (which seems to have been added by the Stoics in order to bring Heraclitus into line with the traditional four-element cosmology handed down by Aristotle), we are left with a cyclical description of the transformations of fire in which the change from one 'element' into the one immediately next to it in the schematism above is metaphorically represented as 'living the death': the disappearance or dissolution of one element or world-mass coincides with the appearance or formation of its successor. Nonetheless, fire is given a pre-eminent role in both Fragment 31 and 90 (the exchange of all things for fire). The fragments already exhibited make it clear, I think, that this pre-eminence is not to be associated ~nth material causality in Aristotle's acceptation, that is, fire is in some way actually transformed into sea, thus beginning the cycle which eventually turns earth back into fire. This 'eventually' cannot .be emphasized too strongly, for it indicates that the process of the tropai pyres takes place in time; in other words, fire is not the actual material constituent of all existent things at all times5); rather, all things are eventually exchanged for fire and, reciprocally, fire is eventually exchanged for all things. (This description, however, does not intend that the complementary processes be understood as consecutive (as in the Stoic £101 (· ~ i.v(ft. ~ interpretation); instead, both the way down (fire-sea-earth) and the way up (earth-sea-fire) are always going on at the same tims.) Consequently, another sense must be found for the pre-eminence of fire in this fundamental cosmo-logical process; this requires a discussion of the cosmos and the logos as revealed in the transformations of fire. Fragment 90, which provides the most explicit statement of the relation between fire and 'all things 0 , furnishes an essential clue to the manner in which a cosmos can come to be out of the -----------~~-~,-~------------- e 5) Fr. 67 ( LI Wf-J. l'a\ ) presents special difficulties; I feel sure, however, that it does not fit in with the notion of fire as the material substratum of all things, even though it might be said to be the substratum of the process which comprehends all things. Cp. also Plutarch, Consolatio ad Apolliniurn 10, 106E (Kirk, op. cit., p. 135) where Fr. 88 is cited. The Heraclitean fire is obviously confused with the image of gold and the moulder of forms in Plato, Timaeus 50 A-C. �- 33 transformations of fire. The exchange of fire and ta panta is analogous to that of gold and goods; in the later case, however, we know that the value .received is equal to the value given, that is, so much gold purchases just so many goods, and vice-versa. · If the analogy is strict, there should al~o be some equality or proportionality governing the transformations of fire. Fragments 30 and 31 remove all doubts concerning the intimations of fragment 90: < > s:)."- tfo~ ")(;, !i,l fl~ ~V I ,. . , o/. fo v i 7_ {,, IJ I () V .. dL~'/Jt:lil A 6<J v ) 1>' Ko ... o s yo ~ . r~Vtc(£r-ZL ~ • /4~1 J\L<i.TfttT~ IT A? o \j J. 8 '- v L "Earth is melted into sea and is measured out in the same proportion as before it became earth." (Fr. 31) "This cosmos (Weltordnung, Diels), which is the same for all, no god nor man has created, but it always was and is and will be, an everliving fire, kindled in measures and extinguished in measures. 99 (Fr. 30) The logos which supports the hen-panta dialectic in the material world-order is thus revealed as metron -~ measure, or, taking into account one of the meanings of logos itself, proportionality. The cycle of the transformations of fire accomplishes itself in a measured, that is, orderly, fashion, so that the dissolution of one world- mass coincides with the formation of an equal or proportional 0 measure 0 of its successor. The 'measuredness' of these transformations is responsible for their unity: because all things are exchanged for fire, and vice-versa, according to the same measure or p~oportion ( .A ~o o s ) and because this process never stops ( rru p ~t f J i.v o" ), the 0 collected 0 whole, which is the successive manifestations of the measured changes of ta panta, is the cosmos, that is, the world, the comprehensive order of all things and the activities they perform. I cannot disagree more with Reinhardt 0 s interpretation of the meaning of cosmos as "ein bestimmter Zustand, eine Phase dieser Welt in Gegensatz zu anderen 1<-.rf6' fA-- o <... , anderen Phasen, vergangenen oder zukunftigen. (Parrnenides, S. 174)." It seems to me that the emphatic eternali ty of the cosmos ( 1<60 fUJ v 1 ;vJf. ) and its uncreatedness establish it as the unique and enduring world order, although within it the manifestations of fire are continually �says: L. ' Fr .. ~ o <:) as \ os that is f, o .s , there s �as the , sea and the measures' opposed to one and, hence, It would be well to s Heraclitus says: is opposed ( together) is and from things that are at variance with one another comes the most beautiful and come to be through (Fr. 8) �s where war is �some sense in a says: way ..., ~ £L L d~ Vd..L ~L~ &-~' ' ' To 'L it / E o vT a.... no I 't. v' cl_ \Td, VIJ.. s -? <..5 u o '7) l'--o\ T ) war is common and come to that have ,., \ sun exceed its if it does the the servants search after it and find it out. The the of the is the intention of this If we take the of the sun the the makes some sense but however, we call to the antithesis ~1 "' J.. I and think of the sun's exceeding its £"urpoV1"'J measure of be to in the summer, the is here the function as to Now that most of the fundamental relationships or this phase have been exhibited, it is to some more specific examples of this 'harmony' of 0 opposites 0 in order to come to an of the oneness of the real that supposedly emerges four of oneness come to in �found the hot grows becomes wet and one and the same. way of the s is motion of the fuller's towards a end con- ��.d-., \<'-a', " -, 0 v ,..~c are whole the concordant oneness comes from all oneness t... c 0 winter and summer is surfeit and 05 war sense of the The the later -- God as the Whether s is the sense when the nature of ' all here and The one and to be called wise the is observances and the god of the and does not Fr. the seems to What Heraclitus seems of most human reof the in some ways �cosmic correspondence it associated Therefore \ 0 says: such a structure which variance and its source in therefore, there must be a wise the cosmos and it is to this name of , if under- arts or is not such as an artisan is, the intelligence that 9 fits things is looked upon, then, as source of human enables man to bring an orderly is this same ""'""',.·" the cosmos. 0 .... �7 0 ever is �wise to cosmos: v was 3 and scornful references ::::. �ETOL J c;\11... occurrences of the connected see .. of most men is together remarkable in -- an of these they do not make them one .. to encounter c. 0 / C' 0 OOd.. 5 men awake escapes the notice of other what do when .. 1) moreover from the What are the consequences of this all men's deeds and the �those who are awake responsive to the is one, common cosmos, but each of those to turns Along with deafness as those who are asleep 11 seems to have been reproach of the The sense here and in fragment is that each man creates in his a world unrelated to the world of the waking.. However, it is precisely this latter world which the maintains and holds together which, not be a if for the 1 Nhat is more, the common cosmos is as a the strife and contrariety which is everywhole in it; on the the world of the sequence of fantasies no inner there is order into the confused images then, the true like the It is not to with the nature hotrever, that necessary if men nature of the world into the nature one the not one will not the since it is not to be searched out and is Fr .. 18) If we must a subject to the seems the wisest choice. IJfuat is meant then, is that tractable and contention of all one to structure which knits whole, s, one would not expect to Consequently, the evidence of the sense is not it one to see some forms of the of opposites, but especially not the most unexpected -- the invi For other clues or �for who have many who to not awareness is the name of the bow is We are not its work is here 1JJi th a mere is an sense s is clear �rather~ the true the of the words To such connexions we must have our utterance one another is no mere it is art cleverness with as a cosmos .. most at variance of on of the same Thus , more than the than the continuous life more summer and Tuanter cohere all Greeks that to all and as With this is and of common become furthermore is common to much the same as the the in the fact that follow what is· common.. the live as (.., with sense it is what is common to all, as the with even greater reliance. For nourished by one For it wi is sufficient for above. Fr .. 11 But �overtake �proper does the 0 fitting-togethera remain hidden; the specific as until it is what is other than it. All of this to me but se to 50 of which manifests itself in with what the attempted to show in the existent itself in and as the �Jessica that wear rags make their that bear their are nowhere pursue what he s not fend a every the basic and he feels and to he is free to express if he On this level, man is and wit necessary to secure and de ... among the �the active element of the is The �nature mutinies; in countries and the bond twixt son the natural unnatural force of evil of own and act wisely in circumstance He must see that, the attainment of that to the mandates of the of his own S to any not the master, own ruin .. on man's fortune: a He must be aware external of nature and .J..l.J....l..!.l.\AlJ,\:iv He wit, the and the rain content 1iiiith his fortunes fit ........"', . . . "" . . . the rain it raineth every day .. With natures is s role in the total To The the substratum, they use to obligations on that nature bond of the act of the ' I of nature �the love Lear to advance themselves as ruthless women of the burden of a them his land, he says: we The name and all th 0 10 to the and Goneril are obligated to father, their king; but they not his knights, he had demanded. of those titles in the necessary and to o-wn wishes. Thus they of nature wi. th the powers allotted them in the substratum under But the dictates of their natures. They are evil of s rejection they are not Goneril sees the of He loved our sister and with what poor hath cast her off Not are they aware of the but they use its weakness to their own advantage .. Lear at the very their own Vision, then, even in men, them to mould their own fortunes. Shakespeare paints such a black portrait of these two Lear off into the out Gloucester 9 s eyes he sees the force of evil in human nature as the most in the Natural Order. of also contributes to the their in the tragedy .. not that vision that rashness: The of Oswald and are also I think, worthy of the Let us as the agent of his mistress 9 s Although it may be argued that, in true to his mistress, he s his ti.on as servant in the suprastratum, one cannot forget the fundamental obligations to honor courtesy, and devotion as a subject that he betrays in siding with , being of less wit and power than Regan, is also an agent of his mistress~s evil nature.. He expresses their evil use of to the order of suprastratum when they are about to Gloucester's eyes �me All with structure have I can 5 the �is one of with the is that when we surfeits of 9 The first mind, but have vision. She the devotion harm she is to circumstance: to Lear in the but rather in a fruitful I out frown s �and Kent seem to me to merry. as a man. in him Kent is on the ••None but the Fool The is the substratum He says to ne 0 er fear to se have that your countenance which I would fain call That se Here we of nature. His vision comments to on �thou art me who I seeks for for to rain in the storm. But I The Gloucester he is action of the If I The as no master work the means To make so much �gods to wanton they kill us for He to a crowd in a blaze his of nature but says I have no way, and ther~fore want no eyes; I stumbled when I saw.3b) master, in character. for the view of the in cuss later has nature although his final is not a blind statement: of time we we feel, not we Let us s this flaw natural forces is of the as trusted it So blind was in of nature beyond that level, that extreme cruelties of and Goneril of the force of in human nature) and the fiercest tempest of the most fundamental power of the the total vision of nature control of his fortune to save him from his fall the moment he fall from the words of his own 0 most small fault, How ugly didst thou in Cordelia my frame of Let �was frame nature in its was in the suprastratu:n of nature He was· King i;\rj_th power and possession of all his country men paid tributes to told him he was up v-Ji th all his whims, and shmrnd respect to every aspect of his being. Ideally he was as because his person was every inch a But idealistic view is a blind one. If all men were good, if they all selflessly used the qualities allotted them in the to support the suprastratum, this idealism would be the But in the of nature, every man is out for and in the natures of men, the force of evil s basic the pampered was not aware of He idealistically, blindly, the natural. course of events.. He had than his other daughters, but when she avowals of he ected the crucial moment, he relied on the outward known and He was to the He was as blind to the ruthless condition of nature as he was to the natures. was his to nature made s so By From Here the of the orbs cease to be,_, care .. J9 prey to foul fortune as natures of Regan Goneril substratum of nature. In the above Gloucester, expresses the 77 excellent by fortune to the orbs above He is fact that the operation from whi 'L he exists and ceases to be is that of nature; the ·virtue of which he is born, the suprastratum in he has t;. and his own individual nature which brings on his t.:T1r::::,: _cin .. to nature is his fault; so nature be U1e ect of his 1 1 of king in the supraoccurs simultaneously in the , he was so fixed in the we have so unaware, that the other by forms could he of nature in , he When he is the cruelty of his to m~n°s sh and evil is l'J2ver instincts, he could not foresee their trcJ.tmont of showed him, he questioning the causes of the always accepted it at face value as genuine respect for his noble He is rather than for his and says: and baffled by Regan's and Goneril's �11J'Or se than murder such of nature s reaction and we know that Doth Save When is Natural any cause in The answer to the is forced and that When the rain came 1rJhen the thunder I �I am even the natural see of fortune that men meet in their natures. the force of I as the cause of nature that We see our nether crimes realizes that there is an nature; he the of force of , well mv-are that is clear when he says Know thou· this that men are as the time i when he is sins see that evil men can of nature the that out in the escape the ~n,=~·~ source. No human Lear says of this: breaks to men for his our us where thee vices cer �and Goneril kill each other on of their deaths is Yet the evil nature is seen However in this when Lear cries out for at the end of the have should a , a horse, And thou no breath at all the as the fe The good are that we must remember vision, if he is both has every to live rat virtue of his if he does is him from the have to live'? I feel ............ . . , ... _,_ ng the Order nature 0 s It is at this If a man in nature not have for bethe form if man were not Lear s blind exi It would have been Lear himself this men - ,, ·:::t•xre have upon in the witl1 the clone by means of the awesome of human existence and action. of man. In - : demands threatened by the challenge of a to be through their understanding Man is a proelements of the substratum of nature. tects the construction of a in nature. existence in the suprastratum raises high above the other creatures of nature. Man is the creature. In his nature is forces of nature. and Goneril s ��22 25 .... 26 28 29 .. II, scene i, Act I, scene i scene ii, 11.. 6 Gentleman, Act scene vi 1 • 20 -203 Kent, Act I scene i 11 155-157 scene 11 26 and 28 . .. I, scene iii, 11. 1 17 scene i, line 158 .. Fool, scene 1 183-185 220 ... 221 , scene 11.. 76. . 83 " .... III, scene 111 1 7 II scene i, 1 .. »~ scene i, 1 .. 11. 9-20 .. v, scene .. 324-325 Act I scene 257-260 . . . 32 33 J4 35 36 .... 37 . 38 3 .... .. .. .. Act Edmund, Act Act I, Lear Kent 58 .... "' scene scene ii, �SOLITARY TOUCHING THE SECRET PROCESSES From Francis Bacon Laurence Berns One question that arises in connection with 'the between the ancients and the moderns , concerns the role of sense in the attainment of scientific to Bacon, and in contrast, accord to One text of Bacon's which I have found in reach some c about the issue is not to be' found in popular selections the market.. It is from the 98 .. The the reliance on reliance on to say that Bacon Aristotle was too , and therefore ; he relied too much on .) The more interes question it seems to me, concerns views of nature and the relatic•:J: of man 1 s mind to nature that underlie these differences. 98. nThe of man hitherto hath been determined the view or ; so that whatsoever is invisible, eitLer in of the fineness of the itself, or the smallneBs of its , or of the subti of the motion, is little yet these be the that govern nature. and without which you cannot make any true dication of the of nature The or that are in all bodies, are scarce known. Sometimes whereas are the most A'~tive take them for air; :Cro!Il ·which , as as water from earth. Sometimes will have them to be ~atural heat, or a of the element fire; whereas some of them are crude and cold. And sometimes will have thme to be the virtues and ties of the which see; whereas they are by themselves. And then, when come to and call them souls. And such , that inward, when • Neither is this a question of words, but infinitely material in nature. For are nothing else but a natural , rarif ied �as were et seen at alL if you you a number of within the before, pass bet'<Teen the as at all handled natures existence is , for no one reason � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The Collegian Description An account of the resource The Collegian began as a student newspaper; became a college community literary publication in 1952 and continued until 1969; became a student newspaper again in 1969; discontinued publication from 1980 until 1989 when it again became a student literary publication. Click on <a title="The Collegian" href="http://digitalarchives.sjc.edu/items/browse?collection=26">Items in The Collegian Collection</a> to view and sort all items in the collection. Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Page numeration Number of pages in the original item. 66 pages Text Any textual data included in the document text Original Format The type of object, such as painting, sculpture, paper, photo, and additional data paper Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Creator An entity primarily responsible for making the resource Lachterman, David (Editor) Title A name given to the resource The Collegian, April 1964 Date A point or period of time associated with an event in the lifecycle of the resource 1964-04 Description An account of the resource Issue of The Collegian. Published in April 1964. Identifier An unambiguous reference to the resource within a given context Collegian April 1964 Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Publisher An entity responsible for making the resource available St. John's College Language A language of the resource English Rights Information about rights held in and over the resource St. John's College owns the rights to this publication. Format The file format, physical medium, or dimensions of the resource pdf The Collegian https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/5d396a77c187887e80d48ff53431fcda.pdf 0c92f3fee6967229f7de2b0067ada9c6 PDF Text Text •• COLLEGV\N ST. JOHNtS CLLLEGE J ANUA;Y, 1964 I �TABLE OF CONTENTS A Sense of Honor·••••••••••••·······•······ Laurence Berns, Tutor 1 Varia on Apollonius •••••••-•••••••••••••••••• Bryce Jacobsen, . Tutor 7 Two Poems •• •• ••• •••• •• ••, •• ~ ••••••• •• ••••••• Beverly \.'ood,;vard, Tutor 11 Civility and the Law ••••••••••••• •·•• •• •· •••• Daniel Carl Schiff '65 Rci>us seau and the Ancients: 1\ Reply~......... 2c-rmeniscus ? 27 -.) .coem • ••••••••••••••••••••••••••••••••••••••• Susan Roberts '66 Comments on Mr. Sparrow's Lecture ••••••••••• Larry Silverman 1 30 66 Essay on Ducks •• ~ ••••••••••••••••••••••••••.• Jeremy Carl Leven '65 Editor •••• De.vid . Lachterman Business hanager ..... David ..asrnussen ,Editorial .·.ssistant •••• Susan Roberts Faculty i\dvisor. • • Eva Brann C The Collegian sends its best wishes to Nrs. Audrey Kempton with hopes for her speedy recovery.) 12 31 36 �lri the editorial · to the September, 1963 Collegian I suggested that members of the community submit questions ·or criticisms to our Friday evening lecturers through the medium of this publication. this month Miss Brann and I · decided to implement this suggestion by inviting a studen~ to review Mr. Sparrow's lecture Rights, Law, end the Right • In the follouing weeks students and tutors will be invited to contribute reflections or challenges concerning future lectures. On occasion more than one review of the same lecture will be published for purposes of comparison and controversy. Lecturers will be invited to submit restatements, counter-challenges and elaborations of their positions. The record of such debates will be, I think, ··~f vclue a~d interest. Mr. Silverman's comments on t-ir. Sparrow's lecture set. an unusually high tone for future contributors; it is hmped that what he began others will continue. Note: The deadline for the Collegian's Short Story Contest has be&n extended until March 15, 1964. ErrDta to this issue: p. 17 cheif should be chief 17 6mvalries and add 'spars' 19 (better called an ) insert ... . }:~ .~. : , . 20 Footnote two = Let II, sc. vi 1. 54 21 Footnote three= Act IV, sc. i 1. 145 25 Footnote one= 27 in seiner add 29 ; ~ct V, scene i, 1. 1-6 /J..1At d\ S promenus should be promeneur; Vaugh~n ed. add pp. 75-86 �El.l:ATA TO JANUAf~Y, 1964 COLLEGIAN the following title was missing from the Table of Contents : The Relationship Between Appositives, Restrictive and Non-~estrictive Attributives •••••••••••••••• Pattie 7urner 1 66 12 The followmog should be added to the secona paragraph on p. 32 ending 11 thus agreeing in part with the traditional liberalism he was trying to undermine." Add: One might counter this objection by remarking that those actions permitted by the laws Dre approved because the actions are right. The square is quite explicit on this point. By calling certain voluntary actions 'right', do not the laws imply that one is duty-bound to do them ? To first p2..ragraph on p. 34 after the 'the peculiar work' " add: ra~o" l-'pyov words "In Plutarch's phrase, �i'\ SENSE OF HONOR Laurence Berns · ur o One of .the .saddest things ·about the lrUrder ·of . President is that we do not realiy or fully kriow what we have lost. So much of what we admired in the I'!lan lay· in the promise he held out .for future greatness: . his intelligence·, 'that is,- his capacity _to learn; his high . style, the grace and fluency of his discourse; his sense ·o·f htimot"• ·the subdued irreverence. of hi.s "deadpan" comic style; his freedom from, or rather, distaste for sentimentality; his sobriety in acknowledging the aspirations of the more articulate members of the polity while at the same time being able to appreciate why the resistance·of the less articulate made it wise to forego the implementation of those aspirations. Being both ambitious and well-bred, he could be at the same time both gracious and hard; ·happy in the exercise of highest authority, he · could be grateful for being provided with the opportunity for living up to, as he. put it~ that ancient .Greek definition of happiness, "the full use of ~our powers along lines of excellence." Like every -. natural politician he greatly desired and greatly sought the esteem .of his fellows, arid 'like any man whose soul has ever been enlightened . by a sense of honor~ - he knew that the .prize was - not worth the winning · if he did not make himself worthy '.of that esteem. II However, the injunction of our new President obliges us to turn our. attention away for a time from memories of John Kennedy to the grim circumstances or conditions .of his assassination, to see if :we might learn anything from them. We shall in all likelihood never know 1:~hat was going on in the mind of the President's assassin, to what extent . he might have been influenced by the atmosphere- of alarm created by the political murders in Birmingham and Mississippi and by the ·rabid charges propagated by political extremists, charges which gain conviction among the dupes of the extremists 'becausG they remain urirepudiated and uncondemned by people ·who know better. , (It is the duty or those who · stand to profit · politic ally from such charges to be the fi~st and the most vigor~us in denouncing them). . · . .' ~ Regar dless of what went on in the assassin·• s mind, the assassinat ion, the murder of Oswald, .the events just referred to, all are signs of danger. Are there any general conditions whi'ch have been contributing to ·this tendency to take direct political action, to take .the law into one's own hands, . thi_s tendency toward the subversion of the rule of law? It .is hard to be verj exact about such matters; but perhaps the knot.Yledge r-e seek does. not yield itself to overexact methods. . It 1 could be that the m~st articulate elements of the nation and the most -· prominent objects of their loathing, the right-wing extremists, have been working in unplanned cooperation to bring about the same conditions. The effects of the extremists and their not so unr.dtting allies, those who have gone into the tawdry business of investigating political opinions in public, are not too . difficult to understand: by sowing �"" 2 - distrust and suspicion they tend to sp r ead the feeling that Americans can no longer trust themselves to behave iike free men, that what is required is extensive official supervision by them and their cohorts. The accusatorial atmosphere not only opens up new fields for bullying but is also expected to reform the situe·tion left us by our traditions and institutions of civil liberty. It does not require a great deal of subtlety to see that what they would reap would resemble most of all, in its political essentials, _ the despotisms admired by their brethren of the far left. To expect any society to be completely free from fanatics, those brooding enthusiasts, to turn Lincoln's phrase, would be unreasonable. The problem is always one of keeping them under control. So long as they present no clear and present danger to free government, it is probably safest to provide · them and their brethren of what they regard as the opposite extreme opportunity for keeping themselves occupied by organizing and meeting peaceably. But wherever unlawful violence breaks out the p~nishment should be swift and severe. Leniency in such matters is likely to function as encouragement. It is equally or even more important than the control of violence that the leaders of respectable opinion in the nation and in the various communities make it perfectly clear how far removed from the serious political life of the nation these people are. This requires, to repea.t, that the primary responsibility for diminishing their influence should fall to those within the pale of respectability who might stand to profit from their efforts. These are the people who are most likely to have some influence on those who are about to join or leave their ranks. It probably is 11 too much to hope" that anythin.s ..migbt · "soften the hearts of those who uould themselves recoil from assassination, but who do · not shrink from spreading the venom which kindles thoughts of it in others." It would be safer to place our hopes in strong anti-toxins. The case of those whom we have spoken of as more articulete, our writers, social scientists, journalists, artists and "communication specialists", is -more difficult and more important. They might be expected to know better. By wallowing in the seamy and sentimental side of Ufe, indulging in and habituating us to that invasion of privacy called pornography, identifying misery with profundity, glorifying vulgar, thoughtless and even perverted passion and preaching its liberation, by all these things our writers tend to undermine the pride, self-respect and self-confidence of the educated public, those who they should be preparing for leadership. One wonders why these supposed divers into the depths of the human psyche never seem to have noticed that it is precisely those peoples who have been most distinguished for self-restraint, self-censorship and selfcontrol, qualities now fashionable to disparage, the English-speaking peoples, who have also been most distinguished for being able to make free republican government work over long stretches of time. Free men need confidence in and encouragement of their ability to control themselves . Our writers and the producers of our mass media drama tic arts for the most part, produce the opposite. To the extend to which free men lose their dignity and self-respect and cease to hehave like free men, like men who deserve free institutions, to that extent the ~ �.. 3 - will to pre.s erve free ins.t i tutions will naturally wane·, to that extent it will seem _ oth right .,and natural to eliminate free institutions. b It is almost pathetic to see hot~, when some of our writers do strive to present something of the noble or heroic, it often turns out to be of the nearly mindless variety, as if deepened understanding can lead only to the low or bestial. This is not too surprising, for when the writers turn to the authorities, the scientists, the "social scientists", for enlightenment, they .find them incompetent, for the most part, to deal w.ith much besides the· low~ the mechanical and the bestial. To speak of the noble and the base, the good and bad, would entail substantiating "value judgments", and that they say has nothing to do with science. This is not .the place to go into the sophisms usually presupposed by those holding to this notion of the science of human things; our concern here i~ primarily with its effects. Yet perhaps it would not be amiss to suggest that if these social scientists believe that the principles of their science cannot be derived from their oim: proper subject matter, that they must .be borrowed by analogy from the more prestigious natural scie·nces, it might be more fitting to go not to physics or mathematics but to a science like medicine which makes qualitative distinctions like that between health and sickne·ss. The effect of the intellectual orientation we have been describing on our journalists manifests itself by the increasing disrespect of respectable periodicals for the privacy of any apparently newsworthy subject. - Ought we to be peering into the astronaut's living room, staring at his wife on the day of his · fiight? . Do we have a right to know as much as possible about the private life of every public figure? Are not the press and media men partly responsible for making the Dallas authorities feel that everyone had a right to know all the details of Oswald's transfer to the county jail? Should the question, as to whether the public's "right to know" should be allowed to jeopardize a man's right to a fair trial, even have arisen? !bes the press have a duty to reveal to us everything they can possibly find out? Or is it not rather the duty of ever;J responsible reporter and journalist to, at least, consiqer whether 1...rhat he says might make his audience better or worse citizens, better or worse human beings? There are times when a personal and private tragedy mecomes an occasion where it is appropriate for the public to know about and participate in the events. Such, of course, were the events leading to the burial of the President. The press and the communications media, for those three days, following that example of proud -and compassionate selfre straint set by the gallant former Fir st Lady and the Kennedy . family, showed us to what heights they are capable of rising by the part which they took in giving the late President the farewell he deserved. One of the incidents for which this writer is most grateful occurred as the dignitaries gathered about just after the President's body had been placed in the Rotunda of the Capital Building. A CBS radj_o reporter, his voice very low and heavy with emotion, noted that most of the mourners were still dry-eyed. Then he said "Just across the way from me s·omeone' s eyes have just filled with tears." He hesitated and finally said,. "I don't think I'll tell you who it is." ·_, �- 4 - Unfortunately, We must now turn to the dreflriest subject o·f this essay. Perhaps the best commentarj that could be made was made during those three days following .the assassination when any person of some sensibiiity knew that this was no time for cheapness, pettiness and triviality, .and consequently that vulgar intruder, · advertising, was expelled from the airwaves. Unfortunately the debasing effects of this continual stimulation of and appeal to every kind of petty desire do not end with periods of national mourning. Assas$inations are not the only occasions for which serious people have a need, perhaps even a right., to ·sustain a serious mood. The advertisers realize their purpose-s most when they imbue their unwary audience, the largest part of their audience, with the feeling that they owe it to themselves, as if by natural right, to gratify the desires the advertisers' products cater to. The American . way of life, American freedom, begins -to s_ m to mean ·the freedom to re amass as many possessions, gadgets and creature comforts as possible. We will not elaborate on the kind of invasions of privacy that are perpetrated by the cosmetics industry. .Can we afford to allow such as they to play so large a role in forming our tastes? To the extent that the advertisers succeed in_forming the national character, it becomes increasingly difficult to insist that we be treated with the dignities befitting free men. What necessity dictates that the communications media should be dominated by predominantly commercial motives? What about the educational role of the media? Ed~cation is gratifying, but, unlike salesmanship, never nattering. The reason is simple: Education involves confronting oneself with minds or spirits that one acknowledges are in some decisive respect superior to one's own mind or spirit, it involves a continual· striving to raise one's own under-· standing to the level of that of one's teachers. Education, then, frequently, if not al~ys, reminds us of our defects in a ·way · that makes us apt to be ashamed of them. It moves us towards overcoming whatever is petty and selfish about ourselves; salesmaship, on the _ other · hand, caters to and ·thereby encourages the selfish and the petty. Educati_n is not likely to rN.in out on the open market. o The problem is complicated by the fact that. a great deal of advertising in this country is not stupid, much of it is rather clever. Obviously considerable artistic, literary and musical talent goes into its production. The sad result is that young budding artists w should be ho .· directed towards working for _he elevation, perhaps even the exaltat tion, of the human spirit become inured to the prostitution of their talents. Someone might reply to the positions we have set forth in this section: Is not our sense of privacy connected to our sense of shame and is not sl;tarn.e·lessre s _ after all, both· the precondition for and the natural s effect of enlightenment? If shame and tact, respect for another's sense of shame, are nature's ways of protecting the intimate, the vulnerable, the· naturally exclusive in man, then sham.elessness would seem rather to be proof of ignorance. Furthermore, i-f our self. respect depends upon sue~ protect:i,.~m and our freedom. depends upon our self-respect, then our freedom depe· ds as well upon our sense of shame, n our sense of privacy. �- 5 III Our -concern has _been .the conditions t~nding towards the subversion of the rule of law. in America. · Throughout our history there has been a tension between a tradition of violence and the · tradition of respect .. for. the la1,i'. Bot the least manifestation of our failure -to come to .grips ad~quately with the probl_e~ of violence is _the widespread · . tolerance for those .home-grown forms ·of military dictatorship called organized and unorganized crime~ On the frontier, it has been said, violence was necessary and the tradition made sense. But now there are too many of us. and we are too close together to tolerate habits appropriate for by-gone conditions. Yet ·it would be truly sad if thanks to the progress of ·medical and technological science we have become so crowded and so inte·rdepertdent that the very physical conditions for -"rugged ,individualism" and privacy are disappearing. However .this may be, · :it ought to be said that the . roots of.·our tradition of violence are not altogether _ ignoble. · Violence is not in itself W:rong, obviously it is required sometimes for the defense ·. of our noblest interests. According to the Declaration of Independence no particular form ,of government· or institution, or adherence or · conformity thereto, is sacrosanct. What is spoken of as sacred ·is honor and certain inalienable rights, properties of separate human beings. Our sacred .honor and our natural rights may.~ under-certain very rare conditions, . require that rebellion and Violence be .undertaken on their behalf againsttha established order. Rather than ·simply deplorin'g all . forms of vioierice our task would seem to be to try to understand the difference between fanaticism ·and noble dedication. The ti:..ro. are often confused t for reasons alluded to earlier and because they do share at least ene iroportant trait in common: they both seem to involve forgetting about one's self. The difference may have something to do with the reaso_ ableness of the goais of each. n 0 ·. The greatest doc.uments f,'rom · .the most v~ried source·s of our tradi tio· , n . _or . example, · the Old and New ·Testarrients, Plato• ·s Republic and the f Declaration of Independence are unanimous in teaching that perfection is not to be _ sought for in particuiar ·institutions, at least human institutions·, that there -will never be heaven on earth. They point to what kind 'of order we ought to aim for, what kind of order should provide us with the standards for improving those actual orders we · find in the w orld, but they w arn us not to expect the realization of our highest hopes. The warning servep not only to prepare us against disappointment and frustration but also to help us from being diverted from the. pursuit.of that perfection "which might be feasible for us: ·'.for example, in t~e ~~p_qbl~.~. Socrates shows ~s why the chances _or f instituting · a perfectiy J~st order aniong a community ·o£men are · completely, or almost completely, ni.1", but he does it. while illustra~ ting in what way it might -be feasible· !or a man to institute such an order, a heavenly ord.e r, namely,- by instituting it "in his own soul. Perfection is not to be ex,pected .in the ordering of other men, but, if at all, in the orderi~g of one's· self. The latter, not the former, might. be in our power. The fanatic does not make distinctions: he does not distinguish between possessing rights and knowing when it is wise to exercise those rights; he does not distinguish between what is �- 6 - good simply,what good is realizable generq.lly and what goodness might reasonably be expected in ·some given situation. The latent fanatic expects heaven on earth. He is inevitably frustrated, and indignation joined to envy and resentment swells into hate against those who he supposes are or represent the causes of his failures. Could one not say that perhaps the nihilism, the bitter resentment against all society exhibited by some of our writers is also a function of their original overexpectation, a function of their divorce from the sobriety of the tradition and hence their failure to consider the immense obstacles that always lie in the path of man's attempts to improve himself and his situation? The fanatic is certainly no respecter of the rule of law, its blessings are far too modest to please him. The rule of law is not going to bring heaven down to earth, the rule of law by itself cannot even produce nobility of soul, though it may go far towards creating the best possible conditions for the cultivation of nobility of soul. Despite these shortcomings a moderate appraisal of those blessings may not be out of order. It might not be too great an oversimplification to say that the trouble ~dth most men is that they tend to bully and or cheat each other. The English-speaking peoples seem to have concentrated on preventing men from bullying each other, and have been blessed with considerable success in this regard, in great part because of those devices and that spirit which we sum up under the head of the rule of lar..r. We must also coneern ourselves with the problem of ·cheating, but not in such -a way that we concentrate so much power in the hands of our protectors that they are then in a . position to bully us. Besides the fact that he who is in a position to bUlly us is also in a position to cheat us, bullying hurts in a much 'deeper way t~an cheating. What should we expect from men? Probably one should save one'.s highest expectations for one's self. One should hope but not expect that men will behave as well as it is possible for them to behave. Yet cynicism is not the answer, for although one should take precautions against the possibility, one should not .expect that men will behave as badly . as it is possible for them to behave. The proper mean probably · lies in_ acting as if one expected men to behave better than they probably will behave. · · What we seem to be recommending is · a kind of puritanism, but puritanism with a sense of humor, for high comedy thrives on · w hat the·. fanatic · · cannot abide, the disparity between human goals and ; human achievements. It mollifies our disappointment and our anger while ·a t the same time it keeps before our minds those goals from which we ·fall short. Let us sum up this paper with a warning and a suggestion. The warning first: If we want to remain free, we cannot allow ourselves,. to become vulgar. The suggestion: If we want to become virtuous without becoming dull, we could perhaps do no better than to consider that highest and . noblest form of "de'adpan" humor, Socratic irony. �- 7 - VARIA ON APOLLONIUS Bryce D. Jacobsen The following article includes three separate items: I II A construction fitting any ellipse into any cone, . The completion of Apollonius' (partial) presentation of circular sections, III A classification of sections with respect to the cutting plane. I Anv Ellipse in Anv Cone Apollonius, in P~op. 52-58 in Book I, solves the problem of generating a cone and cutting it so that a section will be produced whose diameter and parameter are equal to two "given ••• bounded straight lines". He does not, however, address himself to the more general problem of cutting any given cone such that the resultant section is identical to some given section. This form of the problem was mentioned i_n Mr •. Sacks' article in the October, 1963, CollePian, when he said: "For Apollonius, any size conic may -be derived from any given cone and, in fact, any given conic -section may conversely be placed back into any given cone." The problem as stated has solutions for the parabola and the ellipse. No general solution exists for the opposite sections, unless the ratio of transverse side to upright side is given as greater than unity. The solution for the parabola is fairly simple. (We will not present it here, since it makes a nice "original" for those addicted to Apollonius.) We do, however, pr.esent the solution for the ·ellipse. Notice carefully that no restriction whatsoever is put on the cone, o . r on the given section. (The fact that, in our given ellipse, the given diameter is the axis, and the ratio of transverse side to upright side~ greater than .unity ts not a restriction on the possibilities of the given ellipses. For, if the . "figure" of the given ellipse is no-t that of the major axis and its parameter, then by II-47 the major axis may be found, and by I-50 the parameter to this axis is given. So an e~lipse given by any diameter and its parameter is ~given by its major axis and its parameter. And the ratio of major a.xis to parameter is greater than unity.) Problem: In any given cone, to pass a plane through it such that the resultant section is identical to a given ellipse. Let MRN be the given ellipse, with axis MN and parameter MO. �- 8 - Therefore there exists a cirele Nhich can be circumscribed around quadrilateral BAFD. (Converse of Euclid III-22. See Heath's note.) Therefore AB and FD are chords of this circumscribed circle, and E is necessarily outside this circle. Therefore rect. DE, EF equals rect. BE, EA (Euclid III-36). Now And Or Or Or DE : EF :: DE,EF : sq.EF :: BE,EA : sq.EF BE,EA sq.EF comp. BE : EF, EA : EF BE,EA : sq.EF comp. FL : LH, FL : GL BE,EA : sq.EF :: square FL : rect. GL,LH MN : MO:: sq.FL: rect. GL,LH Now let sq.FL : rect. GL,LH :: AB : Z Therefore the section produced by a plane through AK and XX will be an ellipse 1dth AB as axis and Z as parameter (I-13). And since AB : ·z : : MN : MO, this ellipse.~dll be similar to the given one. Now if AB equals r1N, the problem is solved. If AB does not equal MN, then mark off on FL (or FL extended) a line FS equal to MN. From S draw ST parallel to FG. From T draw TW parallel to FL. Therefore TW equals MN. Now let sq.FL : rect. GL,LH : : TW : U And sq.FL : rect. GL,LH : : MN: : MO Therefore MO = U Therefore the section with TW as axis and U as parameter will be identical to the given section. QEF (The proof is the same if the point D is taken on the other side of F, thus reversing D and F. In this case the cutting plane and line FL meet the base of the axial triangle on the other side of the base extneded. Thus it follows that there are at least two sections in the gi ven cone which are equal to some given section.It se.ems probable, moreover, that for any axial triangle in a given cone, a section equal to , a given section can be found. :If this is true, then there are infinitely many so. utions to this problem for a l given cone. I will not try to prove this. I leave this for later consideration, or for others to prove, or disprove! . · II Circular Sections There is a "gap" in the early propositions- of Apollonius. he proves that: In Prop. 5 �- 9 - If a) in an oblique cone the axial triangle is perpendicular to the base b) and the cutting plane is perpendicular to the· axial triangle c) and the section is subcontrary Then the section is a circle. In Prop. 1J ·he proves that, in any cone: If a) some axial triangle is given (not necessarily perpendicular to the base) b) a cutting plane is given whose common section with the base is perpendicular to the ~ of the axial triangle c) and the cutting plane is neither parallel to the base nor sub contrary Then the sec~ion is an ellipse The question naturally arises, "What is the section when the following are given"? a) an axial triangle, as in Prop. 13 b) a cutting plane, as in Prop. 1J, except that it 12. sub contrary It cannot be concluded from Prop. 5 that this section is a circle, since conditions a) and b) of that proposition are absent. Proposition 9 is no help either. The section 1§. a circle, but this is not provable within the Apollonian context. However, a revision of Prop. 5 solves the problem. First note that Prop. 5 is not used at all in proving Prop. 6. Thus we may use Prop. 6 in proving our revised version of Prop. 5. The proof would then be as follows: If a cone is cut by a plane through the axis, and is also cut by another plane such that, on the one hand, its•·common section with the cone's base is perpendicular to the base(or base extended) of the axial triangle, and, on the other hand, it cuts off on the side of the vertex a triangle similar to the axial triangle and lying subcontrari wise, then the section is a circle. �- 10 - Let FGH be any cone, and let axial triangle FGH be perpendicular to the -cone's base. · Through F draw IFJ'.parallel to GH. On IJ take any .point 0, and then take point E such ·that · DE : EF : ; MN : . MO. · On ED at D set up 00 . uch that angle QDE equals angle EFG. Now pass s the circle EPD through points E and D, such that QD. is also tangent to circle EPD. (Only one ·such circle satisfies these 'conditions.) Let circle EPD and lirie FH intersect at ·B• .Join EB, cutting .GF at A. Extend EB;· cutting GH (or GH extended) at K. Through K, in· the plane of the cone's base, draw XY perpendicular to GK. · Through F draw FL parallel to EK~ Join BD. Now angle EDQ equals angle EBD (Euclid III-32). angle EBD equals angle EFG. Therefore angle GFD + angle EBD = two right angles. {And thus angle FAB + an~le FDB = two right angles.) ~erefore ·.: .· . . p .. ' ··'.....,, · -.., ·~ . . ~-" .T ' ., ' - ._,_£_. . --- .~ ~:· :;~-. , ~ I / 1. . ./ /l ,, ,• ' ·ll / . /; I I ! ---- -1 / I . ,. ·'/ / I ~· I '/\ !· I ./... i ~' ,• ,'f I ~·/ / / / ~- . / -... - ;, : / ')!// ' I · 'I L .. �-11 - TWO POEMS Beverly Woodward I wrot~ her a long poem in prose~ I told her that a pomegranate has many seeds. I told her that it was an agreeable fruit, a fruit which ·a man could enjoy. - I wrote that one day when an autumn wind was flowing,not knowing which of the mul ti·col'ored threads she would choose ~s the transmitt~r of her response. I was surprised that she chose ··to play in so many keys. The song was simple, pure, free and melodious, It did not anticipate it could be frozen By the builders of systems Made a foundation For an immobile structure. The song knew ·no rancour It found its new garb very strange And then, it had never been meant To support such a great weight Or to live in darkness under the earth. �- 12 - RELATIONSHIP BETWEEN APPOSITIVES, RESTRICTIVES, AND NON-RESTRICTIVES Pattie Turner Although in the grammar manual apposition and restrictive and non-restrictive attributions are discussed quite apart from one another, it appears to me essential to con~ider all three when discussing any one of them. While apposition is said to be distinct from predication pr attribution; it is also explained to resemble a non-restrictive attributive in signifying a "condensed, additiona],.:pre,d:ication". · So~ obviously, attribution and predication do . play a role in apposition. At the same time that a combination ·o·f signs may be considered in apposition with another sign;- it may also be considered a restrictive attributive as well, although the manual states that it resembles a nonrestrictive attributive. An appositive is described in the grammar manual as a "a substantive signifying the same thing as another substantive ••••• " The substantive which signifies is said also to "resemble a non-restrictive attributive in signifying a condensed additional predication." So, here in this definition, two parts of grammar must be considered. First, the appositive must be accounted for, and secondly, the appositive with .respect to a non-restrictive attributive must be considered. An appositive may be a single word (e.g. a proper noun), a word with a modifier (e.g. noun plus an article or adjective), a phrase in which a combination such as a prepositional phrase is employed and which a combination such as a ·prepositional phrase is employed and which does not indicate a complete thought, or a clause which does signify a complete thought -- a unit in itself. An appositive may be present in a great many forms since it is the loosest form of combination. However', while, it is the loosest form of combination,it is the most intimate c onnection of one thing to another, i . e . identity . It is now in order to look at the different types of apposition. The simplest and perhaps the most frequently used type of apposition is that of one word, e.g. "It is hard to believe that he, Jack Waters, is ten years older than I." In this sentence the proper name "Jack Waters" is in apposition with "he". In other words "Jack Waters" . and "he" are one person, an identity. It would be easy, however, to use both terms "Jack Waters" and "he" in a similar sentence and have each signify two entirely different people: "It is hard to believe that Jack Waters is ten years older than he." In studying these two sentences it becomes apparent that the position of the sign in each sentence is different, thus indicating that position in a sentence plays an important role in apposition. In the first sentence where there is clearly apposition, "he" and "Jack Waters" can interchangeably be the subject of the �- · .. 13 the .'cl. use-..·"that .he. '. .. . . u In the . seco.nd .sentence, . however, only a :t.he proper . n _ime . 0 Jack Waters" is :the subject of :th~ ...analogous : ·clause 11 that Jack .W aters ••• =. 11 • · Thus the position of the apposit~ve is essential to the meaning of the sentence; in point of .fact the appositive is µsu~lly set beside that which, .i t describes or identifies. · This :fact could be considered ·part 0:£ the close relationship between the appositive and that which i t signifies. Similar .to the on~ - word- appositive is ·th~ two word, a noun plus .its :artic.le, : appos:i tive: "We took our pet, a dog, to the veterinarian for ·a sho.t •• ~ · In .t his example there is no doubt that "our .pet" and "a dog" are the same thing, and inde.ed they .are set . nex.t to one another as has been shown to be essential ... Next let examples of .the other. two types of apposition be explo. r:-ed. : First, an appositive phrase: "Donald Duck, a creation .o f Walt Disney, ·i s ·a :favorite cartoon character of children all over the world." Surely, : "a creation of Walt Disney, ••••• n cannot stan<:l al·one, but it c . n replace the name "Donald Duck". a The .phrase as .the subject _ f the sentence would not have as o significant meaning as the proper name (this type of complication wil·l · be discussed later), although each signifies the same. thing. Second, an apposi tiv.e clause: "The realization, that I was now .on my own,. was somewhat :frightening." Again the subject and the claus_ are interchangeable, but the use of one e is· more meaningful than the use o:f the other. Since the purpose of this paper is to observe the correlation between the appositive and the restrictive -and non-restrictive elements . of the ,sentence,, j, t will be necessary to consider the restrictive and non-restrictive as separate elements and then :to examine the connection or subtle ass?ciations between the . three • . Let the restrictive be considered first. The restrictive is an . attributive which is significant and . essential to the meaning of the sentence; without this element the sentence would have little or no meaning • . - The restrictive can be..one word, a phrase, or a clause. An . example of a single word restrictive is: "The extende_ curf~w was . a favour to d those who had to travel a long distance." In this .sentence the word "extended" is essential to the proposition in which i t figures. It indicates , that without the. increase of time allowance, those who had to be~ route for an unusually long . time would not meet the requirements of the plac.e t~ which they were returning • . Surely, withc;:>ut the word - "extended'' the sentence would tend . to sound extremely sarcastic, for i t is not clear that the curfew alone would be of any help {help i~ . adhering to the rules ., that. is) to the t .r avellers. Therefore the word "extended" is essential to the meaning of the sentence,and must be classified as a restrictive attributive. Similarly, in the sentence, - "The prisoners who were . sick were sent . .to the hospital," the clause "who were sick" •• is essential. If . the clause did not appear in the sentence, . the meaning of.- the sentence would change. This new meaning would be that all the prisoners were sent to the hospital. This, however, is not what is . meant, and .the.re:fore the claus~ is restrictive. �-14In t:P,e sen. ence "The girl wearing a pi.nk wig looked . partic.1:J.larly t · p ·e culiar, n it. is obvious that the phrase "wearing a pink wig" is .o:f cardi:nal importance to· the . meaning o:f the sentence. If the phrase were left out, the .. girl would look ·pariicularly peculiar under . any circumstances. . Therefore, ·.necessarily, the phrase ·is restrictive~ Unlike the restrictive, the non-r.e strictive is ne1the·r · ess.ential nor particularly significant to the meaning of' the sentence. Strictly speaking a · non-restrictive is a form of attribution but ,i t may; be thought of as a "cond,ens ed' additional predication." Let us consider the three most common forms of the noh-restricti ve: a singie word, a phrase, and a clause. The one word non-restrictive is often used more .as an attributive which attributes something to .a specific word or defin~s a s~ecific word. Thus it is not meaningful in the light of the sentence, e.g. "The child wore his blue snowsuit outside t ·o keep warm.-" In this sentence the word to be explored i.s -"blue". As an attr.ibutive "blue" indeed defines the snowsuit as far as color is concerned, but · it does not define the . snowsuit in re.g ard to the purpose for wearing the snowsuit i.e. warmth. As a "condensed, additional predication," the word "blue" certainly says something about .' the snowsuit, i.e. that the garment partakes .of the quality blueness. This participation in the quality blueness, however, is irrelevant· in the proposition; whether the snowsuit be blue, red, or .green, the child will keep warm. Thus "blue" in this sentence is a non-restrictive. A phrase also can be a non-restrictive; an example of this is contained in the followirtg sei;itence: . "Having arrived from Boston, the lecturer spoke to us about .anabolism and katabolism." Obviously the lecturer's arrival from Boston had nothing to do with the lecture on the constructive and destructive processes of metabolism. Strictly speaking, one might argue with this statement by saying that unless the man had arrived from Boston, there ·would be no lecture. In this sentence, however, what is important is the material of the lecture. Theref.ore th.e opening phrase is non-restrict~ve in that it is not an intrinsic part · of the sentence , but is rather incidental to the significance of the sentence. A clause too can be a non-restrictive; an exampleof this is c·ontained in .the ·following sentence: "I heard that Mrs. Martin, whom I dislike, will be at :the same party as we." "Whom I dislike" is the non-restrictive clause. It is non-essential to the sentence in that whether I like her or dislike her has no bearing on her anticipated presence at the party. In this sentence the clause is a ma'tter of opinion and is meant as a . passing remark. The relationship o:f restrictives and non-restrictives to apposition shall now be examined.. Three main cases o:f apposition with respect to the restri ·c tive and non-restrictive must be considered with reference to the sttributive and predicative value o:f each. The three different cases to be accounted :for are (1) the appositive as·a non-restrictive, (2) the appositive �- 1 .5 ~ as . a r .estrictive, and .(3) the .appos.itive as a restrictive and at . the same time. . _ n.on~ 'restri'cti ve a · The - appositive ·as a non-restrictive is the most generally accepted case. As a non-restrictive attributive it signifies a "conde:n.sed, additional predication." An example of this is the fo,llowing: ,. Lyndon. B~ Johnson, the President of the United s·tates·~ arrived in Washington yesterday after a Christmas v·a'<~~ttion in Texas.-" . I~ this sentence "the .President of the United States •••• " is . non-restrictive · in that ·it really is · riot essential to the meaning of the proposition in ·which it figures since it is hoped that the name implies the position of the Presidency. In this case, as in attribution, there is present an "assumption of prior unification." Not only does this identity imply attribution by defining the scope of significance, but it also implies predication in that it says something about something else. It asserts the existence of the Presidency as well as naming it. Thus, this is one case of a non-restrictive appositive which includes attribution and predication. An appositive may also be restrictive. The best example of this is one which is similar to an example in the manual: "I, Carole Hocker, do solemnly swear that I have received no help on this exam." Is the name needed to clarify the statement, or could it be omitted causing no detriment to the meaning of the oath? Indeed the appositive is an intrinsic part of the significance of the sentence. If "Carole Hocker" were left out, the sentence would read "I do solemnly swear ••• ," To this sort of oath anyone could fix their signature. Therefore, although the personal pronoun is used, the statement remains too general. On the other hand if "I" were omitted, and the sentence read "Carole Hocker solemnly swears •••• " this indeed would give no indication of personal integrity with respect to the oath, and therefore it would be almost meaningless. Thus the appositive, the name, is of cardinal importance ., and the appositive is restrictive. Under certain circumstances an appositive may be either nonrestrictive or restrictive. An example o:f this appears in the :following sentence from the grammar manual: "lle saw Keane, the actor." In order to label "the actor" as a non-restrictive or restrictive appositive, it is necessary to deviate from the strict practical rules of grammar and consider the reader. Do "We saw Keane ••• ," and "We saw Keane, the actor," indeed signify the same thing? There are several different ways to look at this sentence. Does the speaker of the sentence want to indicate that he saw a man named Keane who happens to be an actor, or that he saw an actor whose name~ oh second thought, is Keane {but the name isn't really important because "it's not what you know, but who you know.")? Perhaps "the actor" was attached :for those people in the same category as I -- those who didn't know that Keane is an actor--. At any rate it seems that the appositive may be either non-restrictive, :for those who are ignorant of the field of acting. It is also both attributive and predicative in that it defines the scope o:f signi:fance of Keane and predicative insofar as it asserts the �- 16 being of Keane not as .a pers o:-.1 , but. as an actor. -It is,. however, more to my liking to think of it as a restrictive since if it is considered otherwise , i t would probably be more appropriately omitted. Thus, I have endeavour~d to .show that there is a link ·which connects apposi ti·on to restrictives and non-restrictives more intimately and intricately than . the manual allows, . and that attribution and predication play. a decided role in the u.se of restrictives and non-restrictives which in turn are an essential part of apposition. �- 17 . CIVILITY AND THE LAW.· · Daniel Carl Schi:f:f · Near the beginning of' the sixth scene of' the s .econd act .of' · The Merchant o:f.: ·venice the ·masked Gratiano tells the masked Salerio·}hat "'11 thin~s that are, are with m?re spirit ·chased than enJ· yed". o By this he means t~at men are more aroused to action and emotions when striving :for something they . desire than when they . are ac·tually enjoying . the thing they · haye sought. It is. when he is in serious conflict :for something that he esteems that .the passions, crafts, and foibles o:f a man become most . evident ., and his inner st:t'engths and weaknesses are rev~aled. In attempting to throw light on the nature o:f life iri the Venetian Republic {and in :fact in all similar societies),. Shakespear has taken some f'i:fteen residents of' that comm~nity · and involved them in a series, indeed a labyrinth, o:f embriolments and conflicts in which ev~ry character . becomes strained ~nd somewhat misshapen. However they are misshapen only insofar ·. as .t h_ eir dominant characteristi~s become more dominant, and thus their natures, f'ree :from - ~he clouds of' Unstudied actions, are mo~e ea~ily seen, analysed ~nd understood. I: .do not. have the time o. r space to .concern myself' .w ith al _ the l encounters, rivalries and described in this play ·· or to :fully deal · wi-th all the combatants. I will _~heif'ly . deal with what is commonly considered to be the central· theme o:f the play, that is, the conflict between the sons and daughters o:f Jacob and . their Gentile hosts in the diaspora. · - ~ntagonisms, The chei:f antagonists in this conflict are Antonio, the "Christian" merchant, and Shylock, the Jewish moneylender. Since both of' these ·men are involved in :fi nance and trade on the Rial to, .it is necessary :for them to have some sort o:f contact. It ~s to be expected that· the Jew would have ill :feelings .to~ards the merchant because Antonio hinders his business by .lending money to his :f~iends gratis. It is strange that the two .would ever ent ·e r into a contract together, and this is only brought about by Antonio's ·desire to supply his :friend Bassanio with enough money so that the latter might become a suitor to fair -Portia, "a lady· ·r ichly le:ft". 2 Thus,' during the :first scene, a p·l ·ea·siilg picture of' Antonio appears. He is a man who, although a trader, . has the·" highest o:f principles and w:i,..1 1 . nly o .deal in usance when · it i ·s necessary to do so :for a :friend' .s benefit, . and at that time will spare no expense or personal e:f:fort · to supply · his · needs. 1 'Act · II, s6ene v, line i4 2 Act II, scene i, line 65 �- 18 - The image o:f Antonio as a Christian gentleman, however, begins to :fade quickly when Shylock reveals the true reason behind his hatred o:f the . merchant. The Jew, in his :first address to the audience, :' gives two major causes o:f his hatred :for Antonio; (although both .o :f them are, :for Shylock _ merely manifestations o:f An'.ton:io' 5· he'ing ·a Christian)~ The :fi_ rst is that Antonio · in his "low ' simplicity ••• lend.s out money gratis and brings doWI1 ·t he t e o :f usanc e here with us in Venice" • 1 The second _i's · tha~ .·;'he ·rails ••• on me my bargains, and my well .w orn . thrift , which he c_alls interest". 2 In other words, Antonio, in his "simpl.: ici.ty" (and :for Shyla ck this is true _ stupidity) ruins ~ · shyloc'li: i· s bu.siness and then, in addi tfon, insults Shylock for ·n ot b· ~ing as :foolish as_ a Christian • . (For Shylock a Christian is, not to mention other more despicable things, one who lends money wi th_ ut .. interest to other Christians.) o r·a When we first see Antonio and Shylock together, Antonio again ·implies (it is the first time· in the play} his contempt of ·· Shyloc~' s money 'policies. by 'telling the Jew that ~ (Antonio), · und~r no~mal cbridition~, - does ~ot borrow or lend by giving or taking excess. Shylock · answers tp.is by :firs_ getting Antonio t involved in the loa.n , by making him state .the term, and then by telling him t:tie. s :tory o:f Jacob and Laban. For Shylock this story implies tha·t · a man can do anything to increase his fortune so long as he does not break the law. Antonio's rebuttal is that Heaven and not Jacob controlled the birth of the lambs and that animals are not the same ~s silvei. The two obviously cannot constructively argue Scripture together .• This interlude once again forces Antonio to ask for the loan and Shylock nowreplies by stating .his deepest grievance against the merchant, namely, that Antonio continually insults and degrades Shylock and his tribe. It is in this passage (Act I, scene iii, lines 195-129) and in a similar one (Act III, ~cene ii,' lines · 56-76} that .Shylock i~ - ~een in his most wrathful and vengeful state. It is the .wrath of an oppressed man who has his :former o·ppressor at .his ·mercy. When he asks "Hath a dog money? 11 J he is._ striding over· the stumbling merc~ant, for Antonio must either adrhi t t ·h at the Jew is not a dog but .a man like himself, or . that. dogs, ewes and lambs have much to do with money., c _ ntrary o tp what he _ had first said. Antonio's reply that he is likely to spit ·on Shylock again, sharpens a question that has been looming since the be.g inning of this scene, i.e. what kind of Christian is this Antonio.? Can a man who so openly hates another man, wi ·thou~ - the least ·sign of compassion, be in any _ sense called .a :followe·r of Christ? Antonio can act in _ love towards his :fellow Christians and not loan money at interest, but in not even attempting to love the Jew, and to see the image o:f God in him, it seems that Antonio is de:finitely :failing · as a Christian. 1 Act II, scene iii, lines J9-41 2 Act I, scene 111, lines 44-46 J Act I, scene iii, line 121 �- 19 . It is this :fail:ing and i 'f;e outward manifestation·, that· is,. Antonio's constant rebuke and open show . a~ hatred . a~ainst the Jew, that has fanned the :flqme.· o:f Shylock's ·hatred;; -. a:nd . this same neglect o:f Antonio's Christian duty is what nearly cost · the good merch~.nt: his li:fe, a li:fe ~ertainly not saved by his own sp"l.:1:so . wit and :fig~ting nature. ' . I:f the animosity betwe~n Antonio. and Shylock can be- laid' ·to Antonio's -not being a true Christian, · it -must also be attributed to Shylock'. s not being a true Jew, or at least ·not acting as a Jew sJ:iould a~t. Shylock _describes the proper role a Jew should :fill when dealing with Christians when he tells Bassanio, "I will buy with you, sell with you, talk with you, walk with you, and so :following, .b ut I _will not eat with you, drink with you, nor pray ·with you •.1" . Shylock realizes that he should only deal with the Chris,tians :for the purpose of pro:fi t. He must, and does, call a Christian good only i:f lie is substantial'· and . bad i:f he cannot pay his debts (-s.ee Ac:G. .J:_,._ _ s _ _ ne,_ iii·-, iines ~1~~2:5}• · oe Shylock shouid,. and even must, ~o-neLa.lJ..i:'21.nsu!.:: t:IS anti b-Ofmp<:liments of' the ·christians that do . not affect his commercial standing with them. This is what he seems· to do at the opening of this scene~ However, as soon as Shylock sees Antonio, he stops talking about money and ~r~:fit and starts recounting his hatred. Antonio's constant railings, coupled with his free loan policy, has struck deep at the old Jew. By taking active offence at the Christian, Shylock is committing a grave sin (perhaps better called an ), :for ~e is engaging a powerfully supported enemy in a method of :fighting :foreign .to Shylock, and is doing it on his opponents ba~tle:field. Shylock. how~ver, has not completely lost his wits. He realizes that if he is ever to avenge himself against Antonio, it must be within the law, in :fact, it must be by the law.- He states from the first that if' he "can catch him (Antonio) once upon · the hip, I will :feed :fat the ancient grudge I bear him.2 " He then enteres .into a contract with Antonio, a contract which can yield no financial ·reward for Shylock, . but only gives him- the hope o:f hurting, indeed destroying, Antonio. To be sure, Shylock's hope is small, but · it is not ·non~existent, as Antonio thinks it is, :for there are, as Shylocl~ himself bas said "land rats and water rats, water thieves and land thieves, I mean pirates; and then there is the peril of the waters, winds and rocks ••• 3" In short, Shylock sees this slim opportunity :for .re.v enge and takes his 'chance. The bond o:f ~a.· pound ()f :flesh is cl_ a'rly not e a "merry boll.d"' and Shylo?k in arrart~i.Hg "ii~-, ~£s '::-~hyt11.irl€f ' but a "gentle Jew ", who, to quote the :fair Antonio, "will turn Christian":-·. becoming so ki_ d. Thus Shylock has done the · seemingly n 1 1 Act I, scene iii, line.s 31--35 2 Act I, scene iii, lines 42-43 2 .3 Act I, scen,e 'iii' line·s . 20- - 3 .. . .. 4 Act I, scen·e ~. 111, lines 180-181 �- 20 - i ·mp.o ssihle. · He has gotten his arch-enemy to legally agree to .t erms which might. . enable ·Shylock to. exact vengence upon him, with the power of the· Republic of Veni .c e helping and protecting the . crafty ·Jew. Before I examine the trial of Antonio and try to understand the wonderful intercession from · Belmont, I feel that it is necessary tp examirie the other major meeting of Christian and Jew iri the play. This, of course, is the marriage of Jessica, Shylock's darighter, and Lorenzo, the Christian friend of Bassanio and Antoriio. (Actually Lorenzo is somewhat lower in station than these two gentlemen.) The first question that comes to mind when trying to understand · this couple is how they originally met. The answer to this question is never given in the play. What we do see is that Jessica is the motivator of the elopement. She is the one who arranges the time and the place of the meeting, procures her disguise, and steals the money from her father. Her reasons for loving Lorenzo are easy enough to understand. His life is young, gay, a ·t tracfive and liberal {with the money stolen from Snylock}. This is in sharp contrast to the dreary, stingy life that Jessica lived while in the Jew's house. What young girl would not · prefer to go with Lorenzo, rather than stay with Shylock? The time she . chooses to depart is when her father is attending a feast at the house of the Christian Bassanio. This is indeed symbolic, for it shows the extent of Shylock's social involvement with the Christians, and it foretells the utter ruin of his household. If Shylock himself begins to associate with the Christians, in non-financial matters, how can he expect his daughter to stay away? It is obvious that the Christian gaiety and vitality would be much more attractive to a young girl than an old man. So if he goes, she must necessarily follow. Why Lorenzo loves Jessica is less clear and certain. He says "She is wise, if I can judge her; and fair she is, if mine eyes be true; and true she is, as she hath proved herself."1 This, coupJ _ed with the girl's willingness and her father's money, make ' Jessica .appear to be quite suitable as a bride for the young man. But Lorcnzio doos more than simply accept ,Jessica, he loves her.2 I believe that there are two main causes of this. Firstly, Lorenzo desires 19ve its~l:f,. and by a clever pun (Act II, scene vi! lines 42~46) indicates that Jessica is obscure, and love is what he is discov_ ring • . Secondly, there is mystery, adventure, e and intrigue in his courtship of, and elopement with, the Jew's beautiful daught.er. Their love, however, once consumated, is ~ot simple, but complicated by the very causes that engendered it. Jessica is a Jew's daughter and Lorenzo is a Christian that has ' stolen her away. Lor~nzo backhandedly admits that he will have to answer 1 A,..+ TT �- 21 - · to the commonwealth for his marriage (Act III, scene v, lines 30-35), thus showing that he realizes that the political unwis dom in his marriage must one day be accounted ·for, and that politics will become involved in his: l ·o ve and home. Jessica, sensing this, shows signs of uneasiness, which are . shown by the constant sparring of wits between her and Lorenzo. (The question as to whether or not their love can survive is dealt with in the final scene. Sin'ce this takes place after the court scene, both according to Shakespear anq my argument, I will deal with it after~ consider the trial in Venice.) · The marriage of Jessica and Lorenzo and the contract between Antonio and Shylock can be thought of as two polar instances of the associa·tions of Jews and Christians, or of a:ry similarly ethnic groups. In the marriage we see the case in which what should be purely emotional and private, has become involved with political considerations. The tragic aspect of this involvement comes about when political strife, and not merely political action, is introduced. The contract between Antonio and Shylock is an example of what can be the result of the passions' becoming involved in what should be a simple civil affair, . such as business. What could have .been a profitable relationship for bqth parties becomes a senseless battle, a battle from which both sides (particularly Shylock, the villian) limp away wounded. At the beginning of the court scene (Act IV, scene i) Shylock is at his strongest. He knows that Venice must uphold the bond, because the commerce on which the city depends is itself dependent on foreigners like Shylock, who carry on trade in the city. The city cannot afford to set the precedent of voiding a legal contract, and therefore the bond can only be broken by Shylock's consent. Thus even though the Duke himself desires Antonio's freedom, he cannot overrule Shylock, but must try to convince him to relent. The Duke appeals to Shylock to show mercy to the downtrodden merchant, and not only 'forgive the forfeiture, but "forgive a moiety of 11he p~incipal". 1 The appeal falls on deaf ears, for Shylock has not yet learned about mercy. He has learned to hear and take offence at Christian insults, but he has not learned to forgive with Christian charity. (Obviously Antonio was not a good teacher.) To Shylock mercy merely means giving up what is rightfully his. He knows that according to the laws of Venice, Antonio is in his power. When Shylock tells Gratiana "I stand here for law. 211 He seems to be remembering only the laws of the commonwealth, and forgetting the Laws of Moses, particularly the sixth commandment. When the appeal for mercy has failed, Bassanio offers Shylcok twice the principal that was originally borrowed.3 (It is 1 Act IV, scene i, line 27 2 Act IV, scene i, line 27 �- 22 interesting, but o:f doubtful signi:fi°cance, to remember that Portia said that she would give Bassanio gold to pay the debt twenty times over, 1 and yet Bassanio, at this time, only o:ffers twice the payment.) Shylock refuses this, and says that he will refuse payment even if twelve times the p~incipal were offered. It must be noted here that this was no small sum (a ducat had a purchasing power equal to seven of our dollars), and that Shylock by himself did not even have ·t he original princip- l of a three thousand ducats when he was first approached by Bassanio and Antonio. 2 Thus Shylock is publicly restating and therefo re re-enacting his first sin 0£ forsaking profit for vengence. The Duke then tries a slightly different tack, and asks Shylock how he hopes to receive mercy if he does not render it. Shylock quickly answers that he does not need mercy, having done no wrong. ~~l ~_xi t l ~u eo~ , Vl ~3~ l After this reply the Duke seems ready to give in to Shylock. Bassanio then tells Antonio that he will defend him by force3, but does nothing, either then or later on, when matters appear blackest for the merchant, to show that this was anything but an idle threat. Perhaps B~ssanio was quieted by Antonio's apparent readiness to die. The good merchant, having long ago withdrawn :from the affair, seems to feel that it is his duty to calm those striving to save him. At this point the Christians seem to be completely vanquished, and only have one hope left. The Duke has sent for Bellario, a learned Doctor o:f the law,5 to help him with the case. The Christians do not seem to realize that this is the wisest thing that they could have done, that is, fight the . Jew with things . h .e understands and fears, e.g. civil and earthly punishments. Bellario, of course, does not come himself, but Portia (Bassanio's wife and Belario's cousin) disguised as a young l awyer comes t o defend Antonio. It is unclear to me how Portia knew that Bellario had been sent for, but the fact is that she does arrive with a letter of recommendation from Bellario. , Portia's first speech. again asks Shylock to show mercy, basing the plea on the majesty and beauty of mercy, and the fact that mercy is necessary for salvation. These arguments are more ignored than rebutted by Shylock, whose lack of charity at this point surpris-es no one. Portia then asks if the Jew will take the principal and forget about th'e -bond. Shy 1t:,··c 'k ohce ~again r ·e mains adamq.nt. So far, it appears as if the young lawyer has no new defense for Antonio, and Antonio himself heartily "beseeches the court to 1 Act 2 Act 3 Act 4 Act 5 Act IV, IV, IV, IV, IV, scene scene scene scene scene i, i' i, i, i, line 86 line 90 lines 114-116 lines 118-121 line 107 �- 2.3 give the judgement.1" However, there is something very strange in Portia's speeches", at the end of each appeal for leniency she tells the court that it is powerless to alter a decree, and that if Shylock does not relent, the court must rule against the merchant. The effect of these lines is to make any relin- ~~~~::~n=h~:s s:~~:c~~~ !::! :::~~e: t~ ~~~!ea~=m~fs:;;~~~ 8 by 2 11. Thus he clearly shows his murderous intent. Portia, seeing that the Jew will show no charity, recedes for a few moments, while Bassanio and Gratiana express their love of Antonio. When they are finished, Antonio explains that he is ready for death, fortune having already impoverished him. Portia, however, is not at ali · ready t~ · see the merchant destroyed In fact, it seems as if her initial appeals to Shylock to show mercy, were in themselves merciful offers, which, having been accepted by Shylock, would save- him from the onslaught to come. Portia's defense is principally composed of restrict.i ons. of Shylock's actions. The first restriction she · cites denies Shylock "one drop of Christian blood3 11 , lest he lose . his land and property. Shylock immediately senses his defeat and asks payment of thrice the principal, the sum Bassanio had previously offered and is still willing to pay. Portia, however, will not leave off and let Bassanio pay. She now tells the Jew that he must take exactly one pound of flesh or he will be killed. Shylock asks for the principal, which Bassanio is still willing to give him, but Portia stops him, claiming that the Jew will have only justice and his bond, and none of the mercy he him- · self did not show. At this po.int Shylock wants to give up th~ suit and leave, not wanting to act against Antonio if there will be reprisal against him for it. Portia and the law, houever, are not through, for they " hath yet another hold on him. " For plotting against the life of a Venetian citizen, Shylock must give that citizen (Antonio) half of his fortune, and forfeit the other half to the state, his life being put at the mercy of the Duke. The Duke, in order to show Shylock what mercy really is like, spares the money lender and only fines him, instead of confiscating half of his fortune. Antonio also shows mercy, and agrees only to use the half of his estate that he will receive, and upon Shylock's death to give it to Lorenzo and Jessica, p r o v ided t ha t Shylock will do the same with the property the Duke has allowed him to retain. Antonio also makes the further merciful stipulation that the Jew become a Christian. Shylock, however, fails to c.omprehend the mercy bestowed on him, and only mourns his 1 Act IV, 2 Act IV, 3 Act · IV, ·4 Ac'i; I_V'' scene scene scene scene i, i, i, i, lines 249-250 line 200 line 320 line 360. �- 24 :fallen poa~~~on • . ~e is not ~om£orted by the f~ct . that he could have fallen further. An analysis of' Shyl·o ck' s· :fall. shows a great deal 0£ ironic justice. The most obvious example is that Shylock, ·the man who claims to champion the law 1 , is destroyed by it. The cause 0£ this is twofold. The :first is that Shylock tries to use the law £or purposes thatwere never intended by the original lawgiver. Thus in trying to ,rise above and master the law, Shylock is trying to s~bvert it, the thing which is most unwise :for a Jew·, whose only defense is the law. So it happens that. when. a wiser person tha~ Shylock becomes his adversary, he is doomed• For this person, understanding the intent of' the law, can show where Shylock is misinterpreting the law. (In t~is · case he is trying to use a law t6 kill someone.) Thus Shylock's only def'.e.n se is to try to justify ·niisinterpretation and if'· he. succ.eeds, by the .saµie arguments• the laws protecting him can be misinter:p.reted. · Thus., _ matched against a thinking · adve_rs~~y • Shy lo ck fs doomed • 0 . · he - se,cond · reason tha·t . Shylock fails before t .h e'· law is that T ·· he .. doesn·1· t fu- .ly un~e~stand:· its · workings. -... Unlike the Mosaic law 1 . (as · Shylock ·sees. it)", .-' the . -V.enetian statutes, and_·in·· fact the Veneti~n stat·e-, · require : a: · cert a.in· amount· of' . "mercy" and nchri s ti an.· charity; " in . order . :for i ~ . to . work .: p·r~pe·rly •. : . . .• 'Social life has become much more complex : since the times 0£ Moses. · I:f every man in Ve~ice were to be prosecu:ted every time he broke a· law, _ the state cou.~d do · nothing .-except arrest its own ' citizens. Commerce and thus the state would soon disappear. · he way to avoid this problem is to be willing to· :forgive T offenders provided .. they · can be useful to the · s.t .ate ~ There is no heed to rigorously press the people as long as everybody is flourishing, and no one . gets too :far out . 0£ hcnd. Legal mercy, then, is the great g~ossing over of' irksome details, the oil that helps the state run smoothly and keeps it prosperous. Sliylock cannot see _ the ' reasons behind it ·, and so must learn them by barsh example~ The play seems _o have rea·c hed its logical end with Shylock's t becoming Christian, and indeed, i:f the problems discussed in . this paper so far 'were :the only problems, or even the central p _oblems o:f the play, perhaps Shakespear would have ended it here r . and ~ot have· his characters :feel compelled to go back to Belmont. Bas,s anio and Gratiana go . because they b-elieve their loves to be tnere.· ·I do not kho: why Portia · and·Narissa chose to return; w perhaps ·it is becau·s ·e it is their home, or perhaps they :feel their love will best flourish there. When the scene shif'ts ' back to ~elmonti we see Loren~o and Jessica, the :fugitives :from Venice, who stayed behind when everyone else returned to the city, engaged in a strange, sad, 1 Act IV, scene i, line 145 �-- , - 25 dialogue.~ · Lorenzo begins .... The mo'o n .shin.es bright: . fri ,such :a ·n ight as this·, When the. sweet wind did: gentl.y kiss the trees And they did· make no no.ise, in ·such a night Tr.oilus methinks mounted the: Trojan .walls, And sigh' d .his soul toward the Gre;cian tents, Where Cressid lay that night. (1) but beauti:ful·lY.· moving ·. ~. ~· It is - no accident that Lorenzo chose thS · tale o:f this couple. For although they loved, they met only :for -a night, not because Troilus did riot want to love forever, but because: the lovers' :fathers were o:f opponents camps. And so they were separated·, and so, uncared for, Cressid proved untrue. Next the stories of Dido, Thisbe and Medea : are told bi the love~s. All o:f these ancients chose their lovers from oppo~ing political camps; all passionately loved :for a while, and all ended sorrowfully. Lorenzo and Jessica then recotint = th~ir bwn love story, knowing the ·pattern it has so :r-ar fo ! lowed, arid sensing that the end that has come :for the others will someday come :for them. They know that when they are alone in Belmont under the bright moon, Jessica, under th.e guise of · a teasing lover, can state her :fears about the truth of Lorenzo's vows of love, and .Lorenzo can forgive h~r. Jessica's fears are not foolish and ungrounded, for . in marrying the Christian : s~e has had to forsake everything else that . she previously had • . Her father's house, people and way of life are closed to her now; she only has Lorenzo and without him she is completely abandoned . It is no ~onder that Jessica fears the end of thi~ b~ight night at Belmont. A:fter each of the lovers has given four speeches, they are interrupted by messengers announcing the return o:f Portia to Belmont. When Lorenzo hears this, he orders the house to prepare :for her coming by :filling the air with sweet music . This music adds further glory to the night at Belmont, and moves Lorenzo to give two b~autiful speeches, one explaining the celestial harmony to Jessica, and the other describing the joys and effect of the less perfect, but still powerful, earthly harmonies. Portia does not return without a gift for the couple, for with her she brings the news that Jessica and Lorenzo are now heirs of Shylock's fortune, and consequently they are no longer dependent on their parents and friends. They are now able to establish their own home where they can rule themselves. They can now live during the day as well as love during the night. Thus with Shakespeare's story of the conflict between Christians and Jews brought to a happy close {only thirteen lines before the end of the play itself), perhaps we should ask what, i:f any thing, can we learn about this struggle :from the play. The happy endings do not :fool us for a moment, :for we realize that the problems were only solved by the mistress of Belmont , wise Portia, a women of singular virtue. But, as I said earlier., �- 26 these two cases are polar extremes, misshapen by embroilment and conflict. In most cases where the emotions become involved in what should be a purely civil affair, a small show of mercy and charity on the one hand, and continence on the other will usually reconci1e the antagonists, provided there is some profi t f'or both • . There is rio need :for a Daniel to judge them, nor for one, if not both of' the opponents, to be destroyed. In the ~ case of lovers, seldom is it necessary actually flee thei~ homes and go to Belmont. their civil problems can be borne through the se.rve to sweet en the love they :feel at night, bright and music prevades the air. · ·, ·~ ·- f'or them to In most cases day and will only when the moon is But what if the civil becomes so strong that i t begins to strangle love, and not.nurse and accentuate it? In that case it seems a pi].gri_ mage to ·B elmont must ~e l" maq~ . •;_--;·~ ffow ?... ·,~: l::e -~- ~t ~~s~est . to ' go wi .th 5trong :f.riends who ' kriow the way, -- · this is " the means Gratiano . uses; . or to come With . oi:ie's love to sojourn there for · a t-ime, -' a 's . fugitiyes from the city - like Lorenzo and Jessica. · Howev.er, , ·some, ... and they · are usually brave princes or noble ·:"ge,µ.t ·l -eme~,. make th~ir way alone, ·driven by a strange sort of iove • .[f ~ this be you.r ' journey, be sure that you do not look . f'or Belmo~t en6ased in iold ~s . di~ t~~ black Morrocco and the ." · ~ld ·shy- ock; ·. or' searc.h :fo:r .i ·t ' . _thi·nkj;n~~( that it is your just l d?.~ e:r-t :and·· na t-liJ:;"al · ~.ue,. ··as · did th~ ., :fo_ ,iish Aragon and the merchant o AnJ;·o nio •. Bu.t ·i:r your .wish is to · be.- ·success:ful like the good Bits~·ani~ :. i ·you .mus.t . b~avely. g:lye · and, haz~rd all you have for this strange ,.something, whose ca.sement ·seem·s quite worthless. And . if' yo.u' g.:~t t ·hat· :far~ · do . ot try" to · make the · right choice by your n . wits aI:o~e, but· li;st~ri. to · the · music in the ·air, for even if' it's spi'rJ.,..t ' do~s no .t move you, .· :it.s inspired rhyme may turn you in the rig~~ di~ection. • I -~- ") . �- 21 · ROUSSEAU AND THE ANCIENTS: A REPLY Parme~iscus (Editor's Note: The following contribution was received through the college mail under the indicated· pseudonym. ·T4e note and the mystery surrounding it intrigued me as editor.although hardly flattering to me as the author of ·the paper under fire. I was tempted to find out more, if not about the true author, at least about the pseudonym. After several unsuccessful attempts, I located the following brief mention in the encyclopaedic Pauly-Wissowa Real Encycloplldie der ~lassiches Altertumswissenschaft, Achzehnter Band, Zweite Halfte, p. 1569 Ferner erz~hlte Semos -in · seiner · , _Parmeniskos sei ein vornehmer Mann gewesen, der einestages in die Trophonsh8hle zu Lebadeia hinunterstiegen. -sei und, als er wieder . heraufkam, die F~higkeit des Lachens verloren hatte. {Loosely translated, the sentence · reads: · "Moreover Semos in his (work on) Delos recounts that Parmeniskos . was · a· · man of eminence who one day decended into• the cave of Trophon at Lebadeia and when he came up again had lost the ability t·o laugh.") · Whether Semos' Parmeniskos is the same as our contributor's, I do not really know; nonetheless, the humorlessness of his piece 'seems to me to be in harmony with the name _and its classical connotations.) Mr. Lachterman's paper, Rousseau and the Ancients {Collegian, Dec. 1963) sins, I should say, more by e~cess than by omission. The comparability of Rousseau and Plato and Aristotle in ~ cardinal aspects of political thought . i~ · established very persuasively through what seems to have been a ·careful reading of the relevant texts. Nevertheless, this one-sided and at times belabored emphasis on the areas of agreement tends to obscure, to my mind, certain more ambiguous notions which enter nonetheless i .n to the total fabric of Roussea'l;l' s political and ethical philosophy. I understand .t he ambiguity of these notions to have arisen from fundamental tensions iri his manner of thinking , the result, perhaps, of his own mode of life·, or, qui t ·e possibly, of an imprecision and ,hesitancy determined ·by the distressing character of the conclusions he seemed led to and for whose modern, more revolutionary offspring he must accept partial responsibility. In order to buttress these critical and, it may seem, unwarranted generalisations, I shall now make brief mention of three: instances of ambiguity that lie close to and thereby imperil the heart of Mr. Lachterman's thesis. Dr. Leo Strauss in his justly celebrated Natural Right and History speaks frequently, in his chapter on Rousseau, of an i~resoluble conflict between society and the individual, that is, society and nature. I would say that, although the conflict seemed to Dr. Strauss irresoluble, Rousseau himself thought that he had found a solution in the elevation of the People to an {unjustifiably) high status, making it at once a collective mystical body and the sophisticated voice of political wisdom. Connected with this god-like status of the People is the idea of public enlightenment which Rousseau never fully abandons despite, as Mr. Lachterman argues, the introduction �d -28of an oDll')iscient Legislator .in B0 ok .II . of ..the Social Contract. To those readers :familiar with classical political thought {and Mr. Lachterman appears to be one o:f these) the myth o:f the People and its corollary o:f Au:fklRrung must seem egregiously out o:f _ place in th~ writings, o:f .a ·. thinke:r so :imbued ·with · and -"inde.b " ted" . to· the · ideas . of the ··anci.ent_ . . In any case :i t seems s -qui t.k 'clear that ·Rousseau's. "assertion _ that · ea·ch ·man in obeying th~ · 1~~ ~ollo~s his . own w~li i~ rooted - ~n the·~dea o:f the People whose g 'e neral· will always_ act in ~ the interests of all citizens • . I The n.ot_ :.Lori of ... a. cont.ract which brings into being this mystic · pody reinairis essential t .c)' the · supsequent ".elabor'ati9ns· of the ·g .e nera1 will and the-. law present.ad_ by Rousseau .in - the· Social Cohtr·a ct', .n otwithstanding ·t .he suggestions · ri · the·· Discourse on 1 · Pali ti cal Economy · that:, mo~al · goodness i .s · t ·he ·true. foundation o~ ~dlitical obligation, suggestions which · Mr. Lacht·erman re- · ceived with ·obvious and unmistakable . rel- sh. i The simiiari ties bet·we~n . R 0 usseau and the ancients in this . .latter work do not, I . daresay~ oirershad_ow' and ultimately negate the principia of · ·the· Social · Con.tract, on the basis of which the - whol·e of Rousseau's w· rk must be evaluated and the relegation o:f which to a seemingo ly· inconsequential place in Rousseau ' __§j political theory is the · mos~ - disturbing element in ·Mr. Lachterman' s thesis. 1 The presence of these two contradictory sympathies--with contract on the one hand and the ancient's conviction of the naturalness and moral ultimacy of the polis on the other-creates 'the £1rst of' the afor~mentioned ambigu~ties in Rousseau's work .. In the next place, however enthusiastically Rousseau (~nd Mr. Lachterman) ·may stress the role o:f reason in the rep-µ.bl-~i'c , - i t is obvious that sentiment rather ·t han reason is. · the· supreme political faculty. Compassi.on--the symp·athy each . of' us should hav'e ' for the miseries o:f our fello~s--is inculcated in man by hattl'.r~ Eihd can only be "extirpat.e d". py the onslaught of' civili zatidrt• Rotisseau' s 0 ideai cons:truction" makes i t possible :for compassion to erltltire wi th:lri. and indee.d hold together society. It Ls thi.s sent:iment which; more thart self-interest; is mean.• , · to support political obligation. In addi ti.on to this, . we ttii'ght merttioh that ih Rousseau~-in marked contrast to Plato a,ritf " Aristotle-•there :1.s rio ~ntologica.l · and_ epistett1ql 9gical jUstif'icatit>n :for t4~ supremacy · of reason i_ri t>oii.tidai society• This ·s hortcoming seriously weakens the £orce of his asser'tion that the ideal society- is. governed by r~l~$.<ln ,..incarnate ih the Law. At most Reason he~e can only mean ~ . k~,~·;.ot sentimehtal wisdom . or the. ·common sense o:f the noq}.e'' 'arid f.iJ..pright peasaht. • 1 . £_ 1 The . :finai issue I should like .t ·o raise with. Mr. Lachterman•s at times ·su~spiciousiy naive interpretatiQrt has . to do with freedotn Let me say first that eneomi~ms of virtue have a peculiarly - unpleasant sound in the mouth of' a mor~l profligate, Maritain's censure of' Rousseau's counterfeited moral rectitude seems t'o me well-intentioned and, an~ moral · responsibility in political - society. 1~ C:f. e~pecial~y Social Con~raci, Bk. ~V, Ch. I, Page 1. �-29as one sees in leafing through the Confessions, accurate, Freedom meant for him (and continued to mean after him) freedom from constraint and authority and·it is on this conception of freedom that the image of the natural and therefore good man is based.I The free spirit can never be successfully integrated into society without sacrificing his cherished liberty- a liberty not in the least harmonious·with the civil and moral liberty that the contractual society makes possible and indeed enforces.2 To ignore, as Mr. Lachterman has done, this side of Rousseau's character and though+.- the side that influenced German and French roman-ticism so greatly - is as much as to close one's eyes to the obvious in the hope of raising the unclear and, it may be, inconsequential to a ra~ that it scarcely deserves. By this I mean . that History saw fhrough Rousseau's pose as the ally · .9 f the ancients, while Mr. L~cht ·~··r~an, if he put forth 'his ·thesis seriously, did. not. ; , , l~ 2· . Cf. Les Revenes du promeneus solitaire {Rousseau's last work!) ~aughan ed of Rousseau's Political Writings) Cf Social Contr.act Bk I Ch VII - a passage frequently referrea to in Mr. Lacnterman's paper. �COMMENTS ON MR. SPARROW'S LECTURE: Rights, ~ ~ the Right Larry Silverman J"n. v-evi ~wi.ng and r .om.me. nting on Mr. Sparrow's lec~ure, it was :necos .~~1·:y :f'.oi~ ~ to paraphrase and deduce from his words. If, in so doing I have distorted any of his opinions, I can only ' . hope that that distortion will lead to a further clarification of them:· The lecture, on Rights, ~ and~ Right, began by pointing out the need for its existence. It contended that the present confusion and conflict among civil, natural, social, and economic rights, culminating in the 'unthinkable' proposition that the same thing is both right and not right, may, in. the future, subject civil society to a turmoil from which " ••• neither the rei t era ti on of once-understood relationships nor man:i :fes ta tions Of good-nat:l.11'0d SP.Tii:i..ment:aJ.i .ty Will be . able to extricate USe 11 This c~isis seemed to stem from the peculiarly modern notion that the "endless worth" and "infinite dignity" of each one of ll.f3 e~t .ii:l.Prl hini to natural and ina. . Pnable rights~ "limitless U in nnmher." Hence the lecture tried " ••• to suggest another :f':t·amework than that of rights within which political issues can be framed and pr_ ori ties assigned •• " i a The suggested framework seemed to rest on the notion of the "common good," rather than a theory of private rights. Hence one. of . the characters of the mythical dialogue expressed surprise and skepticism at hearing that his 'right' to walk down the street was simply the law's approval of that action. This · approval "is grounded on the prior determination that it is all right to do the contemplated act ••• in view of the common good." The skepticism arose from the belief that each individual man had a natural and inalienable right to exercise his individual will, and that the law's approval is only an implement to that right. The lecturer traced this notion back to the philosophy of Hobbes and Locke. According to that philosophy each individual desiring passionately to live, " ••• is conceived as a point surrounded by 'natural rights' some of which he lay down or- transfers to a communal authority when he 'enters' society, but others which he retains because of their inalienable character." It is precisely what Mr. Sparrow called the "inhuman .. necessity" to transfer authority that gives rise to a self-eontradiction within this view. For, if the good is defined as self-preservation, and the right as that which naturally enables or empowers a man to preserve himself, how then can an individual man be so powerless as to be forced to relinquish some ot his powers in order to be enabled by another to exercise these powers? In other words, my right, .or that power which enables me, to walk down the street must he· understood concomitantly with the obligation of others not ·to hinder me. There is n~ entological or historical priority of one over the other. �-32- Mr. ·To clarify this argument, Sparrow made use of something he ..cal·l ed the "square of right." (See next page). He used this device to illustrate with some precision the relationship between: duty and right. In the top half ·of the square, the prohibitions and commands of the law become tbe duti..es and hence the rights of the individual. · This . can· be s .e en .i n . the · cited example of the policeman, who has . a right to direc .t traffic because he was so commanded. At the same time i t is, of course, his duty to direct traffic. f But for all its pre_ cisi.on, · the square contains t~.o _ serious · defects. _ In the :first place, if we are to isola~e dutie·s : in the top half · of t .h e square, what are we to :say of _men who _i.nsist ·upori (.ioing their duty not simply by" obeying commands but by exercising those rights permitted them . by tbe ·law. Socrates " was p~rhaps - the paradigm of such a man_ . If the laws, . for ·a · time, permitted him to speak free·ly, he .spoke fre'ely,, not to -exercise -J1i s rights as an :individual' but . .to do his duty as a ci tiz·en. Indeed, today we are permitted to ·vo.te; is i t not our dtity to do so? Yet · Mr. Sparrow, fo~ the sake _of · lotical rigor, restricted duty to the class of obedience,. thus agre.eing :in par·t; with tlie traditional liberalism he was trying to undermine.· This reply might have validity were it not _for the second ··. __ deficiency in the square. Mr. Sparrow, as was stated earl~er, maintained that the so-called approval of the law was gro.u nded in the prior :determination that an action was ali right. :Wha't .he.· mea:Ilt by 'all right' is still unclear. Here is one of several ·- defini.tions given in · the lecture: " ••• An action which does not harm. or only slightly harms the common good~" But . voluntary acti~ns which are "all right", (e.g., walking down · t.he street; growing a beard, listening to music· , etc.) are '_ very ·d _:fferent - frpm voluntary actions which are right. (e.g. voting, i ·Speaking -One·' s mind fQr the sake of the common good, etc.). It ca.n make no sense, therefore, to cal 1 all permitted ac ti o:ris' 'right', as the square does. We. might overlook the deficiences o:f the square and view i t as an aid in revealing the absurdity of saying that one ·empo·w ered · by nature is constrained by necessity to enter into an artifical bond in orde~ - to obtain · his nat~ral powers. But assuming that that stat ·eme~t is absurd there .are stil-1 many problems before us. For if one's power stems not from natural rights but fro~ the approval of· others, or more to .~he point, of the state, ' how then is tyranny · to qe . avoided? Mr. Sparrow might answer that the approval ~f the laws of - a sta~e are properly based upon . the notion of the _ common good and not the arbit_ rary will of a tyra:rit, .(who i 's of' all men the ,most private); a tyrant is not evil because he de-pri ve.s ·men o.f their 1 -right, ' but because· he makes men incapable of doing the right. Thus, a man is only free insofar as he is free to be a man; and the faculty of manl.iness is the ability to do the right, which is defined as action "tending" towards the good, and the inability to do the wrong, or actions tending towards nothing, and hence "meaningless" agitations. �-JJ...;. THE MODAL .SQUARE OF OPPOSITION UNDERSTOOD AS THE .I . SQUARE OF RI.GHI' . Forbids: Commands: ~ A A MUST Nof"\J3E DONE I have a DUTY not ' to do A; bu~ also · --..J have a RIGHT not to do A; and it,is --;;RIGHT not to do A MUST BE DONE ... . I have a DUTY ·to do . A; but al·so I have . a BIGHT-to do A; and it is RIGHT to do ':; :'%-..>.. ~' 4 : ' ~ ,_ ~/ \~ <v-r- -----. <:, o" ~ L,Q-,~ . ~-- Lt A MAY BE DONE14 . _,... ---·· I have a RIGHT to do . A; and it is RIGHT..... to do A I ~--- ·Perniits: /- --~-· ./ ./ -.... ' -..". LAW, the ~/~ ,, / expression and determination o:f the RIGHT / .JA NEED NOT BE DONE I have a RIGHT not to do A;/and it is RIGHT rtot to do A /PPermits: * * * * * * * * * * * * * * * * * * * * * * STATE OF CIVIL SOCIETY Law, Compulsion, Constraint DUTY (OBLIGATION TO DO) . DUTY (OBLIGATION NOT TO DO) I ~ SOCIAL RIGHT (PERMISSION TO DO) CONTRACT f RIGHT (PERMISSION NOT TO DO) STATE OF NATURE Rights, Liberty, Freedom .l �-34But if' the :faculty o:f manliness is the .ability to do right -and that ability is ·somehow obtained :fro~ an external p:ower, are we not then confronted ~ith the same problem as confronted the partisans of naturals rights? I think Mr. Sparrow· would ;ans~e-r·; "No." The reason :f'or this answer is perhaps tJ:ie most -significant and elusive implication of the entire lecture. For the · community and the law enable a man to .b e · :free not only by authorizing and approving his activities, but _ by pro.vfding a sphere and an object necessary for · those - a ctivities to take place. In Plutarch's phrase, "The pecul.iar work of a good man is political." -* Thus, a community enables a man t-o be a man in a way _ similar to that in whic·h wood and nails enable a carpenter to be a carpenter. There is no longer a contradiction. " ••• Right. and ~ight, necessity and :freedom, mer~e in the depths of the mind and heart o:f a true human person." Near the beginning o:f his lecture, Mr~ Sparrow spoke of the 'common good.' As he went on, he s6mehow managed to unite the common good with the "indiYidual good." Towards the end, the term, "rational supreme good" was introduced. The lecturer seemed to be saying, by implicat1on at least, that in the :final analysis the three were one. .; . This is a most provocative and rather dangerous view. Before we can allow ourselves to dream and to speculate :further we must check the cianger . which _was spoken o:f previously--the danger o:f tyranny. Could not a sixteenth century inquisitor, _ adopting Mr. Sparrow's suggested :framework, justify his -activities by ~ayi:µg, "Some of us have a ·greater knowledge o:f the Supreme Good than o,th_ rs do. e It is our duty, . there:fore, . to conform the political body and the private · man .t o that Good, . with any means whatsoever." How can we repudiate the inquisitor except by replying, No man can be truly :free unless he is :free in the conventional sense to choose or reject the truth.? _ Whether an harmonious_· polity can be established in harmony with the Supreme Good, and whether · the myth of 'i~alienable rights' is necessary :for men in order . to learri arid'. _ to will the Good, must remain, for the time being, open questions~ Just ben eath the surface o:f all the lecturer's remarks was a principle e·x plici-tly :formulated in the last part of the lecture. The terms 'right' and ·• wrong' bear . o action the same relationt ship that 'true' and ':false' b~ar to . sp~ech • . A :false statement is meaningless, and therefore no statement. ·" If' ••• transcending, containing, and ordering human ends is possible," he said, "human acts will be constituted as such precisely to the extent that they tend to or mean, that is are means, to it. And hence, :for them to be truly human actions, :for them to be meaningful, and :for them to be right will be one and the same thing." The so-called wrong actions are meaningless and therefore no actions at all. " ••• What seems to present itself' initially as a wrong action is in :fact mere agitation." Evil is privation of good. * Plutarch, LIVES, Cate, p. XIX, 2 1' �-35This position .caused much comment during the question period. Mr. Sparrow's analysis . was interesting, but it seemed to deal only with one part of the phenomenon of evil. For surely, Adam, after having eaten of the fruit, became aware of his nakedness, or privation. But it is very difficult .to read that story without :feeling the reality and the potency of the Serpent. · Though the lecturer acknowledged that it is " ••• all too possible for members of a divided and discordent humanity not only to mutilate being but ·also to iead others to think that such sub-rational agitations are instrumental to the attaining of the proper end o:f the human person," he did not succeed in expJ.aitiirg how this is possible. taking his view, the problem o:f indifferent or 'all right' actions becomes even more inexplicable. If we are to call the thousands of trivial and harmless acts evi'! because they are meaningless, we had best get a new word to describe those actions which are really evil. Furthermore~ Mr. Sparrow ended his lecture by saying, "However, it is not quite so clear that it is she (dialectic), revealing the ultimate absurdity o:f false actions, that by making us at one with ourselves, can lead us out of the cave and so restore meaning to action. I suspect that instead the cure will have to be :found in another dimension altogether. The Word may . have to be made :flesh and come to dwell, to die, among us." Somehow those last sentences gave the lie to those preceeding them. It is as i:f the lecturer were really saying) "Before the nature of man can truly . be understood as I would have you understand it, before evil can, in truth, be called privation, some Supernatural Being must descend to earth and teach us the two precepts wherein all the law and all the prophets are contained, and say clearly to each man, 'He that is not with me, is against me, and he that is not against me is :for me.'" �-36;.. Ali ESSAY ON DUCKS Jeremy Carl Leven Mr. Lamb notwithstanding, this essay shall deal primarily .with the ·animal "imparatis", for the nobility .o f this animal is most present, not when served, but when serving. I divide my essay, therefore, into six parts: the animal, . himself; his relationship to his peers; his relationship to people and . vice versa; hi~ influence · on the American w~y of life; his humor; and the .duck as an explanation for the order of the kosmos. The work "Duck" is derived from the Old English 'duce" which meant 0 a . diver" • . This shows that evolutionarily the Duck has changed relatively little since the days when he spolte Old English, still diving and sti11 · enjoying it, if one is to judge by appearances. Much like people, the Duck also belongs. to a family, one that is known as "Anatidae", a name that may make a Latin duck quite proud, but currently is so despised · among American ducks that they seldom use it; and it has, therefore, become almost obsolet ·e among all but the most conscientious du.c k-lovers, drakes included. Generically speaking, we must ·refer to the · Duck as an "Anas", a name which undoubtedly has caused many ·a .w eb-footed animal of the family Anatidae to turn her head away in a deep c-r1mson blush. Contrary to the opinion of those who have kept these animals as house-pets, there is no relationship, entomologically or otherwise, between the name of the genus and the alimentary . canal. The Duck has often been described (of. Amer~can Co·llege Dictionary and my friend; Morris' mother) as a "web-footed swimming bird characterized by a broad flat bill, short legs and a depressed body". Perhaps it is better to describe him as a ":feathered biped"; this points with pseudo-Platonic directness to the close link between the Duck and the philosopher, a similarity that becomes even more evident when we listen to the speech of both. This obvious l y brings us to our next topic: the peer r elatio ships of ducks. As a rule ducks get along with each other no better than do men, or, in other words, not at all. For ducks do not share their food with each other, nor do they assist each other in any way. Even a duck who may fancy himself a leader in an activity such as flying South finds that he is constantly overtaken by other ambitious ducks until he :finds himself last in formation. This bureaucracy explains why ducks deem it necessary to start South so early in the F all and do not return until late Spring, this situation is known colloquially as the ducks' "migratory plight". , �-37Concerning his relationship to people and vice versa little can be said, Ducks are· ducks, and p·eople are people. People seem to notice the difference, and the ducks appreciate this perspicacity. Man also seems to revel in the throes o:f bread-tossing, a splendidly humane sport, enabling the Duck to see his way through the cold Winter, while it is a wellknown :fact that Man cannot live by bread alone. While on the relationship of Man and Duck, a human acquainta:nce o:f mine should be mentioned, a brilliant Audubon scholar by the name of Seymour Duck. It appears that he felt the sole advantages of his name were that it was :fairly easy to spell and presented little difficulty with pronounciation. However, Seymour, now the chairman-in-charge-o:f-reevaluating-the-point-systemo:f-classi:fication :for the American Kennel Club, disliked his name so much that he changed it to "Poodle", and, consequently, was asked to turn in his binoculars. The irony of this is that Seymour can best be described as having a broad, :flat bill, short legs and a depressed body. We now approach the subject o:f the influence of the Duck on the American way o:f life, a way of life to which many refer as "ducky", or something to that effect. Ask yourself, what American woman has not wanted to own a genuine duck-feather pillow, a pillow so light that, when one throws it up in the air in delight, it is difficult to realize that what went up can ever be down? And the American Man :finds no sport as enjoyable as sitting in cold marshes from six in the morning until dark, waiting :for his friend, the Duck, to appear so that he may :fill him with virulent lead, a typical show of American affection, a display of human love that calls for ducks to have, if nothing else, a good sense of humor. This brings us to our next subject: Donald Duck, and I am reminded of a comic strip I read last week which went something to this effect. Daisy Duck to Donald Duck, "Who was that cute-looking duck I saw you with last night?" Donald's reply, "That was no duck; that was my wight*!" This points to the wealth of humor the common duck provides for the common man, bordering only slightly on the esoteric. And this naturally brings us to the :final and most essential topic. From this essay the careful reader could not help but be impressed by the :fact that the true value of the Duck, is that in no way does he provide an explanation for the order o:f the' kosmo s. * {Archaic, obsolete) A human being or person � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The Collegian Description An account of the resource The Collegian began as a student newspaper; became a college community literary publication in 1952 and continued until 1969; became a student newspaper again in 1969; discontinued publication from 1980 until 1989 when it again became a student literary publication. Click on <a title="The Collegian" href="http://digitalarchives.sjc.edu/items/browse?collection=26">Items in The Collegian Collection</a> to view and sort all items in the collection. Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Page numeration Number of pages in the original item. 37 pages Text Any textual data included in the document text Original Format The type of object, such as painting, sculpture, paper, photo, and additional data paper Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Creator An entity primarily responsible for making the resource Lachterman, David (Editor) Title A name given to the resource Collegian, January 1964 Date A point or period of time associated with an event in the lifecycle of the resource 1964-01 Description An account of the resource Issue of The Collegian. Published in January 1964. Identifier An unambiguous reference to the resource within a given context Collegian Jan 1964 Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Publisher An entity responsible for making the resource available St. John's College Language A language of the resource English Rights Information about rights held in and over the resource St. John's College owns the rights to this publication. Format The file format, physical medium, or dimensions of the resource pdf The Collegian https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/257a133ecb50a585a7e9260968adb5b4.pdf a7e44dfe2838da6ee6c2918595655d19 PDF Text Text r -.I ! i '•. · ----~: i .tJ l_ ·-.J ~--1 --· r· i I I I I l \_~l [j _\r. '-'r f\' OV, - o h"'·:i Dl?t c,,,I''"''j (. I '1' ~ 3 �C0 NT E N T S Page The Meaning of Regular Solids • • .Robert Neidorf, Tutor . •. 1 • • • • • • Lawrence Feinberg, . '64 • • 9 A Strictly Euclidean Demonstration .Samuel Kutler, · Tutor • • 12 Si tu t'imagines 13 14 A J:trth for Our Times • • • • • • • • • • • Raymond Queneau • • Translation by Christian Harrison, •64 • • Rousseau and the Ancients • • • • David Lachterman, Correction to Page 41 of the October Collegian • • • • • • • 1 65 • • 15 Robert Sacks, Tutor •· ************ Editor • • • • • • • • • David Lachterman Business Manager • • • • David Rasmussen Faculty Advisor .•• . • • • • • Eva Brann . 33 �EDITORIAL In order to encourage excellence in creative writing the COLLEGIAN has established a short-story contest. All St. John's students are invited to submit short stories on any theme to the editor before the end of the first semester. The contributions will be judged on the basis of imagination, coherency of theme and skill in using language to display ideas and characterizations.convincingly. The absolute and not the relative excellence of each contribution will determine the selection of the winning story; if no contribution meets the judges•' standards the contest period will be extended. . The author ~f the winning short story will be awarded a twenty-dollar gi~ certificate for use in the College Bookstore. The judges will-be Miss Eva Brann, Mr. Richard Scofield and the editor. ************* The following is the solution to the mathematics problem in the October COLLEGIAN submitted by Mr. Malcolm 1~att: Proof: Form the longest possible ascending chain, beginning with the first number. Now take the first number ~ in this first chain and form the longest possible ~scending of numbers, none of which appear in the first chain. Then take .the first number which does not appear in either of these two chains, and proceed as above. Eventually· all of the numbers are used up. If any of the chains thus formed have ~ n+v elements, then we are finished. If not, if all of these chains have length ~ n, then there must be at least n+1 of them. Consider the last element of the last chain. It cannot be greater than all of the elements of the next to last chain, or it would have been a part of that chain. In particular, it must be less than the last element of the next to last chain. Again, this element must itself be less than the last element or the last-but-two chains, or it would be an element of that chain. Continuing, we see that the last elements of the chains themselves form a descending chain, and since there are at least n+1 of these elements, we have found a descending chain of length ~ n+1 • �THE MEANING OF REGULAR SOLID Robert Neidorr In the last part of the last proposition of the last book or Euclid's Elements, the claim is made that the so-called regular solids, or Platonic solids, are unique. In the Heath translation he enunciates that ••• • • • no other figure, besides the said five figures, ean be constructed which is cont~ined by equilateral and equiangular figures equal to one another. (SIII, 18) By the phrase "the said five figures" he refers explicitly ·to what are commonly callsd the Platonic or regular solids, which have been con5truoted one at a time in the earlier portions of Book XIII. The fact is that this enunciation is defective; the purpose of this note is to indicate the defect and demonstrate its po·ssible remedies • . 1. The defect - First we review the five ·solids which are said to be unique members of a class. They are: ·( 1) The cube, a six-sided polyhedron contained by mutually congruent squares. Or, in Euclid's terms, the cube i~ contained by six equilateral and equianglllar quadrila~erals, equal to each other. Each vertex~1) of the cube is composed of three faces. (2) The dodecahedron, a twelve-sided polyhedron contained by equilateral and equiangular (i.e., "regular") pentagons, mutually congruent. Each vertex is composed or three races., (3) The tetrahedron, a four-sided polyhedron contained by ;equilateral (and hence equiangular) triangles, mutually congruent. Each vertex is composed of three faces. (4) The octahedron, an eight-sided polyhedron contained by equilateral triangles mutuall congruent, Each vertex is composed of four faces · (5) · icosahedron, a tvrenty-sided polyhedron contained by equilateral triangles mutually . congruent. Each vertex .is composed of five faces~ ' .' ' ' We see that each of these solids . is bounded by a set of mutually congruent faces or surfaces, which are in every case regular polygons. A so, i t i s clear that ·Euclid's entire discussion is intended to be l ·restricted to .convex polyhedra, which can be defined -as those polyhedra which lie entirely to one side of the plane of any face. Hence we are ternpte.d to characterize a regular or Platonic ·solid as a convex p·o lyhedron bol.lnded by . mutually congruent regular polygons. Although Euclid does not use the terms "regular" or ttp1atonic'', it is evident from the cited enunciation that he means to describe some general class of (convex) polyhedra, of which the · listed five constitute unique exemplars. The enunciation .itself is the only clue as to' how he intended to describe that general class;· taken literally, the class it 'describes !,! the ciass or (convex) polyhedra bounded by regular ("equilateral and equiangular") (1) By vertex I mean the. point, a.nd the reg'iort around it, on .the surface of a polyhedron where three or more faces and edges converge. Euclid calls this a solid angle. �- 2 - polygons mutually congruent ("equal to one another"). The defect in all this rests on the fact that there are more than five such figures. As a preliminary to further discussion we require a fixed terminology. We shall say that solids bounded by congruent regular polygons are rhoregular: Df. A rho-regular solid is a convex polyhedron whose faces are mutually congruent regular polygons. Next, we will use the term "Platonic solid'~ to refer specifically to the five polyhedra listed above; thus the term "Platonic solid" should be understood as having only a denotative meaning. The terms "regular solid'~ will be defined later. Euclid's enunciation amounts to the assertion that the Platonic solids are the only rho-regular solids, and it is this assertion which turns out to be false. We begin by restating Euclid's uniqueness proof. From Book XI, 21 it is known that the sum of the plane angles which meet at a vertex must be less than 360 degrees. It is also evident that a vertex on a solid must be composed of a minimum of three faces joined together; hence at least three plane angles must meet at the vertex. It follows at once that there is no rho-regular solid contained by regular hexagons, since the interior angle in a regular hexagon is 120 degrees. Similarly, no vertex could be constructed with regular polygons of more than six sides, for their interior angles are even greater. The angle in a regular pentagon is 108 degrees, so a vertex can be built up with three and only three such figures; this corresponds to the dodecahedron. For a similar reason a vertex can be constructed from three and only three squares; this corresponds to the cube. Since the angle in an equilateral triangle is sixty degrees, a vertex can be constructed from three, four, or five such triangles, corresponding to the tetrahedron, octahedron, and icosahedron. On the basis of these considerations, Euclid concludes that "no other figure • • • can be '.~ constructed which is contained by equilateral and equiangular figures equal to one another." Or, as we should say, no other rho~regular solids exist. But his conclusion is false. If we seek to build a polyhedron(2) with a stock of -regular pentagons or squares, it is clear enough that each vertex must be formed from a junction of just three such faces; hence the dodecahedron and cube are the only such figures possible. If we now tty building polyhedra from a stock of equilateral triangles by joining three together at each vertex we get a tetrahedron; by joining four at each ·vertex an octahedron; five an icosahedron. But suppose we try to build a· polyhedron from equilateral triangles in which some vertices are composed of three faces and some of four, (2) Or better: the surface of a polyhedron. �- J - . or some of four and so~ae of five, etd This possibility is left ·out of account in Euclid's text. And if we try such constructions we shall succeed; that is, contrary to Euclid's explicit assertion there exist rho-regular solids other than the Platonic figures. We describe two such -solids: First, lay two identical tetrahedra face to face. The result is a rhoregular figure of six sides. Two of its vertices consist of a junction of three ~aces, and three of its vertices consist of a junction of four faces. Second, take two equal squares laid together sandwich-wise. Around their common center rotate one with respect to the other 45 degrees (Figure 1). Keeping them parallel, separate them by some con- Figure 1 Figure 2 venient di stance. From each corner of the upper square drop lines to the two nearest corners of the lower square; in this way a belt of eight triangles will be formed around th~ squares (Figure 2). If the distance of separation is suitably chosenlJJ the lines joining the squares will have the same length as the sides of the squares, and the triangles will be equilateral and congruent. Next, on top of the upper square and under the lovrer one erect four-sided pyramids with triangles of the same size (each of the pyramids will be half of an octahedron) ) The resulting figure, having a total of sixteen sides, is rho-regular.{ 4 Two of its vertices consist of a junction of four faces; the rest of five faces. 2. The first remedy - To escape this difficulty, we need to tighten the specification of the class of solids of which the Platonic figures are to be unique members. The easy and obvious way to do this is to add the stipulation that every vertex ·must be composed of a junction of the same number of faces; or, briefly, that the vertices must be homogenous in type. We shall use the ' term "Regular solid 0 to refer to rho-regular figures which have this property. Rho-regular solids with heterogenous (J) .,'Ir-:~ It must be S/ v 2 , where S is the length of the side of the square. (4) It is not obvious (unless one builds a model) that this polyhedron is convex; specifically, one worries about the dihedral angles between the triangles around the: .b elt and the faces of the pyramids top and bottom. However, calculation shows that these are convex junctions; -the angle is about 154 degrees. . . It should be noted that here and at other places throughout this discussion we are assuming the validity of Euclid's Df'. 10, Book SI, which states in effect that a convex polyhedron is uniquely determined by the size and shape (and arrangement) of its faces; this proposition is notoriously provable. �- 4 - . vertices (such as the two described above) will be called "quasi-regular." Df. nr. A Regular solid is a rho-regular solid with homogenous vertices. A quasi-regular solid ;s)a rho-regular solid with heterogenous vertices.~5 It is clear from Euclid's uniqueness proof that the Platonic solids are indeed the only Regular solids. To render the text precise, we should add the property o.f possessing homogenous vertices to the characterization contained in Euclid's uniqueness enunciation. Using his terms, it might read as follows: No other figure, besides the said five figures, can be constructed which is contained by equilateral and equiangular figures equal to one another, and in -.which every solid angle is contained by the same number of planes. It may be argued that this is obviously what Euclid meant. Perhaps. Indeed, I shall argue below that if we are to polish up Euclid's text at all, it should be altered in this way rather than in other equivalent ways which will emerge shortly. But in any case we are here concerned with the logical accuracy of what is written, not with what may or may not have been meant. Furthermore, one often hears it said that the Platonic solids are the only "regular" solids, and that a "regular" solid is (merely) a polyhedron bounded by identical regular polygons: ~ assertion is wrong. J. Other remedies - One of the significant properties of the Platonic solids is that they can all be inscribed in a sphere, as Euclid shows throughout Book XIII. It is also evident that the two quasi-regular solids described above are not spherically symmetrical. This raises the following question: Among rho-regular figures, are the Regular solids the only ones which can be inscribed in a sphere, or not? To answer this question we need to establish two theorems. (A) In any rho-regular solid which can be inscribed in a sphere, the dihedral angles between adjacent faces are ,all equal. Proof: Suppose first that the polyhedron is contained by equilateral triangles. Consider a pair of adjacent faces ABC and ABD (Figure 3). The edges AB, BC, CA, AD, and DB have the ~aine length, say s. Point (5) I have avoided the natural term "semi-regular" solid, as this already has two other meanings. It ·has been used to refer to~ ..polyhedra whose vertices and edges are identical, and whose faces are regular polygons of different types; these are also called Archimedean solids. The term has also been used to indicate polyhedra whose vertices and edges are identical, but whose faces are not regular polygons. �- 50 is the center of' the circi.lmscribed sphere;. herice the lines joining it to the vert.ic·e s at ·A, B, C, and D are all equal; let them have length R. M is the mid-point of AB, and lines MC, MO, and MD are filled in. 0-:-.:\ /,/;/'I ',,.,... ,.1 ,// / II1I I ""'' . . .- C~- . . ~ \ " '. \ " .. I I f-1--- ' b ~~I · ,/' \ I' . '\ ; . ~ . I I ))-~·-:-.- ··7 0 7 . I I ......... ·" -- _,/ \\/ :~/ ; ;V . Figure 3 Since ABC is an equilateral triangle, CM is perpendicular to AB; for the same reason, IM is perpendicular to AB. Since AOB is an isosceles triangle, OM is again perpendicular to ABo Hence CM, OM, and DM lie in one plane (cf. Book XI, 5). (~~xt, we see that the lengths of CM and IM are un~q~ely determined by s. J Also, OM is uniqueiy determined by S and R.~7) Finally, OC and OD have length R, so the sides ~f the triangles OMD and OMC are uniquely functions of S and R, and therefore so are the angles in those triangles. Furthar, the triangles have been shown to lie in one plane, and the angle IMC is a measure of the dihedral angle between the faces (cf. Df. 6, Book XI). Hence that angle(8) is determined by S and R, and since S and R are the same for all faces, all the dihedral angles must be the same. An exactly similar argument will hold if the faces are squares or regular pentagons. The second theorem we need is the following: (B) In any rho-regular figure whose dihedral angles are equal throughout, the vartices are homogenous in type. · Proof: Take any two vertices on the surface of such a figure, and think of them as juxtn.p~sed in such a way that two of their faces are coincident (or ·"superimposed'°) • ·Since the dihedral angles are equal all around both vertices, and the angles in one equal to ~he angles in the other, all the faces inthe one must coincide with all :the faces in the other. Hence there must be the same number o.f faces ·.'in. each.· .. Combining theoDems· -A and B:- we can now assert: (6) . · CM = IM = · S -y:J/2· ·: _, ~ . .·, (7) OM = ( 8) Angle Il-m I '\ l/4R2 - s2/2 • = 2~r·c~s (S/ '{ 12R2 - 3~2) • �6 be the �constructed which is contained by equilateral and equiangular figures egual to one another, and in which every solid angle is contained by the same number of planes. Although this is logically adequate in .conjunction with the proof which follows it, it is not ideally elegant; specifically - we can show that it is partly redundant. For the phrase "equal to one another", which is underscoped above, evidently means that the faces of the polyhedron in question must be mutually congruent. But it happens that if a convex polyhedron is bounded by regular polygons of the same type (all triangles, or all squares, etc.) then it follows that the faces· are of the same size as well (and hence congruent or "equal to one another.") - · To prove this we first establish a lemma: Lemma: In a convex polyhedron there can be no junction. between polygonal edges of unequal lengths. Proof: Suppose there were a polyhedron with adjacent faces F and G having unequal edges meeting along the line AB. In the Figure 4 it is convenient to imagine that the surface of the polyhdron is viewed from the outside, and that F lies in the plane of the paper. Figure 4 Since the entirety of a convex polyhedron must lie· to· ·one side ·of the plane or· any face, the whole of the polyhedron in question will be on or behind the plane of the paper. Consider the vertex · at V. If this were formed .by a junction of just three faces, the third face would have to be . n .the plane formed by _ i the lines AV and VC; but this is also the plane of G. Hence there must be more than three faces meeting at V, and there must be one or more additional lines (i.e., edges formed by the junction of two faces) radiating out from v. One of these lines, call it VD, forms with VC two sides of the face adjacent to G along the line VC. We will call that face H. Next consider the line of intersection between the plane of H and the ~ F. Since that line passes through the �- 8 - point V, it must fall in some direction across the face F.(9) Hence part of F will lie to one side of the plane of H, and part to the other side, which is impo: sible in ·a convex polyhedron. . This establishes the s lemma. Now suppose there exists a convex polyhedron contaiped by regular poly• gons o·f ·the· same type, but of at least. two different sizes. The faces wuld· be ·mutually similar; hence if two differ in size, all of the edges. of one would be shorter or longer than all of the edges of the other. ,Thus somewhere on the surface of such a polyhedron there would have .to be a junction between two·polygonal edges of differing lengths. But the lemma shows that this is impossible. Thus if arry convex polyhedron is bounded by regular polygons of the same type, they must be mutually congruent as well. With this result it ·emerges that the conjunction of properties indicated by Euclid's phrase ."equilateral and equiangular figures equal to one another" is partly redundant; it could be replaced by the weaker phrase "equilateral and equiangular figures having the . same number .o f sides": it would· follow that the figures (i.e., the faces) would be "equal to ... · one another." ·:.: I certainly do not mean to suggest that this refinement ought to be packed into some ideal and ideally-perfect edition of the Elements, for it is not really relevant ·to the spirit and purpose of Euclid's effort in Book XIII. On the other hand, for Euclid and for Greek mathematics generally, the important conception of a "regular" geometrical solid turns on the notion of a convex polyhedron bounded by regular polygons; I am arguing that in the spirit of that approach a "regular solid" is most adequately and economically defined as a convex polyhedron bounded by regular polygons of the same type, with the same number meeting at eech vertex. It follows that there are only five such figures, and that each one has identical faces, equal dihedral angles, and spherical symmetry. (9) Unless it is coincident with AB. But then AB would be the line of intersection between the planes of F and H, and also between the planes of F and G. Then, as the planes of F, G, and H would all meet in the line AB, AB would also be the line of intersection between the . planes G and H. But VC is that line • . �- 9 A MYTH FO!t OUR TIMES Lawrence Feinberg · Not long ago, I had a few conversations with someone who swore me to secrecy concerning them. For reasons which shall become evident, I may not reveal who that ·person ~s·f·. however I have his permission to relate to the public the substance of a particular conversation we had, shortly before we parted. It was· not actually a conversation, for he spoke and I listened without replying to him in any way. Times are such that I can no longer refrain from telling others what he told me, and though hesitant about the veracity of his words and the public's credulity, I here make that conversation known to all, to stand or fall by rrry reader's judgment of it. · An elder of a certain tribe, a~er much deliberation, decided that he and his fellows should end their existence as speedily as possible. Why he reasoned thus, or what his moral promptings were, no one knew; as a wise man amongst his people, his counsels were sought in all important matters, consequently it was not without much concern and fear that the people learned of his grave decision. Now the persons of this tribe were not of the same species as you and I, or if in fact they did bear us some resemblance, it will never be known. For each individual lived, from birth to death, in a box the dimensions of whose. each and every- side was 10 feet by 10 feet. Their way of life mµ~t se$.m a mystery to beings like us -- certainly there could not have been propogation of the species as we understand it. Yet every individual of that tribe was most intimately acquainted with li- e in all its formal f aspects: When movement was heard outside of one's box where no movement had been heard before, this was a sign that life had been generated; likewise the cessation of external sound was construed as the · cessation of life. In this way every one knew that his fellows would regard hinias no longer existing when they failed to perceive movement in the place t~ which he was. ~The knowledge of .one's own death was nonetheless in- · comprehensible, for how could there be cessation of movement without· its being perceived? Language as We know it was unknown to this tribe. Their grammar of ~xpression. consisted of certain knockings and rappings -on the wa.lls of the box. Since. there was no direct confronting of one individual with another like himself, this language could· not be taught in the same way that our language is taught. But only those of the feeblest intelligence failed to learn the language; for the most part i~ was only a~er one was . advanced in years that he was able to grasp how the various knockings · ware used and ·himself. put into practice that ·knowledge of knockings which ' had taken so l .ong to . acquire. The language in its · comprehensible forms of expression was Qhieny· used by the tribal elders, and consequently th~re existed the possibility of knowledge of ·a · state of affairs • . (Let us not . forget this point.) - · -· . · · .The techniqu_ for using .·knockings ·was intricate: e various shade·s of meani_g n ��- 11 ' . - . .was no such thing as outside. that every individual was inside.. But, he argued, this was not a blessing but a cu..t·se, for communication lost all meaning when it was shown that there wq:: nothing that could be communicated. The knock~language was a sham. Thus for the first time the tribe learned of its coll~ctive existence, but it further realized that this existence was being threatened by the very person who had informed everyone. Hatred for that one spread through the entire people -- pandemonium was everywhere. Without exception the tribal members furiously threw themselves against the sides of their boxes, to reach and destroy their betrayer. Boxes, undergoing such violence, collapsed, and their occupants were at once annihilated. The frightened philosopher at last began to stir when he perceived that all motion had stopped -- everyone was dead but he. Thereupon, the philosopher was seized by a fit or tncontrollable laughter, and he then trod over every inch of his dwelli~g •. rapping gently over the smooth surfaces~ '. · · · · · .t ,'.t �AC a are to AC For (X. G be the as E is to F. measures E G make G measures E therefore as P is to F so is But as to as P is to Q so is AB let K and L be the square numbers for the and as K is to L so is the square on AB to the squares on the square on AC is BC are square on , the square on of the square is the of Since Q is the of K is also an even P is the since p Q is also an even since L of the p is an even of are each are a since E and measures E F. Q. * , we an from �- ! 3 Raymond Queneau Si tu t'imagines si tu t'imagines fillette fillette si : ~tu t' imagines xa va xa wa xa va durer toujours la saison des za · la saison des za saison des amours ce que tu te goures fillette fillette ce que tu te goi.ires Si tu crois petite situ crois ah ah. que ton teint de rose ta gaille de gu~e tes mignons biceps tes engles d'email ta cuisse de nymphe et ton pied leger si tu crois petite xa va xa va xa va durer toujours ce que tu te goures fillette fillette ce que tu te goures les beaux iours s 1 en vont les beaux jours de fete soleils et planetes tournent tous en rond mais toi ma petite tu marches tout droit vers sque tu ne vois pas tres sournois s'approchent la ride veloce u " la pesante graisse le menton triple le muscle avachi allons cueille cueille les roses les roses roses de la vie, et que leurs petales soient la mer. 8'tale de taus les bonheurs allons cueille cueille si tu le fais pas ce que tu te goures fillette fillette ce que tu te goures �- ff Translated by Christian Harrison Chinka chuhng, chinka chuhng, chinka chuhng chuhng chuhng Hey sweet thing in that skintight sweater, If you think your lovin 9 s gonna keep gettin' better, If you don't think that time is runnin' out instead, Then baby you been misled, misled, baby you been misled. Hey sweet thing with them high priced hats, Try your painted toenails in these two-bit flats, Take them nylon stockins off your sexy legs, And see how they fit in size 28 pegs, Cause if you don't think that they will fit in time, Then honey you cornmittin' yourself a crime, You been waitin' around for a rich man's bed, But baby you been misled, misled, baby you been misled. Yes sweet baby you are off the beam, Thinkin' party party party is an unendin' theme, Cause the sun and the moon may go round and round, But you are headin' straight for a hole in the ground, And there ain't a single man irt the world can save Your sweet round fanny from the cold cold grave. Chinka chuhng, chinka chuhng, chinka chuhng chuhng chuhng And before you get there, rrr:J good lookin' friend, You gonna rest your lips on a big triple chin, You'll get a mighty lot o' wrinkles in that rosy cheek, And need a big, strong girdle cause your muscles are weak, Now baby you been savin' a sweet, red rose, And its ripe for pickin', says the man with a nose, So baby lets you and me pluck it today, And let the petals fall baby, where they goddam may, Cause if you don't baby, ,yes if you don't, If you keep messin' round with will and lilTOn't, If you keep on waitin• for that rich man's bed, Then baby you been misled, misled, baby you been misled. �- 15 . ;. ROUSSE~U ·i~ltNIHTHE ·~ANGIENTS~ David Lachterman Introduction The central problem of modern political philosophy, political philosophy a~er Machiavelli, is power. As the doctors of the Church wrestled-valiantly with the enigmas of the Trinity trying to elucidate the identity, individuality and province of the Persons. so political philosophy -- secular theology -- must determine the relationships between prince and people, ru1er and ruled, or more generally' part and whole. It is the common . assumption of theorists such as Hobbes, Locke and Rousseau that civil society is artificial, a work of man and not of nature. In passing therefore from his original, a-political state into civil society, the individual sacrifices whatever privileges he may have enjoyed in the state of nature for the sake of something new although not necessarily something better. It becomes the task of political philosophy to understand the new relations that do or ought to arise among men in a political community: relations involving the distribution and legitimacy of authority and power and the ·obligations of the ruled to their rulers. Thus, a complete account of . society · must include a description of man in the state of nature ~- his rights and responsibilities -- and an eXplanation · of what man acquires or loses by deciding or being forced to unite in political societies. Rousseau is chiefl.y concerned with explaining this transition without introducing the exercise of constraint or force on the part of any individual. · His problem is to describe a form of politieal association in which the advantages of the civil order fully recompense the loss of natural independence. However, his intentions go beyond the solution of this particular problem. In several, central aspects of his political thought, Rousseau reveals an indebtedness or, more accurately, a sympathy with the ancients .i .e., Plato and Aristotle. This sympathy requires a break with the tradition of modern political theory laid down - y Machiavelli. b His political construction is in part determined by an understanding of the true end of civil society that few of his contemporaries, especially Hobbes, would . share. At the same time, ·however, he accepts the notion · of consent or contract, the meqhanism invented by the moderns to explain the transition from nature to society. He is .concerned, as, perhaps, Plato and Aristotle were not, not only with What the city ought ·to be but also with how it can come to be. In brief, Rousseau seems to adopt modern means to achieve ancient ends. Perhaps his greatest concession to the modern spirit would be the admission that the ends. justify the meari's. In presenting And interpreting the thought of Rousseau· I shall make use of five categories which represent the fUnqamentai issues that he investigates: the state of nature, the social contract, sove• reignty, ·the general will, and law. Underlying .each is the .idea of liberty; it is in the gradual unfolding o~ the meaning of th~s· idea that the spirit of Rousseau• s thought can best be . grasped. · · �- 16 Section I The .most cursory examination of the Discourse on the Origins of Ineguality and the first book of the Social Contract will undoubtedly reveal what seem to be two fundamentally opposed understandings of the state of nature. In the first treatise we find a panegyric on the nobility of savage man; in the second, an eulogy of social man. In the first, the establishment of society entails the depravity of the human race; in the second, it brings about its ennoblement. If however we attempt to distinguish Rousseau's purpose in each TNriting it may appear that the t"WO understandings correspond to two different intentions. As long as we adhere to Rousseau's advice to "lay facts aside", that is, to refrain from confounding history and philosophy,(1) we should be able to regard each understanding not as a categorical assertion of (historical) fact but rather as an hypothetical construction designed to serve a particular end. In the Discourse the abusive, tyrannical condition of present day political society is given; philosophic inquiry must account for this condition, it must explain the origin of society and its evolution until the present, a present in which "we see around us hardly a creature in civil society who does not lament his existence". The main argument of the Soc1al Contract, on the other hand, does not assume the existence of any politi~al society; it proceeds within the realm of the ought not the i[.(2) Rousseau's initial remarks in the first vhapter of the Social Contract explain the connection between the two TNritings quite clearly: "Man is born free and everywhere he is in chains • • • how did this change .come about? I do not know. What can make it legitimate? I think I can answer this question." The first sentence is historical, that is, it represents as an actual fact the enslavement of man in society.()) It echoes the description, in the Disc.ourse~ of the final degradation of society -- the conversion of the legitimate into arbitrary power or, more briefly, the state of slavery.(4) However, Rousseau does not intend to examine the root s of t his condition , he forsakes the historical method used previously in the Discourse. The last sentence could be misleading if we regarded Rousseau as proposing to render 1egi timate the actual condition of . civil society, the condition of master and slave. Rather, he must be understood as bringing into question civil society itself, apart from any historical instances of it; the form of his question is thus what can make any civil society legitimate, taking man as he is and the laws as they might be? The anstrer proposed in the Social Contract is meant as an ideal basis for the founding of any society. ( 1 ) Cf. esp. Introduction to the Discourse on the Origin 1 ·o f Ineguality (hereafter D:iscourse)(Everymans ed.) p. 175-176; also, end of First Part p. 206. (2) In violation, it should be noted, of Machiavelli's basic methodological restriction, cf. Prince, Ch. X:V. (3) "all ran headlong to their chains as hopes of securing their liberty.'' Discourse. p. 221°. (4) Discourse p. 2)1 �- 17 I have devoted so much space to this distinction because most of the confusion regarding Rousseau's political theory seems to arise from the failure to separate the Discourse and .Social Contract as I have done. Indeed, the preoccupation with the "noble savage", the conviction that Rousseau intended man to return to the idyllic and pristine state of nature(5) must ·have been the result either of a hurri~d and superficial reading of the text ( 6) or of a misapprehension of Rousseau• s purpose in the Discourse: an in. ictment of d the illegitimate e~ercise of political power. Despite the difference in purpose between the two writings, the axioms of the •science of mankind•{?) set forth in the Discourse are for the most part unchanged in the Social Contract. Since Rousseau claims that his political construction is based on the "nature of man" it will be essential to our purpose to uncover these axioms tdthin. the description of the "state of nature". In doing so we are likely to discover that nature in these two phrases has two distinct meanings. Man in the state of nature must be stripped of every d1sposition, passion and talent that he possesses and can only possess in society. It was Hobbe's mistake to attribute to his natural man passions, pride and the fear of violent death for example, which only came about as the result of communal life. Thus hi·s identification of the state of nature :with the. state of war is spurious; man's life prior to any form of association or mutual dependency is "simple, uniform and solitary". All intercourse with other men, even the most casual, is rare .and insignificant. The ·great store of nature supplies each individual's physical needs without demanding to;l Man thus lives only in the consciousness of the present,~8~ an animal like other animals stronger than some, weaker than others. What of the moral and metaphysical side of savage man? Rousseau believes that he has discovered the two fundamental operations of the human soul prior to reason: the desire for self-preservation and compassion at the sight of the pain or death of any sensible being, particularly of any human being. Man in the state of nature is ."destitute of every. species of intelligence". Since "the y.nderstanding is greatly indebted to the passions" and viceversa~ 9) and since the passions originate in our wants of which we mu.st . have some idea or· to which we are· directed instinctively, it follows that savage man's desires are limited to physical wnnt. Man is a creature of pure . sensat~ons, lacking .evenythe most sifilI:?le knowledge. · (5) Cf. for example, Otto Gurbe, Natural Law and the Theory of Society 1500-1800 (Eng. tr. Cambridge, 1950) P• 109 (6) Cf. esp. the concluding paragraphs of the Discourse 1~ere Rousseau -imptites this interpretation for his adve~saries. (?) Prefac_ t .o. Discourse, p. 168 e . . . · .-··· · ·· ·(s) Disc· m.·50 ·~. 187 · ·· · · a ~ This ·re.ciproeity :Hobbes -.see~s tc) have ignored. Cf. :.Leviathan Bk•· ·I, Ch • . J, 6. . -. . · . <9). �- 18 However Rousseau makes . two crucial distinctions between man and the ·other beasts. The beast is directed solely by nature -instinct -- whereas man in his capacity of a free agent can choose between acquieseing and resisting the impulse of nature. Man's liberty, his free will, is natural and his consciousness of this liberty "displays the spirituality of his soul". Secondly, man in contrast to the beast possesses the faculty of self-improvement , of almost unlimited perfectability. It is this faculty that gradually "draws man out of his original state,produces his discoveries and errors,his vices and virtues". Nevertheless, it is important to recognize that in the state of nature, before any inconveniences or deficiencies impel man to improve his faculties,(10) these two distinctive qualities are largely irrelevant. Man's free will is rarely exercised; his perfectability is, in a true sense, pure potentiality. Here we begin to see two possible meanings of t~e word 'nature': pristine, original state and essence or entelechy -- the full actualization of possibility. It is in this latter sense that the nature of man is understood in the Social Contract -- what man ought to become if he is to be a true man. (11) I mentioned previously the. feeling of compassion which Rousseau attributes to man in the. state of nature; this innate sentiment is the spring of all social virtues -- generosity, clemency, humanity, friendship~ Although Rousseau denies that men in the state of nature have any "moral relations or determinate obligations one with another" the natural virtue of compassion i's the one feeling that contributes to the preservation of the species. This virtue is especially interesting bec~use its analogue in civil society is conformity to the law(12J and because the corrupt and debased civil society thrives on the profit every man foresees in the misfortunes of his neighbors. The development of existing seciety entailed the extinguishing of compassion; the success of the rational society requires its preservation in a different form. It is because of this virtue that Rousseau can characterize man as naturally good (although actually wicked) -- a fundamental axiom of the science of mankind. I have tried in the preceding paragraphs to collect the essentiaal descriptions of ·man in the state of nature as given by Rousseau in the Discourse on the .Origin of Ineguality. His own words can ·best summarize what I have been arguing: "Let us conclude then that man in a state of nature, wandering up and do'Wll the forests, without industry, id thout speech, and td thout home, an equal stranger to war and to all ties, neither standing in need of his fellow creatures nor having any desire to hurt them, and perhaps even not distinguishing them one from another; let us conclude •- .. (10) "· •• the fortuitous concurrence of many for·e ign causes." Discourse, p. 205 . ( 11 ) Cf. Social Contract, Bk. :t, Ch. VIII -- "1' instant qui fit un etre intelligent et un homme." ( 12) As . Rousseau· puts it: "it supplies the place . of law; morals and virtues." �- 19 that, being self-sufficient and subject to so few passions, he could have no feelings or knowledge but such as befitted his situation; that he felt only his actual neceqsities, and disregarded everything he did not think himself immediately concerned to notice, and that his understanding made no greater progress than his vanity." One thing more: liberty is the noblest faculty of man. Whereas in the account of the transition from the state of nature to civil society given in the Discourse, man's faculties and natural goodness are progressively debilitated, the Social Contract (Book I, Chapter 8) views the change as an ascent that affects a radical change in the nature of man or, rather, as I have argued previously, fulfills for the first time tha nature of man. "file passage from the state of nature to the civil state produces a very remarkable change in man, by substituting justice for instinct in his conduct and giving his actions the morality they had formally lacked. Then only, when the voice of duty takes the place of physical impulses and right of appetite, does man, who so far had considered only himself, find that he is forced to act on different principles, and to consult his reason before listening to his inclinations. Although, in this state, he deprives himself of some advantages which he got from nature, he gains in return others so great, his faculties are so sti..~ulated and developed, his ideas so extended, his feelings so ennobled, and his whole soul so uplifted, that, did not the abuses of this new condition often degrade him below that which he left, he would be bound to bless continually the happy moment which took him from it forever, and, instead of a stupid and limited animal, made him an intelligent being and a man." Here civil society is represented as the highest good man can achieve and the state of nature, as a positive hindrance to the fulfillment of all that is best in man. At this point the distinction made previously between the Discourse and the Social Contract is valuable:because all existing states have from the beginning been wrongly constituted, it was necessary in the Discourse to commend the state of nature by- way of rebuke;(1J) now that civil society is to be constructed anew and properly, it is possible to regard the state of nature in its true form. The two accounts are connected by the parenthetical qualification regarding the frequent abuses of a ci'vil society that is not founded on legitimate authority. Perhaps the most significant substitution brought about by the transition to civil society is that of. civil and moral liberty for natural liberty. Rouseeau has argued that liberty belongs to -man by nature in one sense, namely, that man in his original condition enjoyed the freedom to follow his own inclination; it will be his task to show how rational. civil society guarantees this natural liberty in another sense -- as the actualization of man's possibiliti.e s as man. (1J) P• 176 This is explicit in the final paragraph of fhe Introduction, �- 20 - Section II Turning now to the construction of a rational civil society,(14) Rousseau attempts to discover the legitimate conventions that support political obligations -- the duty owed by citizens to their rulers. Two terms in the foregoing must be explained. 'Legitimate 0 is opposed by Rousseau to 'arbitrary' and has two complimentary connotations: first, that to which a man can freely acquiesce and second, that which conduces to the good of those whom it serves. A convention is an artifice, a work of man not of nature. However, the conventional in Rousseau is··.htgher than the natural, in the sense of original, becaus-._ it testifi· s e to the perfectability of man, to his ability to escape the· thralldom of instinct. From what has been said -about legitimate it should be clear that force or compulsion can have no part in the institution of the rational political society. A major part of the first book of the Social Contract is consequently a refutation of political theories that set up force as the origin and prop of political obligation. Such theories invoke alleged rights -- the right of the strongest, the right of the conqueror -- which Rousseau shows to be meaningless: a right is based on a reciprocality of interests and responsibilities; where one party is all powerful the other · yields to him not because he ought to obey but because his desire for self-preservation forces him to obey. The •ought' in this analysis will become increasingly important; for the .· moment it ·need only be pointed out that duty is always a moral, never a physical obligation. Hence the exercise of physical power can n~ver legitimize itself; right can never be established by fact.~15J If right, not force, is the legitimate foundation of society, the ·end pf the political community must be adjusted accordingl¥, Under a despot the people are said to enjoy tranquility,l1°J violen~e is curbed and men°s lives are no longer threatened from every side; but, Rousseau asks trenchantly, "What do they gain i f the very tranquility they enjoy is one of their miseries? Tranquility is also found in dungeons, but is that enough to make them desirable places to live?" As long as a people is merely · subject to a ruler, as long, that is, as the interests of ruler and ruled are not the samE}, power will be arbitrary and obedience a physical necessity.l17J What is to be secured in {14) In the Discourse, civil· society is largely a matter of chance,· er. ·p. 223 ( 15) ·An additional instance of the 'idealism' of Rousseau's endeavour, in contradistinction to Machiavelli's. See also Grotus, The Laws of War and Peace, Bk. I, Cn. III, paragraphs VI-VII for the argument of the supremacy of the prince over the people. (16) Compare Hobbes Leviathian, Part II, .Ch. XVII-~ . o.egi.nning (17) ·See the discussion in the Social ·contract, Bk. I." Ch ·~ · II of whether the rulers rule for the sake of their subjects·~ : �- 21 civil society is not merely life, but rather. ~hat freedom which man.enjoys by nature. Alienation of liberty can never be the basis of society, Eobbes notwithstanding. "To renounce liberty is to renounce being a man." Civil society must be a refiection of the essential nature of man (and liberty is the essence of man) or else it will inevitably degrade and debase man until he will look back with regret on his primitive, a-political conditions. Rousseau finds man equipped for nobler ends than mere physical existence; .i ndeed his freedom, which is one with his perfectability, makes it possible for man to become a moral being,in a sense which I shall explain later. Therefore the crucial question for Rousseau is how the existence of the state can be compatible with human freedo~ g~• in other words, how can liberty and obedience be reconciled.~1 J . In the Discourse the gradual diminution . . of nature's abundance and the concoraLtant" . rise of agriculture and industry combine to introduce equality between men: inequality of possessions and resources and thus of power. The state of primitive society(19) became a "horrible state of war". The rich, desiring to secure their lands and wealth, conceived the "profoundest plan that ev~r entered the mind of man": to ally themselves with their enemies in order to use the combined force of their attackers for their own protection. If it was the profoundest, it was also for Russo the most treacherous plan that man has conceived. "All ran headlong to their chains, in hopes of securing their liberty" -- a ~ope soon and irremidiably disappointed. In the Social Contract natural incommodities outweigh the resources of each individual for his self-preservation; man then realize that it is only by combining their individual forces that they can save themselves from perishing. No ment.ion is made of property, wealth, compulsion or the state of war. What form should their association assume? · If any individual is dependent on the will of another individual, hi~ liberty is abridged; if any indi vid.ual retains for himself privileges which his associates. do not share, their liberty is abridged. Each associate must remain as free as before yet with the advantage of h~ving the common force of the whole for his defense. The fulfillment ·o r these two requirements is provided for in the terms of. the .cont. act: "each of us puts his person and all his r power in .COI1lnlon, under the supreme direction of the general will, and, .in ou:r corporate capacity, we :'r eceive each member as an indivisible part of the whole." Although the ·c entral term -- (18) Cf. Social Contract, Bk. III, Ch. XIII (p. 80 in St. John's text): "l' essence 4es .corps politique, etc." · · · ( 19) Not civil ·society in the full sense, ., but ah aggregation of ~amilies living on the same land. �- 22 - the general will -- will only become meaningful in the subsequent discussion,, it's possible to see how this contract eliminates personal dependency: the individual alienates his person and possessions to the corporate whole, the "moral and collective body" created by the act of association and not to any other individual; each individual in turn is accepted as a part of this whole, enjoying its protection and having a voice in its deliberation. The social compact however -- and this is possibly Rousseau's most striking contribution to the political dialogue -- does not establish arry particular regime, any constitution such as a democracy or monarch. The' discussion of government is postponed until the third book. Having united themselves into a public person, the people do not constitute or even elect b¥ this act a government; instead, they create the Sovereign.(20) Section III Rousseau commends the philosophers, especially Hobbes, who realized that to understand the foundation of p9litical society it is necessary to go back to a state of nature.l21) In this sentiment, at least, be is firmly leagued with the moderns against the ancients. Political philosophy, then, must not merely describe how the best or most stable order might be constructed; i~s real task is ta justify the authority exercised. by the ruler. This modification of the fundamental political question is the outcome of the modern conviction that man is not by nature (i.e., originally) a political animal. Rousseau's Sovereign satisfies this requirement ina9much as all who submit to any act of authority are its authors;l22) they will be, in effect, obeying themselves. Rousseau expresses this reciprocality of authority and obedience in the distinctions given in the Social Contract Bk. I, Chapter 6: the active aspect of the body politic is called sovereign (this activity is legislation); t he passive aspect , state . Citizens are those who share i n the sovereign power: subjects, those who are under the law; but, every member of the people is both citizen and subject and hence those who obey the law ,w rite the law.(23) · By justifying political authority in this way. Rousseau can quickly dismiss that ancient question: whether the ruler rules for his own sake or for the sake of the ruled. · If 'the sovereign is a collective body whose members are the citizens, it clearly cannot have interests opposed to theirs, for the same reason that a living body can maintain it( 20) Compare the wording of the covenant in Hobbes, Leviathan Part II, Ch. 18 (p. 143, Liberal Arts text). Especially noteworthy is the notion of rep'r esentation, expressly rejected by Rousseau. ( 21) Discourse, p. 175 · (22) Compare Hobbes' account of authors and actors, Leviathan, Part I, Ch. XVI (23) Cf. Social Contract, Bk. III, Ch. XIII: "the words subject and Sovereign are identical correlatives, the idea of which meet in the single word 'citizen'. �- 23 self only when all the members, or organs are healthy.(24) "Merely because the sovereign exists,(25) it is always what it should be". There should be, in the rational society, some legitimate source of authority; that is, as I have explained above, a· source of authority that it is right for men to obey; but this is what the sovereign is, for it is always rj,:ght for a man to · conform at his own will and thus the proposition, is true. In spite of the perfection of the relation between sovereign and citizen, there is a grave difficulty in establishing an equally successful relation between subject and state. An individual may think it proper for him to enjoy ''the rights of citizenship" without being pre-pared to fulfill "the duties of the subject", in short, he may refuse to obey the authority of the sovereign. Consequently, the body politic must reserve the right to compel a man to obey. In Rousseau's paradoxical formulation this means that a ·man is forced to be free. The full significance of this statement will hopefully become apparent in the discussion of Law (Section V). Before passing on to the general will, I should like to clarify two issues involved in the notion of sovereignty. Sovereignty is a central concept in Hobbes, too; but how great is the difference between its meaning there and in Rousseau! For Hobbes, by the original covenant every man alienates his right to govern himself, that is to obey what his own will . dictates. Instead, a single man or a body of men (or the whole body of the people, an alternative rarely considered) is given the right of acting in the name of each, while each in turn "owns and · acknowledges himself to be the author of whatsoever he that so bears their person shall act or caused to be acted". "He that ·carries this person is called smtereign. t1( 26) In every case the will of the sovereign is what Rousseau~ calls .a particular will, to which each .must submit his own will. In the Social Contract, quite to the contrary, it is to dissolve the body politi·c to promi9e to obey any particular will, that of a monarch, for example.t27) Sovereignty can never be transferred to an individual or a group of individuals for by its very nature it belongs to all the people; indeed, it is the very bond that makes a people, a single corporate being, , out of an aggregation of men. Consequently, government, which is the bearer of sovereign power in Hobbes, becomes in the Social Contract an intermediary between the sovereign and its subjects. · For the determinations O ·f< the sovereign, that is to· say, the will of the people, to be put into action by every individual, some force is necessary; this · · ( 24) Rousseau himself recognizes the partial.. iriaccuracy of this comparison. See Disco~rse ~n .Po.liti?,al E?on?mv p. 252 (Everym~s ed.), for the reservation and comparison in :i,ts fullest fo. m. . . r (25) Translating "par cela seul qu'il ·est''' existentially, . not . qualitatively. (26) Cf. Hobbes Leviathan Part II, Ch. XVIII, P• 142-143 (27) Cf. Social Contract Bk. II, Ch. II �- 24 force, which brings the people as subjects into conformity with its or,m will as citizens, is the government or supreme administration. (28) Its power is executive while the~ of the sovereign is legislative. From this it follows that the only valid or proper acts of the sovereign are laws; whereas the government acts legitimately when it takes steps to secure "mutual correspondence between subjects and sovereign", in other words, when it makes every man free. Under closer scrutiny sovereignty is made to reveal its essence: the general will. It is this that we now have to examine. Section IV The social compact creates a moral body, moral, because it has a will: an act forced upon us, one9 that is, that we do not will to do, can never be mora1.(29) The terms of the compact name this will: it is the general will, to whose authority every associate entrusts his person and his possessions. In what follows I shall be primarily interested in assembling and explicating the various notions Rousseau associates with the general will; in the final section a discussion of the positive acts of the general will (the laws) will introduce an analysis of the fundamental issue in this paper: the similarity between Rousseau• s political ideas and those of the ancients. While the social contract provides a quasi historical, and sovereignty a political, explanation of social obligation, the notion of the general will presents a psychological or philosophical account. Sovereignty is the active aspect of the city; Rousseau illuminates what this means when he writes that the sovereignty "is only the exercise of the general will".(30) The ground of political rights and duties thus becomes not so much something that is explicitly political (as institut.ions, constitutions) as something corresponding to a pschyological faculty. Rousseau in the preface to the Social Contract declares that his purpose is to reconcile right and interest, justice and utility. The immediate or explicit argument behind the notion of the general will does just that; it demonstrates that submission to the conditions one imposes on others is both equitable and in one 9 s .own- ~in~·~re-st. For the general will must be distinguished both from the will of all and the will of the majority. The generality of the general will is fundamentally qualitative, it is defined less by the number of votes then by the nature of (28) This administration takes the collective name of 'Prince' -surely a 'r eference to Machiavelli. ( 29) Social Contract, Bk. I, Ch. III (JO) ibid. Bk. II, Ch. I �- 25 the voter and the object of the vote. The voter is ~ citizen, a member of the sovereign and is therefore concerned with the · "public advantage" rather than with his private interest. (Rousseau wants to show, of course, that public advantage and private interest coincide.) Thus a unanimous decision reached by a body of men each guided by his private interest is not a decision of the general will; nor, a fortiori., is a majority decision under similar circumstances. The g·e neral will considers only · the common good and hence is al ways right -- because it pursues the object for which it was instituted. The question ·p roperly -gut to each citizen is "is it to the advantage of the state?" f31donsequently whenever the issue relates to some particular or determinate object -- an individual, a profession, · a class of subjects -- the will of the people lo·s es its generality, ·f or in that case each citizen is judging. something foreign to himself·; particular interests replace the common interest that unites all. The mathematical metaphor that Rouseeau uses to describe what happens when the general will is being determined is somewhat obscure!J~rom the footnote we gather .that the common interest must make itself felt in ·opposition to various private interests. Two voters with contrary private interests meet together ·in the perception of the .·common interest; thus cancelling their private interests. Every valid act of the sovereign recognizes all citizens .collectively and all actions in the abstract. The carrying out of these acts is- obligatory because each has agreed to fulfill the cenditions he has imposed on others; as long as the will remains general, each man is obeying, in effect, his own will. Now the special effect of this arrangement according to Rousseau is that "in fulfilling them (social obligations) we cannot work for others without working for ourselves"!33Every man by nature prefers himself above all other men; in casting his vote every man thinks of himself -- each man constantly wills the happiness of all because each man thinks of himself when thinking of all. Certainly no man will impose burdensome or injurious conditions on others which he himself is obliged to fulfill. Every sovereign act is thus legitimate because based on the social compact, equitable, because common to all and useful, because it has the general good as its object. Notwithstanding the . absolute power of the body politic over all of its members, the general will cannot impose fetters that are useless or even harmful to the community. · In exchanging the state of nature for the social order man gives up his natural independence for civil liberty, _liberty limited .. ·· (31) (32) ibid. ibid. ( 3).) . ibid. Bk. ,IV, . Ch. .I Bk. II, Ch. II Bk. II, Ch. IV �l - 26 only by the general will. Far from being a renunciation this is an advantageous exchange, "instead of an uncertain and precarious way of living they have got one that is better and more secure, instead of the power to harm others security for themselves," and instead of their strength, which others might overcome, a right which social union makes invincible". I stipulated previously that the foregoing presentation contained the explicit argument of the social contract. The obscurities of that argument along with its utilitarian tone might provoke questions concerning the fundamental character of the society and the people who are to live under the social contract. 1rJhy does the general ~dll refrain from imposing useless fetters on the community? Why will particular 1dlls meet in the recognition of the common interest? Is self interest, the preference each man has for himself, the sole ground of political obligation? If so, is this interest noble or base? Is civil liberty --freedom as Hobbes understands it -- the only fruit of social existence? In short, what is the relationship between the general will or, more precisely the law and the moral character of those subject to it? The very fact that Rousseau attempts to answer these questions, both in the Social Contract and in the Discourse on Political Economv, testifies to his deviation from modern political thinking, or, at least, from the conviction of the true end the city common to the moderns. Whether his answers correspond to the solutions given by the ancients is the subject of the next section. Perhaps I have prejudged the issue by calling Rousseau's political construction a rational civil society. Section V By way of prea,mble I should like to exhibit two passages from Rousseau's writings which demonstrate his awareness of the quarrel he tried to reopen. "As nature has set bounds to the stature of a well-made man, and, outside those limits, makes nothing but giants or dwarfs, similarly, for the constitution of a state to be at its best, it is possible to fix limits that will make it neither t'oo large for good government, nor too small for self maintenance." (Social Contract, Bk. II, Ch. 9) "I shall. suppose myself in the Lyceum of Athens, repeating the lessons of my masters, with Plato and Xer.ecrates for judges and the whole human race for audience." (Discourse, Introduction) The first reminds us of Plato 0 s similar discussion in the Republic; more significantly, it suggests that the context or field of Rousseau 0 s political c..011~t.:ruction is the ancient polis, not the modern Leviathan. (34) The seconcf'advises us that (34) This suggestion is already implied in the demand for periodic assemblies of the whole people and is strengthened by the contents of the 'Dedication to the People of Geneva' at the head of the Discourse. �- ·27 ,.. Rousseau's arguments here and, by extension, elsewhere, are to be judged according to the principle$ of the ancients. In Rousseau as in Plato and· Aristotle virtue and reason are inseparable and, · moreover, necessarily involved in any determination or the best political order. Rousseau adds to these the concept of liberty. for him the essential element of the nature of man. To see how these three ideas are interrelated and how their interrelationship established Rousseau's allegiance to the ancients is the principle task of this section. Any decree of the general will which relates the whole people as sovereign to the whole people as subject -- any decree general both as to subject and ·object -- is a law. The state governed by laws, whether it be a democracy, aristocracy or monarchy is a republic; in 0th.er words, a state in which t.he public things or interests are foremost. Ho-wever Rousseau forsees the possibility of conflict between the general will and the particular wills of the subjects; will compulsion alone bring the subjects into ·· conformity with the decisions thev themselves made as . citizens? More generally, what ·kind of men ought to live in the republic? Government does of course compel obedience in exceptional cases; nevertheless there is reason to suspect that the majority of subjects obeys not out of fear of punishment but from some positive sentiment or .consciousness. We recall that the political equivalent of compassion is obedience to the laws. Obedience can no longer be associated exclusively with self interest; some form of' fellow-feeling some real consciousness of the common interest as common must be taken into account. What kind of men might be expected to have such a consciousness? It would .be instructive to compare Aristotle's description of homonoia (being of the same mind) with Rousseau's General Will, without necessarily trying to locate the roots of .the latter notion. "The citizens are said to have homonoi a ·'trJhen they t 'h ink similar thoughts about what is advantageous to them and choose after .deliberation the same things and carry out the things opined in common. Thus when all the citizens thi9k the._ public offices ( d/~ ,(_a(L ) ought to be elective ( rlfL/ETo( l) there 'is said to be homonoia between them" • . And here some light is shed on the arithmetic metaphor Rousseau uses to explain .the operation of the _ general will: "For it -is not .to be of one mind whenever each thinks the same .thing but when each thinks the same thing in relation to the same thing; for instance when both the demos and the nobles think the best should rule." Unless I am mistaken, . this is ,a·· reasonably clear description of what Rousseau envision.ed in his explanation of the general will. Aristotle goes on to qualify the notion of homonoia: it can �- 28 -' / only exist between good men (£.7Tt £t t(fr'- v) "for they are of one mind with themselves and with one another ••• the desires of such men are constant and do not flow back and forth like the tide; they desire just and advantageous things, which they aim at in common. The base on the other hand cannot be of the same mind .except to _ small degree. "(J.5) a If it is only good m.en who can perceive the common interest and vote for it, surely only good men will conform to their own decisions. Compassion was for Rousseau the sign of man's natural goodness; through obedience to law and what it presupposes -consciousness of the common interest -- that goodness persists in the rational civil society although it has been fully extirpated in the existing civil societies. This conclusion, however, is of little value until we have discovered how Rousseau understood the good man or if, indeed, he admitted his existence or, perhaps, his indispensability. For Plato the account (logos) one gives of the just city corresponds to the account of the just man; for Aristotle ethics and politics are inseparably correlated. It is indeed a constant and distinguishing feature of ancient political thought that the virtue or depravity of a man's character has an essential influence on the character and permanence of any regime. The best ~tate should therefore contain the most virtuous men. The best state, furthermore, is the one in which wisdom -- either living (the philosopher-king) or traditional (the laws) -- rules: the rational state; just as the -best man is the rational man, the man whose actions are directed by reason. Virtue is a kind of knowledge, a mode of activity determined by and in conformity with reason. The terminology of the Social Contract (moral liberty, moral equality), the .-1·daa of the legislator and the entire Discourse on Political Economy (Rousseau's answer to the Prince of Machiavelli) make it abundantly clear that Rousseau ~ concerned with ethics and in much the same way as Plato and Aristotle were. Consequently we might expect Rousseau to understand the good man, the citizen of tae Republic, -as the man whose actions are guided by reason -- his own or that of another. The general will always wills the public good but sometimes it fails to understand what that good is that it is willing. Likewise, the individual · always desires his own good and in acting thinks that he is doing what is good but he too is often deceived.(J6) Both ·are in need of guidance. And here I may allow Rousseau to speak for himself: (35) Aristotle, Neomachean Ethics 1167a26 ff. (My translations) (J6) This argument may be examined in greater detail in Plato's Gorgias 466E-468E �- 29 .. "The individuals ·must be. compelled to bring· their wills into conformity·with their reason: the general .'Will must be taught·· to know what it 1,-.71.lls. If that is done, public enlighte?IrBnt. leads to the union of understanding and will in the .social body; the parts are made to work exactly together, and · the whole is · raised to its highest power. · This makes a legislator necessary." Here we have reached a crucial point: Rousseau accepts the ·modern notion or contract with its denial that man is by nature political, yet he also accepts the ancient requirenient that reason. direct .the. operations of the city-- a situation only brought about by the intervention and supervision of a god-like, sublimely wise man who sets up the laws for a people. Contract alone is insufficient to assure the foundation of a rational' . political society, for the many are not wise or enlightened, they are blind. The legislator -- and Rousseau is thinking of Solon, Lycurgus, and perhaps r;:IJ.µnahl?o~ilius-- at the appropriate time in the history of a people{J?) establishes a system of laws based on a thorough understanding of their passions, capacities and circumstances. The laws he sets up impose continual deprivations, the advantages of which no young people can be expected to understand. Hence he cannot appeal to reason in order to have his laws accepted; reason is the product of his enterprise. Nor does he have any actual authority, being neither sovereign or prince -"his office has nothing in common with human empire". Therefore he · prefers the authorship of the laws to the gods, hoping that the ·greatness of his own soul will certify the "miracle of his mis·s~on"(J8) and in'. .this he '"persuades Nithout convincing". The lawmaker thus creates the social spirit; he in effect recreates · men through the laws bringing it about that "each citizen is nothing and can do nothing without the rest". The city comes to be what it ought to be through wisdom~ · · · .. · · It is presumed that after the departure of the legislator t he people as a whole will be sufficiently enlightened by his precepts and example _to . ~11 its true good therea~er; at the same time each subject will recognize the coincidence of his private interest and ~ the interests of the city. The law becomes the form of rationality in the city,(it)is _ the depository of the (divine) wisdom of the legislator. 39 .(37) . Rousseau i discusses this point in Bk • . 2 1 Ch. 1-0 . of the Social Contract. . (JS) Plato's ·Laws begins. with the· question: "Do you take a god.:· or some man as· the cause of the setting up- of your laws?" ( 39) The secularization. of the Divine becomes complete when Rouse.eau calls the '~voice of the People" the "voice of God". The imminent God of Spinoza's metaphysics -"" the laws of nature -corresponds in Rousseau to the general will of the city. Adequate reasons, I think, for calling political philosophy secular theology, as I did in the Introduction. 1 �- 30 When each subject acquires the habit of obedience to the laws, when each is educated to "the due observance of what is proper",(40) the state becomes solid and lasting for then the law merely "assures, accompanies and rectifies · the natural relations . between men" -relations ·. that are grounded in the practice of virtue. . The word 'nature' has certainly taken on a· new and intriguing meaning which I shall explore in the concluding paragraphs. A manual for rulers, a new and totally revised version of Machiavelli's Prince, must make the rulers aware of the supremacy of law and the necessity of obedience to the law.(41) Such a manual Rousseau wrote under the title A Discourse on Political Economy. In it he makes explicit what was beneath the surface of the Social Contract. The general will is found here both under its own 9ame and under the name of the "Public Reason, Which is the Law''.~42) It is to this Public Reason that the Prince must listen at all times, for government here is legitimate and popular, having for its object the good of the people (which the public reason determines). "The power of the laws depends still more on their own wisdom than on · the severity of their administrators and the public will derives its greatest weight from the reason which has dictated it." Thus the Prince who is under the law and not its master must be its guarantor and must use every means of inspiring the love of it. "The first· law is to respect the law." Thus the government, the legitimate executor of the general Will, can pres·e rve peace and order in the Republic, assure tranquility and respect for law in the state. To what other ends could it properly aspire? Does not the fulfillment of the being of any civil society consist in these very ·.,-things? Certainly, for Hobbes, for those who took the "satires of Machiavelli" seriously. (40) Compare Aristotle, E.N. X, ix 1179b20-1180a15. (41) Rousseau's interpretation of Machiavelli is quite fascinating. I shall quote his most significant remark and· most of the appended footnote: "He (Machiavelli) professed to teach kings; but it was the people he r~ally taught. · His Prince is the book of Republicans." "Machiavelli· was a proper man and a ·good citizen; but, being attached to the court of the Medici, he could not help veiling his love of liberty in the midst of his country's oppression." · Note also .that he calls the writings of Machiavelli "the Satires.'' · · (42) This Public Reason determines the rule of what is just and unjust in the city. We might preo.ipitately suppos.e that justice is relative to each state. · Hence, it is well to recall Rousseau's statement in the Social Contract: "Doubtless there is a universal justice emanating from Reason alone." Rousse~u is perhaps think~ ing of the lex naturalis of the Roman juris consults or even. of Cicero's "Law is transcendent Reason, implanted in nature, commanding what should be done · and forbidding what. should not be done." (Cicero, de Legibus !-,. 6) · �- J1 ·"But if .noth_ng ~ore is done, there will be· in all this i more appearance than reality; for that government which confines itself ·to mere obedience will find difficulty in getting itself obeyed. For if it is g·ood to know how to deal with men as they are, it is much better to make them what there is need that they should be. The most absolute authority is that which penetrates into ·a man's inmost being, and concerns itself no less with his will than with his actions." The virtues of the subje.c t of the Leviathan are extrinsic, they do not involve his inner character, nor do the laws or acts of the Sovereign attend to this inner character. Rousseau stands here quite resolutely with the ancients. "This was the great art of ancient governments, in those distant times when philosophers gave laws to men and made use ·or their authority only to render them wise and ·happy •. ·• . • • .~ ·BUt··,our •mocle:rm .. gove1mments, which imagine they have · done everything when they have raised money, conceive that it is unnecessary and even impossible to go a step further." The government which seeks to have the general will accompli~hed -and this is the true office of any legitimate government -- must bring all the particular wills into conformity 'With it, "in other words, .as virtue is nothing more than this conformity of the particular wills with the general will, establish the .. reign of virtue" -- true virtue, the hapit of acting in accord with what reason declar.e s to· be good. (43) Government, by enforcing the law, teaches men not to be inconsistent with themselves; it teaches men to be free. "It is not only upright men who know how to administer t .h e laws .; but · at bottom only good men know how to obey them. tt Civil liberty is not all the city gives to men for they also acquire in their renunciation of their natural (i.e., original) independence moral liberty ·"which alone makes a man truly master of himself; for the mere impulse of appetite is slavery, while obedience toa law which we prescribe to ourselves is liber t y" .( 44) What could be more in the spirit of Plato and Aristotle? The greatest good of all, which is the true end of legislation, consists in liberty and equality. The former we have described already; the latter is not, according to Rousseau, uniformity of wealth : ancl power, but ·the proper use of these: "power shall never be great enough for violence -- and no citizen shall e. er be v wealthy enough to buy another nor poor enough to be .forced to sell (43) It must be noted, however, .that in most cases this Reason does not belong to the particular wills as such; in p·raising .. Socrates and Cato ._:R.ousseaU: "·say13 "we should be taught by the one (Socrates) and led . by the other; and : thi·s ·a lone is enough to ,determine which to prefer: for no people has ever been made into a nation of philosophers but it is not i~possible to make a people happy." The people of Rousseau's Republic will be "upright and si~le." (44) Social .Contract, Bk. I, Ch. VIII ' ��- J3 CORRECTION TO PAGE 41 COLLEXJIAN, October 1963 Robert Sacks And now the Central Sections Let the same things be set oat as before, but let it no longer be required that angle A'AO be a right angle; and again let it be required to find the relation of the parameter to the lines involved. Now since BN,NC = NQ2 and AA 0 1 AN therefore AA' A'N Or NQ2 BN ,AD : : NA 0 ... AD : NC . ... BN,.AD BN,NC . A'A As in the case of the parabola, if I draw AF making angle ADF equal to angle BAN meeting AA' at F triangle ABN will be congruent to triangle ADF, and thus BN, AD = AN, AF therefore NQ2 AN,AF :: NA 0 A'A Or NQ 2 AN,NA' :: AF 0 AA Thus AF is the parameter. But this formulation is useless because its construction is dependent upon the choice of the point N. Therefore Apollonius draws the line OK parallel to AA', meeting BC at K. And since AD AF : : AN NB and AN NB : : OK KB AD AF : : OK KB We now have, etc. � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The Collegian Description An account of the resource The Collegian began as a student newspaper; became a college community literary publication in 1952 and continued until 1969; became a student newspaper again in 1969; discontinued publication from 1980 until 1989 when it again became a student literary publication. Click on <a title="The Collegian" href="http://digitalarchives.sjc.edu/items/browse?collection=26">Items in The Collegian Collection</a> to view and sort all items in the collection. 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Creator An entity primarily responsible for making the resource Lachterman, David (Editor) Title A name given to the resource The Collegian, November-December 1963 Date A point or period of time associated with an event in the lifecycle of the resource 1963-11 Description An account of the resource Issue of The Collegian. Published in November/December 1963. Identifier An unambiguous reference to the resource within a given context Collegian Nov-Dec 1963 Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Publisher An entity responsible for making the resource available St. John's College Language A language of the resource English Rights Information about rights held in and over the resource St. John's College owns the rights to this publication. Format The file format, physical medium, or dimensions of the resource pdf The Collegian https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/03ade866f1978cd280162c486d1deb57.pdf 2b6f0b837de5491d2817e0dd141b8b0b PDF Text Text CONTENTS Page t . Montaigne: Ecco Homo · •••• • • • • Jeffrey ter Meulen, 64 • • 1 Two Portraits of Augustus • • • • • • • • Richard Freis, •63 • • (University of Cci.l.ifornia, Berkeley) 7 A Note on Ptolemy's Tacit Assumption. • • • John Hetland, •65 • • 8 On the Theatetus • • • • • • .• • • • Neal Weiner, 1 64 • • 10 Three French Poems • • Translated by Christian Harrison, •64- • • Sonnet For Helen A Cassa.ndr.e Ma Boheme (Fantaisie) 24 A Note on the Golden Section • • .,. Sanford Feman, 0 •64 • • 30 Bruce Collier, •65 •• 35 Introduction to the 11th, 12th and 13th Propositions of-Apollonius., ••.• • .Robert Sacks, Tutor • • 36 Humpty Imnpty on ntscourse Un-Matrix • • • • • • • • • • • • R. Thomas Ing, • • • • • • • • • • On the Metaphorical -- Postscript • • 1 67 • • 42 Charles G. Bell, Tutor •• *********** Editor ~ • • • • • • • • • David Lachterman Business Manager • • • • • - Da\rld .RasnratJsen Faculty Ad~sor • • • • • • • • • • Eva Brann 43 �Malcolm has suggested the fol An an ordered L:. chain. Simila Given there is fixed order n tted The appear the order of 1 elements �MONTAIGNE: ECCE HOMO An Essay on Method Jeffrey ter Meulen If I wrote a book called The World as I Found it, I should have to include a report on my body, and should have to say which parts were subo~dinate to my will and which were not, etc., this being a method of isolating the subject or rather of sho~dng that in an important sense .there is . no subject, for it-. alone could not be in · mentioned _ that book. The subject does not belong to the world: rather, it is the. limit of the world -Ludwig Wi ttgensteil'.l In the end one experiences only oneself -Friedrich Nietzsche I In his essay On Presumption, Montaigne announces for the first time that he is making a study of man. He has already denied in an earlier essay the possibility of knowledge, since the inve~tigation of nature had been rendered vain in his eyes tn)that "so · many interpretations disperse the truth and shatter it. n( 1 There was nothing in that inquiry which was of use to man -- there was no truth, no being in it. "• •• there is no existence that is constant, either of our being or of that of objects. And we, our judgment and all mortal things go on flowing and ~olling unceasingly. Thus nothing certain can be established about one thing by another, both the judging and the judged being in continual change and motion."(2) . And so MontaiBne turned his inquiry in toward himself. What is the purpose of these essays? I propose to investigate that question from the vantage point of the pr~fatory note To the Reader. It is Montaigne himsel·f who tells us that ••each particle, each occupation of a man betrays him, and reveals him, just as well as any other. n(J) The writing of these essays is his occupation. He betrays his most intimate self to us. In contrast to other authors, he is the first, he tells us, to communicate_with people by his 'entire being 0 as ¥nchel de Montaigne, not as a grammarian or a poet or jurist. He is himself the matter of his book, he writes in the preface; we would be unr~asonable to spend our time on so frivolous and vain a sub,ject. ~ suspect here we are the victims of his �- 2 ·- Democritean mockery,his gay science. "You can tie up all moral philosophy with a cummon and private life, n he writes in the essay On Repentance a "Each man bears the entire form of man's estate. n{ 4) His 'respect ·for the pubbic' he says, inhibited him from showing himself naked. Yet his announced aim is a private one -- the tb.i os He has made the Socratic precept -- Know Thyself -- his own. And if the world complains that he speaks too much of himself, he would complain that "it does not even think of itself."(5) He thinks "the worst c9gdition of man is when he loses knowledge ·and control of himself."~ ) • ')I( The ld£OS must be kept intact, a.~d because he cannot keep his subject still, he· must keep constant watch, and spy upon it in different circumstances. Our aim is to find ourselves, not to lose ourselves, he says. It is primarily from these considerations that he develops his method. "1-tr history needs to be adapted to the moment; ":-.lae -writes, "I ·take hold of my subjects in different circumstances and aspects. n(7) II I wou)..d like to leap ahead for "The whole)difficulty is as to itself. n~b To speak in these inquiry and, in a sense, to go that it is relevant in dealing experience with Montaigne's. a minute to repeat Kant's statement: how a subject can inwardly intuit terms is to change the level of our beyond Montaigne. But I believe with the subjective to match our own It is to be understood that Montaigne is looking at him-self, whatever is concealed beneath the word. As he himself describe'S it, (in the essay On Renentance), his method is a ceaseless listening to the rhythm of his inner being. His essays vary in tone as do his moods between reticence, self-satisfying ·irony, and an emphatic seriousness. It is an artless method which obeys no preconceived plan, but adapts elastically to the changes of his oivn being. We see here that this strictly experimental method conforms so aptly to his subject. His acknowledged ignorance and indifference to 'things' is part of his method; he seeks in them only himself. He tests his subject by its responses to many things: books, ideas, people, or situations; he illuminates it from every direction; he dances around it. The result is not, however, a mass of unrelated snapshots, but a spontaneous apprehension of the unity of his person emerging from the multiplicity of his observations. The totality of his becoming is his being. He may contradict himself, but the truth he never ·c·o ntradicts. It is the fluctuating subject which he contradicts' 'never the truth of the subject • . Let us try to explicate in more lucid terms, using our own experience. Men, man-made objects, and products are perceived as bodies within the spatio-temporal world, but they differ from rocks and trees in �- 3the expression of a soul. A bed, for instance, is more than a simple body; what it is can only be understood by a reference to men. An individual is not perceived in quite the same way as a rock; there is involved an interpretation of what is perceived which indicates the presence of a soul, the subjective, the 'I' or 'self' of this other thing. This 'self' is understood through the interpretation of simple perceptions, as being 'in' or 'with' what has simply been perceived as a body. But what more can we say about this •self'? What is it? Where is it located? It is evident from the examination of Montaigne's method that the subject is rather elusive. The necessity for his method of irony lies in the nature of the subject • . The self, however, is no-where to be found. Montaigne, _as his method indicates, sees it only in the world of things, the circumstance, the .things of the world. What happens in the world points to something which we might call our self. What we call the activities of self come to represent the 'I'. Every· act betrays and reveals the self, Montaigne said. The pronoun 'I' seems to be a sign of this perception of centeredness. We predicate the actions which we think of, to this subject. In this perception of centeredness, we separate ourselves from the world and think of the center as something apart from the world of which it is the center; the world, after all, is only what we perceive. "Things do not lodge in us in their own form and essense or make their entry into us by their own power and authority," Montaigne writes, "external ooj~cts surrender to our mercy, they dwell in us as we please." (CJ J In the everyday world -- before we begin to philosophize -- none of us is a private self confronting a world of external objects. The mine-ness of my existence does not consist in the fact that there is an I-substance at the center of my circumstance, but that this mine-ness permeates the whole field of my being. ~ being is spread over a field which is the world of its care and concern. This is why Montaigne's method is successful. In a sense, the 'I' or 'self' is our only world. Montaigne made himself the subject of his physics and his metaphysics. Being, however, still escapes us; though we are porn to seek out . truth, "to possess it belongs to a greater power." It is knowledge that Montaigne calls the foundation of being. But with the world in · its cqnstant .f lux, it is impossible to know anything about it. We only have experience, which the "infini t~ confusion of opinions that is seen among philosophers themselves"~10) demonstrates .to us is capable of endless interpretation. Interpretation, rememb~r, is the mode of subjective understanding. hlhat has happened, and what was done is for Montaigne always capable of being understood in a new way. As things are interpreted anew, they become a new reality; temporal existence can never correctly be understood by man. The mask of being, the appearances, is all that we see in the world. Even reason cannot penetrate beyond experience, . into the essence of �III �•. - 5I relate this because it conveys Montaigne• s .a ttitude, one of humorous superiority. His scepticism did not lead to despair, but to freedom. He has broken the chain connecting himself with the outer world. This freedom has given him his privacy -- the of which I spoke before. Self-knowledge or this freedom relieves the individual from a forced obedience to imperative moral command. The individual .must a?sume the responsibility and dignity of becoming a self, and '15,os • Montaigne looks upon self-possession as the highest good. The aim of self-knowledge is to keep the intact. Montaigne avoids anything which might jeopardize his selfpossession. The limits of the individual are dissolved in tragedy, ecstacy, drunkenness, madness, selflessness and excessive self-love. What Nietzche called the Dionysian element is avoided by Montaigne. Conversely to what he considers the highest good -- "the worst condition of rnan is when he loses knowledge and control of himselr,n(13) His essay On Drunkenness is a cautious discussion of this Dionysian urge. It is to Apollo that Montaigne commends himself for the protection of his 1 gay and sociable wisdom. 0 ·"The most beautiful lives to my mind are those that conform to the common humCJl:l pattern with order but without miracle and without eccentricity."~14J This is the closing thought· 6f his last essay, On Experience. 1 Montaigne disparages the seriousness of the world which would like to push ''us out of ourselves and drive us to the marketplace for the benefit of public safety.n(15) The problem of virtue then is to find a w~v "to give myself to others without taking myself from myself.-"~'16) It is this which is the source of his moral philosophy. Integrity is self-possession. A man limits his freedom, which is synonomous with self-possession, when he loses himself to possession by something other. "They want to get out of themselves and escape from man," he says "that is madness: instead of changing into angels, they change into beasts; instead of raising themselv~s, the~ lower themselves. These transcendental humors frighten me.n{17) For Montaigne, if man were to know his nature and live naturally, he would be moderate. ! IV I have come a long way from the .prefatory note To the Reader, which in its simplicity told us only that we were being given a man in his natural form. But the thought which I have just outlined is only an introduction to something which seems to me ver;J important. It is something which may have a connection with what we call Morttaigrie's •scepticism'. His scepticism is a natural doubt in view of the endless debate of philosophers. "What rule rrry life belonged to, I did not learn" he writes "until a~er it was ~ompleted and spent. A(new figure: an unpremeditated and accidental philosopher." 18 J. In his philosophy "we have emancipated ourselves from nature'~ rule to abandon ourselves to the vagabond freedom of our fancies."~19) �P• 220 p p p 1 �Statues of Art New York Dick Freis construes Likeness in these featured the same reveals The subtler difference. The man Eludes us: nor may s be true About this man who wa.s so many, So see this: Grandiose: Man-and-a-half gazers Who crack their necks toward his indifference, And fall back until his focuses. Grace of stone: intell~~-~··-~ moves Nowhere power balanced Down.on whose muscles groan Beneath of Rome; intell.~~··~~ Moves nowhere; nor does blood beat toward the heart. Arms are in stilted pose: arm Lucretian skies, slate and God ... hollow; left rests toward the hollow eros tacked on at his feet is Amor Hard ; other swords rise the blind death His breas tells it all center, Rome victorious , and Dmm and Deu; Earth rests at his side ride Sun and Moon; The universe: all circled round Beautiful; but one is comfortable: The senseless comets of Vesuvius Fall seem as human. Then see this: An older face -asides Fires of ambition The brows Necessi Now thin The chin relaxes into Here and mercy live: and therefore steel To hold the Roman peace ceaseless war; But no more the fiercer battle flares, himself a man from The unknown �- 8 - NOTES ON ·PTOLEMY 9 S TACIT·ASSUMPTION John Hetland On page 16 of the sophomore Ptolemy manual it is stated that, for any outer planet, the angular movement in longitude plus the angular movement on the epicycle is equal to the angular movement of the mean sun. From this fact, three consequences follow, provided a certain assumption is made. It is with this assumption, and the proofs of the consequences, that I ~tlsh to deal. The assumption, as stated in the manual, is that at ~ time the planet, the mean planet, the mean sun, the apogee of the eccentric, and the earth are all in one line. However, r;the apogee of the eccentric" should" be struck from this list, as will be shown. The importance of this seemingly trivial correction will be seen when the strange coincidence is accounted for by Copernicus. For while it is necessary, in the heliocentric theory, that the. planet, the mean sun, the earth, and what will correspond to the mean planet, be all on a straight line at some time, it by no means follows that what will correspond to the apogee of the eccentric be on this line. Thus the coincidence would still be unaccountable under the heliocentric theory, were it not for this change. Although the demonstration becomes more complex, the three consequences still follow, as is shown below. In the diagram I have drawn the earth E, the mean sun S1, the mean planet M1, and the planet P1 all on the -same line, but the apogee of the eccentric not on this line. I have shown these points at any later point in time as E, Sz , M 2• and P2 • I have also drawn the line through the equant Q and the mean planet M1 to the mean apogee F and called the angle in the epicycle between the mean and true apogees angle b. I wish to prove the first consequence given in the manual, that the line joining the earth and the mean sun is always parallel to the line joining the planet and the mean planet; that .is, that ES 2 is parallel to M P2 • _ 2 As the epicycle moves from M to M 2w:lth speed 1, the planet moves 1 from P1 to P2 , that is from G to P2 (for. the planet is moving with respect to tne line through the equant and the mean planet,* and it has moved through the angle b before crossing this line), with speed a, and the mean sun moves from s 1 to s 2 with speed .i + a. I have drawn EJ parallel to ' QH; therefore angle M EJ equals angle 1 M:1NM2 • But angle I11NM2 is the exterior angle of triangle MQN, and tnerefore it equals L :f. b. Therefore angle M EJ equals L 1+ b. 1 Now angle s1Es 2 equals ~t+ a. Subtracting from it angle M EJ equal 1 to £ + b, we have as the remainder angle JES 2 equal to a - b. But ·in the epicycle, angle HM2P2also equals a - b, and EJ is parallel to QH. Therefore ES 2 is also parallel to M P2 , which was to be proved. 2 The consequences involving opposition and conjunction now follow directly from what I have proven. * See Ptolemy, ~x.6, para. 4. �- 9 - . I ; r. A briefer and less rigorous .demonstration is as follows: Since QM rotates with spe,ed J2. with respect to any stationary point, and MP rotates in the same direction as QM with speed a with respect to any stationa~; poirit, ·av . therefore MP rotates with speed ,R t.Jwith respect to any stationary point. But ES also rotates with speed + a with respect to any stationary point, and Es is at some time in a line with MP. Ther efore ES . is always parallel to MP. .e �..... is �- 11 - . trying to use that rough whole it will be found that it requires modification and refinement. We then arrive at a dialectical interplay between whole and part-between theory and fact, which moves towards the completely unified goal. This paper is only the first timorous step in that direction. ******** *** Let us begin this examination by looking at the character Theodorus. Theodorus, a mathematician from Cyrene, reveals in his very first words that characteristic of his which will most concern us. Socrates has asked him to tell of young men of Athens who are devotl.Q.g themselves to mathematics "or any other fonn of philosophy".{lJ Theodorus replies: "Truly Socrates, it is well worth while for me to talk to you about a splendid young fellow, one of your fellow citizens, whom I have met. Now if he were handsome I should be very much afraid to speak lest someone should think I were in love with him."(144) What is revealed here is a high sense of respectability in Theodorus. Al though what he had to say may have been perfectly true and merited by Theatetus 0 s conduct, he would not have said it if it were to involve him in the shameful act of chasing a beautiful young boy, even if there were no more truth to the involvment than idle gossip. Let us compare this to a passage from the great myth of the Pheadrus:(2) "Now when the charioteer beholds the love-inspiring vision and his whole soul is wanned by the sight and by the ticklings and prickings of yearning, the horse that is obedient to the charioteer, constrained as always by modesty, control9 bimself and does not leap upon the beloved."(254A)~3J It would seem that both the modesty imaged here and that brand of respectability evidenced by Theodorus are the offspring of t he . same shame. In his first words Theodorus identifies himself with ·the white horse, and hence with that sort of respectability that is opposed to eros. But perhaps this is because he sees nothing beautiful in Theatetus' s face. · This is not so. Throughout the first third of the dialogue, Socrates endeavors to get Theodorus into the conversation. At one point in his attempts he compares the conversation at hand to a Spartan wrestling school. Theodorus is accused of sitting in the bleachers watching the naked wrestlers and refusing to strip himself. He replies: "Why not, if I could persuade them to let me do so?"(162B) What a preposterous perspective he has on the whole affair. He, an old man, most likely with an aging body, i ·s going to try to persuade the beautiful bodied youths not to drag him into the ring. Now clearly the wrestling itself is symbolic of the sexual act. How then, can Theodorus sit there �2 • • were �- 13 - . the cave, it is Socrates, the philosopher, who is doing the pushing. Hence, the "philosopher" described here is repudiated by Socrates in his own actions, and certainly Socrates is a true philosopher. Then who is- this "philosopher"? Thales, the prime example., was an astronomer who fell into the ditch ·while studying the stars. According to Pindar the "philosopher•• is . "above the sky, measuring the stars (astronomy) and the surface of the earth ( 7:G.Lv~tr1:>&;'ryci'-) (175E) • These are the things which Theodorus is said to be able to teach Theatetus. Also, early in the dialogue it is mentioned that Socrates knew Theatetus's father (of his own generation and hence his neighbor) while Theodorus did not, but ·knew Theatetus. Again, at the end of the digression it is curiously asserted by Socrates that the "philosopher" knows "how to drape his coat like a freeman". This· is certainly out of place with the rest of the description; why should one above the sky care about his cloak? There are two reasons. First, the cloak represents clothing in. general and by contrast, nakedness. Secondly, the proper draping of the cloak evidences the social respectability of the wearer through indicating his social position. So again we find the image of nakedness occurring in conjunction with respectability. Again we must point our fingers at Theodorus as this "philosopher". But the quotation from Pindar points to him in his capacity as mathematician. It is as mathematician that Theodorus shares the activity of the "philosopher". The relationship of eros to knowledge here begins to reveal itself -can we now find some relation between these two images: -mathematics and nakedness? At an earlier point in the dialogue (165A), when Theodorus is begging off from entering the conversation, he offers the following excuse: I , . ' "As for me, I turned too soon from tµ 1.. A c:.J v /\ ~ ;rov to geometry." . a Now ? l ,..\ c:.: .S -J/ in its primary sense means naked. In itself this is nothing new; Theodorus is once again fleeing the erotic. But in this case, the· nakedness is directly opposed to geometry, or in general -- mathematics. We now have two reasons for associating his lack of eros with mathematics, the one just given and the fact that it was Theodorus as mathematician who was the "philosopher" of the digression. There remains one more incident with which to weld the link. As stated before, Theodorus spends the first third of the dialogue avoiding involvment, but at 169 he finally enters. Why? Immediately after Theatetus defines knowledge as perception and Socrates tells the Protagorean _p- J{;, .s , Theodorus asks if all this is wrong after all. Socrates, forgetting Theatetus, addresses bis next question to Theodorus. He begins by pointing. out that if this is true, then one man cannot be any wiser than another man (160A) or even another animal, but to illustrate his point Socrates relegates himself and Protagorus to the dogs. At this point Theodorus admits that Protagoras is wrong, but since they were friends Theodorus will not contribute to Protagoras 0 s defeat. Socrates then makes a speech, in the name of Protagoras, saving the f'- :,.;"'6os from his own criticism! �- 14 - Next Socrates refutes it again by means of memory and again tries to involve Theodorus -- . again he fails. So, Socrates makes another speech in Protagoras's name, reinstating the ~:()()o:, • In that speech Protagoras demands that he have a man, not a child. to defend him. Again Theodorus refuses, but here a new twist occurs. Socrates again brings up the question of ·the equality of men's wisdom as a result of the tu-v'6o , but this time he illustrates it differently 5 "But come my good .man, follow the discussion just until we decide whether you must be the measure in respect to diagrams, or whether all men are as sufficient unto themselves as you are in astronomy and the other sciences in which you are alleged to be superior."(~) 'Fheodorus yields. He will enter the discussion long enough to answer this and only this question. Clearly it was for this purpose that Socrates kept defending the µJ{J os after he had refuted it. Only if Theodorus's position of r~spectability and security which he had as one knowledgeable in mathematics were threatened would he ever enter the argument. Hence, Socrates could not allow the ;-t:Jt5)o s to be defeated until he entered. So then, Theodorus comes in in order to save his mathematics from the chatter of the many and chaos of the fl.ux. As long as his mathematics was secure from assault he would not enter, but realizing the threat posed by Protagoras, he condescends to the "act of love", Socratic dialectic, in order to save it. The chain is complete, and his main characteristic, lack of eros, is bound to his mathematics. We must endeavor to come to(sQme understanding of this discovery. A passage from the Symposium 5J will be helpful here: "The truth is this: no god seeks after wisdom or desires to become wise -- for wise he is already: nor does anyone else seek a~er wisdom if he is wise already." (203) The most curious thing about this entire dialogue is that two mathematicians (men of science-scientia-episteme ) are unable to define knowledge. !bes not the above quotation tell us why? it is a part of Diotima 9 s explanation of love to Socrates, the conclusion of which will be that eros is a lack. It is an emptiness and a seeking a~er what one lacks, and hence it is the lover and not the beloved who embodies eros. So then, since we confirmed the connection between Theodorus's lack of eros with regards to both sexual and intellectual objects, would it not then seem reasonable to say that, for the same reason that the wise seek not wisdom, he seeks not knowledge? This is not to say that he has knowledge in the full meaning of the word , but he thinks he does. We said that Theodorus, as mathematician, was the "philosopher" of the digression, hence it is as mathematician that he should have knowledge. It was only when the Protagorean doctrine of universal flux threatened his status as mathematician that he entered into the conversation. Why, in non-dramatic terms, is this? If there is one thing we know �- 15 - . about mathematical works, it is that they stand still. The words, the theorems, the whole system stands in its beauty to be understood, appreciated, and remembered. It is in virtue of this stasis that mathematics is here threatened by the flux. When he describes the Ephesian followers of Heraclitus, Theodorus rails against them because they let nothing stand still, and says we should not act as they do, but we should treat the problem mathematically (181) (as was previously demanded by his friend, the sophist Protagoras). The mathematical course fails, but the interesting thing is that Socrates makes a left-handed defense of the Ephesians by saying that they teach their real doctrines in private and only to their friends who are explicitly not Theodorus but implicitly Socrates. As further evidence for this, we noted earlier that Socrates assumed the guise of Protagoras in his defense of the j)- v&l .s • It must be because in some way they are alike. Now if we look at the dialectic as presented in the dialogues we see that it never stands still. It is a constant search, never affirmtg~ but always denying and refUting while moving towards the truth.l J Just as Socrates' diamon only says "no", so works the dialectic, only refuting but never affirming. And yet we have Socrates's own words at 15JC: "And what of the soul? I:bes it not acquire information and is it not preserved and made better through practice and learning which are motions?" Because of the identification of Socrates and Protagoras, we cannot take this as simple irony. Is it not true that he who learns ler~rns about that of which he is ignorant? i.e., that which he lacks (the question of knowing and not knowing will be taken up presently). The analogy between loving and l ·e arning is complete. Both the lover and the learner must lack. But just as Theodorus is unnaturally continent, so has his mathematics staais given him a false sense of knowing (stasis). Thus we see why lack of eros was picked as his characteristic. The only time he expresses any interest, eros, is in order to hear about Parmenides who denied the existence of motion, which negates dialectic and hence is a negative sort of eros. We shall return to mathematics later and a more detailed examination of its relation to knowledge will be taken up then, but now we must turn to Theatetus. Since we began with Theodorus's first words, let us try to be consistent and do the same with Theatetus. Socrates asks him whether we should inquire about Theodorus 9 s qualification for saying that he and Socrates look alike. His reply is "We should enquire" ( 145E). Whatever we say about him, we cannot put him in the same class of non-erotic beings in which we put his teacher, for by his reply he seems the exact opposite. VJhereas in the case of Theodorus there may have been some doubt about eros's being his defining characteristic, there can be none in the case of Theatetus -- it is timidity. Throughout the entire dialogue Socrates is constantly trying to arouse courage in him, and he sinoerely wants to be courageous (187B). The whole midwife tale is �- 16 - . told for the purpose of inducing Theatetus to have confidence in his own abilities through belief in the devine authority of Socrates as an infallible destroyer of bad children and deliverer of good ones (151D), When Theodorus disdains being made equal to the dogs, Theatetus is too modest to accept equality with the gods (161c-162A and 162C respectively). To list all or even more of these examples is needless, it is the exceptions and ·variations that will interest us here. As we went to th~ Symposium before, let us now take a passage from the Laches.(7J "Then the answer which you have just given, Nicias, includes only a third part of courage, but our question extended to the whole of courage, and according to your view ••• courage • . •• is knowledge of nearly every good and evil • • • ". ( 199C) However playful we may take this to be, we must recognize the problem contained within this passage. Action in ignorance cannot be called courageous, but only foolhardy. Our old saw about fools and wise men does not mean that the latter are cowards, but that unless one acts with knowledge of the circumstances under which one is acting, no matter what the situation,- he cannot be courageous. It is easy to walk off a cliff into a net five hundred feet below, if I do not indeed know that I am at the edge of a cliff. And yet, if I am in ignorance of the situation, but nevertheless realize that there is danger ahead, I must fear. There ·may b.e fear without courage, but not courage without knowledge. Now Theatetus does fear, but is he ignorant? Clearly the answer is no, at least no more than he should be. He is, a~er all, a young boy and should not be expected to hold forth on great matters. But more than that, he is extremely capable. Several times throughout the dialogue he exhibits his ability to deduce from given pre~~es by seeing the conclusions that will follow from what he says.~ 0 J Also, in the example of correct definition at the beginning of the dialogue, he exhausted an infinite group of numbers into a general theorem, while all Theodorus was evidently able to do was to give a few particular examples by drawing them. In addition to all this innate capacity, he has heard all the prevalent theories of the day concerning the subject of knowledge and, although they may ultimately hinder him from reaching his own understanding of the questions, he is not only talented, but "learned". How then, if what we said before is true, can he lack courage? The point is, I think, that however good he may be, he nevertheless thinks himself ignorant. In all honesty he admits that he did not understand the Protagorean doctrine, and the third definition he offers is one he had. heard, barely remembered, and certainly did not understand. The only definitions of knowledge that are his own are that it is many things and that it is right opinion. The first of these was quickly shown to him to be worthless, and the second is self-defeating, and he only mutters it half-heartedly. Before offering the last definition he says that he is dry and has no more �- 17 - . to offer (148E), even a~er Socrates has told him that he is pregnant. Unlike his teacher, he does not lack eros, for we are told that he wondered about this question even before he met Socrates, and Socrates quickly fills him with more (155C), but his eros stems from his thinking himself totally ignorant. It is for this reason that he is too modest to approach these grand questions -- he feels himself inadequate and unworthy of them. Can a connection be made between his timidity and his position as mathematician? A~er Theatetus has given the first mathematicarexample, the dialogue.sgoes as follows: Soc. Most excellent, my boys, I think Theodorus will not be found liable to action for false witness. Theat. But really Socrates, I cannot answer the question of yours about knowledge, as we answered the question about length and square roots. And yet you seem to want something of that kind. So Theodorus appears to be a false witness after all. Soc. Nonsense! If he were praising your running and said he had never met a young man who was so good a runner, and then you were beaten in a race by a full grown man who held the record, do you think his praise would be any less truthful? ( 148B-D) Theatetus, although he was able to find the unity in the manyness of these mathematical objects, fears that he cannot apply the same method to non-mathematical objects. Socrates does not deny the distinction, ~ut he adds that the study of mathematical objects is a prelude and training ground for the more important and ultimate objects of knowledge. And this is the crux of the matter; Theatetus is aware of a special nature belonging to objects of his inquiry, whatever it in fact be, and hence is fearful of his ability in other fields and therefore thinks himself ignorant and incapable. Socrates affirms a similarity between the two fields- but Theatetus does not understand it. , So both Theodorus and Theatetus feel the limitedness of their positions as mathematicians, but in the former case this feeling resulted in complacent acceptance ·of it, and in the latter it resulted in a desire to go beyond but also a fear of treading outside of the mathematical limits. Thus we perceive the ·similarity and difference of their characters. The · study of mathematics has given to Theodorus an attitude of knowing, but to Theatetus an attitude of not-knowing or ignorance. Now in a dialogue about knowing we W"Ould expect this dramatic antithesis to reveal something about the nature of knowledge. If we look at the center of the dialogue (9) we find this same antithesis expressed in other terms. At 181 the Pa:mnernoean philosophy· is opposed to that - f the ~hesians. Wa cannot go into his doctrine in any o detail, but it is clear that they are opposed to Protagoras in respect to motion and rest. the former maintaining universal flux, the latter universal stasis. The former of these presented itself as a challenge �- 18 - to the possibility of knowledge since nothing could be know because nothing was, all was becoming. If then, we posit complete stasis, there should result complete knowledge, for if non-being does not exist how can there arise the possibility of error? Now the possibility of error is perhaps the main theme of this dialogue. Each of the three definitions as they are expounded reflects upon this question. In the second definition, it is found that if we assume that a thing is either known or not known, there is in no way the possibility of error. There will result either complete knowledge or utter ignorance of the object in question. At the end of the third definition of error, interchanged opinions, this assumption is rescinded, but nevertheless we have not seen the end of it. At the end of the aviary analogy (which began as a distinction between having and possessing as a way to solve the problem of knowing or not knowing) Socrates comes to the following conclusions: Soc. Then a~er our long wanderings we have come round again to our first difficulty. For the real reasoner will laugh and say 'Most excellent sirs, does a man who has both knowledge and ignorance think that one of them which he knows is another which he knows? Or knowing neither of them he is of the opinion that one which he does not know is another which he does not know? • • • Or will you go on and tell me that there are kinds of knowledge of the kinds of knowledge • • • and thus lead us in an endless circle? (200B-C) The assumption that we either know or do not know is clearly then still with us. A~er this, the search for false opinion is dropped and a new definition of knowledge is advanced, that it is right opinion plus . .\ 6 - o.s • The question immediately becomes what ;s r ,.\,J ~f'OS • Now in the first of the three definitions of A6ff 0 ':3' offered by Socrates it is called "making one's own thought ( fu( volv<) clear through talking by means of verbs and nouns, imaging the opinion in the stream that flows from the lips as in a mirror or water" ( 206D). This is probably the true one of the three (Socrates has said (206C) that one of the three is true) since previously ( 190C) ~ t.~ vvc...d.. was defined as silent ...\D'/[.;:iS • But why is it then rejected? It is rejected because under those circumstances there would be "no possibility of right opinion apart from knowledge". Hence we are driven back to the possibility of false opinion and thence to the problem of knowing and not-knowing. We see then that his question of knowing and not knowing casts its shadows over the whole of the last two definitions of knowledge and prevents any conclusion from being attained. Just as this question is the prevailing one in the arguments, so is it reflected in the dramatic characterizations of the two interlocutors, Theodorus representing, as we have seen, knowledge; Theatetus, ignorance. This then provides a means for viewing the sequence of the arguments through the entire dialogue. As we said, the bulk of the time given to the consideration of knowledge as perception is given to Theodorus or is at least for his sake, while once knowledge is �- 19 - de fined a-s right opinion he never speaks again. Taking that point as a natural dividing place, we note the following: in the first part the definition is viewed as reducing all opinion to equally true sensations, but it is always true to someone, never in itself. Hence the threat is that there be no knowledge at all, for truth must be simply true. However, .in the second part, the assertion to be denied is, as we have seen in both its definitions, that there is no false opinion, or, in other words, that all is knowledge. In accordance with the principle that one must either know or notknow, the dialogue is then divided into two parts the first of which corresponds to not-knowing and the second, to all-knowing. As we said, Theodorus is the chief interlocutor of the first part, and this ·is because he must be shaken from his knowing mathematical security, and so he is by the Protagorean (.).$ • On the other hand Theatetus thinks himself ignoran • and he therefore is in the half where the possibility of ignorance is quest_oned. Finally, at the i end of the dialogue (205A) he gains courage, and this is because the atmosphere of the whole section suggests that none can be ignorant. c· (.) V But if the mathematicians are at the opposite ends of the stick, where is Sdcrates the philosopher? Let us return to the same passage from the Symposium. "Then who are the philosophers, Diotima, said I, if those who seek after wisdom are neither the wise nor the ignorant? That's clear enough even to a child, she answered, they are the ones between these two, as love is. You see, wisdom is one of the most beautiful things, and love is of the beautiful, so love must be a philosopher, and so being he must be between wise and ignorant.ff(203) So then, the philosopher, as the truly erotic man, stands necessarily between knowing and not-knowing. \rJhat about courage? Soc. • • . ~~ You seem not to remember that our whole search from the beginning has been for knowledge because we did not know it. .•· : . • r~ • • Theat.Oh yes, I remember. Soc. Then is it not shameless to proclaim what it is to know, when we are ignorant of knowledge? But really, Theatetus, our talk has been badly tainted with unciearness all along; for we have said over and over again 'we know' and 'we do not know' and 'we have knowledge' and •we have not knowledge', as if we could understand each other while we were ignorant of knowledge; and at this ve-ry moment have used the terms 9 be ignorant• and 0 understand 0 as if we had any right to use them if we are deprived of knowledge. Theat.But how will you .converse, Socrates, if you refrain from these words? Soc. •I Not at all being the man (;;(v -17f) I am • • • • (196D-197A) �- 20 :JI . ~1 ~/ f) . The use of the word ~If 1f ( "1'· n • ~ v o;> ~-s) for man inst~ad of a<ll /)-i..urro~ indicates Socrates as the possessor of courage ( ~v6 f ~ L-°' )• So then, Socrates, as the courageous man, also stands between knowing and not-knowing. In both cases he is the mean between the extremes, Theodorus and Theatetus. This whole affair is summed up in his well known epigram oil>~ al,,1< o~~~ • (" o,-'- But does this dramatic characterization reveal, as before, something about the arguments as a whole? !bes the addition of Socrates the philosopher tell us something more about knowledge, something that could not have been told by Theodorus and Theatetus? If we return once more to the early part of the dialogue (145E) we find Socrates and Theatetus engaged in the following conversation: Soc. And does this differ at all from knowledge? Theat. Ibes what differ? Soc. Wisdom. Or are not all people equally wise in that of which they have knowledge? Theat. Of course. Soc. Then knowledge and wisdom are one? Theat. Yes. Soc. Well it is just this that I am in doubt about and cannot fUlly grasp by my own effort -- what knowledge really is. 1J'e tend to forget that the purpose of the whole dialogue is here . r clearly shown to that of distinguishing knowledge from wisdom. The word "wisdom" occurs many times throughout the rest of the dialogue, but no direct mention is ever made of the question again. Now I would like here to quote Aristotle on the subject of wisdom.(10) "Such and so many are our notions, then, which we have about wisdom and the wise. Now of these characteristics that of knowing all things must belong to him who has in the highest degree, universal knowledge, for he knows, in a sense, all things which fall under the universal. Now these things, the universals, are the hardest for men to know, for they are the farthest from the senses. And the most exact of the sciences are those that deal with first principles." (Metaphysics Bk. 1, Ch. 1, 982a 20-25) How does this bear on this dialogue? To see this we must look for a moment at the divided line in the Republic. Th~ mathematician is the pictured in the section whose faculty is <5 '-r;( v uL.d-, • He uses the objects of perception in his sc~ences, but not as things in themselves, but as images of other things about which he is truly concerned. However, there are principles which underly his own �- 21 - . science -- his suppositions which he does not consider as objects of inquiry, but which he uses, assu.ming them to· be understood. These suppositions are the first principles of Aristotle's introduction to metap~sics just quoted. So when Theatetus is afra~Ei to examine one of these, viz., knowledge and all that it involves,t11) he reaffirms the Republic's image of the mathematician. Mathematics ( a<..:_ Voul(.) is, as we have seen before, the subject of this- dialogue~ Let us take one more look at the whole in the light of the divided line. First we must recall the; sequence of the divided line. The first division has as its object of cognition images ·or perceived things; the second, the perceived things themselves'; the third, the perceived things as images of more intelligible entities. But · the fourt and highest, the realm of highest knowledge and hence wisdom, has as its objects of inquiry those principles which underly the &1_r/ vo<-cA of the third section. Now let us look at the sequence of the three definitions of knowledge in the Theatetus. The first presents perceived objects as the results of unseen motions which are the "realities", as images of them. The second, on the whole, deals with perceived obiects, ~aken simply as such, and their identification or being mistaken.~12) The third deals with the logos and its nature, and, as we have said before, the A.6,ro s is a reflection of £ <-~ 11 OL°' , the mathematical mode of coghition in the Republic. These two sequences parallel each other exactly! It is all there .but for the absence of an analogue to the fourth · part of the divided line. But the whol~ dialogue is climbing up, the divided line by means of the .,16 S P1 the dialectic itself. It therefore must stop at the third part because the interlocutors are mathematicians and cannot go beyond dt.o("v'o1..o<. to the realm of the eye of the mind (which in a curious way leads us back to perception as knowledge).(13) Just as this dramatic sequence must here stop, so are we unable to reach any conclusions about knowledge until we examine· the various objects of knowledge.-0 which requir~. s a study of being as such. The very statement of ot~ .1\ 0n_ o ~ K. 0 ~~ ·)!., implies ontological necessities which stagger the imagination.(14) Also then, just as the dramatic action is carried into the Sophist, so does the subject of the Sophist, being-non-being, same-other, motion-rest, correspond to the fourth part of the divided line. And so, because we must identify dialectic with A 5 0 H and ~' 1.. . ~ "''.) 1-')(, in the realm of discursive reasoning, Socrates will sit silently in that dialogue, yielding to the god-like Eleatic stranger. ro 1 1 And yet there is still a greater reason for all this. At the end of the dialogue Socrates goes to the porch of the king "to answer the suit· which Meletus has brought against me." Throught this entire dialogue he has known that he is about to die. What will becom of e philosophy when he is dead? hlhat will become of Athens without its greatest politician? In his first words Socrates demands to be introduced only to a promising Athenian youth. When he looks with his old eyes on the youthful Theatetus and perceives his own form proportionately scaled down, he finds hope in Theatetus's possibly being the one to replace him as the gadfly of Athens. But Theatetus must be saved from Theodorus and mathematics before he becomes as hopelessly dead as his teacher. In death Socrates takes the role of midwife to give life to �o. �- 23 - . Aristotle to interpret Plato, but I feel justified in this particular case. There seems to be no reason for this King first of all that Aristotle disagreed with his teacher on this point. Secondly, it is a common enough notion of wisdom for us to have arrived at it on ouw ovm. Thirdly, the fourth part of the divided line should correspond to wisdom as it is clearly the highest knowledge. 11 • Namely the principles enumerated in the lists of 18~187 plus motion and rest which p~rvade the whole dialogue. 12. It is only in the aviary analogy that we begin· to deal with what may be called "pure noetic entities". In the wax tablet attempt they are dismissed as being somewhere else above the tablet, and are not mentioned again. This is hinted at at the end of the aviary affair where the result is that is must be discarded for it would require kinds of knowing of the kinds of knowing. The aviary was conceived for the purpose of explaining 7 plus 5 equals 11, after the wax: tablet had taken care of mistakes as regards perceived objects. The mis-addition is not of dogs or · other obj'ects of sense, but is of the pure monads and their ;;f"&f-"- ot. . • Hence, what is required is an explanation of a mean ·between knowing and not-knowing the "pure noetic entities". The obvious conjecture is, why not a wax tablet for them also, only let it be different in its "matter" than the first? This leads immediately to the question of how and where the no·e tic imprints were gotten, and suggests the eye of the mind and the myth of recollection. The existence of non-being, and resultant degrees of being are two such implications and reflect the symmetry between knowing and the thing known, i.e., between epistemology and ontology. 13. 14. �sous ombres Vous serez au Pierre �bones nor �- I ��crevees ��2 = :::: = = ��- 33 - . HUMPTY DUMPTY ON DISCOURSE Bruce Colli.ar "The question is," said Humpty Dumpty, "which is to be master." APOLOGY I intend to take two or more statements from Lewis Carroll's works Alice• s Adventures in Wonderland · and Through the · Looking-Glass and explicate them in terms of discourse. This is not meant as an interpretation of the t.Jorks in question, or even of the passages from which the statements come or of the characters who utter them, al though I may speak as if it were. I may seem to be reading too much into the statements, but I should rather be thought of as reading out of them. I make no claim to follo1dng the thoughts of the author. The problems of discourse are deep; and the pitfalls and byways are many. rwould hope to avoid the pitfalls and byways, and, perhaps, to hint at a few of the problems. I will be saying rather more or less than I mean. *********** In confronting the problem of discourse, we are presented with three focal points for our inquiry i the sounds that make up speech, the thoughts that give rise to speech, and the objects in the world about which our thoughts are formed. We shall- be concerned with the relationship between these three points. The following two statements will be taken as representative of two points of view on the nature of discourse. The Illchess: Humpty Dumpty: "Take care of the sense, and the sounds will take care of themselves." "When I use a word, it means just what I choose it to mean, neither more nor less.·~ The approach of the Ihchess is an innocent one. What is primary here is the sense: the meaning or thoughts in the mind of the speaker. The sounds uttered in speech are a reflection of these thoughts: a means of conveying them from a speaker to a listener. The actual form that the sounds take is unimportant, merely conventional; insofar as they are more than noise, they ~ the thoughts that they represent . In a sense, they pass out of the picture entirely. In many ways they are similar to the electrical impulses used in radio transmission. But the world is left out. Words re-present thoughts about objects in the world, but not the objects themselves, Since, then, thoughts can be formed about these objects only on the basis of sensation, words are no more than expressions of sensations and opinions• The Dlchess �are. the �- 35 nature and structure of discourse reflects the nature and structure of the world, and can lead us to see it; indeed, there is no other way. Humpty Dumpty tells Alice that he pays words for using them. We too must pay them -- pay them reverence •- submit to their mastery before we can master the problem of discourse and rise to the truth. EPILOGUE It may seem strange that Humpty D.nnpty, realizing the problem of rhetoric, should himself be so poor a rhetorician as to impress Alice as highly •0unsatisfactory." .Alice herself resolves this problem in commenting on the fact that Humpty llunpty is alone. Since the power of rhetoric consists in knowing the available means of persuasion, it cannot be developed in isolation from other men. Humpty Dlrnpty, shrewd as he may be, cannot fully pursue rhetoric or dialectic, outside of a political context. "~ can°t, perhaps", said Humpty Dumpty; "but . ~ can." �is us have been aware of from the were more the ones he of human also be even cases from the most from any conic may be be conic back cone. was not For the case have known in fact be conic cone from any to case in which the of the nature section was the vertex at the vertex is a Thus be a the and if ( to one the cone. were as in the of the the cone if acute , the the section be a be an �- 37 "Apollonius the geometer, my dear Anthemius, nourished at Perga in Pamphylia during the time of Ptolemy Euergetes, as is related in the life of Archimedes written by Heraclius, who also says that Archimedes first conceived the theorems in conics and that Apollonius, finding they had been discovered by Archimedes but not published, appropriated them for himself, but in my opinion he errs. For in many places Archimedes appears to refer to the elements of conics as an older work, and moreover Apollonius does not claim to be giving his own discoveries; otherwise he would not have described his purpose as •to investigate these properties more rul.ly and more generally than is done in the works of others.• But what Geminus says is correct: defining a cone as the figure formed by the revolution of a right-angled triangle about one of the sides containing the right angle, the ancients naturally took all cones to be right with one section in each -- in the right-angled cone the section now called the parabola, in the obtuse-angled the hyperbola, and in the acute-angled the ellipse; and in this may be found the reason for the names they gave to the sections. Just as the ancients, investigating each species of triangle separately, proved that there were two right angles first in the equilateral triangle, then in the isosceles, and finally in the ocalene, whereas the more recent. geometers have proved the general theorem, that in any triangle the three internal angles are equal to the two right angles, so it has been with the sections of the cone; for the ancients investigated the so-called section of a right-angled cone in a rightangled cone only, cutting it by a plane perpendicular to one side of the cone, and they demonstrated the section of an obtuseangled cone in an obtuse-angled cone and the section of an acuteangled cone in the acute-angled cone, in the cases of all the cones drawing the planes in the same way perpendicularly to one side of the cone; hence, it is clear, the ancient names of the curves. But later Apollonius of Perga proved generally that all the sections can be obtained in any corte, whether right or scalene, according to different relations of the plane to the cone. In admiration for this, and on account of the remarkable nature of the theorems in conics proved by him, his contemporaries called him the "Great Geometer." Geminus relates these details in the sixth book of his Theor:v of Mathematics. The precise method which the pre-Apollonians used is not known, but something like the following proofs has been suggested by Archimedes' Conoids and Spheroids; Prop. 7. �the Given a to a and L. 0 I say that the base the cone at I Draw Draw Now ::::::: I may BAG and BCG are both a as G BN = bisects But is to = AF = = FG = NG and NG = = and since BN = = �- 39 I say that NQ2:AN,2AL::A'N' :A'A or that the original equality has been contorted* by the ratio A'N:A 0 A, and I call 2AL the parameter. The ratio A'N:A'A I call the ratio of contortion of the section. Let the construction be as before Now But ; .t. ::. . ; Thus But And And NQ2 = BN,NC (Euclid VI, 13) BN,NC = AN,NG (part 1) .1'jQ~ = AN ,NG NQ :NG: :AN :NQ NG : AF : : NC : AD: : A' N: A' A NG:AF: :A 0 N:A 0 A NG:AF, AN:NQ comp. A'N:AA', AN:NQ NG:AF, NQ:NG comp. A'N:AA', AN:NQ 0 NQ:AF comp. AN,NQ, A'N:A'A** Or NQ:AF, NQ:AN comp. A'N:A 1 A NQ 2 :AF,AN::A'N:A'A But AF=2AL and 2AL is the parameter . NQ 2 :AN,2AL: :A 0 N:A 0 A * Contorted, i.e., the original ratio of equality which existed between NQ2 and 2AL has been changed to the ratio A'N:A'A. ** This comes from the general proposition that if AB:CD comp. EF:GH, JK:LM Than EF:GH comp. AB:CD, LM:JK It can be proved as follows: Given AB:CD .comp. EF:GH, JK:LM AB:CK, LM:JK comp. EF:GH,JK:LM,LM:JK EF:GH comp. AB:CD, LM:JK (Q.E.D.) Thus the noble root of the ugly word "invert". G �us to OD AN is and to :BN DA: = NC :BN BN = is the F that this to the a line dralNTl section was dralNTl the base of section the vertex of the drav.m the third to the vertex .............. r-. .......... 11 and the base formulation so to the cut. that he can :CO or on construction of I the �- 41 - . The central sections: (Sarne setting out as previous central sections, page 39 • ) Construct OK parallel to AA' Continue AD to K Since and But OK is drawn parallel to A0 A OK:BK:AN:BN OK:CK: :A'N:CN: :A' A: :AD •• CK2:BK,CK::AN,A 0 N:BN;CN . N. CN =·NQ2 . . ·· . ·· B OK~:BK,CK::AN,A 0 N:NQ2 0 also .OK2:BK,CK::AN,A A:AD,BN Thus NQ2:AN,A 0 N::AD,BN:AN,A'A Or NQ2:AD,BN::A'N:A'A Construct AF such that AN AF = AD,BN NQ2 :AN ,AF: :A'N :A" A .And thus AF is the parameter. A This formulation, however is ~:::::-t----1~--------=~~ useless because it depends K upon the point N: but AD:AF: :AN :BN and AN:BN::OK:BK Thus AD:AF: :OK:BK 0 \ \ We now have the first solid G'~A formulation for the parameter. If it is compared with the first formulation of the para-meter of the parabola (AD:AF: :OD:AD or AD:AF:OC:BC), we see that the only difference is that the ratio OC:BC has been replaced by the ratio OK:BK, or that in some sense the point C has been replaced by the point K. Finally, Apollonius finds a way of fitting into the formulation those lines which are not relevant to the cut. Sine But But or OK:BK:AD:AF OK2:BK,CK comp. AD:AF,OK:CK OK:CK::A 1 A:AD OK2:BK,CK comp. A'A:AD, AD:AF OK2:BK,CK: :A',A:AF which is the final formulation Note on the construction of AF Since and AD is parallel to Be OK is parallel to AN Angle DAF = Angle OKB Now if I let Angle FDA = Angle BOK The two tx-ian·g le s FDA and BOK will be similar Thus OK:BK::AD:AF ~Q.E.F.) (Q.E.D.) �- 42 - UN-MATRIX R. Thomas Ing Hold in thy mind's eye an image of sand held high in tightened fist that slowly falls in formless drift. See it caught, grain by grain, by wind which cares not what the sand intend. But for an instant are there grains, · and then the sand is nothing more than proof of standing on the shore where sea does wear upon the earth. Stay as rock, do not venture forth. Go, thou moment of my knowledge -into the earth from whence thou came. Ib not linger, nor show thy name. For wind might tear it from my sight, and names become but more·of earth. �- 43 - . ON THE METAPHORICAL -- POSTSCRIPT Charles G. Bell To have spoken badly might be a boon if the stammer necessitated writing. No doubt it was hard to make out what was being said in the lecture on Metaphor, ~h and ~bol. · One had to search for a theme underlyiµg sallies of argument, fragments of example. Conceive a spectrum without limits or clear divisions, the continuum of rese~blance in which we operate. ·As always, groupings have defined themselves, species, separating from others, have received tpe stamp of tradition. These are the basic modes of discourse, which we .try to hold distinct (or they, against dissolving currents, reassert their distinctness): mathematics, the sciences, philosophy, fiction, poetry. Cutting across these, subgroupings have appeared, peculiarly related to the main thrust of symbol: mythology, prophecy, allegory, the symbolic poem, the symbolic novel. Devices of analogy have been refined and named: personification, metaphor, simile. Over all such fruitf'ul specialization rides the unifying drive which is the creative life of man. Within each range of the phenomenon we may draw out poles, which merge again in particulars, interbreeding as we try to hold them apart. Thus with the distinction which dominated the question period: the symbol as precise actuality (Mr. Allanbrook' s Beatrice), or as the stripped abstract (number). Tempting as it is to call the specific symbol poetic and the generalized symbol philosophic and mathematical, and so clarify the realms, I cannot help seeing a catenary between these opposites, including such abstractions as the points of the compass, the mythopoetic numbers, the circles and gyres, each of which, loaded with all kinds of content, can take up procreative residence in mathematics, science, history, religion, poetry, art. Even the irreducible particular -- that woman, Beatrice bears a somewhat doubtful relation to the powers released, may only precipitate what has been concentrating, perhaps for centuri es, in the cultural ambiance (so Beatrice out or Virgin worship and Troubador love; D:m Quixote, Hamlet, Lear, out of who knows what). In such an inwrought field I seized on ·.a group of illustrations, which when fitted together adumbrate a kind or history: Even number, in the first ecstacy of its outcropping (Pythagoras), carried with it the whole complex or what was later separated into poetry, religion, philosophy, science. So, too, in Renaissance science (Leonardo), or wherever imagination, with germinal images, working like a spermatic logos in the rational ear, gives evidence of penetration into the stricter disciplines: thus with Harvey and Kepler, with Newton• s "Nature as a perpetual circulatory worker", with Kekule•s molecular snake swallowing its tail -- thus necessarily with current attempts to make sciences out of history, mythology, the psyche. �- 44 - . It is true that there are places in nature where the under-girdil]g of number lies at the surface -- crops out -- like crystal rocks from the so~er mottlings of earth and vegetation. It was here that the ancients seated gods and angelic intelligences, in the divine regularity of the stars. Here, too, mathematical· physics contrived the triumphs of determinism (the Principia and return of Halley's ·comet)which have lured all sciences in their wake, and which the naive . are always taking as a paradigm of knowledge -while the uni verse of the Bible and War and tragedy and flesh and marriage eddies around us and in us -- how simple -- the paradigm of knowledge. What opposes it is not only the cry of romantic poet_ : What s 'Whitehead called Wordsworth's "modal presences", make' s: "God forbid that knowledge should be limited to mathematical demonstration"; it is what we know to have happened in the strongholds of physics and mathematics. As if, in the suspensions of the organic universe we had managed to dry up a small area of causal rationalism, . only at the expense of deepening at its center those paradoxes of space, time, iogic and number, which the present century has revealed. (And how late what we call reason appeared on the scene, after some organizing w.ill or blind chance had already put the world and us, with that reasoning mind itself, together.) So it is not surprising if in our time we have auguries of a new convergence (Thoreau• s bug boring out of the dry table), the sciences gaping again, pregnant w.ith the philosophic and mythopoetic (Whitehead, the Jungians, Teilhard de Chardin), while poetry and fiction (Blake to Yeats; Hawthorne and Melville to Joyce and beyond) hint at a new world-epos. I doubt if the need that set the symbolic powers in motion has yet reached fulfillment, even in Yeats, though his may be the most burning achievement yet available. That which is to be fulfilled rests upon us as a creative COillJltand, a divine obligation, drawing us on -The half-read wisdom of daemonic images, Suffice the aging man as once the growing boy. � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The Collegian Description An account of the resource The Collegian began as a student newspaper; became a college community literary publication in 1952 and continued until 1969; became a student newspaper again in 1969; discontinued publication from 1980 until 1989 when it again became a student literary publication. Click on <a title="The Collegian" href="http://digitalarchives.sjc.edu/items/browse?collection=26">Items in The Collegian Collection</a> to view and sort all items in the collection. Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Page numeration Number of pages in the original item. 44 pages Text Any textual data included in the document text Original Format The type of object, such as painting, sculpture, paper, photo, and additional data paper Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Creator An entity primarily responsible for making the resource Lachterman, David (Editor) Title A name given to the resource The Collegian, September 1963 Date A point or period of time associated with an event in the lifecycle of the resource 1963-09 Description An account of the resource Issue of The Collegian. Published in September 1963. (Front cover missing.) Identifier An unambiguous reference to the resource within a given context Collegian Sept 1963 Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Publisher An entity responsible for making the resource available St. John's College Language A language of the resource English Rights Information about rights held in and over the resource St. John's College owns the rights to this publication. Format The file format, physical medium, or dimensions of the resource pdf The Collegian https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/3c3d0ae928a0a806429a1e666b32ead5.mp3 cdf40187adb39f3411c29c35ff75d911 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College. During the Fall and Spring semesters, lectures are given on Friday nights. Items include audio and video recordings and typescripts. For more information, and a schedule of upcoming lectures, please visit the <a href="http://www.sjc.edu/programs-and-events/annapolis/formal-lecture-series/" target="_blank" rel="noreferrer noopener">St. John's College website</a>. Click on <a title="Formal Lecture Series" href="https://digitalarchives.sjc.edu/items/browse?collection=5&sort_field=Dublin+Core%2CDate&sort_dir=d">Items in the St. John's College Formal Lecture Series—Annapolis Collection</a> to view and sort all items in the collection. A growing number of lecture recordings are also available on the St. John's College (Annapolis) Lectures podcast. Visit <a href="https://podcasts.apple.com/us/podcast/st-johns-college-annapolis-lectures/id1695157772">Apple Podcasts</a> or <a href="https://open.spotify.com/show/6GDsIRqC8SWZ28AY72BsYM?si=f2ecfa9e247a456f" title="Spotify">Spotify</a> to listen and subscribe. Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Title A name given to the resource St. John's College Formal Lecture Series—Annapolis Identifier An unambiguous reference to the resource within a given context formallectureseriesannapolis Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data audio cassette Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 00:37:03 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource What Is a Philosophic Question? Description An account of the resource Audio recording of a lecture delivered on April 28, 1989, by Samuel S. Kutler as part of the Formal Lecture Series. Creator An entity primarily responsible for making the resource Kutler, Samuel S. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Date A point or period of time associated with an event in the lifecycle of the resource 1989-04-28 Rights Information about rights held in and over the resource A signed permission form has been received stating: "I hereby grant St. John's College permission to: Make an audiovisual recording of my lecture, and retain copies for circulation and archival preservation in the St. John's College Greenfield Library. Make an audiovisual recording of my lecture available online. Make a typescript copy of my lecture available for circulation and archival preservation in the St. John's College Greenfield Library. Make a typescript of my lecture available online." Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Subject The topic of the resource Aristotle. Metaphysics Philosophy Language A language of the resource English Identifier An unambiguous reference to the resource within a given context LEC_Kutler_Samuel_1989-04-28_ac Friday night lecture Tutors https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/28bacd6f3058043922aa21ad0b7115c6.mp3 81629ce2576173fdc42273dbdeb2a4e8 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College. During the Fall and Spring semesters, lectures are given on Friday nights. Items include audio and video recordings and typescripts. For more information, and a schedule of upcoming lectures, please visit the <a href="http://www.sjc.edu/programs-and-events/annapolis/formal-lecture-series/" target="_blank" rel="noreferrer noopener">St. John's College website</a>. Click on <a title="Formal Lecture Series" href="https://digitalarchives.sjc.edu/items/browse?collection=5&sort_field=Dublin+Core%2CDate&sort_dir=d">Items in the St. John's College Formal Lecture Series—Annapolis Collection</a> to view and sort all items in the collection. A growing number of lecture recordings are also available on the St. John's College (Annapolis) Lectures podcast. Visit <a href="https://podcasts.apple.com/us/podcast/st-johns-college-annapolis-lectures/id1695157772">Apple Podcasts</a> or <a href="https://open.spotify.com/show/6GDsIRqC8SWZ28AY72BsYM?si=f2ecfa9e247a456f" title="Spotify">Spotify</a> to listen and subscribe. Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Title A name given to the resource St. John's College Formal Lecture Series—Annapolis Identifier An unambiguous reference to the resource within a given context formallectureseriesannapolis Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data audio cassette Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 00:38:53 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Friendly, Unfriendly Universe Description An account of the resource Audio recording of a lecture delivered on September 10, 1982 by Samuel S. Kutler as part of the Formal Lecture Series. Creator An entity primarily responsible for making the resource Kutler, Samuel S. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Date A point or period of time associated with an event in the lifecycle of the resource 1982-09-10 Rights Information about rights held in and over the resource A signed permission form has been received stating: "I hereby grant St. John's College permission to: Make an audiovisual recording of my lecture, and retain copies for circulation and archival preservation in the St. John's College Greenfield Library. Make an audiovisual recording of my lecture available online. Make a typescript copy of my lecture available for circulation and archival preservation in the St. John's College Greenfield Library. Make a typescript of my lecture available online." Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Language A language of the resource English Identifier An unambiguous reference to the resource within a given context Kutler_Samuel_1982-09-10 Friday night lecture Tutors https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/c84ebb2ceea1c524b0194146adb63862.mp3 d1b366827cfa211642ef2e0f5b9f3597 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College. During the Fall and Spring semesters, lectures are given on Friday nights. Items include audio and video recordings and typescripts. For more information, and a schedule of upcoming lectures, please visit the <a href="http://www.sjc.edu/programs-and-events/annapolis/formal-lecture-series/" target="_blank" rel="noreferrer noopener">St. John's College website</a>. Click on <a title="Formal Lecture Series" href="https://digitalarchives.sjc.edu/items/browse?collection=5&sort_field=Dublin+Core%2CDate&sort_dir=d">Items in the St. John's College Formal Lecture Series—Annapolis Collection</a> to view and sort all items in the collection. A growing number of lecture recordings are also available on the St. John's College (Annapolis) Lectures podcast. Visit <a href="https://podcasts.apple.com/us/podcast/st-johns-college-annapolis-lectures/id1695157772">Apple Podcasts</a> or <a href="https://open.spotify.com/show/6GDsIRqC8SWZ28AY72BsYM?si=f2ecfa9e247a456f" title="Spotify">Spotify</a> to listen and subscribe. Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Title A name given to the resource St. John's College Formal Lecture Series—Annapolis Identifier An unambiguous reference to the resource within a given context formallectureseriesannapolis Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data audio cassette Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 00:54:43 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Desiring What is Beautiful Description An account of the resource Audio recording of a lecture delivered on November 19, 1993 by Samuel S. Kutler as part of the Formal Lecture Series. Creator An entity primarily responsible for making the resource Kutler, Samuel S. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Date A point or period of time associated with an event in the lifecycle of the resource 1993-11-19 Rights Information about rights held in and over the resource A signed permission form has been received stating: "I hereby grant St. John's College permission to: Make an audiovisual recording of my lecture, and retain copies for circulation and archival preservation in the St. John's College Greenfield Library. Make an audiovisual recording of my lecture available online. Make a typescript copy of my lecture available for circulation and archival preservation in the St. John's College Greenfield Library. Make a typescript of my lecture available online." Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Language A language of the resource English Identifier An unambiguous reference to the resource within a given context Kutler_Samuel_1993-11-19 Friday night lecture Tutors https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/dfc816c55f411d28883e01b7919f77f6.mp3 70ac6788f4253b6178dc75e1dfdf7907 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College. During the Fall and Spring semesters, lectures are given on Friday nights. Items include audio and video recordings and typescripts. For more information, and a schedule of upcoming lectures, please visit the <a href="http://www.sjc.edu/programs-and-events/annapolis/formal-lecture-series/" target="_blank" rel="noreferrer noopener">St. John's College website</a>. Click on <a title="Formal Lecture Series" href="https://digitalarchives.sjc.edu/items/browse?collection=5&sort_field=Dublin+Core%2CDate&sort_dir=d">Items in the St. John's College Formal Lecture Series—Annapolis Collection</a> to view and sort all items in the collection. A growing number of lecture recordings are also available on the St. John's College (Annapolis) Lectures podcast. Visit <a href="https://podcasts.apple.com/us/podcast/st-johns-college-annapolis-lectures/id1695157772">Apple Podcasts</a> or <a href="https://open.spotify.com/show/6GDsIRqC8SWZ28AY72BsYM?si=f2ecfa9e247a456f" title="Spotify">Spotify</a> to listen and subscribe. Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Title A name given to the resource St. John's College Formal Lecture Series—Annapolis Identifier An unambiguous reference to the resource within a given context formallectureseriesannapolis Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data audio cassette Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 00:49:59 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource On Complex Numbers Description An account of the resource Audio recording of a lecture delivered on December, 09, 1983 by Samuel S. Kutler as part of the Formal Lecture Series. Creator An entity primarily responsible for making the resource Kutler, Samuel S. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Date A point or period of time associated with an event in the lifecycle of the resource 1983-12-09 Rights Information about rights held in and over the resource A signed permission form has been received stating: "I hereby grant St. John's College permission to: Make an audiovisual recording of my lecture, and retain copies for circulation and archival preservation in the St. John's College Greenfield Library. Make an audiovisual recording of my lecture available online. Make a typescript copy of my lecture available for circulation and archival preservation in the St. John's College Greenfield Library. Make a typescript of my lecture available online." Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Language A language of the resource English Identifier An unambiguous reference to the resource within a given context Kutler_Samuel_1983-12-09 Friday night lecture Tutors https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/3d4a67f1830d28336d570b7db681578f.mp3 844051e33d4cd77d620e40f16acde015 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College. During the Fall and Spring semesters, lectures are given on Friday nights. Items include audio and video recordings and typescripts. For more information, and a schedule of upcoming lectures, please visit the <a href="http://www.sjc.edu/programs-and-events/annapolis/formal-lecture-series/" target="_blank" rel="noreferrer noopener">St. John's College website</a>. Click on <a title="Formal Lecture Series" href="https://digitalarchives.sjc.edu/items/browse?collection=5&sort_field=Dublin+Core%2CDate&sort_dir=d">Items in the St. John's College Formal Lecture Series—Annapolis Collection</a> to view and sort all items in the collection. A growing number of lecture recordings are also available on the St. John's College (Annapolis) Lectures podcast. Visit <a href="https://podcasts.apple.com/us/podcast/st-johns-college-annapolis-lectures/id1695157772">Apple Podcasts</a> or <a href="https://open.spotify.com/show/6GDsIRqC8SWZ28AY72BsYM?si=f2ecfa9e247a456f" title="Spotify">Spotify</a> to listen and subscribe. Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Title A name given to the resource St. John's College Formal Lecture Series—Annapolis Identifier An unambiguous reference to the resource within a given context formallectureseriesannapolis Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data audio cassette Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 00:46:43 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Poetry and Mathematics Description An account of the resource Audio recording of a lecture delivered on October 2, 2009 by Samuel S. Kutler as part of the Formal Lecture Series. Creator An entity primarily responsible for making the resource Kutler, Samuel S. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Date A point or period of time associated with an event in the lifecycle of the resource 2009-10-02 Rights Information about rights held in and over the resource A signed permission form has been received stating: "I hereby grant St. John's College permission to: Make an audiovisual recording of my lecture, and retain copies for circulation and archival preservation in the St. John's College Greenfield Library. Make an audiovisual recording of my lecture available online. Make a typescript copy of my lecture available for circulation and archival preservation in the St. John's College Greenfield Library. Make a typescript of my lecture available online." Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Language A language of the resource English Identifier An unambiguous reference to the resource within a given context Kutler_Samuel_2009-10-02 Friday night lecture Tutors https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/7d2b32dbb0bd78a7e0db9027e6a4c52d.mp3 15f38a4dabe9a06174e82a04c0068478 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College. During the Fall and Spring semesters, lectures are given on Friday nights. Items include audio and video recordings and typescripts. For more information, and a schedule of upcoming lectures, please visit the <a href="http://www.sjc.edu/programs-and-events/annapolis/formal-lecture-series/" target="_blank" rel="noreferrer noopener">St. John's College website</a>. Click on <a title="Formal Lecture Series" href="https://digitalarchives.sjc.edu/items/browse?collection=5&sort_field=Dublin+Core%2CDate&sort_dir=d">Items in the St. John's College Formal Lecture Series—Annapolis Collection</a> to view and sort all items in the collection. A growing number of lecture recordings are also available on the St. John's College (Annapolis) Lectures podcast. Visit <a href="https://podcasts.apple.com/us/podcast/st-johns-college-annapolis-lectures/id1695157772">Apple Podcasts</a> or <a href="https://open.spotify.com/show/6GDsIRqC8SWZ28AY72BsYM?si=f2ecfa9e247a456f" title="Spotify">Spotify</a> to listen and subscribe. Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Title A name given to the resource St. John's College Formal Lecture Series—Annapolis Identifier An unambiguous reference to the resource within a given context formallectureseriesannapolis Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data audio cassette Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 00:56:24 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The Republic of Letters and the Republic of Numbers Description An account of the resource Audio recording of a lecture delivered on September 29, 2006 by Samuel S. Kutler as part of the Formal Lecture Series. Creator An entity primarily responsible for making the resource Kutler, Samuel S. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Date A point or period of time associated with an event in the lifecycle of the resource 2006-09-29 Rights Information about rights held in and over the resource A signed permission form has been received stating: "I hereby grant St. John's College permission to: Make an audiovisual recording of my lecture, and retain copies for circulation and archival preservation in the St. John's College Greenfield Library. Make an audiovisual recording of my lecture available online. Make a typescript copy of my lecture available for circulation and archival preservation in the St. John's College Greenfield Library. Make a typescript of my lecture available online." Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Language A language of the resource English Identifier An unambiguous reference to the resource within a given context Kutler_Samuel_2006-09-29 Subject The topic of the resource Comprehension Friday night lecture Tutors https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/a01ab61d333e6ca848160c3be8e4bccf.mp3 467cae9f49777b5db231d907c5eb1b0e Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College. During the Fall and Spring semesters, lectures are given on Friday nights. Items include audio and video recordings and typescripts. For more information, and a schedule of upcoming lectures, please visit the <a href="http://www.sjc.edu/programs-and-events/annapolis/formal-lecture-series/" target="_blank" rel="noreferrer noopener">St. John's College website</a>. Click on <a title="Formal Lecture Series" href="https://digitalarchives.sjc.edu/items/browse?collection=5&sort_field=Dublin+Core%2CDate&sort_dir=d">Items in the St. John's College Formal Lecture Series—Annapolis Collection</a> to view and sort all items in the collection. A growing number of lecture recordings are also available on the St. John's College (Annapolis) Lectures podcast. Visit <a href="https://podcasts.apple.com/us/podcast/st-johns-college-annapolis-lectures/id1695157772">Apple Podcasts</a> or <a href="https://open.spotify.com/show/6GDsIRqC8SWZ28AY72BsYM?si=f2ecfa9e247a456f" title="Spotify">Spotify</a> to listen and subscribe. Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Title A name given to the resource St. John's College Formal Lecture Series—Annapolis Identifier An unambiguous reference to the resource within a given context formallectureseriesannapolis Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data audio cassette Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 00:51:59 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource What Does Mathematics Have to Do with the Way We Lead Our Lives Description An account of the resource Audio recording of a lecture delivered on November 7, 1997, by Samuel S. Kutler as part of the Formal Lecture Series. Creator An entity primarily responsible for making the resource Kutler, Samuel S. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Date A point or period of time associated with an event in the lifecycle of the resource 1997-11-07 Rights Information about rights held in and over the resource A signed permission form has been received stating: "I hereby grant St. John's College permission to: Make an audiovisual recording of my lecture, and retain copies for circulation and archival preservation in the St. John's College Greenfield Library. Make an audiovisual recording of my lecture available online. Make a typescript copy of my lecture available for circulation and archival preservation in the St. John's College Greenfield Library. Make a typescript of my lecture available online." Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Language A language of the resource English Identifier An unambiguous reference to the resource within a given context LEC_Kutler_Samuel_1997-11-07 Subject The topic of the resource Mathematics Friday night lecture Tutors https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/f9219b72c780337ed9d6338c639ee6cc.mp3 b482491829b66340f9595f92f88a709f Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College. During the Fall and Spring semesters, lectures are given on Friday nights. Items include audio and video recordings and typescripts. For more information, and a schedule of upcoming lectures, please visit the <a href="http://www.sjc.edu/programs-and-events/annapolis/formal-lecture-series/" target="_blank" rel="noreferrer noopener">St. John's College website</a>. Click on <a title="Formal Lecture Series" href="https://digitalarchives.sjc.edu/items/browse?collection=5&sort_field=Dublin+Core%2CDate&sort_dir=d">Items in the St. John's College Formal Lecture Series—Annapolis Collection</a> to view and sort all items in the collection. A growing number of lecture recordings are also available on the St. John's College (Annapolis) Lectures podcast. Visit <a href="https://podcasts.apple.com/us/podcast/st-johns-college-annapolis-lectures/id1695157772">Apple Podcasts</a> or <a href="https://open.spotify.com/show/6GDsIRqC8SWZ28AY72BsYM?si=f2ecfa9e247a456f" title="Spotify">Spotify</a> to listen and subscribe. Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Title A name given to the resource St. John's College Formal Lecture Series—Annapolis Identifier An unambiguous reference to the resource within a given context formallectureseriesannapolis Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data audio cassette Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 00:53:00 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Memories and Memory Description An account of the resource Audio recording of a lecture delivered on September 22, 2000, by Samuel S. Kutler as part of the Formal Lecture Series. Recording is not complete. Creator An entity primarily responsible for making the resource Kutler, Samuel S. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Date A point or period of time associated with an event in the lifecycle of the resource 2000-09-22 Rights Information about rights held in and over the resource A signed permission form has been received stating: "I hereby grant St. John's College permission to: Make an audiovisual recording of my lecture, and retain copies for circulation and archival preservation in the St. John's College Greenfield Library. Make an audiovisual recording of my lecture available online. Make a typescript copy of my lecture available for circulation and archival preservation in the St. John's College Greenfield Library. Make a typescript of my lecture available online." Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Language A language of the resource English Identifier An unambiguous reference to the resource within a given context Kutler_Samuel_2000-09-22 Subject The topic of the resource St. John's College (Annapolis, Md.) Friday night lecture Tutors