1
20
105
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/88b364f0f37e6b7bf534883b5edcb43a.pdf
f0cf1b77988d8c4befbb7282cd34e655
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
paper
Page numeration
Number of pages in the original item.
25 pages
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
William Blake : a beginning
Description
An account of the resource
Transcript of a lecture given on July 7, 1993 by Stephen Van Luchene.
Creator
An entity primarily responsible for making the resource
Van Luchene, Stephen R.
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
1993-07-07
Rights
Information about rights held in and over the resource
Meem Library has been given permission to make this item available online.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Subject
The topic of the resource
Blake, William, 1757-1827 -- Criticism and interpretation
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
24000215
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/687a2f5ea81153a3c2ddfaf656313fca.pdf
475f45ae3bea6ffa4783000b6cc26a39
PDF Text
Text
William Donahue, November 5, 2021
What would Kepler say to Einstein?
The more I think of this topic, the more appalled I am at my hubris in
proposing it. “Really, Mr. Donahue (you might be thinking), “aren’t you just using
Kepler’s name to throw out your own rash thoughts into an arena in which you
have no business contending?”
As for the arena, I have no defense, other than that here at St. John’s we
routinely contend in contests where, by standards accepted elsewhere, we have no
formal qualifications. We do it anyway, unapologetically. We do not expect to
establish new truths, nor to overthrow established theories. But we do hope that in
this rather mad undertaking we may gain for ourselves a little more
understanding, both of the amazing universe we live in and the powerful thinkers
and their remarkable insights into that universe.
In presuming to offer advice to Einstein (ostensibly in the persona of Kepler), I
am on much shakier ground. The theme of this lecture sprang from a lecture I
gave some years ago at Johns Hopkins, which had to do with Kepler’s
introduction of physical principles into astronomy. In the question period, I was
asked what could be learned (if anything) from Kepler’s views on what constitutes
a good hypothesis. I replied that, more than merely accounting for phenomena, it
would have to be based in physical reality. This occasioned much rolling of eyes,
presumably at my extreme naïveté, which led me to ponder, in the ensuing years,
what Kepler might have understood by “physical reality,” and whether a similar
mode of understanding might be applicable to more recent physical theory.
At length I thought about Einstein’s replacement of Maxwell’s “luminiferous
ether” with a postulate asserting the constancy of the speed of light in all possible
inertial frames of reference. This seemed to me a theoretical move about which
Kepler would have been extremely skeptical. The idea of the ether, after all, was
�introduced into electromagnetic theory in order to provide a medium for
electromagnetic waves, whose existence could be demonstrated. These waves
were conceived as being similar to sound waves in air and other fluid media, and
the idea of a wave in nothing, without any medium or substrate, was (I believe)
calculated to raise objections from the community of physicists. And with them, I
believe, would be Kepler, were he still around to express his views.
Now of course Kepler did not say anything about waves, so I must say first of
all what criteria he used in evaluating a scientific hypothesis, and why I think he
would have raised an objection to Einstein’s rejection of the ether.
One of the best examples of Kepler’s thinking about the physical reality behind
the apparent motion of the planets is found right near the beginning of Astronomia
Nova, in Chapter 2. This chapter is a deliberate reprise of Book I Chapter 3 of
Ptolemy’s Almagest, but Kepler takes the argument a giant step farther, asking
how each of the two kinds of hypotheses (epicyclic and eccentric) could be
understood to function in physical reality. And for Kepler, the limits of physical
reality are very broad, but are also clear and definite. So here is where we have to
begin.
Kepler takes it as demonstrated that there are no real, rigid epicycles and
deferent circles or spheres in the heavens. The question, then, is how the moving
power (supposed for the moment to reside in the planet) could make the planet
move in a circle. Like Ptolemy, Kepler considered both the eccentric hypothesis
and the concentric-with-epicycle hypothesis; in the interest of time, we will
consider only the former.
This power, whatever it is, would have two jobs: it would have to be strong
enough to move the body of the planet, and it would have to know where to move
2
�it. This would require knowledge or perception of the circular path in the
unmarked aethereal air—a function of intelligence and not physics.
Now a circle, Kepler says, even for God, is nothing but
equality of distance from some point. Citing Avicenna, he
says that the planet will either have to “imagine for itself
the center of its orb and its distance from it,” or use some
other property of a circle to establish its distance and
direction. Here’s the example Kepler gives. The planet’s
assignment will be to move on the path γεδ, with center β (not occupied by any
body or perceptible mark). This center is fixed in position with respect to point α,
which is occupied by a body. To make this motion possible, the planetary mover
might be endowed with some means of perceiving the angular size of the body α,
as it should appear, in succession, at γ, ε, η, δ, continually adjusting its distance
from α while moving with uniform speed on the resulting circle. Somehow, it will
have to match continually the distances γα, εα, ηα, δα, and so on (calculated by
observing the apparent size of the body at α), with the angles γαε, γαη, γαδ,
respectively. It will also have to know the direction of the line of apsides, αγ, in
the sphere of the stars. As Kepler says, “The planet’s mover will thus be occupied
with many things at once,” his implication being that this arrangement makes no
sense.
“To escape this conclusion,” he continues, “one must assert that the planet pays
attention to the point β, entirely empty of any body or real quality, and maintains
equal distances from that point.”
“Body or real quality”: that is the kind of thing that Kepler was looking for. It
would have to be perceptible, somehow, even if it were not the same as objects
that we interact with every day. Objects of pure mathematics, such as points and
3
�lines, would not do. Beyond that, Kepler was extraordinarily open to analogies,
both mechanical and living. Earlier in the chapter, he considers at length the
operations of muscles in the human body, and concludes that animal locomotion
would not be a good model for planetary motion;
nonetheless, he is very much open to a role for minds in the
heavens.
In passing, it is fascinating to note that Newton, too,
found that the supposedly “natural” circular motion could
not be produced by central forces alone. The moving body
would have to take into account, in addition to its velocity and its distance from
the center of force, the chord from its present position through the center to the
opposite side of the circle. Thus it would have to know where it would be on the
other side of its orbit!
We are now prepared to imagine what Kepler might think while reading
Einstein’s world-changing paper “On the Electrodynamics of Moving Bodies.” He
would certainly want to know what this “electrodynamics” is. Maybe the best
explanation would be to display the two experimental demonstrations that
Einstein describes, which (I hope) are now familiar to all seniors.
[show two videos, one with a hand moving a magnet through a coil of wire, the
other with a hand moving the coil with the magnet fixed.]
Kepler might be surprised to see the magnetic needle wiggle, whether the
magnet moves inside the coil of wire or the coil moves along the magnet. He
might be more surprised to learn that, despite the identical effects of the motion
upon the magnetic needle, the theory requires that there be two very different
accounts of what is happening. Here is Einstein’s description:
4
�“If the magnet is in motion and the conductor at rest, there arises in the
neighborhood of the magnet an electric field with a certain definite energy,
producing a current at the places where parts of the conductor are are situated.
But if the magnet is stationary and the conductor in motion, no electric field
arises in the neighborhood of the magnet. In the conductor, however, we find
an electromotive force, to which in itself there is no corresponding energy, but
which gives rise—assuming equality of relative motion in the two cases
discussed—to electric currents of the same path and intensity as those produced
by the electric forces in the former case.” [emphasis supplied]
(This is clearly shown by the deflection of the needle in the two cases.)
Kepler would, I think, be reminded of the physical difference between
planetary motion as conceived by the geocentrists and the motion as seen by the
heliocentrists. The phenomena as observed by the astronomers would be the same
in both cases, but the physical reality would be radically different. I imagine he
would say, “Well, two contradictory accounts cannot both be true, so one of the
descriptions must be the correct one.”
Einstein, however, went off in a completely different direction. Ignoring
conventional ideas of space and time in Newtonian physics, he adopted two
principles, which he initially raised as conjectures, but immediately (in his own
words) “raised to the status of postulates.”
1. The same laws of electrodynamics and optics will be valid for all frames of
reference for which the equations of mechanics hold good.
2. Light is always propagated in empty space with a definite velocity c which
is independent of the state of motion of the emitting body.
A “postulate,” we should recall, is how we usually translate the Greek Ἀιτήμα,
meaning “demand, request.” It is a statement that the author requests us to accept
5
�as true, without proof, as a basis for the demonstrations that will follow. Einstein
states that he will use these postulates to attain “a simple and consistent theory of
the electrodynamics of moving bodies based on Maxwell’s theory for stationary
bodies.” In other words, he proposes to fix the inconsistency between the
theoretical accounts of electromagnetic induction, noted above, not by adopting
one or the other account as “true,” but by a radical reformation of the foundations
of all of physics, by adding these two postulates to Newton’s three “Axioms, or
Laws of Motion.”
Now I do not think that Kepler would be troubled, in principle, with the idea of
a radical reform of physics. But I do think he would be troubled by Einstein’s next
sentence. Einstein wrote, “The introduction of a ‘luminiferous ether’ will prove to
be superfluous inasmuch as the view here to be developed will not require an
‘absolutely stationary space’ provided with special properties.” Since Kepler
believed that God had created a finite, spherical universe with the sun at its center,
he clearly was an advocate of a stationary space with special properties, such as
privileged places. But aside from this, I will argue that, theology aside, Kepler
would have philosophical or methodological objections to abandoning the
Maxwellian ether. To do this, I will have to make an excursion into the
considerations that led Maxwell and other physicists of the nineteenth century to
espouse the ancient idea of an ethereal medium filling all space.
The excursion I propose will lead us into some rather elementary physical
considerations. These may be the sort of thing that Einstein would think of as
“superfluous,” but this is exactly the kind of inquiry that Kepler enjoyed. So let us
invite him to join us in considering the “simple” pendulum.
[video of pendulum]
[video of two loosely linked pendulums]
6
�[video of a number of loosely linked pendulums]
[video of Bell wave machine, first with all but one bar clamped, then with the
clamp removed so as to create a wave]
Kepler was very good at constructing mathematical models of physical actions,
but (as we saw in Astronomia Nova) he wanted more from a sound physical
explanation. In several places in Astronomia Nova he set out a three-leveled
structure: the observational evidence, geometrical modeling of the observations,
and a physical account that could underlie the geometry. As for what could serve
as a “physical account,” Kepler was open to a very wide range of examples:
animal joints and muscles, magnetism, whirlpools and other examples of water
flow (such as Heron’s fountain), amusement park
rides, oars and paddles, and so on. He seems to
have sought examples that would be generally
acknowledged as physically real and that could be
understood as constituting an analogue to a
phenomenon that is felt to be in need of
explanation. In proposing such examples, he was
often not claiming that the analogy provided a full and adequate account of the
phenomenon, only that the physical reality might be somewhat like the example.
Sometimes he combined two different analogies in a single diagram, as here (from
Astronomia Nova Chapter 59), where a magnetic planet (the big black circle) with a
vertical axis (note the arrowhead at the top) is alternately attracted and repelled by
the sun, but is also being propelled by a boatman with a pole (or perhaps an oar).
He also candidly admits the provisional or conjectural nature of some of his
analogies: for example, in Ch. 57 he writes, “I am satisfied if this magnetic
example demonstrates the general possibility of the proposed mechanism.
7
�Concerning the details, however, I have doubts...There may be absolutely no
material, magnetic faculty that can accomplish the tasks entrusted to the planets
individually…” His point is, that wherever possible, it is preferable to invoke a
physical force or power such as magnetism or weight, but if all else fails, it is
permissible to invoke mental or animate powers.
So let’s think about what Kepler might hope for as a generalized physical
metaphor or underpinning for wave phenomena. To help us, I’d like to bring back
the Bell wave machine.
[Bell video]
We may think of the machine as an assembly of linked pendulums. Each
crossbar is attached to a longitudinal torsion bar that runs the length of the
machine. If the crossbar at the end is displaced and released while the bar next to
it is clamped in place, it oscillates while all the other crossbars remain at rest.
[video with just one bar moving]
It is acting as a torsional pendulum. When its displacement is at its maximum,
the twisting force is also maximum, and the bar, when released, moves in the
direction of the force, towards its rest position. But when it gets to its rest position,
it has acquired some speed, which carries it past the rest position to a new
maximum displacement.
If we remove the clamp from the next crossbar, the motion of the first bar twists
the torsion bar, which then imparts that twist to the next crossbar, which in turn
adds a twist to the torsion bar, and thus the twisting motion is passed on. It’s
much like those pendulums we saw earlier, that were connected by springs.
So it appears that in order to have a wave in some medium, two things are
needed.
8
�1. Individual places in the medium have to be able to move like pendulums:
when given an initial push, they will depart from their position, but they then
experience a restorative force in the medium that pushes back towards whence
they came;
2. The pendulums have to be linked so that the swinging of each of them is
communicated to neighboring pendulums. When they are tightly linked, the wave
move quickly through them; when loosely linked (as the pendulums were), the
motion is communicated more slowly.
This can be applied to
electromagnetic waves in a general
way quite directly. Consider this
simple assembly of two collinear
pieces of metal (called a “dipole”),
colored red, connected to the output of a device (the transmitter) that makes
electricity slosh back and forth between the two sides of the dipole. The
transmitter/dipole assembly is our initiating pendulum: the natural tendency of
the electricity is to create an equality of tension or “potential” between the two
sides of the dipole; the transmitter provides the pushes that keep the electricity
oscillating. What happens, as we know from Faraday, is that the electric and
magnetic forces generated by this assembly ripple out through the surrounding
space. Electric tension builds up in space, and as the tension is released by the
restoring force of the medium, this release constitutes an electric current, which
generates a magnetic force, which grows and decreases in a similar way,
generating an electric displacement current, and so on. This action continues, and
constitutes what we call a “radio wave.” And there is strong evidence that light,
too, is just such a wave.
9
�So the wave metaphor, built up out of linked motions that act like pendulums,
evidently applies to electricity, magnetism, and light too, in a direct and
comprehensible way. In the face of such evidence, Kepler might say, is it not a
retrograde step to dismiss as “superfluous” the medium in which the actions
foundational to the observed phenomena take place? Isn’t adopting the
“postulate” that the speed of light in a vacuum is constant too much like
postulating (as astronomers had done for thousands of years) that all celestial
bodies move with uniform circular motion?
This, then, is what I think Kepler would say to Einstein. And I could stop here,
but in all fairness, we need to let Einstein respond.
I think Einstein would point out two problems, one cosmic in scale and one
inherent in electromagnetic theory as it was then formulated.
The first is related to the ether itself: are we moving through it, or is it moving
along with us? If we are moving through it, then waves would seem to us to be
moving faster in some directions and slower in others. But when we measure the
speed of light, it seems to be pretty much the same in all directions. It gets worse:
the speed of electromagnetic waves, as deduced from Maxwell’s equations, is
determined by the ratio of the two fundamental electrical and magnetic constants.
These are the physical constants that seem, both conceptually and experimentally,
to be independent of coordinate systems, and that determine the “springiness” of
the medium. In junior lab, every spring, we do a lovely experiment that gives us a
number for this ratio, and our number (perhaps surprisingly) is pretty close to
what the textbooks say it should be. And further, when (in a second beautiful
experiment) we measure the speed of light directly (using a tape measure and a
tuning fork), the number we get is not too far from the ratio in the previous
experiment. So we are left with a dilemma: either the fundamental electromagnetic
10
�constants are somehow coordinate-system dependent so that they match the
measured motion of our coordinate system through the ether, or there is a
preferred coordinate system for the entire universe in which we happen to be
absolutely at rest. It’s hard to imagine what the first horn of the dilemma even
means, while the second horn basically throws out all cosmological thinking since
Copernicus. We can call this problem the “ether wind” problem.
The other thing Einstein would say is that in his view, this whole dilemma
associated with the ether is just a side issue: his concern, which he thinks was a
much more fundamental problem, was the asymmetry in the way Maxwellian
electrodynamics applies to magnets and wires. As Einstein put it, in the passage
quoted earlier in this lecture,
“If the magnet is in motion and the conductor is at rest, there arises in the
neighborhood of the magnet an electric field with a certain definite energy,
producing a current at the places where parts of the conductor are situated.
But if the magnet is stationary and the conductor in motion, no electric field
arises in the neighborhood of the magnet. In the conductor, however, we
find an electromotive force, to which in itself there is no corresponding
energy, but which gives rise—assuming equality of relative motion in the
two cases discussed—to electric currents of the same path and intensity as
those produced by the electric forces in the former case.”
Restating this in a slightly different way, if you have an observer who sees the
magnet moving through the coil of wire, she sees an electric field in the space
surrounding the magnet, and this field embodies a definite amount of energy.
Another observer, moving uniformly along with the magnet, will see no electric
field, and the same space will now be devoid of energy. But energy is a conserved
entity. So we have a theory that has lost what later physicists have called “local
11
�reality”: for one observer some real thing is there which according to the laws of
physics is not there for the other observer. Thus, the whole idea of objective reality
has broken down, which, you may imagine, is a big no-no for a physical theory.
So Einstein’s response is to adopt the two “postulates,” mentioned earlier, by
which he will solve both the ether wind problem and the objective reality
problem. His claim is that abandoning the idea of a physically real medium in
which electromagnetic waves occur is a price worth paying for saving the claim of
physics to represent objective reality.
So, Kepler, what might you have had to say in response to Einstein’s powerful
reply?
It is clear from our brief look at Astronomia Nova Ch. 2, and from many other
places in the book, that Kepler believed it was wrong to allow a physically
unsupported postulate (uniform circular motion) to overrule principles, even if
they are provisional or conjectural, that are supported by physical arguments. As
he put it, we need “a body or real quality” as a foundation. Although he was open
to a wide range of examples and analogies that would constitute “real qualities,” a
simple rule lacking such support, such as the constancy of the speed of light,
would not do. Although Kepler would have acknowledged it as a clever and
ingenious solution, it would remind him too much of the many astronomers of his
day who rejected his “celestial physics” (a term featured prominently on the title
page of Astronomia Nova) and reverted to the circular tracks and angelic movers of
the old astronomy.
What would have to be done instead, Kepler would say, would be to solve the
ether wind problem and the local non-reality problem without abandoning the physical
12
�basis of electromagnetic radiation. This would surely be a difficult task. But would it
be more difficult than establishing a sound physical basis for planetary motion?
I’ll finish this lecture by saying a few things in support of Kepler’s advice. Not
that I consider Kepler’s position is in need of support—it seems to me one of the
really deep questions—but to show that, despite what seems to be the unanimous
acceptance of Einstein’s two postulates, there has been, and continues to be, a
quiet but respectable undercurrent among physicists, a willingness to wonder
whether despite the remarkable success of Einstein’s relativity theory, its
unsupported second postulate might turn out to have been a mistake.
The first direct attempt to determine a possible motion of the earth through the
ether was carried out in 1881 by A. A. Michelson, in Potsdam, Germany.
Michelson nonetheless noted that the problem had already been approached by
Stokes (in 1846) and later by Maxwell (1878). The more famous Michelson-Morley
experiment, which used a much larger instrument, followed in 1887. These purely
experimental results, which showed no measurable motion, set a problem for the
theorists to solve.
The approach that appeared most promising at first was that some of the ether
was being dragged along by the earth; however, no one succeeded in
demonstrating such a phenomenon. A competing account was suggested by
Oliver Heaviside’s conclusion, on the basis of Maxwell’s electromagnetic theory,
that electromagnetic fields contracted along the direction of their motion. In 1895,
H. A. Lorentz published an article in which he proposed that, like
electromagnetism, the forces that hold the particles of material bodies together
also contract in the direction of motion. He writes,
13
�“Thus one would have to imagine that the motion of a solid body (such
as a brass rod or the stone disc employed in the later experiments) through
the resting ether exerts upon the dimensions of that body an influence
which varies according to the orientation of the body with respect to the
direction of motion. …
“Surprising as this hypothesis may appear at first sight, yet we shall
have to admit that it is by no means far fetched, as soon as we assume that
molecular forces are also transmitted through the ether, like the electric and
magnetic forces of which we are able at the present time to make this
assertion definitely.”
In 1904 (the year preceding Einstein’s Special Relativity article), Lorentz published
a more thorough treatment of the same basic idea. In the interim, Poincaré had
argued that electromagnetic forces alone were insufficient to produce Lorentz’s
contraction, and added an additional hypothetical force. However, as Lorentz
notes, Poincaré also objected to this piecemeal approach. Lorentz writes:
“Poincaré has objected to the existing theory of electric and optical
phenomena in moving bodies that, in order to explain Michelson’s negative
result, the introduction of a new hypothesis has been required, and that the
same necessity may occur each time new facts will be brought to light.”
Lorentz believed that by “starting from the fundamental equations of the theory
of electrons,” he could “treat the subject with a better result.” The article that
followed is a tour-de-force of Maxwellian analysis, packed with equations dealing
with such matters as the electromagnetic inertia of electrons.
At this point, Lorentz’s fundamental revision brought his theory, which
avoided Einstein’s second postulate, into agreement with Einstein, as far as the
observations were concerned. But by a strange turn of events, a series of
14
�experiments by Walter Kaufmann (involving the mass of high-speed electrons
rather than motion through the ether) appeared to show that the Einstein/Lorentz
predictions were wrong, and that rival theories of Max Abraham and Alfred
Bucherer produced more accurate results. Lorentz conceded that Bucherer’s
theory was “decidedly unfavorable to the idea of a contraction, such as I
attempted to work out.” Einstein, on the other hand, acknowledged that the
Abraham and Bucherer theories fit the data better than his own, but wrote, “they
have a small probability of being correct since they produce complicated
expressions for the mass of a moving electron.” In other words, theoretical
simplicity trumps agreement with the data!
But now Planck entered the fray, with a meta-analysis of Kaufmann’s numbers,
which tipped the balance back in favor of Einstein and Lorentz. And in 1914,
refined experiments by Günther Neumann (using Kaufmann’s own equipment
with some modifications) appeared to favor Einstein decisively. The curious result
of this was that, even though Einstein’s and Lorentz’s theories were essentially in
agreement in most of their predictions, Einstein’s ether-free approach came to be
viewed as the victor.
Nevertheless, attempts to find an “ether wind” continued. The most extensive
work was by Dayton Miller, a prominent American physicist who was president
of the American Physical Society in 1926. His measurements extended over nine
years and, by one account, comprised over five million individual measurements.
His primary aim was to show a difference between the ether drift at low
elevations (essentially none) and at the summit of Mt. Wilson. He claimed to have
found that the solar system is moving towards the constellation Dorado through
the ether at a speed of 227 km/s, a result that was similar to independent
measurements by the French astronomer Ernest Esclangon and the Swiss
15
�astronomer Leopold Courvoisier. These results have been questioned on various
grounds, but were well-received at the time and have never been adequately
repeated, according to one scholar whom I know personally and whose work I
respect. It appears that the remarkable success of both the special and general
theories of relativity have made a search for ether-drift an unattractive career
move.
However, questions have more recently crept in from an unexpected source:
quantum mechanics. Physicist John Bell, in 1964, came up with a purely
mathematical theorem that established certain numerical limits to the relatedness
of states of certain particles (in this case, polarizations of so-called “entangled
photons”). The assumptions upon which the theorem was based were, first, that
the particles involved really and actually possess the properties involved (the
criterion of “reality”), and second, that communication among the particles cannot
occur at speeds faster than light. Naturally, this set a challenge for experimenters:
violate Bell’s theorem! The definitive experiment, by Alain Aspect, came along in
1982. It violated the conditions of Bell’s theorem, while remaining entirely
consistent with quantum mechanics. In practical terms, this meant that one or both
of the assumptions that Bell made would have to be abandoned or modified.
Physics would have to give up the idea of local reality, or of what Einstein called
“spooky action at a distance,” or perhaps both.
One is reminded of what Kepler wrote in Astronomia Nova when he showed that
the classically formulated hypothesis of Chapter 16 is inconsistent with the
Tychonic observations. He wrote,
Therefore, something among those things we had assumed must be false.
But what was assumed was: that the orbit upon which the planet moves is a
perfect circle; and that there exists some unique point on the line of apsides
16
�at a fixed and constant distance from the center of the eccentric about which
point Mars describes equal angles in equal times. Therefore, of these, one or
the other or perhaps both are false, for the observations used are not false.
So, in light of the Aspect experiment, what gets thrown out? A variety of
solutions have been proposed, but I will conclude this lecture with what John Bell
himself said, in a discussion with BBC producer J. R. Brown and physicist P. C. W.
Davies.
Question:
Bell’s inequality is, as I understand it, rooted in two assumptions: the first is what we
might call objective reality—the reality of the external world, independent of our
observations; the second is locality, or non-separability, or no faster-than-light signaling.
Now, Aspect’s experiment appears to indicate that one of these two has to go. Which of the
two would you like to hang on to?
Bell:
Well, you see, I don’t really know. For me it’s not something where I have a
solution to sell! For me it’s a dilemma. I think it’s a deep dilemma, and the
resolution of it will not be trivial; it will require a substantial change in the way we
look at things. But I would say that the cheapest resolution is something like going
back to relativity as it was before Einstein, when people like Lorentz and Poincaré
thought that there was an aether—a preferred frame of reference—but that our
measuring instruments were distorted by motion in such a way that we could not
detect motion through the aether. Now, in that way you can imagine that there is a
preferred frame of reference, and in this preferred frame of reference things do go
17
�faster than light. But then in other frames of reference when they seem to go not
only faster than light but backwards in time, that is an optical illusion.
Question:
Well, that seems a very revolutionary approach!
Bell:
Revolutionary or reactionary, make your choice. Behind the apparent Lorentz
invariance of the phenomena, there is a deeper level which is not Lorentz
invariant.
Question:
Of course the theory of relativity has a tremendous amount of experimental support,
and it’s hard to imagine that we can actually go back to a pre-Einstein position without
contradicting some of this experimental support. Do you think it’s actually possible?
Bell:
Well, what is not sufficiently emphasized in textbooks, in my opinion, is that
the pre-Einstein position of Lorentz and Poincaré, Larmor and Fitzgerald was
perfectly coherent, and is not inconsistent with relativity theory. The idea that
there is an aether, and these Fitzgerald contractions and Larmor dilations occur,
and that as a result the instruments do not detect motion through the aether—that
is a perfectly coherent point of view.
Let me finish by briefly summarizing the main points of this lecture.
Kepler strove mightily throughout his life to oppose the prevalent idea that
astronomy must be founded on hypotheses, and that the fundamental and
indispensable hypothesis is the principle of regular, uniform circular motion of all
18
�heavenly bodies. He proposed instead that all attempts to understand the cosmos
must be founded in some way upon physical reality. As to what constitutes
physical reality, we must use familiar examples to try to understand what is less
accessible to us, and we may be led to consider accounts or examples that may at
first seem far-fetched. But it is a mistake to limit the range of possible explanations
within the boundaries of arbitrary postulates. This is what I believe is the advice
he would most want to give Einstein. And this is advice that, despite the rolling of
eyes at Johns Hopkins, may retain a degree of cogency today.
19
�
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
Word doc
Page numeration
Number of pages in the original item.
19 pages
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 would Kepler say to Einstein?
Description
An account of the resource
Transcript of a lecture given on November 5, 2021 by William Donahue as part of the Dean's Lecture and Concert Series. The Dean's Office provided this description of the event: "Einstein once wrote that there is little value, other than the satisfaction of intellectual curiosity, in studying scientific works of the past. Kepler would not agree, and were he alive today he would criticize Einstein for repeating errors of early science. This lecture
will begin by exploring some of Kepler’s views on past failures, and then will apply Kepler’s criticism to Einstein’s views, especially his rejection of the ether. The inquiry will then consider the alternative account proposed by H. A. Lorentz, showing how the
contraction of bodies at high velocities was deduced from Maxwell’s electrodynamics without abandoning the ether. The lecture then concludes by considering remarks of more recent physicists, most notably John Bell, on the possibility of reviving the ether to solve the local reality problem in quantum physics."
Creator
An entity primarily responsible for making the resource
Donahue, William H.
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
2021-11-05
Rights
Information about rights held in and over the resource
Meem Library has been given permission to make this item available online.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Subject
The topic of the resource
Kepler, Johannes, 1571-1630
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
SF_DonahueW_What_would_Kepler_say_to_Einstein_2021-11-05
Friday night lecture
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/433170265cb61a133a1cd8d0c840eab4.pdf
89b0a8fc6dc76aa3170892a7ff0b03e2
PDF Text
Text
1
What is the Measure of Electricity? 1
Howard J. Fisher
What is the measure of electricity? The question itself raises questions. For not all
things are susceptible to measure; and even when they appear to be, it is not always
clear whether “measure” applies to them as wholes, or only in certain respects. For
purposes of this talk, let me propose that a measure of something must, at minimum,
enable us to speak of that thing in terms of more and less. Faraday inherited an
electrical vocabulary that appraised electricity as more and less in two respects: first,
in quantity; and second, in intensity. At the outset of Faraday’s researches, neither he
nor anyone else had been able to state just what these two characteristics were, nor to
explain how they related to one another. On the other hand, everybody had some rough
and practical idea of them, as we may gather from Faraday’s unassuming
characterization in the Third Series:
The term quantity in electricity is perhaps sufficiently definite as to
sense; the term intensity is more difficult to define strictly. I am using
both terms in their ordinary and accepted meaning. [360, note]
If Faraday regarded the term “quantity” as relatively straightforward, it is probably
because at the time he began his researches, the conventional idiom of electrical
thinking was that of electric fluid, a special kind of substance, thought to be endowed
with the power to attract or repel other portions of electric fluid. Electric fluid was either
vitreous, like that which could be evolved upon glass surfaces, or resinous, like that
which could be produced on rubber, gum, amber, and similar materials. Portions of
unlike fluids attracted one another; portions of like fluids repelled each other; and the
more fluid there was, the stronger that attraction or repulsion would be. It is easy to
know what we mean by “quantity” if electricity is a fluid. But is it a fluid? And how can
we know?
In contrast, as Faraday implies, the fluid language fails to offer a similarly clear
image of intensity. What can it mean for a fluid to be more or less “intense”? Faraday
will seek, and perhaps he will find, a clearer understanding of both these terms.
As the Third Series opens, we find Faraday in almost the same position as Socrates
of the Meno; for how can we hope to know the properties of electricity unless we first
know what electricity actually is? We well remember Meno’s reply when Socrates
asked after the “what” of virtue:
1
Meno. “There will be no difficulty, Socrates, in answering that. Take
first the virtue of a man: it is to know how to administer the state, in
which effort he will benefit his friends and injure his enemies, and will
take care not to suffer injury himself. A woman’s virtue may also be
easily described: it is to order her house, and keep what is indoors,
Lecture delivered 23 February, 2024 at St. John’s College, Santa Fe
�2
and obey her husband. Every age, every condition of life, young or old,
male, or female, bond or free, has a different virtue....” [71]
Meno is positively exultant as he contemplates the rich variety of virtues! How
disheartening is it, then, to consider that the electrical science of Faraday’s time,
though professing to seek a unitary account of electricity, can offer little more than a
Meno-like catalog of “electricities.” These include:
Voltaic electricity, which is evolved by devices like Alessandro Volta’s “cups.”
Faraday will study voltaic action extensively in the Seventh Series and will show there
its relation to chemical combining power.
Magneto-electricity, obtained through the
relative motion of magnets and conductors, and
which Faraday had already studied in the First
Series.
Thermo-electricity, produced when the junction
between two different metals is exposed to heat.
A.
B
Animal electricity, which is produced by several fascinating families of both
freshwater and saltwater fishes. Faraday will study the wonderful electric eel in the
Fifteenth Series, one of the most engaging of all his researches. And, finally...
Common or ordinary electricity. This is what we
now call “static” electricity: the electricity produced
primarily by friction—for example, by rubbing a
resinous rod with wool, or a glass rod with silk. But how
often do we undertake such highly specialized activities
as these, except in a classroom or similarly contrived
setting? In our day there would seem to be nothing at
all “ordinary” about the electricity that arises from
friction; but I assure you that when I was a child, rugs,
sofas, and especially automobile seats, could easily give
you a very unpleasant jolt if you carelessly walked across a carpeted room, or slid out
of an upholstered piece of furniture, and then touched a doorknob or a water faucet.
Today, many fabrics contain antistatic materials which greatly reduce the frequency of
�3
such experiences; so for us, the terms “common electricity” and “ordinary electricity”
are no longer apt, and they are consequently no longer in common use.
Unfortunately, today’s more familiar term, “static electricity,” is misleading in its
own way; for many of the signs that alert us to the presence of static electricity occur
precisely when that electricity is not static! Those unpleasant shocks which lurked in
my family’s home and automobile, patiently awaiting their opportunity to strike,
represented the discharge of electricity which had previously been built up by friction:
they were instances of electricity in motion, not electricity at rest.
Faraday’s efforts to demonstrate the identicality of this “swarm” of electricities
occupies the first and longer part of the Third Series. Only then does he set out upon
the second part, where the topic is measure—and particularly the measure of quantity.
Readers may notice a distinctive suppleness in the language Faraday adopts for this
discussion: while he does not reject the imagery of electric fluids outright, he never
crafts his descriptions in a way that depends on that imagery.
Now, one way we can estimate quantity—whether of electricity or anything that is
evolved or produced—is to identify a repetitive element in the process that produces it;
then, presumably, each repetition of that action will produce an equal amount afresh.
Faraday obtained common electricity from a frictional “plate machine,” in which a large
plate of glass was rotated against a fixed
“rubber”—which was usually made of silkwrapped leather, rather than what we now
call rubber. The appliance shown here is a
smaller version of Faraday’s enormous
machine, which featured a glass plate of
fifty inches diameter—nearly four times as
large as this one. 2
At several points in the Third Series
Faraday treats each turn of his machine as
developing the same quantity of electricity.
You can see why such a supposition is
reasonable; for it is easy to make sure that all revolutions of the crank are
accomplished with uniform effort and speed. And to the extent that individual turns
are identical to each other, there is no obvious reason why successive turns would not
produce identical results.
2
Photo courtesy London Science Museum. The glass disk is 35 cm in diameter.
�4
This “same-again” principle of reasoning is familiar to us in other contexts, such as
grinding pepper in a mill. Indeed, in the case of grinding we are rewarded with a clear
image of “quantity” in the form of a heap of the ground
substance, as shown here. But when Faraday cranks
his plate machine, no “heap” of electricity is produced.
Is electricity even the sort of thing that possesses
“quantity” in the sense of a heap, a pile, or a mound?
Once again we are reminded of Socrates’ lament to
Meno: “If I do not know the ‘what’ of something, how
can I know the ‘such’ of it?” 3 In our present case, if we
do not know the “what” of electricity, is it really
meaningful to ask the “how much” of it?
When Faraday remarked that the term quantity
was “perhaps sufficiently definite as to sense,” he
meant to acknowledge that we habitually think of “quantity” through images of
accumulation or gathering up. But do not overlook the note of reservation suggested
by his word “perhaps.” Faraday is far from confident that electricity is really amenable
to such imagery. We regularly use such language for electricity without a second
thought; but can we point to any body of experience that gives real content to that
language?
If electricity does not manifest its quantity directly in experience, might it do so
indirectly? Sometimes, for example, we think it natural to express the magnitude of
something in terms of the power it exercises. Galileo offers a memorable instance in
the Two New Sciences; Sagredo is speaking:
“Thus a vast number of ants might carry ashore a ship laden with
grain. And since experience shows us daily that one ant can easily
carry one grain, and it is clear that the number of grains in the ship is
not infinite, but falls below a certain limit, then if you take another
number four or six times as great, and if you set to work a
corresponding number of ants they will carry the grain ashore and the
boat also. It is true that this will call for a prodigious number of ants...”
[67]
That delightful phrase, “a prodigious number of ants,” seems to employ the imagery
of number; but its rhetorical burden is rather the sheer magnitude implied by the
ability to move “the grain and the boat also.” The phrase expresses huge
undifferentiated totality, whose greatness is known primarily by what it can
accomplish. It is an indirect representation of quantity.
3
71A
�5
Frictional electricity, too, seems to express quantity only indirectly. When a rubber
rod is stroked with woolen cloth, it acquires the power to attract a small ball of cork or
I
\
pith. We say that the rod has been electrified, or charged with electricity; and in the
left-hand sketch, the electrified rod has succeeded in drawing the ball aside through a
moderate angle of perhaps 9 or 10 degrees. But after receiving additional strokes with
the wool, the rod is able to urge the ball to a greater angle—perhaps as much as 18 or
20 degrees, as shown on the right. Is it not reasonable to believe that the rod on the
right exerts more attractive force precisely because it has acquired more electricity?
But this is conjecture, not direct experience. Any notion of quantity we can gain
from this experiment is limited to what we can surmise from the angle of the
suspended pith ball. But angle is no image of “muchness,” and it shares none of the
straightforwardness of such eminently legible figures as heap, mound, or—in the fluid
case—puddle.
If not the pith ball, then, might some other electrical instrument offer a more
immediate experience of electrical “quantity”? The distinctive power of electrified
bodies to attract or repel other electrified bodies is the principle of several electric
indicators that are considerably more refined than the pith ball.
Two early
instruments operate on the principle of mutual repulsion. The leaves of the gold-leaf
electroscope, pictured here on the left, diverge from one another more or less,
.
depending, partly, on how many times the rubber rod has been stroked. On the right,
Henley’s electrometer calls even sharper attention to angle by incorporating an obvious
pointer and protractor in its design; when the instrument is mounted on the electrified
conductor of a plate machine like Faraday’s, the pointer is repelled from the body, just
like the leaves of the electroscope. With its angular scale, the Henley instrument
emphatically announces its rhetoric of numerical measurement—and hence its name
“electrometer” rather than “electroscope.” But what, exactly, does it measure? The
�6
angle of the pointer, even when expressed numerically, still seems far removed from a
direct image of quantity.
In fact, one of Faraday’s experiments in the Third Series suggests that the
electrometer is better understood as indicating some other electrical attribute—an
attribute rather different from quantity, though it may be related to quantity. Faraday
describes that experiment in paragraph 363 of the Third Series. It involves an array,
or “battery,” of fifteen identical Leyden jars, like this one. You see that the central
conductors, which are connected to the jars’ inner coatings, are all joined together.
Within the wooden container, the outer coatings rest upon a conductive plate that is
connected to the flexible chain B, which in turn is connected to the earth.
Faraday will charge these jars using the plate electric machine. Notice the Henley
electrometer mounted on the prime conductor; this was one of the chief applications
of the Henley device.
At first Faraday connects only eight of the jars, charging them by thirty turns of the
plate machine. This causes the electrometer to rise to some position A. Does that
position represent the quantity of electricity supplied to the jars? Certainly that
quantity must be considerable, since Faraday noted that merely one revolution of the
plate will, in his words, “give ten or twelve sparks from the conductors, each an inch in
length.” 4
At a later stage of his experiment, Faraday charges all fifteen jars, again by thirty
turns of the machine. This time, he reports,
The Henley’s electrometer stood not quite half so high as before...
4
Paragraph 290.
�7
Obviously the electrometer is not measuring quantity! For the quantity of
electricity was the same in both cases—the result of thirty turns of the machine. Yet
with a greater number of jars, the electrometer reading was lower by more than half.
What electrical characteristic was it, then, that the electrometer measured when it
registered that striking reduction?
In hopes of answering this question, let us conduct an experiment of our own. Recall
that Faraday noted the generous number of sparks produced with each turn of the plate
machine. This should give us pause: why does the machine produce a series of sparks
rather than one continuous spark?
To study the conditions under which spark develops, I will use an electrometer of
still greater refinement—one which, although invented long after the Henley device,
does not differ greatly from that instrument in the essentials of its operation. The
electrostatic voltmeter operates on the principle of attraction rather than repulsion. On
C
the left is a photograph of our meter. It dates from the 1950s, and is therefore
calibrated in units whose defining assumptions would have had little meaning to
Faraday. But we can regard the scale divisions as arbitrary units of attractive force; let
me explain this.
On the right is a much-simplified diagram of the meter’s internal mechanism. A
movable plate B is mounted on a pointer which pivots at C and is held in an equilibrium
position by a very light spring. Plate A is fixed in place. When the plates are oppositely
electrified, they attract one another; and plate B will move upward until its force of
attraction is balanced by the spring. The pointer’s angle of displacement then reflects
the amount by which the spring has been stretched, and therefore, also, the force of
attraction between the plates. The scale divisions are so marked as to represent,
broadly, equal increments of that force. 5
We will connect the electrometer’s plates to a Wimshurst machine. I have separated
the machine’s terminals by about a millimeter or so (VIDEO BEGINS).
This is not really accurate, since true volt-meters must take into account both the plate separation and
effective plate area, both of which vary as the reading increases. But in the meter we are using, the
correction can be ignored for our purposes.
5
�8
Next, I will slowly crank the machine—and notice that the meter rises until a spark
develops, at which point the needle suddenly falls. As I continue to crank, the meter
repeatedly exhibits this pattern of rise to a maximum, followed by abrupt descent when
the spark passes. The maximum is not always the same; but there always is a
maximum, and the subsequent descent always coincides with the spark.
The regular association between the meter’s descent and the spark suggests a more
pointed question: “What is the condition between the terminals just before the spark
passes?” Whatever that condition is, it evidently results in spark each time it occurs.
And since the electrometer consistently develops a maximum reading just prior to each
spark, it seems very likely that the electrometer is indicating precisely that condition
which, when it reaches a certain degree, results in spark. What, then, is the nature of
that condition?
Faraday thought of the spark—and, for that matter, all instances of electric
discharge—as the breakdown of an antecedent state of stress in the region where the
discharge takes place. Faraday calls that region, or the material which may occupy it,
the “dielectric.” Here is his description in the Twelfth Series:
All the effects prior to the discharge are inductive; and the degree of
tension which it is necessary to attain before the spark passes is
therefore ... a very important point. It is the limit of the influence
which the dielectric exerts in resisting discharge; it is a measure,
consequently, ... of the intensity of the electric forces in activity.
This golden passage finally lends imaginative content to the term “intensity,” which
seemed so questionable to Faraday at the outset of the Third Series. The chief
manifestation of electrical action is a condition of tension in the region between two
surfaces, and that action is said to possess intensity commensurate with the degree of
that tension. “Intensity,” then, characterizes the action; “tension” the region or
material that experiences that action.
The distinction between intensity and tension is a subtle, but a natural one. We find
a comparable distinction in two descriptions of Odysseus’ great bow in Book 21 of the
Odyssey. The suitor Antinous knows the bow in terms of its own strength, which makes
stringing it so difficult. He warns the crowd: 6
6
Homeric passages translated by Gilbert Murray.
�9
“For not easily, I think, is this polished bow to be strung.”
(line 90)
(The image in this slide is that of a fifth-century Theban coin.) But once the bow is
strung and in action, it is known by the thrum of its string, the sign of surpassing
tension: 7
And Odysseus held it in his right hand, and tried the string, which sang
sweetly beneath his touch...
(line 408)
Just as Odysseus’ stout bow reveals its strength through the superlative degree of
tension it creates in the string, so electric action reveals its strength, or intensity, in the
form of tension in the material between oppositely-charged electrodes. Intensity and
tension are two different rhetorical aspects of electrical action: “intensity”
characterizes the action itself (corresponding to the bow); “tension” characterizes the
material or region which experiences that action (analogous to the bowstring). Do not
underestimate the scientific importance of such metaphorical images as those of string
and bow. Without them, or something like them, our understanding of natural powers
would degenerate into a merely formal correlation of numbers with numbers. But any
reader of Faraday quickly discovers that Faraday has little interest in symbols,
numerical or otherwise. Faraday is constantly alert for legible images that convey the
essential character of nature’s beings and powers. What is so remarkable about
Faraday’s experimental practice is how much of it consists in allowing the phenomena
to reveal their own images. 8
7
8
Illustration: detail from an etching by Theodoor van Thulden, part of a series produced in 1632–33.
Fisher, Howard, “The Great Electrical Philosopher,” The College, XXXI,1 (July 1979).
�10
Faraday’s interpretation of electrical discharge as being essentially a release of
antecedent tension departed sharply from the then-accepted account, represented here
on the left. Conventional thinking posited a buildup of opposite electric fluids on the
0 5 ed buildup of
.
Sup_
1• e (+) electric fluid
pos11v
sed buildup of .
Supo. (-) electric fluid
negative
j Tension
surfaces between which spark took place. As those fluids accumulated—or so the
account maintained—the inherent repulsion of like portions of fluid, combined with
the mutual attraction of unlike portions, would eventually propel the electrical
substances across the gap to combine with and nullify one another. Notice that the
conventional view recognizes no role for the space or material between the charged
surfaces; all action is ascribed to the electrical fluids.
Faraday’s view—represented on the right—reverses the order of priority by
focusing on the gap rather than the bodies which it separates, ascribing tension to the
gap, but assigning no causative role to the adjoining bodies, nor to any supposed
buildup of electricity upon them. If the dielectric material occupying the gap is capable
of sustaining high degrees of tension, it constitutes what we call an “insulator”; but all
known insulators, including air, have a limit to the tension they can sustain, and when
this limit is exceeded, they break down, electrically speaking. The release of tension
associated with that breakdown is disruptive discharge, or spark. In contrast to
insulators, the materials classed as “conductors” are incapable of withstanding any
tension at all; they break down under the slightest degree of electrical tension, and the
condition of continuous breakdown under tension is how Faraday understands
“current” in a conductor.
Thus the electrometer’s pattern of rise and sudden fall in our spark experiment
gives us reason to believe that the electrometer measures that very tension—or its
rhetorical counterpart, intensity. 9 How does it do so? If you recall our earlier diagram
of the electrometer’s inner workings, you will remember that the needle’s
Throughout the Eleventh and Twelfth Series we find Faraday using the terms “tension” and “intensity”
almost synonymously.
9
�11
displacement indicated the degree of extension of the internal spring, and hence the
force on the moving plate—or, rather, the tension in the region between the plates. But
of course the condition of the electrometer’s own plates is not what we are interested
in! If the electrometer is to function as a measuring instrument, the pointer’s
displacement must tell us about some other object—the object whose condition we
wish to measure. How is that possible?
Consider, from the standpoint of tension, what must be the case when the
electrometer plates are connected to the terminals of the Wimshurst machine. When
C
the machine is operated, electrical tension is established in the air between its
terminals D and E. I say that equal tension must therefore develop in the region
between the electrometer plates A and B; for if the tensions were not equal, the
conductors DA and EB would together have to bear the difference between those
tensions. But recall that, for Faraday, a conductor is incapable of sustaining electrical
tension. Thus the tension between A and B must be equal to the tension between D
and E; and the needle’s displacement will therefore reflect not only the tension
between the electrometer plates but the tension between the Wimshurst terminals as
well.
Have we gained any fuller understanding of those troubling electrical terms,
quantity and intensity? Faraday’s study of the forms of electric discharge, especially
spark, led to the idea of electric tension; and that image of tension, in turn, does indeed
seem to offer a firmer notion of intensity, namely, the action producing a certain level
of tension in a dielectric.
But what about quantity? Initially, we looked to the electroscope as an indicator of
quantity; but successive refinements of that instrument brought us, not closer to, but
farther and farther away from the expected imagery. All our attempts to find, in
experience, the imagery that a material substance would ordinarily demand—a
localized heap, mound, or puddle—have led us instead back to tension. Why do the
phenomena of static electricity seem to lead us so persistently away from “heap”
imagery and toward the vocabulary of tension? Might that be a sign that tension is
actually more fundamental than quantity?
In fact, Faraday already has ample grounds for this view; for if electrifying a body
really represents the accumulation of electric substance upon it, we ought to be able to
�12
electrify a body “absolutely," that is, without relation to any other body—just as we can
fill a glass with water regardless of whether or not we fill any other container with
water. But Faraday’s famous Cage Experiment, along with other investigations,
showed definitively that no body can be in a “charged” condition at all except through
a mediating relation with some other, oppositely charged, body. This means that there
is no such thing as a quantity of electricity in itself. Every instance of electric charge is
but one element of a mutual relation to which Faraday gives the name “induction”; and
in a striking passage in the Eleventh Series he explicitly elevates the relation over the
things related:
All charge is sustained by induction. All phenomena of intensity
include the principle of induction ... All currents involve previous
intensity and therefore previous induction. INDUCTION appears to be
the essential function both in the first development and the
consequent phenomena of electricity. [1178]
Furthermore, since all of what Faraday calls the “phenomena of intensity” involve
tension in a dielectric, then it is the dielectric, not the so-called “charged” body, which
is to be counted as the principal entity in static electricity. In Faraday’s words,
In the theory of induction founded upon ... action of the dielectric, we
have to look to the state of that body principally for the cause and
determination of the ... effects. [1368] 10
If the dielectric is indeed the principal entity in static electric induction, it is easy to
see why Faraday devoted so much of the Eleventh Series to studying the dielectric
specifically. To that end, he designed the special “inductive apparatus” illustrated here.
The appliance on the left is an historical reproduction; 11 Faraday’s own diagram
appears on the right. Today we would call this contrivance a spherical capacitor; but it
In an omitted term Faraday characterizes the action in question as “molecular.” By this he merely means
action at the level of small portions of the dielectric. He does not refer to chemical molecules of the sort
propounded by atomic theory—as readers of his 1844 paper, “A Speculation touching Electric Conduction
and the Nature of Matter,” will appreciate. See Experimental Researches in Electricity, Vol. II (1844), p. 284.
10
Photograph generously supplied by Dietmar Höttecke; see Höttecke, Dietmar, “How and What Can We
Learn From Replicating Historical Experiments? A Case Study.” Science & Education 9, 343–362 (2000).
11
�13
is essentially a Leyden jar consisting of an outer and an inner conductor, with electrical
connection to the inner conductor established by a conductive wire terminating at the
little sphere on top. Faraday’s experiments established for all time the pre-eminent
role of the dielectric in induction.
We can emulate Faraday’s induction experiments. 12 In place of his spherical
capacitors, we shall use a pair of our adjustable plate capacitors, set to equal plate
separations and thus electrically identical.
Faraday placed his two identical inductive devices on a grounded metal work
surface, so that their outer conductors were permanently connected to the earth while
their inner conductors remained free. We will use a heavy copper wire for the same
purpose by connecting it to the earth. The righthand plates of our capacitors are joined
to it, and are thus in permanent electrical contact. The lefthand plates will be isolated
from one another, except when I briefly connect them later.
To measure the electrical tension that developed when his devices were charged,
Faraday employed a sensitive torsion balance, pictured here on the left. That fine
instrument balanced the tension between two electrified spheres against the elastic
twist of a slender thread—just as our modern electrometer, as in the diagram we saw
earlier, balances the tension between two electrified plates against the elastic stretch
of a spring. Both instruments, therefore, serve to measure electric tension.
12
Faraday describes this series of experiments in paragraphs 1208–1214.
�14
Faraday possessed only a single balance with which to measure both his inductive
devices; but we have the luxury of using two electrometers, one for each capacitor,
A
B
To earth
designated A and B, respectively. Let me first outline the procedure we shall be
following; then I’ll show some videos of the actual experiment.
Faraday began by charging only one of his devices. Similarly, I will connect the
Wimshurst machine to capacitor A alone, and crank it until the electrometer
B
To earth
approaches its full scale reading. Capacitor A will thus sustain a definite tension,
indicated by the electrometer. Capacitor B, of course, will remain uncharged and will
sustain no electric tension.
Next I will momentarily join the ungrounded capacitor plates. Now, think about
To oatlh
what must happen when I do that. The joining wire is a good conductor, so it cannot
�15
sustain tension; therefore when contact is made, the electrical condition of both
capacitors should instantly change to make their respective tensions equal, and we
should expect both electrometers to read the same. That will constitute the first part
of our experiment; so now, let us carry out the steps I just described (VIDEO BEGINS).
Here is the setup. The copper wire that is appearing on the left will connect
capacitor A to the Wimshurst machine... Now I am cranking the machine, and you can
see the electrometer rise almost to its full scale.
And here is a closeup view of the electrometer; it shows that Capacitor A is
sustaining a tension of 2.80 units. I could not fit the second electrometer into this view,
but it reads zero—as of course it must, since Capacitor B was not charged.
Now I join the capacitors momentarily ... and the tension in Capacitor A falls; we’ll
take a closeup look at the electrometer to see the new value...
The tension in Capacitor A has fallen to 1.37 units, while the tension in Capacitor B has
risen to the same amount, as it must—though, again, I could not include both meters in
the same view.
Now, this change in tension took place when I allowed Capacitor A to share its
electricity with Capacitor B. But since the capacitors are identical, they ought to divide
that electricity equally—so that each capacitor should now embrace half the quantity
of electricity that resided originally in Capacitor A alone.
And the tension in both capacitors is 1.37 units, that is, almost exactly half the initial
tension of 2.80 units. Thus as the quantity of electricity in Capacitor A diminished to
half, so too its tension diminished to half. Evidently tension is here proportional to
�16
quantity! But doesn’t this contradict what we saw in the Third Series? For there, when
Faraday charged first eight Leyden jars, and then fifteen, with the same quantity of
electricity, his Henley electrometer gave two different readings; and obviously if one
magnitude can take on two different values while the other remains unchanged, those
magnitudes cannot be proportional.
This reasoning, though, overlooks a critical difference between the two
experiments. In the Third Series, Faraday was comparing the tension of a fixed quantity
of electricity distributed first over eight jars and then over fifteen jars, as illustrated
here. The electrometer readings are indeed very different, just as Faraday reported.
But our experiment, like Faraday’s in the Eleventh Series, compares the tensions of
different quantities of electricity in one and the same capacitor. The two experiments
are not comparable, because in the earlier exercise the physical environment
underwent significant change—from a smaller number to a greater number of jars—
while in the later experiment the environment did not change: the electrometer
measured the variation of tension in one and the same capacitor.
Clearly, the physical environment affects how much tension a given quantity of
electricity will develop. This should not surprise us, since that environment includes
the dielectric; and we have already seen how central is the role of the dielectric,
according to Faraday’s thinking.
The next step in Faraday’s experiment, and in ours, will confirm that central role by
showing that different dielectric materials develop specifically different tensions.
Faraday filled the air space in one of his devices with various substances; and we shall
do the same to our capacitor B by inserting a sheet of glass between its plates. Then
we will run through the same experimental sequence as before; but remember that this
time, our capacitors will no longer be identical.
(VIDEO BEGINS.) You see I have mounted a glass sheet between the plates of
Capacitor B.
�17
And again we connect Capacitor A to the Wimshurst machine, and charge it to an
initial tension.... Its electrometer reads 2.83 units, nearly the same as before, while of
course the other electrometer continues to read zero.
Again I briefly join the two capacitors together; and the electrometers once more
display equal deflections—as they must, since the tensions have to be equal. But notice
that this time the tension is not equal to half the original tension... Instead the tension
is only 1.02 units, roughly one-third of the initial tension. How shall we understand
this?
When Faraday obtained a similar result with his spherical capacitors, he concluded
that the apparatus containing a solid dielectric had, in his words, “a greater aptness or
capacity for induction” than the apparatus whose dielectric was air. To see what he
means by this phrase, let us analyze our results in the same way that Faraday
interpreted his. When I joined the two devices, the charged capacitor gave some of its
electricity to the uncharged capacitor. Specifically,:
The capacitor with air dielectric lost a certain quantity of electricity, and
its tension decreased by 1.81 units.
The capacitor with glass dielectric gained that same quantity of
electricity, but its tension increased by only 1.02 units—a much smaller
amount.
Air dielectric-greater change
Glass dielectric-smaller change
I~ \
I~ \
by 1.81 units
by 1.02 units
�18
Thus one and the same quantity of electricity is associated with lower
tension when the dielectric is glass, and higher tension when the
dielectric is air.
Evidently, then, “greater capacity for induction” means the ability to sustain the
same quantity of electricity at a lower tension. Or, equivalently, it denotes the ability
to sustain a greater quantity of electricity at the same tension.
We could go on, as Faraday does, to show that a dielectric’s “capacity for induction”
depends on its dimensions as well as its specific material. But the main point is clear:
where static electricity is concerned, our only access to electrical “quantity” is
indirect—through the measurement of tension, 13 taking account of the medium’s
capacity for induction. And thus we must regard electrical quantity as only an
alternative rhetorical expression for tension—a special figure of speech. Recall
Faraday’s earlier remark, that we have to look principally to the state of the dielectric
for the determination of the electric effects. In contrast, he described the supposedly
“charged” conductors in this almost dismissive way:
The conductors ... may be considered as the termini of the inductive
action.... [1361]
Charged bodies, then, are merely the boundaries of electrical action, not its cause!
To say that a body is “charged” no longer labels it as the source of electric effects, but
merely the place where a medium that does sustain tension switches to a medium that
does not. With this characterization, Faraday has effectively turned the conventional
order of causal priority on its head. Charge is no longer prior to tension; rather, tension
is prior to charge. Whatever else this may mean, it fatally undercuts the notion that
“charge” is the name of an electrical substance, for—to use an Aristotelian formulation
that would have been quite foreign to Faraday: “How can a non-substance be prior to
a substance?” 14
I hope I have conveyed how thoroughly Faraday’s account of electricity inverted the
conventional understanding. At the same time, I hope it is clear that Faraday did not
arrive at his unorthodox view through polemic or disputation. He did not marshal
evidence so as to refute the established conceptual scheme. In fact, at least in the
Experimental Researches, Faraday hardly ever engages in “collecting evidence,” any
more than he engages in symbolic mathematics. Instead, he looks directly to nature
showing itself.
Classic doctrines of scientific “method” emphasize putting hypotheses and
conjectures to the test, establishing a preponderance of evidence for or against them.
For electricity undergoing discharge, as Faraday shows, the ballistic galvanometer offers an alternative
measure of quantity. But while it might seem obvious that when electricity discharges, its quantity in
discharge must be the same as its quantity prior to discharge—when it was still static—the problem of
correlating the measures of static and dynamic electricity would prove to be a knotty one. It would
eventually become the problem of relating the electrostatic unit to the electromagnetic unit, the problem
that would lead Maxwell to his electromagnetic theory of light.
13
14 Aristotle, Physics, Book I (189a34) tr. Cornford. In the present case, how can tension (not a substance) be
prior to electric fluid (a substance)?—implying that electric “fluid” is not actually a substance after all.
�19
Such an approach is suited to an alien world, a world indifferent to human
understanding, a world in which, as has been said, “nature loves to hide.” 15 Faraday’s
world, on the contrary, shows itself in forms that may challenge our understanding;
but they are not incommensurable with it. Faraday’s science flourishes in a world that
is fit for us, a world that is preeminently knowable.
How did Faraday manage to nourish a scientific outlook so little influenced by
conventional scientific doctrine? A customary answer to this question singles out
Faraday’s lack of a conventional education. To be sure, Faraday had little formal
education and was largely self-taught; but the materials of his self-education were
steeped in established knowledge. As a bookbinder’s apprentice, he read volumes of
the Encyclopædia Britannica while engaged in binding them. By his own account he
benefited greatly from Jane Marcet’s Conversations in Chemistry, a lovely book which,
however, reliably held to established and accepted teachings. 16 Through the
generosity of a friend of his employer, Faraday was able to attend lectures by
Humphrey Davy, an establishment figure in science if there ever was one. I do not think
it was ignorance of established science that explains Faraday’s relative indifference to
it. Much of his practice in “reading the book of nature” 17 points instead to his religious
tradition.
Faraday belonged to a very small Christian denomination, the Sandemanians, a
dissenting offshoot of the Church of Scotland. Sandemanians eschewed theology and
had no established clergy; instead, the Bible was the central source of guidance in every
aspect of their lives. Reading the Bible demanded no special credentials, for it was
written in human language for the sake of human understanding. 18 Similarly, they saw
the natural world as having been created as a gift and a fitting home for mankind. Like
the biblical text itself, the created world was seen as a channel of God’s communication
with the human race.
You can see how such views concerning nature could inform Faraday’s methods of
natural investigation. If natural phenomena show themselves in terms we can grasp,
they will not need to be expressed mathematically—or, for that matter, through any
other external symbology. We see from Faraday’s own example that the study of
nature requires patient and prolonged labor—but much of that labor stems not from
nature’s recalcitrance but from our own sluggishness to put familiar thought patterns
aside—what Faraday once called “mental inertia” 19—and allow the phenomena to
speak to us directly. For Faraday, at least, the means for cultivating an ear for nature’s
15
Heraclitus, B123
Jane Marcet never sought to break new scientific ground; but by composing instructional texts that were
explicitly directed to young women, she conspicuously broke new social and educational ground.
17 Geoffrey N. Cantor, “Reading the Book of Nature: The relation between Faraday’s Religion and his
Science” in Faraday Rediscovered: Essays on the Life and Work of Michael Faraday, 1791–1867. The
Macmillan Press, Ltd. (1985).
16
See David Gooding, Michael Faraday, 1791–1867: Artisan of Ideas. http://www.bath.ac.uk/~hssdcg/
Michael_Faraday.html, 15 June 2002; accessed 4 September 2023 through the Wayback Machine.
18
See Faraday’s “Observations on Mental Education” (1854) in Experimental Researches in Chemistry and
Physics (1859), p. 463
19
�20
dialect and an eye for its forms are practical rather than analytical. Before he asks
questions in speech, he asks them in practice; such are Faraday’s experiments.
Nevertheless, while Faraday’s mode of experimenting clearly reflects central
elements of the Sandemanian outlook, it would be a mistake see him only as dutifully
putting the Sandemanian creed into action. Faraday just doesn’t write as though he
were feeling the weight of doctrinal obligation. His prose, both in his laboratory Diary
and in the published Researches, is simply too fresh, too lively, too responsive to what
just happened. There is a palpable difference between being open to nature and
observing a code of being open to nature. I invite you to think about that difference—
the difference between responsiveness and responsibility 20—and how it plays out both
in consciousness and in speech. But for now let us return to the terms “quantity” and
“intensity,” the two candidates for electrical measure; for as regards their lucidity, I
think we will have to acknowledge that the terms have effectively exchanged places.
The term intensity, which Faraday initially found “more difficult to define,” has
gained considerable clarity, since Faraday has been able to assimilate to it the figures
of speech associated with tension; and we may now understand electrical intensity as
commensurate with the degree of tension developed in a specified region. But the term
quantity, which Faraday previously thought “sufficiently definite as to sense” has
instead become highly questionable. For the “definite sense” of that term rested on the
image of heaping up or accumulation of electrical substance; and we have seen how
that image has repeatedly failed to find any grounding in experience. Moreover, now
that Faraday has identified the primary electrical entity as being the dielectric under
tension, not the so-called charged body, any idea of “quantity of electric substance” can
only be regarded as a merely verbal one—a figure of speech. Under such
circumstances, would it not behoove any responsible thinker to avoid the term
“quantity of electricity” altogether? And yet Faraday continues to speak of “quantity of
electricity” throughout the remainder of the Eleventh Series, and in the Twelfth,
Thirteenth, and Fifteenth Series. Why would he do this?
Faraday nowhere speaks directly to that question as regards electrical terminology;
but he does address a similar one in connection with the language of atoms. Some of
you have read, and some of you will read, his 1844 paper, “A Speculation touching
Electric Conduction and the Nature of Matter.” 21 In that essay, after having reviewed
his many reservations about the theory of atoms, and hence also the atomic language
that takes their existence for granted, he nevertheless admits,
I feel myself constrained, for the present hypothetically, to admit them
[that is, atoms], and cannot do without them.
Here, then, is another instance where Faraday feels obliged to make at least
provisional use of a terminology that has not been grounded in phenomena. A
doctrinaire purist would have avoided such a compromise; but Faraday’s openness
Contrast, for example the Knight of Faith in Kierkegaard’s Fear and Trembling with the rule-inferring
“insomniac” who, reflecting on Abraham’s willingness to sacrifice Isaac, confidently deduces, “Oh, I see how
it works: you raise the knife, and then suddenly there’s a ram!”
20
21
Experimental Researches in Electricity, Vol. II (1844), p. 284, esp. page 289.
�21
extends to language as well as to experience, for each of these must evolve along with
the other.
Natural phenomena show themselves in forms and images that human beings can
apprehend; and those images continually try to shape a language that is anchored in
the phenomena. But such a language requires discovery, interpretation, and
adeptness; and these in turn require time, patience, and love. As we do not expect to
take in a dialogue, or a drama, on first reading, we must not expect to “perform”
experiments once only and then set them aside. We must live with them, enter into
them, and try them again and again. The idea is less to get the right answer, than to
capture the right idiom. The book of nature deserves multiple readings; and no two of
those readings are likely to be quite the same.
�
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
paper
Page numeration
Number of pages in the original item.
21 pages
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 the Measure of Electricity?
Description
An account of the resource
Transcript of a lecture given by tutor Howard Fisher on February 23, 2024 as part of the Dean's Lecture & Concert Series. The Dean's Office has provided this description of the event: "Faraday made use of numerous electrical measuring instruments; but what, exactly, did they measure? What properties of electricity are "measurable" at all? Faraday's efforts to identify these properties raised a question which Meno would have recognized: how can we know the properties of electricity unless we first know what electricity actually is?"
Creator
An entity primarily responsible for making the resource
Fisher, Howard J., 1942-
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
2024-02-23
Rights
Information about rights held in and over the resource
Meem Library has been given permission to make this item available online.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Subject
The topic of the resource
Electricity
Faraday, Michael, 1791-1867
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
SF_FisherH_What_is_the_Measure_of_Electricity_2024-02-23
Friday night lecture
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/c4519c1d1e2473f5c5d1e8c084fca69c.pdf
7dcbb75d872f9571b1b7d7deb003e365
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
paper
Page numeration
Number of pages in the original item.
23 pages
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 discussion?
Description
An account of the resource
Transcript of a lecture given in September 1991 by Kent Taylor.
Creator
An entity primarily responsible for making the resource
Taylor, Kent
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
1991-09
Rights
Information about rights held in and over the resource
Meem Library has been given permission to make this item available online.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Subject
The topic of the resource
Discussion -- study and teaching
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
24000268
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/c71f19b6a666d027cec9ed79d9f07cc9.pdf
227d8a113c280f3566a8ce7ae6e4d15c
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
pdf
Page numeration
Number of pages in the original item.
33 pages
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
Truths about quantum mechanics
Description
An account of the resource
Transcript of a lecture given on November 30, 2018 by Bernhardt Trout as part of the Dean's Lecture and Concern Series.
Creator
An entity primarily responsible for making the resource
Trout, Bernhardt L.
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
2018-11-30
Rights
Information about rights held in and over the resource
Meem Library has been given permission to make this item available online.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Subject
The topic of the resource
Quantum theory
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
Trout, B. Truths about Quantum Mechanics
Friday night lecture
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/ce789f61b6997180ca1dc76624bae159.pdf
96136207810f761155e2fa715e4dd4fa
PDF Text
Text
Trigonometric Interpretation of Complex Numbers
Grant Franks
June 3, 2019, revised September 16, 2019
Dedication
Caspar Wessel (1745 - 1818)
Let us pause for a moment to remember and give thanks for Caspar Wessel, Norwegian mathematician
and cartographer, who conceived the idea that complex numbers might be usefully portrayed on a
map.
Introduction
We ended the last talk with Rafael Bombelli staring at Cardano’s Formula as applied to the cubic
equation
x 3 - 15 x - 4 = 0
The formula gives for a solution:
x=
3
2 + 11
-1
+
3
2 - 11
-1
which at first glance appears to be nonsensical since it consists of two terms both containing the cuberoot of expressions involving the square-root of negative one, which mathematicians in other contexts
�2 ���
2 Trignometry and Complex Numbers.nb
agreed meant that no solution was possible. But Bombelli knew that there was a solution to this
equation. In fact, he knew what at least one of the solutions was: the integer + 4 solves the equation.
43 - (15) (4) - 4 = 64 - 60 - 4 = 0
Bombelli figured out by some combination of guesswork, deduction and just inspired staring that there
are expressions which, when cubed, give 2 + 11 -1 and 2 - 11 -1 . They are 2 + -1 and
2 + -1 , respectively. Hard as it is to find them, it is easy to confirm that they work. All you need do
is to multiply them by themselves three times, keeping in mind the one rule we know about -1 ,
namely, that when multiplied by itself it gives -1.
-1 = 4 + 4
2 +
-1 2 +
2 +
-1 = 2 +
2 -
-1 2 -
2 -
-1 = 2 -
3
-1 - 1 = 3 + 4
2
-1 2 +
-1 = 3 + 4
-1
-1 2 +
-1 = 2 + 11
-1
-1 = 2 - 11
-1
And
-1 = 4 - 4
3
2
-1 2 -
Knowing the cube roots of 2 ± 11
x=
3
2 + 11
-1
-1 - 1 = 3 - 4
+
3
-1 = 3 - 4
-1
-1 2 -
-1 allowed Bombelli to solve the particular problem facing him:
2 - 11
-1 = 2 +
-1 + 2 -
-1 = 4.
However knowing the answer to this problem doesn’t show us how to deal more generally with other
numbers involving -1 .
The answer to that problem is the subject of tonight’s talk.
Arithmetic of Complex Numbers
Their Real and Imaginary Parts of a Complex Number
The most evident problem with -1 is that it doesn’t stand in a relation of “more” or “less” with
regard to other numbers we have come across. That feature more than any other makes -1 seem
especially weird. When one takes the step from the whole numbers to fractions (that is, to positive
rational numbers), things like “one-half” or “five and a quarter” could be related as greater or less than
whole numbers we were already familiar with. Later, for all its undefinable strangeness, an irrational
like 2 at least sat snugly between rational numbers, greater than some and less than others. (That,
in fact, is how Dedekind defined irrational numbers, namely, by identifying which rationals each was
�2 Trignometry and Complex Numbers.nb
���
3
greater than and which it was less than.) Even the very strange negative numbers aren’t as peculiar as
-1 . If your idea of a number is that it should respond to counting something or measuring something, negative numbers are nonsense because there is less than nothing there to count or to measure.
But if you can get over that problem, at least negative numbers still stand in greater-and-lesser relations to one another.
Not so -1 . It is neither greater than nor less than any real number. That much is pretty clear: the
square of every real number is positive, or at least “non-negative.” The -1 is not anywhere on the
real number line that stretches from enormous negatives to enormous positives. So, if we are to
imagine it at all, we have to picture it being “somewhere else.”
Caspar Wessel set the imaginary numbers apart from the reals on an axis of their own at right angles to
the real number line. He thus established a complex number plane. One axis represents the real
numbers, the other the numbers that include -1 . A real number and an imaginary number together
form a two-part entity called a “complex number.” Each point on the complex number plane represents a single complex number. A complex number can look like:
2+
-1 or
-3 + 9
-1
or
0 +5
-1
or
-4 + 0
-1 .
The first and second examples have both a real and imaginary part. The third has only an imaginary
part; the real part is zero. The last example has only a real part; the imaginary part has a zero coefficient.
At the risk of seeming overly pedantic, I want to note here that “having an imaginary part with a zero
coefficient” is not quite the same thing as “being a real number.” The complex number “-4 + 0 -1 ” is
not quite the same thing as the real number “-4.” The reason for this hyper-technicality and squeamishness about nomenclature is not at all clear at this point and it won’t become clear until the last lecture.
It’s not unusual to overlook this distinction and, for now, doing so won’t cause any problems. It is
common, even convenient, to skip over the zero terms and to write “-4 + 0 i” as just “-4” I mention this
not-yet-developed distinction only so that, when it comes back again in the final lecture, I can say “As I
have already said …”, and you will all nod sagely in agreement.
The real and imaginary parts of a complex number stand in different orders and, when they are added
or subtracted, they act independently of one other. In modern parlance, one might say that a complex
number can be represented as a vector on a plane with a real axis in one direction (generally, le�-right)
and an imaginary axis orthogonal to it (up-down). That’s not how Caspar Wessel spoke because the
term of a “vector” wasn’t introduced until the middle of the 19th century, decades a�er Wessel died.
But the fundamental idea is there: a complex number is a two-part object whose parts add independently of one another. That idea had been around for years, at least since Isaac Newton had analysed
motions into components towards and parallel to the sides of a parallelogram.
�4 ���
2 Trignometry and Complex Numbers.nb
5i
4i
3+4i
3i
2i
-3 + 2 i
i
-5
-4
-3
-2
-1
0
1
2
3
4
5
-i
-4 - i
-2 i
-3 i
-4 i
2-4i
-5 i
Here, then, is the representation of four complex numbers on a complex number plane: 3 + 4 i, 2 - 4 i, -4
- i and -3 + 2 i. So far, this is just a picture. Its value appears as we see how it is used.
Addition and Subtraction of Complex Numbers
In addition, the real and imaginary parts act separately. So, if one adds
-3 + 2 i
to
4+2i
one gets
(-3 + 4) + (2 + 2) i
The real parts add ordinarily, and the imaginary parts do too, thanks to the (formerly implicit, now
explicit) understanding that “distribution of multiplication over addition” works for the number i as it
does for other numbers, so that we have:
2 i + 2 i = (2 + 2) i = 4 i.
This procedure is just what one would do with components of a vector or of a decomposed Newtonian
force or velocity. Graphically, as Wessel proposes envisioning complex numbers, the result looks like
this:
�2 Trignometry and Complex Numbers.nb
���
5i
4i
1 + 4i
4+2i
3i
2i
-3 + 2 i
i
-4
-3
-2
-1
0
1
2
3
4
-i
Multiplication by a real number (or real part of a complex number) is like ordinary
multiplication
Multiplying a complex number by a real number amounts to multiplying each of the real and complex
parts of the complex number as you would expect. For multiplication by positive integers, the result is
just like repeated addition of the vector representing the complex number. Multiplication by negative
numbers is like repeated subtraction.
5
�6 ���
2 Trignometry and Complex Numbers.nb
5i
4i
Multiplication of 3 + i by 3 + 0 i
3i
2i
i
-2
-1
3+i
0
1
2
3
4
5
6
7
8
9
10
-i
-2 i
So far, so good. The graphic representation hasn’t yet shown us anything novel about complex numbers or given us new, but there is more and better yet to come.
The Crux of the Problem: Imaginary Multiplication
Next we have to deal with complex numbers times other complex numbers. This is where things get
interesting. It’s not immediately clear what that means graphically, but we do have an algebraic
understanding. The one thing we know for sure about -1 is that when you multiply it by itself, it
gives -1.
Let’s go back to the example we have already seen: Bombelli’s discovery that 2 + i is the cube root of 2
+ 11 i. As we showed already, we can multiply 2 + i times itself:
(2 + i) (2 + i) = (2 ⨯ 2 )+ (2 ⨯ i )+ (2 i ⨯ 2) + ( i ⨯ i) = 4 + 2 i + 2 i - 1 = 3 + 4 i.
�2 Trignometry and Complex Numbers.nb
5i
(2 + i)(2 + i) = 3 + i
4i
3i
2i
i
-2
-1
2+ i
0
1
2
3
4
-i
-2 i
So far, this is not too revealing. Multiply the product by 2 + i again:
5
���
7
�8 ���
2 Trignometry and Complex Numbers.nb
12 i
(2 + i)3 = 2 + 11 i
11 i
10 i
9i
8i
7i
6i
5i
4i
(2 + i)(2 + i) = 3 + i
3i
2i
i
-2
-1
2+ i
0
1
2
3
4
5
-i
-2 i
What sense does that make?
The meaning appears more easily with Polar Coordinates
So far, we’ve been writing complex numbers like points on a plane using Cartesian coordinates. For
some purposes, it is a LOT easier to understand what is going on if you use polar coordinates. (Trust
me.)
To start, consider a circle with a radius of one centered on the origin. This is the “unit circle in the
complex plane.”
�2 Trignometry and Complex Numbers.nb
���
�
������
1.51 = A
1.11841 = θ
A (cos θ + i sin θ)
Now if you choose any angle θ, the point (Cosine(θ) + i Sine (θ)) will necessarily fall on the unit circle.
As the angle θ goes through the complete cycle from 0 to 2 π -- we measure angles in radians, which is
easier for all sorts of reasons once you get used to it; if you are thinking in degrees, say “0° to 360°” -the point (Cosine(θ) + i Sine (θ)) goes around the circle. If the angle continues to grow, the point spins
endlessly around the unit circle.
If you want a point, that is to say “a complex number,” inside or outside the unit circle, multiply the
result by some constant A. If A is greater than one, the corresponding point (complex number) will be
outside the unit circle; if it is between zero and one, the point (complex number) will be inside the
circle. Any point on the complex plane can be designated with a pair of numbers A (for length) and θ
(for angle).
In complex-number-speak, the angle of the complex number expressed in polar coordinate form is
called the “argument”; the length is called the “modulus” of the number.
9
�10 ���
2 Trignometry and Complex Numbers.nb
Multiplication of Two Arbitrary Complex Numbers
Try multiplication again with two arbitrary complex numbers, this time expressed in polar form. Let
the two numbers be:
A B (cos(θ) cos(ϕ) - sin(θ) sin(ϕ) + i (cos(θ) sin(ϕ) + cos(ϕ) sin(θ))
“Okay,” you say. “How has this helped me?” The answer to that would be clear if you had been careful
about memorizing trigonometric identities, in particular, the identities for the sine and cosine of the
sum of two angles. On the off chance that you don’t have those identities burned into the forefront of
your minds, let me show you what you need to “remember” or, as Socrates might say, “recollect.”
Digression: Trigonometric Identities for Sine and Cosine of the Sum of Two
Angles.
Consider a portion of a unit circle with center at O. From center, draw a line OA at any (acute) angle;
call the angle ϕ. Drop a perpendicular AF to the horizontal diameter of the circle. The right triangle
formed as lengths that represent cos ϕ (horizontal OF) and sin ϕ (vertical AF). Now draw a line OB,
creating another angle, θ, on top of the first one. Drop a perpendicular BC to OA, the hypotenuse of the
first triangle. The segments OC and BC represent cos θ and sin θ, respectively. Drop perpendicular CH
to the original diameter OA. Also, drop a perpendicular from BD at the top of angle θ down onto the
original diameter. The segments thus created, OD and BD, represent cos (ϕ + θ) and sin (ϕ + θ) respectively.
Note draw a horizontal CE from C to the line BD. In triangle BEC notice that angle EBC is equal to ϕ.
Since segment BC is equal to sin ϕ, we conclude that BE = sin θ cos ϕ and that EC = sin θ sin ϕ.
Meanwhile, since OC = cos ϕ, we conclude that CH = cos θ sin ϕ and that OH = cos θ cos ϕ.
Examination will show that:
BD = sin (θ + ϕ) = BE+ EC = sin θ cos ϕ + cos θ sin ϕ ; and
CD = cos (θ + ϕ) = OH - DH = OH - EC = cos θ cos ϕ - sin θ sin ϕ.
�2 Trignometry and Complex Numbers.nb
���
11
������ ����� ϕ
��� ����� θ
������
���-��� ������
�������� ���� ���
����������
B
Sin
θ
Sin θ Cos ϕ
ϕ
A
G
θ
C
Sin ϕ
sθ
Co
Sin θ Sin ϕ
Cos θ Sin ϕ
E
ϕ
O
D
Cos θ Cos ϕ
Cos ϕ
H
F
Now look back at the product that we just obtained in multiplying two complex numbers.
A B (cos (θ) cos (ϕ) - sin (θ) sin (ϕ) + i (cos (θ) sin (ϕ) + cos (ϕ) sin (θ))
cos (θ + ϕ )
sin(θ + ϕ )
The collection of trigonometric terms associated with the real portion of the expression is cos (θ + ϕ).
The collection of trigonometric terms associated with the imaginary portion of the expression is sin (θ +
ϕ). The numbers associated with the lengths (modulus) are multiplied; the angles (arguments) are
added.
The significance of the imaginary multiplication is now visible:
In multiplying two complex numbers, whether written as A (cos(θ) + i sin(θ)) and B (cos(ϕ) + i
sin(ϕ)) or as a + b i and c + d i, graphically speaking what happens is that one
�12 ���
2 Trignometry and Complex Numbers.nb
(i) multiplies the distances of each number from the origin of the plane (the moduli), and
(ii) add the angles (arguments) made between the positive real axis and the line from the
origin to the point representing the number.
In short, again: in complex multiplication, distances from the center (moduli) multiply; angles
from the center add.
All sorts of neat things follow from this observation.
Raising Complex Numbers to Powers Causes Them to Spin!
If you raise a complex number to a (real) power, the argument (angle) of the result will grow continually
as the distance from the center grows (if it begins outside the unit circle) or shrinks (if it begins inside
the unit circle). Raising complex numbers to real powers therefore causes the results to trace spirals in
the complex plane. Here is the exponentiation of a complex number represented by a point a little bit
outside the unit circle:
�������
{Modulus =, 1.0435}
If we reduce the modulus (the “length”) so that the point falls inside the unit circle, the spiral will go
inwards because increasing powers of a length (modulus) less than one will shrink.
�2 Trignometry and Complex Numbers.nb
���
13
Between these two cases is the balanced point, where the modulus is one and the point lies on the unit
circle. Then, increasing powers of the complex numbers will result in a representative point that spins
forever around the circumference of the unit circle.
The investigation of complex numbers is a vast field. Thick textbooks are devoted to “functions of a
complex variable.” The Mandelbrot set, which lies at the beginning of complexity studies, exists in the
complex field. (It is defined as the set of complex numbers c that do not diverge when the function
fc (z) = z2 + c is iterated from z = 0.)
All this would be subject matter for an immense study. However, for the present , I want only to point
to two results that are relevant to the particular path that these talks are taking toward their goal,
constructing the heptadecagon.
Taking Integral Roots
First, now that we understand how complex numbers are multiplied and raised to powers, we can
easily find how to find integral roots of any complex number and thereby develop a general solution to
the problem that faced Rafael Bombelli. His great triumph, recall, was finding the cube root of one
complex number, 2 + 11 i, which he did by a combination of great genius, immense labor and fabulous
luck. (Almost any other complex number would have been much harder for him to deal with.)
However now we can see how easily to take the cube root of any complex number. Remember, to cube
a complex number, you cube the real number that is its modulus and triple the angle (argument). So,
�14 ���
2 Trignometry and Complex Numbers.nb
to take the cube root of a number, all you need do is to (i) take the cube root of the length (the
“modulus”) and (ii) and divide the angle (the “argument”) by three.
The Cube Root of 2 + 11 i
The particular problem that Bombelli faced was finding the cube root of 2 + 11 i. To take its cube root
the new way, first calculate its modulus (length) and argument (angle). The length of the vector from
the origin to (2 + 11i) we can get with the Pythagorean Theorem:
length (modulus) =
22 + 112 =
4 + 121 =
125
If we allow ourselves some trigonometry, the angle is easy enough, too:
= 1.39094 radians (79.7 degrees).
angle (argument) = ArcTan 11
2
To take the cube root, take the cube root of the length (modulus). In this case, we are assisted because
125 = 53 :
3
125 =
3
125 =
Take the angle and divide by three:
5.
1.39094
3
= 0.463648 radians.
So we get:
5 (Cos(0.463648) + i Sin (0.463648) )
= (2.236) (0.894427 + i 0.447214)
=2+i
Just the result that Bombelli arrived at by genius, sweat and divine guesswork.
Here, for comparison, are the values Bombelli worked on plotted atop the graph of the spiral
z = (2 + i)n
�2 Trignometry and Complex Numbers.nb
���
15
12 i
(2 + i)3 = 2 + 11 i
11 i
10 i
9i
8i
7i
6i
(2 + i)n
5i
4i
(2 + i)(2 + i) = 3 + i
3i
2i
i
-2
-1
2+ i
0
1
2
3
4
5
-i
-2 i
The Roots of Unity
When the Modulus Equals One
We have seen that when complex numbers whose representative points lie outside the unit circle spiral
outward when squared, cubed, or generally raised to powers greater than one. Those that lie inside the
unit circle spiral inward.
Those that lie on the unit circle -- those with a modulus that is exactly equal to one -- spin around the
unit circle with out moving inward or outward. These are very interesting, very handy numbers.
Because the cosine of a given angle and the sine of the same angle can form the sides of a right triangle
whose hypotenuse is equal to one, we can write these complex numbers with modulus one in the form:
z = cos θ + i sin θ
�16 ���
2 Trignometry and Complex Numbers.nb
We have seen that multiplying two complex numbers adds their angles (arguments) and multiplies
their lengths (moduli). In the case of these numbers, the modulus is one, so multiplying it any number
of times leaves it unchanged. For these numbers, multiplying means just adding the angles. So, if we
take a number and multiply it by itself, we get:
z2 = (cos θ + i sin θ) (cos θ + i sin θ) = (cos 2 θ + i sin 2 θ)
If we do it again, we get:
z3 = (cos θ + i sin θ) (cos θ + i sin θ) (cos θ + i sin θ) = (cos 3 θ + i sin 3 θ)
And in general,
zn = (cos θ + i sin θ)n = (cos n θ + i sin n θ).
If two different modulus one numbers are multiplied, we get:
z1 z2 = (cos θ + i sin θ) (cos ϕ + i sin ϕ) = cos (θ + ϕ) + i sin (θ + ϕ)
�2 Trignometry and Complex Numbers.nb
���
θ
ϕ
Cos ϕ + i Sin ϕ
Cos θ + i Sin θ
Cos (θ + ϕ) + i Sin (θ + ϕ)
In this operation, multiplication of the complex numbers is tightly bound up with addition of the
angles. Such tight linkage of multiplication and addition is characteristic of exponentiation and logarithms, and in fact it is a very short step from what we have seen here to a formula expounded by
Leonhard Euler in his work Introduction to the Analysis of the Infinite that identifies the two:
ei θ = cos θ + i sin θ.
(A few years ago I gave a whole lecture on this identity; I’ll see about having it available on the library
web-site alongside this one.)
For now, we will be especially interested in a subset of these numbers that bear the intriguing and
evocative name, the “Roots of Unity.”
Roots of Unity
The “Roots of Unity” sounds like a New Age metaphysical treatise or the name of a theologically
inclined folk-rock ensemble, but in our present context it means something rather different and more
precise. It refers to numbers that, when raised to integral powers come to the result 1. Numbers like:
17
�18 ���
2 Trignometry and Complex Numbers.nb
2
1,
3
1,
4
1,
5
1 … etc.
To put the matter slightly differently, we are talking about numbers that are the solutions of equations
like:
x2
x3
x4
x5
-
1
1
1
1
=
=
=
=
0
0
0
0
or in general,
xn - 1 = 0
Based on what I learned in high school, these equations are not hard to solve. For x 2 - 1 = 0, I know
that there are two solutions, + 1 and -1. For x 3 - 1 = 0, there is only one solution, +1, because
(-1)3 = -1. That pattern continues down the line, with even numbered powers having two solutions
and odd numbered powers having only one. That understanding works so long as one considers only
the real numbers. But in the complex number field the answer is more complete, more interesting and
in some ways more satisfying.
Take x 3 - 1 = 0 for example. We are looking here for a number which, when cubed, is equal to one,
that is, the cubed root of one. Easy! One, when cubed, is equal to one. That’s fine, but it’s not the full
story. Consider the number on the unit circle whose angle is 120°: when squared it is still on the unit
circle and its angle is 120° × 2 = 240°; when cubed, it is still on the unit circle and its angle is 120° × 2 =
360° = 0°. That number is +1 + 0 i. Thus, the complex number at 120° on the unit circle is also a
cubed root of one! So, for that matter, is the number on the unit circle at 240°: squared, its angle is
480° = 120°; cubed, its angle is 360° = 0°. There are, in fact, three cube roots of one, and the points that
represent them form an equilateral triangle in the unit circle.
�2 Trignometry and Complex Numbers.nb
���
2i
The Cube Roots of Unity
i
1
- , + i ,
2
-2
3
2
0
-1
1
3
- , + i ,
2
2
{1, + i , 0}
1
2
-i
-2 i
The Algebraic Approach
The graphical approach to the cube root of unity is simple: take the 360° of the circle and divide them
by three. One can also take a strictly algebraic approach which is a little more intricate but which
reaches the same result. Begin with the equation:
x 3 - 1 = 0.
As you noticed at first, the integer 1 (or, better, the complex number, 1 + 0 i) is solution. Therefore, we
expect that this polynomial will be divisible by the linear factor (x - 1), as indeed it is:
x 3 - 1 = (x - 1 ) (x 2 + x + 1) = 0.
The new factor, (x 2 + x + 1), can easily be broken down into two linear factors by applying the
quadratic formula to the equation x 2 + x + 1 = 0 :
x=
-1 ±
1 - 4 (1)
2
=
-1
2
±i
3
2
.
So, the complete breakdown of the equation x 3 - 1 = 0 into linear factors is:
(x - 1), x - -12
+ i
3
2
, x - -12
- i
3
2
19
�20 ���
2 Trignometry and Complex Numbers.nb
You can verify this result by multiplying any of the solutions -- 1, -12
3
2
+ i
or -12
3
2
- i
-- by itself three times and seeing that you get the result 1 + 0 i.
More Roots of Unity
It should not surprise you to learn that the equation x 4 - 1 = 0 gives four fourth roots of unity: +1, -1, +i
and- i. And the equation x 5 - 1 = 0 gives five fi�h roots of unity, like this:
2i
2i
i
-2
-1
i
0
1
2
-2
-i
-2 i
-1
0
1
2
-i
-2 i
And so forth. Generally speaking, there are always n nth roots of unity. This tidy fact is a special case of
a more general result proved by Gauss and called the “Fundamental Theorem of Algebra” which states
that in the complex number field a polynomial equation of the nth degree always has n solutions. That
is a wonderful result, but we don’t need its full generality for our task-at-hand.
Look again at the polynomials that define the n roots of unity. We can see from the graphic representations that the number 1 + 0 i is a solution of each of these “roots of unity” equations. Consequently, we
can divide any of them by the factor (x - 1), just as we did with the cube-root of unity equation:
x 4 - 1 = (x - 1) (x 3 + x 2 + x + 1) = 0
x 5 - 1 = (x - 1) (x 4 + x 3 + x 2 + x + 1) = 0
and generally:
x n - 1 = (x - 1) (x n - 1 + x n - 2 + … + x 2 + x + 1) = 0
The increasingly lengthy remainder terms are of special interest to us. The solutions to the corresponding polynomial equations
�2 Trignometry and Complex Numbers.nb
���
21
x3 + x2 + x + 1 = 0
x4 + x3 + x2 + x + 1 = 0
and generally:
xn - 1 + xn - 2 + … + x2 + x + 1 = 0
are precisely what we need in order to find the vertices of regular polygons inside a unit circle. For
fairly evident reasons, these equations are called collectively the cyclotomic (that is, “circle-cutting”)
polynomials.
They will be the subject, not of the next talk, but the one a�er that.
Oh, By the Way … One More Thing to Note About Roots of Unity!
Before closing, I want to note one more feature about the roots of unity that will show up in a later talk.
It is this: for any whole number n, the sum of the nth roots of unity comes to zero.
This can be seen pretty easily by looking at the case of the four fourth roots of unity:
2i
i
-2
-1
0
1
2
-i
-2 i
The four roots are +1, +i, -1 and -i. It is evident (isn’t it?) that when these four are added together, the
sum is zero. A�er all, the pair + 1 and -1 add to zero, as do the pair +i and -i.
�22 ���
2 Trignometry and Complex Numbers.nb
Only a little less evident is what happens with the three third roots of unity:
2i
i
-2
-1
0
1
2
-i
-2 i
The two red vectors are parallel and equal to the blue vectors to the two complex third roots of unity.
Placing the three vectors end-to-end in the usual way for vector addition gives a closed triangle, beginning and ending at (0, 0).
Similarly for all nth roots of unity: their sum always comes out to zero.
As a quick corollary, if one takes all nth the roots of unity for any n, the whole collection excluding the
number +1 sum up to -1. This follows easily from the fact that all the roots of unity sum to zero; if one
excludes +1, the rest must sum to -1 so that all of them together come to zero.
These facts will be used repeatedly in what follows. If you don’t remember them, I’ll remind you of
them when they come up again.
Conclusion
So these are the fundamentals of the arithmetic of complex numbers. The next talk will concern itself
with another topic altogether, the algebraic difference between points that can be constructed and
those that can’t. In the fourth lecture, these two topic will come together to demonstrate how algebra
can decide whether a construction is possible or not; we’ll look at two classical problems -- the trisection of an angle and the doubling of the cube -- and then at a new problem: the construction of the
seven-gon. The fourth Tuesday lecture will bring all that has been said to bear on Gauss’s surprise, the
construction of the seventeen-gon.
�
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
pdf
Page numeration
Number of pages in the original item.
22 pages
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
Trigonometric interpretation of complex numbers
Description
An account of the resource
Transcript of a lecture given on September 17, 2019 by Grant Franks.
Creator
An entity primarily responsible for making the resource
Franks, Grant H.
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
2019-09-17
Rights
Information about rights held in and over the resource
Meem Library has been given permission to make this item available online.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Subject
The topic of the resource
Numbers, Complex
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
SF_FranksG_Trigonometric_Interpretation_of_Complex_Numbers_2019-09-17
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/ec85bdc7b9d94de0b94e3b276b46d4a9.pdf
b176bca3c0e75e831c733124f6274bd1
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
paper
Page numeration
Number of pages in the original item.
24 pages
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
Translation as a liberal art : notes toward a definition
Description
An account of the resource
Transcript of a lecture given on November 14, 1975 by William Darkey.
Creator
An entity primarily responsible for making the resource
Darkey, William A.
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
1975-11-14
Rights
Information about rights held in and over the resource
Meem Library has been given permission to add this item to its collections.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Subject
The topic of the resource
Translating and interpreting.
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
24000385
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/d415186b7e840ea65d8b1a36c06d2ab3.pdf
37282093349aa636f0e2b7a98c24f11c
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
paper
Page numeration
Number of pages in the original item.
45 pages
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
Thucydides : circa 460-400 B.C.
Description
An account of the resource
Transcript of a lecture given on May 19, 1986 by David Bolotin.
Creator
An entity primarily responsible for making the resource
Bolotin, David, 1944-
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
1986-05-19
Rights
Information about rights held in and over the resource
Meem Library has been given permission to make this item available online.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Subject
The topic of the resource
Thucydides
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
24000273
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/276fcab53ac6bda03f8f9c082241f562.pdf
a90817733c79ae5b32787ac08bc9d5f6
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
paper
Page numeration
Number of pages in the original item.
17 pages
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
Three love poems by Robert Frost
Description
An account of the resource
Transcript of a lecture given on September 02, 1994 by William Darkey.
Creator
An entity primarily responsible for making the resource
Darkey, William A.
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
1994-9-02
Rights
Information about rights held in and over the resource
Meem Library has been given permission to add this item to its collections.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Subject
The topic of the resource
Frost, Robert, 1874-1963.
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
24000297
Friday night lecture
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/b3ed8b906e47a4df431da8e6fc052130.pdf
f0b8efc414011a82dfd71934e35b6ebf
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
paper
Page numeration
Number of pages in the original item.
24 pages
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
Thoughts on canto 33 of the Inferno
Description
An account of the resource
Transcript of a lecture given on February 6, 2013 by Cary Stickney as part of the Dean's Lecture and Concert Series.
Creator
An entity primarily responsible for making the resource
Stickney, Cary
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
2013-02-06
Rights
Information about rights held in and over the resource
Meem Library has been given permission to make this item available online.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Subject
The topic of the resource
Dante Alighieri, 1265-1321. Inferno.
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
24004147
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/0f20ac4b2e691cb4e0de1e6b41d12e5a.pdf
ab6b8d9906dddf78b4882f73e4a34158
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
paper
Page numeration
Number of pages in the original item.
20 pages
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
This senseless course of human things : 'one of Professor Kant's most cherished ideas'
Description
An account of the resource
Transcript of a lecture given on August 26, 2005 by David Levine as part of the Dean's Lecture and Concert Series.
Creator
An entity primarily responsible for making the resource
Levine, David Lawrence
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
2005-08-26
Rights
Information about rights held in and over the resource
Meem Library has been given permission to make this item available online.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Subject
The topic of the resource
Kant, Immanuel, 1724-1804.
Education, Higher
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
24003173
Friday night lecture
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/f446ba6075bd1132d9f8c4a279efad2a.pdf
ffe0aa06cb240aa8344fbefa776fc1ed
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
paper
Page numeration
Number of pages in the original item.
27 pages
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
Thinking on thinking
Description
An account of the resource
Transcript of a lecture given on April 14, 2010 by Robert Richardson as part of the Dean's Lecture and Concert Series.
Creator
An entity primarily responsible for making the resource
Richardson, Robert Allan, 1937-
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
2010-04-14
Rights
Information about rights held in and over the resource
Meem Library has been given permission to make this item available online.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Subject
The topic of the resource
Upanishads.
Thinking
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
24003873
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/a3fc70ca03f71775bb5e22cdeb4dfb7f.pdf
680dc4b0110f538fecee3a62dd4264de
PDF Text
Text
The Winnowing Oar: Odysseus’ Final Journey
Claudia Hauer
St. John’s College, Santa Fe, September 21, 2016
Abstract
In book 11 of the Odyssey, in the underworld, Teiresias describes to Odysseus a final
journey that he must take to propitiate Poseidon when his labors on Ithaka are
concluded. Teiresias tells Odysseus he must walk inland with an oar until a
wayfarer mistakes the oar for a winnowing fan. There, Teiresias says, Odysseus
must build a shrine to Poseidon and plant the oar as a dedication. In this talk, I will
explore various interpretations of this puzzling description of Odysseus’ final
journey.
Paper
In book 11 of the Odyssey, called the Nekuia, the book of the dead, Odysseus goes to
the underworld in order to question the ghost of Teiresias. Teiresias describes a
final journey that Odysseus must make after he kills the suitors:
But after you have killed these suitors in your own palace,
either by treachery, or openly with the sharp bronze,
then you must take up your well-shaped oar and go on a journey
until you come where there are men living who know nothing
of the sea, and who eat food that is not mixed with salt, who never
have known ships whose cheeks are painted purple, who never
have known well-shaped oars, which act for ships as wings do.
And I will tell you a very clear proof, and you cannot miss it.
When, as you walk, some other wayfarer happens to meet you,
and says you carry a winnow-fan on your bright shoulder,
then you must plant your well-shaped oar in the ground, and render
ceremonious sacrifice to the lord Poseidon. (Lattimore, 11. 119 – 130)
The most obvious intent of this journey will be the propitiation of Poseidon. i While
Athena and Zeus love Odysseus ‘terribly’ (1. 265), Poseidon’s anger over the
blinding of the Cyclops Polyphemus remains a menace to the ‘long, peaceful life’ that
Teiresias next prophesies Odysseus could enjoy.
Poseidon, we learn later, did not intend to prevent Odysseus’ homecoming, but the
god wanted Odysseus to suffer more than he did, and arrive at Ithaka without
treasure. In book 13, after Odysseus has been ferried safely to Ithaka by the
Phaeacians, Poseidon complains to Zeus about the potential loss of status he will
incur with respect to his thwarted intentions for a more difficult homecoming for
Odysseus. Poseidon says:
Father Zeus, no longer among the gods immortal
Shall I be honored, when there are mortals who do me no honor,
the Phaiakians, and yet these are of my own blood. See now,
I had said to myself Odysseus would come home only after
much suffering. I had not indeed taken his homecoming
�Hauer 2
altogether away, since first you nodded your head and assented
to it. But they carried him, asleep in the fast ship, over
the sea, and set him down in Ithaka, and gave him numberless
gifts, as bronze, and gold abundant, and woven clothing,
more than Odysseus could ever have taken from Troy, even
if he had come home ungrieved and with his fair share of the plunder.
(Lattimore 13. 128 -139)
Having been thwarted with respect to his intention for Odysseus, Poseidon’s wrath
now turns to the Phaeacians. Poseidon takes Zeus’ suggestion that he teach the
Phaeacians a memorable lesson by turning the returning ship to stone in sight of the
Phaeacian harbor. Odysseus is safe… for now. The fate of the Phaeacian ship has
the intended effect. Alcinous recalls that his father prophesied that Poseidon would
cover the Phaeacian city with a mountain of rocks after turning a ship to stone, and
the Phaeacians immediately begin fervent propitiations to the god.
The forgotten prophecy suggests that the Phaeacians have let their diligence lapse
with respect to Poseidon – Alcinous only belatedly recalls the prophecy that
Poseidon would one day be angry with them because of their indiscriminate offers
of convoy. Yet Odysseus told the Phaeacians the story of the blinding of the Cyclops,
a story which included Polyphemus’ declaration that he is the son of the “glorious
earthshaker” (9.518). Nonetheless, the Phaeacians did not recognize that they were
risking Poseidon’s wrath in showering Polyphemus’ mutilator with gifts and giving
him convoy.
Odysseus’ propitiation of Poseidon on his final journey will, similarly, attempt to
compensate for any lost honor (τιμή) that the god may suffer over the ease of
Odysseus’ homecoming. But Odysseus’ propitiation will lack the anxiety that
characterizes the renewal of the Phaeacians’ worship of the god. Teiresias mentions
no future threat or calamity, and predicts a sleek old age for Odysseus once this final
labor is complete.
The propitiation will permit Odysseus to render proper observances to all of the
divinities. After describing the final journey, Teiresias orders Odysseus to return
home and “render holy hecatombs to the immortal gods who hold the wide heaven,
all of them in order” (Lattimore, 11.132-3). Odysseus’ propitiation will restore
Odysseus’ own relationship with the pantheistic order and permit him in the future
to perform the ceremonies of piety with Poseidon in his proper position of honor.
The task of the final journey is to establish a shrine to Poseidon in an inland region
far from the sea. The propitiation will extend Poseidon’s domain and his worship
cult into a community whose inhabitants do not recognize the dedicatory oar as
such, and who have never even seen the sea. What does this mean? Why is
Odysseus’ founding of an inland shrine with an unrecognized talisman an
appropriate propitiation of the god of the sea?
�Hauer 3
Whatever oar Odysseus will take on this final journey, it will not be from the ship on
which he left Troy. Lattimore (1965), Fagles (1996), and Mitchell (2013) all
translate Teiresias’ command as a reference to “your well-shaped oar.” Yet the
Greek could also mean “a well-shaped oar,” and this must be its correct meaning, for
all of the oars from Odysseus’ ship were lost when Zeus destroyed it in the blast that
kills the companions. At the end of book 12, we learn that Odysseus, the only
survivor, washes up on Kalypso’s isle with the aid of “two long timbers” (perimήκεα
dοῦra), the only surviving pieces of the lost ship.
The last we hear of those ship’s oars, they are associated with impiety, since the air
around them is permeated with the odor of the cooking meat from the forbidden
cattle of the Sun God on Thrinakia. Odysseus, who had been praying to the gods by
himself, apart from his men, describes his return to the scene of the crime as
follows: “as I was close to the oar-swept vessel, the pleasant savor of cooking meat
came drifting around me, and I cried out my grief aloud to the gods immortal”
(Lattimore, 9.368-9). The contrast between the pleasant odor of the meat, and
Odysseus’ grief highlights the distinction between the worldliness of the
companions’ human appetite and Odysseus’ otherworldly concern with honoring
the gods. Odysseus’ words condemn the vessel and its oars, now tainted by the odor
of impiety.
From the first 4 books of the Odyssey, the Telemachia, we have learned that the
other surviving heroes from Troy, Nestor and Menelaus, are both very careful to
render proper sacrifice to the gods. Yet neither of these two surviving heroes is
“home” in the way Odysseus intends to be home. The anger-prone Nestor tells
endless war stories about the glory days, to the chagrin of the youth of Pylos. In
book 15 (191ff), Telemachus asks Nestor’s son Pisistratus to conspire to let
Telemachus skip a goodbye visit to Nestor, since the old man’s clinging company is
so tedious. Pisistratus immediately sees the sense of the request. “How overbearing
his anger will be,” Pisistratus rues, yet urges his new friend to sail immediately.
Menelaus, on the other hand, whose memories might be somewhat more bitter, is
drugged into mindless joviality by Helen.
Odysseus, as we have learned, is active, engaged, and responsive to mortals and
immortals alike, and we must assume that this is indicative of the way he wishes to
be “home.” Odysseus must restore his good standing with Poseidon, but the way he
is ordered to do it will require that an oar be mistaken for a winnow-fan. Does this
have something to do with Odysseus’ particular homecoming, his νόστος?
Teiresias describes the dedicatory oar as “well-made,” from eὐ + ἀραρίσκω. The
verb ἀραρίσκω means “to join” – so the adjective describes something crafted to be
“well-joined.” The fin and handle of the oar will have been “put together” by a
craftsman who knows how to fit the joints to be strong and seamless. There are only
two other uses of this adjective in the Odyssey. Elpenor’s oar is also named as “wellmade,” as is the ax with the olive wood handle that Kalypso gives Odysseus to use as
he builds his getaway raft. Certainly, the gods’ tools would be well-joined. Perhaps
�Hauer 4
there are other well-joined oars in Odysseus’ palace at Ithaka, or perhaps Odysseus
himself will make the oar. Odysseus is well-skilled at joining – he is τέκτων, a homo
faber, a ‘maker-man,’ able to control his environment through the use of tools. ii
Odysseus’ talent as a craftsman is made clear in the two descriptions of his
handiwork in the Odyssey: the construction of the raft in book 5, and Odysseus’ own
description of his construction of his marriage chamber and marriage bed in book
23. Homer uses ἀραρίσκω to describe how Odysseus “joins” the deck boards as he
constructs the raft on which he will flee Kalypso’s island (5.252). In this passage,
Odysseus emerges from a night of extra-marital love-making with Kalypso in the
innermost chamber of her cave (μυχός), then joins the planks of the raft well to
make a sea-worthy vessel with which to return to his wife. In book 23, the joining,
by contrast, consecrates the innermost bedroom of man and wife. Odysseus himself
uses the verb ἀραρίσκω as he describes to Penelope how he “joined” the doors to
their bedroom around the stump of the olive, before he constructed their marriage
bed. In this passage, the well-joined doors seal off the marriage chamber (θάλαμος)
from the rest of the house. These doors have also so far protected Penelope from
the suitors, as she retires each night to the marriage chamber of Odysseus. The
dedicatory oar, insofar as it too is well-joined, references Odysseus’ prowess in both
adventure and return, such that he can craft the tool of passage across Poseidon’s
domain to and from war.
The oar that Teiresias has in mind will have been made to row “ships whose cheeks
are painted purple” (foinikoporeίoi). The Cyclops, Odysseus has already explained to
the Phaeacians in book 9, have no such ships (“red-cheeked ships” - miltopareίoi)
nor carpenters to build them (9.125). Although Poseidon is god of the sea, his son
Polyphemus and the race of Cyclops have no knowledge of ship-building and the art
of joining, and do not venture onto the sea. The people whom Odysseus will
introduce to Poseidon-worship on the final journey will also have no direct
knowledge of the craft that could take them into Poseidon’s domain. Moreover,
Teiresias does not instruct Odysseus to correct the wayfarer who thinks the oar is a
winnow-fan, so the expansion of the Poseidon-cult apparently does not require that
the new worshippers understand the oar as talismanic of Poseidon’s actual domain.
Teiresias tells Odysseus that the wayfarer’s mistake about the oar will be the “sign”
(sῆma) that he is in the correct place to perform the propitiation. Teiresias uses the
word “sign” to denote a “signal” or “mark” by which Odysseus can recognize the
sacred place. Yet the word can also be used to denote a grave-marker, as the shade
of Elpenor has used it just 50 lines earlier.
Recall that Elpenor fell off the roof just as Odysseus and his companions left Circe’s
island for the underworld. Since the body of Elpenor was left unburied, he does not
seem to have to drink the blood to speak to Odysseus in the underworld. He
approaches even before Odysseus has spoken with Teiresias, and begs Odysseus to
return to Aiaia, and
�Hauer 5
burn me there with all my armor that belongs to me, and heap up a grave
mound (sῆma) beside the beach of the gray sea, for an unhappy man, so that
those to come will know of me. Do this for me, and on top of the grave
mound plant the oar with which I rowed when I was alive and among my
companions. (Lattimore, 11.74-8)
Here the oar on the tomb signifies the grave of a man who rowed. Such a sign would
be an appropriate grave-marker for any of the companions, but the others will have
no graves, for they have offended Helios in eating the cattle of the Sun God, and are
lost at sea when Zeus (interestingly not Poseidon) blasts their ship after their
departure from Thrinakia. The oar that Odysseus is to plant at the new shrine to
Poseidon will also be a sign, but one that only Odysseus as its human founder will
fully understand, as the local community will mistake it for a threshing tool.
How might Odysseus understand the significance of the winnowing oar? Given the
proximity of Elpenor’s request and Teiresias’ instructions, perhaps Odysseus will
understand the oar as a death-token of his lost companions. Perhaps the oar will
signify his lament for the companions’ human frailty, their inability to refrain from
the flesh of the forbidden cattle, an inability which Homer and Poseidon describe at
the very beginning of the Odyssey as the companions’ characteristic “wild
recklessness” (1.7, 34). This is a single word in Greek, a word that shares the root
with the word ἄth, a delusion about the cosmos that leads to ruin caused by the
gods.
Odysseus himself is apparently free from this kind of cosmic folly. Yet it can be hard
for a contemporary reader to appreciate this, for Odysseus’ relationship with his
men often strikes contemporary readers as merely instrumental. As Robert
Hollander writes in his commentary on Dante’s Inferno Canto 26, which I will return
to below, “the companions are Ulysses’ oars” (Hollander, 493). By this reading, the
oar is, in a sense, any man who rowed for Odysseus, and it will be unnamed and
unrecognized, just as the companions are.
Another possibility is that Odysseus will understand the transformation of the oar
into winnow-fan as a transition from sea to land, or from war to peace. Teiresias has
prophesied for him a long life into “sleek old age.” Odysseus’ life will be peaceful and
prosperous, focused on animal husbandry, fruit trees, and the growing and harvest
of grain. We don’t know what form this death will take, but it will come not “on the
sea” but “from the sea,” as something foreign to Odysseus’ pastoral life on Ithaka.
The Odyssey tells the tale of a man who figured out how to get home from war – not
just back to his native place, as Nestor and Menelaus have also done, but home, at
home, alive to the present, not haunted as so many veterans are. The poem shows us
that homecoming in this fullest sense (νόστος) is not simply a transition from
‘violent’ war to ‘non-violent’ domestic life. Odysseus makes himself at home in a
world that he understands to be full of forces beyond the power of humans to
control. The transformation of “a well-made oar” into a winnow-fan is not simply
�Hauer 6
the transformation of a tool of the stormy sea into a tool of a peaceful agrarian life.
Olson (1997) points out that the word Teiresias uses for winnow-fan is a hapax
legomenon, a word that only appears once in surviving Greek literature, a composite
of ἀleίqei (ears of corn) and loigόj (destroyer). Homer’s poem avoids the two names
of this tool in daily use – ptύon and qrίnax - in order to call attention to the way this
agrarian tool destroys the objects on which it is used.
Odysseus is, similarly, a wrecker of men. By this reckoning, the Odysseus we get to
know in the Odyssey may strike a reader as a strange figure to be charged with an
evangelical function with respect to Poseidon. The final journey will have little in
common with Odysseus’ adventures on the way home from Troy. Odysseus has only
to walk inland until the oar across his shoulders is mistaken for a winnow-fan. Yet
Odysseus’ survival to perform this final propitiation has come at considerable cost, if
not to Odysseus, then certainly to others. Odysseus has broken the goddess
Kalypso’s heart, lost his own comrades, brought about the death of the sailors on the
Phaeacian ship that bore Odysseus safely to Ithaka, and potentially led to the
obliteration of the rest of the Phaeacians as well. Odysseus has mercilessly
slaughtered the suitors and the maidservants who slept with them, including
Eurynomos, characterized by Homer as a decent, god-fearing man. He has all but
killed Irus, the local beggar.
For a reader who has never gone to war, it may be hard to accept that none of these
actions have earned Odysseus the enmity of the gods. Rather, Odysseus is hated by
Poseidon because he mutilated Poseidon’s monstrous son Polyphemus in order to
escape from the Cyclops’ cave. Obviously, the contemporary code of universal
morality condemns this. The only reason Odysseus was in the Cyclops’ cave in the
first place was because he was eager to acquire gifts and glory. Yet any concerns a
contemporary reader has about the moral quality of Odysseus’ motives will have to
be set aside. In the Homeric context, gifts and glory constitute a warrior’s honor,
much as medals, decorations, and ceremony do today. Yet the markings of honor do
not always go to the most worthy men. Indeed, there is much dispute in the classical
literature about whether Odysseus deserves the honor he received at Troy, since he
utilizes cunning in combat, and does not rely on naïve strength and physical power.
Yet Odysseus’ methods work. His strategems at Troy won the war for his side. Lives
were saved, as we would say in the American military context today. Homer’s gods
are not much concerned with the methods that mortals employ in the deadly
business of war.
With respect to the gods, Homer lacks what we would call moral theism. Moral
theism is the notion that the proper worship of the gods entails the wish for moral
purity, serves as the basis for moral redemption, or requires repentance of sin.
Worship of the Homeric pantheon is transactional, and, as we shall see, relational,
but it is not based on any stable moral theism.
The idea that a character like Odysseus could successfully propitiate the angry
Poseidon requires many contemporary readers to suspend something of their
�Hauer 7
theistic expectations. Against the backdrop of my own peaceful life context and
relatively unchallenged theological commitments, much of what Odysseus does
appears morally deficient. Odysseus himself, for example, in telling the tale to the
Phaeacians, calls his boast to the mutilated child of Poseidon a “taunt” (κερτόμιος).
However savage the Cyclops were, they kept to themselves, and Odysseus only
visited their island because he wanted to “find out about these people, and learn
what they are” (Lattimore 9.174), in the hopes, as always, of gaining status in the
form of gifts (dῶra), honor (timή), and fame (klέoj). Such passages challenge a
reader to ask herself: what do we really mean when we insist that the tenets of
moral theism apply to soldiers at war?
Occasionally in translation, Homer is made to suggest that divinities engage in
something like moral retributive justice. When Poseidon complains to Zeus that
Odysseus has had an effortless landing on Ithaka, Zeus tell him “if any humans
should insolently deny you your proper honors, of course you must feel free to
punish them in whatever way that you want to” (Mitchell, 13.141-2). The Greek here
for ‘punish,’ tίsij, means something like “extract payment”. Yet it is not the primary
offender Odysseus who is characterized as owing a debt to Poseidon, it is the
secondary offenders the Phaeacians who owe this “payment” for their lapse in the
diligence of their worship. Poseidon’s turning the ship to stone alerts the Phaeacians
to their debt, which they attempt to pay by their fervent sacrifices. Poseidon does
not pursue any principled retribution against Odysseus, in fact Poseidon is
(presumably) placated at the very moment that Odysseus is free to undertake his
machinations against the suitors. Poseidon does not “punish” in the morally theistic
sense of giving mortals what they justly deserve. Rather, the word tίsij implies a
transactional lapse; the Phaeacians “owe” Poseidon something if the god is not to
lose honor and status among his fellow immortals.
Homer’s narrative captures this through the technique of caesura in line 13.187.
Caesura, or “cut,” is a sense break inserted in or after the 3rd of the 6 feet of a line of
dactylic hexameter. In book 13, line 187, the 1st three feet are translated: “standing
around the altar”, a phrase that finishes the description of the ceremonies of
sacrifice that the Phaeacians diligently undertake after witnessing the ship turning
to stone, and after Alcinous’ belated recollection of the prophecy about Poseidon’s
wrath. That is the last we will hear of the Phaeacians. We do not find out if their
propitiation succeeds. Homer’s narrative now turns to Odysseus and his attempt to
slay the suitors. The final three feet of this line of dactylic hexameter shift the scene
with: “But now great Odysseus wakened.” Homer intends us to view these separate
scenes as simultaneous, so that Odysseus’ forthcoming strategems on Ithaka are
framed by the urgency with which the Phaeacians have learned that because of their
good treatment of him, they are in danger of suffering Poseidon’s full punishment,
which would obliterate their city and presumably themselves under a mountain of
stone.
This narrative device of caesura draws attention to the fact that Odysseus will not
have to “pay” Poseidon the price of the many sufferings the god intended for him.
�Hauer 8
Poseidon, it turns out, does not reckon his accounts that way. Poseidon’s wrath is
kindled for partisan reasons: when his own kindred are wronged or have done him
wrong. He is angry that the Phaeacians did not keep him properly in mind, and he
wants to take revenge on the man who mutilated his son. Poseidon’s vengeance has
tribal meaning, but not moral meaning. Those who expect gods to be moral will have
trouble appreciating this curious feature of Homeric divine retribution. Homer’s
world is a theistic world, in that there are gods, and they take a personal interest in
the affairs of mortals, but it is not a moral theism. Yet there is much to learn from
this theistic tribalism, in particular the way it appears to reflect something timeless
about the way warriors respond to violence.
To illustrate this contrast between the heroic model of the warrior and the Christian
model of moral theism, let us return to Dante, the 14th century author of the
Commedia. Dante knew of Homer’s Odysseus not directly, but from references in
Latin poetry. In his Inferno, Dante offers a retelling of the Odysseus story that is not
willing to absolve Odysseus merely on the grounds that Homer lacked a context of
moral theism. Dante’s language illustrates that Dante understood that this context
was absent in Homer’s pagan world, and yet condemned Odysseus anyway.
Dante’s Ulysses is guilty of irredeemable impiety. Dante’s story overlaps with
Homer’s up until the departure from Circe’s island, when Homer has Odysseus enter
the underworld, but Dante has Ulysses abandon the homecoming in order to
attempt a hubristic assault on the Mountain of Purgatory. Condemned to hell for his
unrepentant false counsel, Ulysses speaks of the Christian God dismissively, as
merely “another,” a god with his own petty tribal reasons for wanting Ulysses
destroyed.
Dante’s account is true to Homer’s in one respect: Homer’s Odysseus does not show
any hint of repentance or concern for redemption. Much has been made of the
transition in Greek narrative from the ‘shame culture’ of Homeric times to the ‘guilt
culture’ in place by the 5th century before the Christian era. E. R. Dodds (1951)
points out that Homeric characters do not exhibit much anxiety about triggering
fqόνος, the “jealousy” of the gods, or, more theistically, their “righteous indignation.”
Dodds writes, “It is plain…from the uninhibited boasting in which Homeric man
indulges that he does not take the dangers of fqόνος very seriously: such scruples
are foreign to a shame culture.” iii Yet Dante’s Christian theism requires him to
condemn this attitude categorically, and Dante places Ulysses in the 8th circle of hell,
among the false counsellors.
Dante’s Ulysses is driven by a love of experience that is also familiar to readers of
Homer. Ulysses tells the pilgrim Dante: “not tenderness for a son, nor filial duty/
toward my aged father, nor the love I owed/ Penelope that would have made her
glad/ could overcome the fervor that was mine to gain experience of the world and
learn about man’s vices, and his worth” (Hollander & Hollander, Canto 26. 94-99).
This passage seems to reference the opening lines of the Odyssey: “Many were they
whose cities he saw, whose minds he learned of” (Lattimore, 1.3), as well as
�Hauer 9
Odysseus’ misguided insistence upon landing on the Cyclops’ island, to “go and find
out about these people, and learn what they are….” Dante’s language also references
The Aeneid of Virgil, Dante the Pilgrim’s beloved guide, who wrote his own epic hero,
Aeneas, to exhibit filial piety as his most characteristic and reliable virtue.
Dante’s treatment of Ulysses in Inferno has other interesting references to Homer’s
Odysseus that can help us see how foreign Homer’s notion of propitiation is to a
theistic understanding of redemption. In Canto 26, the shade of Ulysses uses the
following language as he boasts to the pilgrim Dante how Ulysses exhorted his men
to cross the straits of Hercules in search of the realm of Purgatory:
“O brothers,” I said, “who, in the course of a hundred thousand perils, at last
have reached the west, to such brief wakefulness of our senses as remains to
us, do not deny yourselves the chance to know – following the sun – the
world where no one lives. Consider how your souls were sown: you were not
made to live like brutes or beasts, but to pursue virtue and knowledge”
(Hollander & Hollander, 26. 112 – 120)
Dante’s passage has much in common with the opening of Homer’s Odyssey book 10,
where Odysseus’ companions lie disheartened, having just landed on the shore of
Circe’s island after being unable to control their impulses with Aiolus’ precious bag
of winds. Odysseus exhorts his disconsolate men as follows: “Dear friends, for we do
not know where the darkness is nor the sunrise, nor where the Sun who shines
upon people rises, nor where he sets, then let us hasten our minds and think,
whether there is any course left open to us…” (Lattimore, 10. 190-193). The
overlap between this passage and Dante’s continues, as Circe will turn the first
group of Odysseus’ men into pigs, a reference that Dante also intends us to get, with
his “you were not made to live like brutes or beasts.”
Dante focuses on the way Odysseus’ insatiable curiosity and love of experience
constitutes an irredeemable failure to grasp the nature of original sin. Although
Homer condemns the companions and saves Odysseus, Dante insists that it is
Odysseus himself who is damned. His comrades, and the hapless Phaeacians as well,
could be reframed by Dante’s theist paradigm as simple sinners who
wholeheartedly repent of their impious follies once they learn of their
consequences.
Homer’s passage, by contrast, draws attention to the vice that Odysseus’
companions exhibit with the bag of winds, the lack of control that Homer and
Poseidon have noted. The companions’ recklessness with the bag of winds is the
reason Odysseus has to go to the underworld to talk to Teiresias, who will give
Odysseus the instructions for the propitiation of Poseidon. Odysseus himself draws
attention to the connection between the companions’ fatigue at their oars and the
current crisis that requires the intervention of Teiresias, calling his companions
after their expulsion by Aiolus as “teίreto… eἰresιηj,” (worn down by rowing –
�Hauer 10
10.78). iv Thus Teiresias’ name itself phonetically evokes the companions’ inability to
sustain themselves at the oars of Odysseus’ ship.
Odysseus’ exhortation here, “let us think, whether there is any course left open to
us,” has its own etymological reference to an important difference between
Odysseus and his companions. The ‘course’ that Odysseus asks his companions to
seek is the Greek word μήτις – ‘cunning’ or ‘strategem’, a word that has a homonym
in the word μητις – the indefinite ‘not anyone’ that Odysseus counts on when he
tells Polyphemus his name is οὐτις – ‘nobody.’ This is the trick that prevents
Polyphemus from getting help from the other Cyclops after Odysseus has escaped
and barricaded Polyphemus in his own cave. The sound of the word μήτις is thus
associated with the action that triggered Poseidon’s wrath. In book 12, Odysseus’
companions will not find a stratagem, they will fall under Circe’s spell. Odysseus,
however, with the intervention of Hermes, will find an effective stratagem. Odysseus
will be able to master Circe, while his comrades will have to suffer living “like brutes
and beasts” until their leader can save them.
From Dante’s perspective, it is precisely the lack of repentance that condemns
Ulysses to hell. While this aspect of Dante’s portrayal is true to form, in that
Homer’s Odysseus does not repent, Homer’s world does not have a theistic context
in which Odysseus could be condemned morally for this absence.
Since the tenets of moral theism don’t apply, Odysseus’ identity as propitiate on the
final journey has to be understood in some other way. Many scholars speak of the
anthropomorphic patron-client relationship between mortals and immortals in
Homer’s world. Odysseus is able to get home and slay the suitors because he is
favored by Athena, and Odysseus is favored by Athena because he has the kinds of
qualities that will enable him to get home and slay the suitors. Odysseus can utilize
deception and cunning to attain his ends, and does not see these means as entailing
scruple. In fact, this very freedom from scruple may be the key to Odysseus’ piety,
for deception and cunning require a certain type of self-discipline, and this selfdiscipline constitutes what Homer intends as Odysseus’ salvific virtue, the capacity
that distinguishes him from his reckless companions.
Nowhere is this more evident in Homer’s epic than in the tale Odysseus tells of what
happened on the island of Thrinakia, where only Odysseus has the self-control to
refrain from touching the Cattle of the Sun. Olson (1997) explores the relationship
Teiresias draws between the oar and a winnowing fan by connecting Teiresias’
prophecy to the upcoming ‘winnowing’ that will happen to Odysseus’ men in
retribution for their offense of eating the cattle on Thrinakia. v On Thrinakia, the
‘pure’ Odysseus is separated from the ‘impure’ chaff that his men represent in their
reckless unwillingness to endure hunger rather than touch the sacred cattle. Olson
notes that the name Thrinakia comes from qrinax, a common word for a winnowing
shovel (literally “three-toothed” or “trident,” which references the symbol of
Poseidon). Olson reads the entire legend of the Cattle of the Sun as a winnowing test
for the kind of self-control that Odysseus alone is capable of. Odysseus passes the
�Hauer 11
test himself, but cannot control his companions’ desire for the forbidden food,
because he has left them to go off to “pray to the gods” that they might show him a
road (ὅδος) homeward, “but what they did,” Odysseus says, “was to shed a sweet
sleep on my eyelids” (Lattimore, 12.338). vi
Throughout Homer, sacrifice is associated with the odor of the meat of oxen, cattle,
sheep and rams, an odor that spreads out into the air, conveying honor to the god to
whom it is dedicated. The meat of the different animals yields distinct odors, and
these odors seem together to make up the completeness of propitiation. Teiresias,
for example, tells Odysseus that the sacrifice to Poseidon on the final journey must
consist of a ram, a bull, and a boar. The herds of Helios though, are sacrificially
taboo, for Helios does not want to smell the odor of their flesh, since this god
“delights” in watching these animals with his eyes (χαίρεσκον, 12.380).
Although Odysseus reports to the Phaeacians that Lampetia told Helios that “we”
killed his cattle, Helios only asks Zeus to punish “the companions of Odysseus.”
Odysseus ordered his companions to swear to him before they left the ship that they
would not touch the herds on Thrinakia, but once the food from the ship has run out,
the men forsake their oath during Odysseus’ pious absence and slaughter the
animals. Although Helios has to be told that the animals have been killed, he
somehow knows to absolve Odysseus, whose salvific self-control includes the ability
to fast and pray when necessary, apparently without consequence to his strength
and endurance.
Since the events on Thrinakia are narrated by Odysseus to the Phaeacians, a reader
might be suspicious of Odysseus’ self-portrayal of his singular purity, especially as it
includes the actual conversation between Zeus and Helios. Odysseus, however,
makes a point of telling the Phaeacians that he heard this dialogue narrated by
Kalypso, who in turn heard it from Hermes. We are to understand the conversation
between Kalypso and Hermes as having taken place during Hermes’ visit, described
in book 5, to convey the message to Kalypso that Zeus has decided Odysseus must
finally get home. It was during this visit that Hermes presumably narrated to
Kalypso the conversation between Helios and Zeus which led Zeus to administer a
“just” punishment to Odysseus’ comrades for killing the forbidden cattle of Helios on
Thrinakia. vii
Once Odysseus has lost his comrades, his choice for killing the suitors is no longer
between “treachery, or openly with sharp bronze” as Teiresias told him. Now the
suitors must be overcome only by treachery, or trickery, as Homer’s word δόλος can
also mean. And so Odysseus does overcome them this way, which brings us to the
final scene of the Odyssey, in which the families of the suitors, seeking tribal
vengeance for their sons’ deaths, put on armor and march openly on Odysseus,
Laertes, Telemachus and their small armed band of loyal supporters.
Here we have an event which Teiresias did not mention in his prophecy, perhaps
because it is not left to Odysseus to find a way to accomplish it. At first it appears
�Hauer 12
that open warfare will break out. Laertes throws himself into the conflict, and exults
at the way “My son and my son’s son are contending over their courage” (Lattimore,
24.515). With Athena’s help, Laertes throws the spear that kills the father of
Antinous, who was, appropriately, the most vicious of the suitors. But as Odysseus
and Telemachus throw themselves into the front line, Athena stops them. “Hold
back,” she says, “from grievous war, so you can most swiftly come to a decision
without blood.” When Odysseus is slow to withdraw, Zeus sends a warning
thunderbolt. The epic closes with the phrase “And pledges for the days to come,
sworn to by both sides, were settled by Pallas Athene, daughter of Zeus of the aegis”
(Lattimore).
Because of the deus ex machina character of this passage, Athena’s role in this scene
could be compared to the end of Aeschylus’ 5th century tragic series, the Oresteia,
where Athena brings in judges to make a decision about Orestes, and then
persuades the furies to transform into the kindly ones and bless the Areopagus. In
Aeschylus’ play, the presence of the goddess accounts for Athens’ shift from the
customs of tribal vengeance to the rule of law. Zeus’ and Athena’s motive for not
permitting the battle between Odysseus and the families of the suitors can perhaps
be similarly characterized at the end of the Odyssey in terms of the gods’ righteous
concern with overcoming the tribal cycle of violence, a morally theist concern that
appears anachronistic in light of my argument here.
Yet this is not the only way to interpret the Odyssey’s final passage. The other
Homeric epic, The Iliad, ends with a similar stasis, as the hostilities of the Trojan
War are interrupted for the “burial of Hector, breaker of horses,” which restores
order to both the Achaian and the Trojan communities, similarly disrupted by
Achilles’ berserk mutilation of Hector’s corpse. The end of Iliad book one offers
another example, this time of hostility between the divinities, restored to stasis from
the brink of violence by the quick words and obsequious capering of the impulsive
god Hephaistos. Many Shakespearean tragedies, including King Lear, Hamlet, and
Julius Caesar, end with comments that draw attention to a cycle in human affairs in
which a broken community acknowledges the restoration of intelligibility and stasis.
This cycle need not be interpreted through a moral lens. Michel Foucault (1970)
suggests that such a cycle is intrinsic to a human sociology, as a pre-moral model in
which human beings emerge from conflict by “finding a solution that will – on one
level at least, and for a time – appease their contradictions.” viii If this is true of the
Odyssey, perhaps there is a connection after all between Athena’s and Zeus’ abrupt
ending of the hostilities and the contractual, not moral, piety I have attributed to
Odysseus.
In book 23, after the slaying of the suitors, Odysseus repeats Teiresias’ instructions
for the propitiation word-for-word, in the first conversation he has with Penelope
after their reconciliation. The word-for-word repetition is a mnemonic technique of
oral composition, but it also serves here to establish Odysseus’ piety with respect to
the importance of this task of propitiating Poseidon. The epic could have ended
without further reference to Teiresias’ mention of the final journey. The iteration
�Hauer 13
suggests that Odysseus’ intention is to follow the instructions precisely, and further,
to enlist his wife’s support and understanding of the significance of this crucial task.
This, I suggest, might be the most profound meaning of Odysseus’ νόστος.
Odysseus’ earnest piety takes additional meaning from the final passage, in which
the conflict between Odysseus and the families of the slain suitors is resolved by
divine intervention. The Odyssey ends with a divine mandate that the people of
Ithaka live in peace. Odysseus’ need to propitiate the one remaining hostile deity
now emerges as an urgent prerequisite to his fulfilling his political potential as a
wise and engaged combat veteran, capable of rendering due observances to a
complex hierarchy of divinities. Odysseus, Homer suggests, will not permit the kind
of lapse that exposes the religious and political immaturity of Poseidon’s existing
mortal kin, the Phaeacians. Perhaps it is Poseidon’s frustration with the
forgetfulness of those who should by their kinship be his most zealous celebrants
that leads the god to wish that Odysseus extend the Poseidon-cult into the
landlocked agrarian communities of what scholars assume was Arcadia, a name that
calls to mind Virgil’s eclogues, which are pastoral idylls of shepherds and wood
nymphs. The pastoral pleasures certainly await Odysseus, once he has seen to it
that his own religious observances will be acceptable to all the gods, even Poseidon,
in their due pantheistic order.
Hansen, William F. “Odysseus’ Last Journey.” Quaderni Urbinati di Cultura Classica. No 24. 1977. 27 –
48.
ii Pucci, Pietro. The Song of the Sirens: Essays on Homer. Roman & Littlefield, 1998.
iii Dodds, E. R. The Greeks and the Irrational. University of California Press. 1951. 30.
iv Dimock, G. E. Jr. “The Name of Odysseus.” The Hudson Review. Vol. 9, No. 1. 1956. 11.
v Olson, S. Douglas. “Odysseus’ “Winnowing Shovel” (Homer. Od. 11. 119-37) and the Island of the
Cattle of the Sun.” Illinois Classical Studies. Vol. 22. 1997. 7-9
vi It is worth noting here as a digression that Odysseus’ landing on Ithaka is characterized by his
being in a deep sleep.
vii Burkert, Walter. Greek Religion. Harvard, 1985. 129-30.
viii Foucault, Michel, The Order of Things. Pantheon Books, 1970. 357.
i
Additional Bibliography
Adkins, Arthur. Merit and Responsibility. Midway Reprint, 1975
Burkert, Walter. Greek Religion. Harvard, 1985.
Kirk, G. S. The Nature of Greek Myths. Overlook Books, 1975.
Lord, Albert B. The Singer of Tales. Harvard, 2000.
Parry, Milman. Ed. Parry, Adam. The Making of Homeric Verse. Oxford, 1971.
�
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
paper
Page numeration
Number of pages in the original item.
13 pages
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 winnowing oar : Odysseus' final journey
Description
An account of the resource
Transcript of a lecture given on September 21, 2016 by Claudia Hauer as part of the Dean's Lecture and Concert Series.
Creator
An entity primarily responsible for making the resource
Hauer, Claudia
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
2016-09-21
Rights
Information about rights held in and over the resource
Meem Library has been given permission to make this item available online.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Subject
The topic of the resource
Homer. Odyssey.
Homer -- Characters -- Odysseus.
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
24004562
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/004f7729cdec06b265fff02cd54e4566.pdf
a488c86ca1ae76b539ad54db3feca46e
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
paper
Page numeration
Number of pages in the original item.
33 pages
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 tragedy of Oedipus at Colonus
Description
An account of the resource
Transcript of a lecture given on April 29, 1994 by Janet Dougherty as part of the Dean's Lecture and Concert Series.
Creator
An entity primarily responsible for making the resource
Dougherty, Janet
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
1994-04-29
Rights
Information about rights held in and over the resource
Meem Library has been given permission to make this item available online.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Subject
The topic of the resource
Sophocles. Oedipus at Colonus.
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
24000294
Friday night lecture
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/c31006301547cf74718332e6c2c7b0bf.pdf
5476000426cfcb7dd3df21a39e549846
PDF Text
Text
In the year 610 of the Christian Era, a merchant of the prominent Quraysh tribe sat
meditating in a cave on Mt. Hira near Mecca. He heard a voice saying,
Recite: In the Name of thy Lord who created.
created man of a blood clot.
Recite: And thy Lord is the Most Generous,
who taught by the Pen,
taught Man that he knew not. [96.1-5]
Thus began the youngest of the major world religions and one of the most successful
lives in world history. As a religious, political, and military leader, Muhammad
(570-632) is without equal. Only Moses comes close, but Moses was not allowed to
enter the Promised Land, while Muhammad returned to Mecca as a victorious conqueror.
We are. moreover, fortunate to have better documentation for his life than for that of
Moses, Jesus, or the Buddha. On any reckoning, Muhammad’s biography is one well
worth studying. If you read the Qur'an, you may want to read along with it the most
important early biography, the Life of Muḥammad by Ibn Isḥāq.
Today, however, our primary goal is to become acquainted with the Qur’an. While
some light may be shed on this great book by a fuller knowledge of its historical context,
nothing replaces study of the text itself. Thus, most of my talk will focus on the primary
text, though I will first discuss some of the major events and issues that form the
background of the Qur’an.
Muhammad was an orphan. His father died before he was born and his mother when
he was six years old. His grandfather took care of him for two more years before he died
#1
�as well. Thereafter his uncle Abu Talib, head of the Banu Hashim clan, assumed
guardianship of the boy. Thus Muhammad grew up as something of an outsider within
Meccan society. Although he did belong to its most prominent tribe, the Quraysh, he was
a weak and vulnerable member of it. He rose to prominence, however, due to his skills as
a caravan trader, as well as for his reputation of honesty. When he was 25, the wealthy
widow Khadija, rather impressed, asked for his hand in marriage, was accepted, and
became his first wife.
Mecca was a major hub of the Arabian caravan trade routes that connected the
Byzantine Empire in the north with the spice-exporting Yemen in the south. The Quraysh
not only dominated Meccan trade but also were custodians of the Kaaba, the central
shrine for the still largely pagan Arab tribes. The word Kaaba, related to our word
"cube", refers to the cubical structure enclosing the Black Stone, a sacred object
traditionally venerated by the pagan Arabs and possibly of meteoric origin. Mecca and
the Kaaba were already sites of pilgrimage before Muhammad's time, the time that
Muslims refer to as Jahiliyya, or the time of ignorance.
During their sojourn there, the Arabs would hold fairs, including competitions in
poetry, still a largely oral art. Several of these pre-Islamic poems survive. Some of them
are known as the "Hanging" or "Suspended" Odes and were supposedly hung up in the
Kaaba as a token of honor.
Although Arab polytheism still flourished at its major center of Mecca, monotheistic
religions were common not only in the surrounding areas but even with Arabia itself.
Orthodox Christianity was the official religion of the Byzantine Empire, while the
#2
�Sassanid Persian Empire supported Zoroastrianism, arguably a monotheistic faith,
although a highly dualistic one. Many Christians, of various sects, were spread througout
Arabia, and there was a sizeable Jewish community in the city of Yathrib.
Thus when Muhammad brought forward his monotheistic message, he had many
enemies. Although he had hoped to find a receptive audience among the “People of the
Book”, i.e, Jews and Christians, in this hope he was largely disappointed. The fiercer and
earlier struggle, however, was against the leaders of his own city and tribe, the polytheist
Quraysh, for Muslims, like Jews and Christians before them, not only believed in the
existence of one God, but held that God to be a jealous god, a god who would “have no
other gods before him.” Polytheism was not simply mistaken, but even a direct affront to
God and could not be tolerated.
Polytheism is more tolerant than monotheism. The chief god of the Arabic pagan
pantheon was Allah, or "the God." "Allah" simply comes from a common Semitic root
for "god" and is cognate with Hebrew Elohim and Ugaritic El. The pagan Arabs had
traditionally associated other gods with Allah and worshipped these other divinities, in
particular Allah’s daughters (al-Lat, Manat, and al-Uzza). The polytheists could well
accept that Allah was the one supreme god; they could not, however, accept that he was
the only god or the only god to be worshipped. Particularly offensive to this traditional
tribal society, however, must have been the claim that their ancestors, by worshipping
associates alongside of Allah, were now burning in hell. Moreover, Muhammad’s attack
upon polytheism was a direct threat to their domination of the Meccan trade and shrine.
#3
�The polytheists challenged Muhammad to prove his apostleship by performing a
miracle. He replied that it was not in his power to perform miracles, but only in God’s
power to do so, and that the Qur’an itself was the miracle. A noble, elevated discourse
spoken through an illiterate merchant, the Qur’an impressed both believers and nonbelievers alike. Muhammad challenged his opponents to sit down and produce
something like it. If they could not do so, the argument goes, then the Qur’an must be a
work of greater than human creation.
Besides the Qur’an itself, there is one other miracle involving Muhammad that
cannot be passed over in silence, since it is the basis of the Muslim claim on Jerusalem as
a holy city. It is reported that one night as he was sleeping in Mecca, Muhammad was
transported by the fabulous winged beast Buraq to the Temple Mount in Jerusalem,
whence he was allowed to ascend the seven heavens and discourse with Abraham, Moses,
and Jesus. Thence he was brought back to Mecca the same night. More than half a
century after the Muslims conquered Jerusalem from the Byzantine Christians, the
Umayyad Caliph Abd-al-Malik, had the Dome of the Rock constructed on the Temple
Mount, known to Muslims as Haram es-Sharif.
The hostility of the Quraysh leadership could well have led to the murder of
Muhammad, if it had not been for the protection of his still pagan uncle Abu Talib. The
killing of somebody under tribal protection would have led to a blood feud. So instead of
attacking Muhammad directly, the polytheists persecuted his followers. Despite
persecution, Islam grew, attracting in particular many of the alienated members of
Meccan society, such as freedmen and slaves. When Abu Talib died, however, (619) and
#4
�the new leader of the Banu Hashim, Abu Lahab (another uncle of the prophet) withdrew
protection from him, Muhammad looked for another home for the Muslim community.
When an opportunity for refuge and alliance presented itself in nearby Yathrib, he and his
Muslim followers migrated there. This migration, or hijra, is the beginning of the
Muslim epoch.
Up to this point, Muhammad had been a religious leader. Now he became a political
leader by founding the nascent Islamic state in Yathrib, now known as Madinat an-Nabiy,
that is, the City of the Prophet, or Medina. The revelations of the Medina period show a
much greater concern for political matters and laws relevant to the foundation of a state.
The hostility between the Muslims and the polytheists of Mecca did not end then,
however. Muhammad insisted that the Muslims be allowed to worship at the Kaaba,
which he claimed had been originally a monotheist shrine founded by Abraham and his
son Ishmael. The Meccans had also confiscated Muslim properties in Mecca and the
immigrants to Medina turned to the Arab tradition of caravan raiding to make a living.
This hostility broke out into open war when Muhammad led the Muslims in a raid on a
Meccan caravan at Badr (624). Engaging with reinforcements from Mecca and
outnumbered by more than three to one, the Muslims won a decisive victory. After
further battles with mixed results, Muhammad entered Mecca as a conqueror in 630,
pardoned nearly the whole population, and purified the Kaaba of its idols.
Muhammad only lived for two more years. In that time he completed the conquest
and conversion of Arabia and unified the Arab tribes for the first time in history, a
unification made possible perhaps by religion alone. He thus provided the basis for the
#5
�astonishing Arab military expansion that was to explode onto the world scene shorty after
his death. He had no surviving sons, however, and his only significant failure as a leader
was that he did not appoint a clear successor or establish a clear policy of succession.
This failure resulted in a series of civil wars after his death and in the schism of the
Islamic community into Sunni and Shi’ite sects that has remained of fateful importance
even to the present day. The majority sect, the Sunnis, accepted Abu Bakr as the caliph
or successor to Muhammad, whereas the Shi'ites believed that Muhammad's nephew and
son-in-law 'Ali should have been recognized as the first caliph.
Even if Muhammad had only united the Arab tribes, he would be remembered as an
eminent political and military leader. But his importance as not merely an Arab leader,
but also as a world leader rests on his prophetic mission. For although the Qur’an is in
Arabic and addresses Arabs most directly, its message is of universal import. From the
beginning, Islam, like Christianity, has seen itself as having a universal mission. So
without further ado, let us turn to the Qur’an.
When we first encounter with the Qur’an as Westerners, we are likely to be puzzled.
This is not a book like the books we are familiar with. It does not tell a story like the
Iliad or War and Peace. Although it has many themes in common with the Bible, it lacks
the narrative frame that organizes many, if not all, of the books of the Bible. Although it
has chapters, or suras, there is little or no apparent connection between a given chapter
and the one that comes before or after it. Even within a given sura, one can encounter a
bewildering mixture of prophetic warnings, stories, and legal stipulations. So our first
question is, “What kind of book is the Qur’an?”.
#6
�Just as the Bible is not one book, but a collection of many books, so too the Qur’an is
not a single revelation but a collection of several revelations. If one were to sit down and
read the entire Bible, one would be rightly puzzled if one were to find the book of Joshua
next to the Gospel of Matthew, the Song of Songs next to Paul’s Letter to the Romans. It
is not surprising to find diversity within the Bible, a collection of texts spanning some
thousand years, written by different authors, addressing different audiences in widely
divergent circumstances. Since the Qur’an, however, was all revealed within a span of
some 23 years, and to one man, Muhammad, we might have expected a high degree of
uniformity, and while there is more uniformity in the Qur’an than in the Bible, there is
still a surpising amount of diversity, as we shall see.
When I say that the Qur’an was revealed to Muhammad, I do not wish to take a
stance on the question of divine authorship, but I do want to emphasize that Muhammad
did not compose or write this book. According to all accounts, both those supportive of
and hostile to him, Muhammad spoke forth individual suras while in a kind of trance or
ecstatic state. Some believed that he was receiving communication from the angel
Gabriel, others that he was possessed by a genie or demon. The former, of course, took
him to be the latest prophet and became his first followers; while the latter accused him
of being a “poet possessed,” alluding to the traditional Arabic view of poets as being
possessed by some divine or demonic spirit. The Arabic word for "crazy," majnun
derives from the same root as jinn or genie.
While some thought that he spun old wives’ tales, there is no contemporary
accusation that he was simply “faking” an ecstatic state for some ulterior motive, e.g., a
#7
�political one. This, I have no doubt, is how Machiavelli sees Muhammad, thus joining
him with Numa and Moses as political leaders who feigned divine communication in
order to bolster a political order. But telling against this view is the fact that when the
Quraysh offerred Muhammad political leadership in exchange for ceasing to preach
monotheism, he refused.
Muhammad spoke forth individual revelations or suras when he fell into an ecstatic
trance. He and many of his followers were illiterate, so although some may have been
written down by his literate followers, by and large the revelations were passed on by
word of mouth, until they were all written down and collected by the third caliph
‘Uthman (c.656). Although traditions had passed down some information about when the
various suras were revealed, in particular whether during the Meccan or the Medinan
period, ‘Uthman did not attempt to arrange the suras chronologically. Instead, by and
large, and with the exception of the first sura, the suras are arranged from longest to
shortest.
It turns out that the Meccan suras tend to be shorter than the Medinan suras, so the
Qur’an roughly moves in a backwards chronological order. Thus the traditional Muslim
way of learning the Qur’an in Arabic—beginning with the end of the book—also makes
chronological sense. A concern with chronology, however, is a largely Western concern,
for Muslims would deny that there is any change or development in the message revealed
in their holy book, whereas Westerners are always looking for development, even where
there is none to be found. Although I would argue that there are interesting differences
between the Meccan and Medinan suras, it is still debatable how significant those
#8
�differences are. The Meccan suras tend not only to be shorter, but also often use beautiful
natural imagery to discuss the coming Day of Judgment. The Medinan suras, by contrast,
are not only longer, but often deal with many of the social and legal issues that needed to
be addressed by the nascent Islamic state in Medina.
So the Qur’an is not a composition, if by “composition” we mean an arrangement
ordered according to a certain principle, so that it would be impossible to move pieces
around and still have the same thing. Exodus cannot come before Genesis, the death of
Patroclus cannot come before the anger of Achilles, Proposition I.47 of Euclid cannot
come before proposition I.1. Nothing is lost, I would argue, by reading the Qur’an
backwards. This is another way of saying that the Qur’an is a collection rather than a
composition.
But perhaps a more important point to emphasize is that each sura is meant to stand
on its own. The longer suras, one might argue, are even meant to present the whole truth.
Thus to go from one sura to another in sequence is not like adding pieces together to form
a whole picture but is like revisiting the same truth again and again, sometimes from a
slightly different angle. Thus a key feature of the form of the Qur’an is repetition. While
this may be tedious for a Western reader who is used always to encountering something
new in the next chapter, this formal feature also reinforces one of the central points of the
content of the Qur’an: human beings’ central failing is that they are forgetful. Prophets
come to remind us of the truth that we have forgotten or that we would like to forget.
And as anybody knows who has tried to learn a foreign language, repetition is the key to
remembering.
#9
�To fend off the accusation that Muhammad was just another “posessed poet,” the
Qur’an itself is claimed not to be poetry, altough it does make use of many poetic
techniques. The suras are composed of verses and make extensive use of end rhyme. I
will now play for you a recitation of the first sura, “Al-Fatihah”, or “The Opening.”
Notice the end rhyme on “-im, -in.”
I hope this excerpt, even through the medium of a foreign language, gives you a sense
of the beauty, power, and appeal of the original. These features of language, in particular
of poetic language, suffer the most in the process of translation. Nor are they thought to
be extrinsic to the essence of the Qur’an. For the Qur’an tells us more than once that it
is written in clear, noble Arabic. The incomparable beauty of the language is the main
argument for the Qur’an being a divine revelation. The verses are called ‘ayāt,’ which
literally means “signs.” Just like the beautiful and powerful cosmic signs such as the sun,
the moon, and the stars, the verses of the Qur’an are taken to be signs that point to the
power, goodness, and wisdom of the Creator who made them.
Having touched briefly on the form of the Qur’an, I will now turn to its content. The
first and most essential part of this content is the theology. A concise statement of its
theology is provided by sura 112:
Say: ‘He is God, One
God, the Everlasting Refuge,
who has not begotten, and has not been begotten,
and equal to him is not any one.’
#10
�Thus God is one and without associates. That he neither begets nor is begotten not
only rules out the Arab polytheist beliefs that he has daughters but also the Christian
trinitarian doctrine. He is eternal and absolute. Elsewhere we are told that he is allknowing and all-powerful. He created everything, not only inanimate things like the sun
and moon, stars and earth, but also the different orders of living things—the angels, the
jinn, and human beings and plants and animals. God is not only just but also
"compassionate and merciful." He commands human beings to do good and resist evil,
but is compassionate towards those who turn to him and ask for forgiveness. On the Day
of Judgment, human beings will be resurrected and summoned before God. Their good
and evil deeds will be recorded and weighed in a balance. Those whose good deeds
prevail will be rewarded will eternal life in Paradise. Others will be cast into the pit of
Hell to suffer eternal torment.
When God created Adam he commanded the angels to bow down before him. All did
so except for Iblis (Satan), who thereby became man’s bitter enemy. Adam and Eve were
cast from the Garden for eating of the fruit of the tree of life, contrary to divine
prohibition. There is no Islamic doctrine of original sin, however. We are not being
punished now for the sin that Adam and Eve committed. We have, however, inherited
their forgetfulness. In particular, human beings get caught up in pursuing their individual
self-interest, such as accumulating wealth, and forget divine warnings. We will all die
and cannot take our wealth with us. We will all be judged and our wealth will not help
us. We are commanded to provide for the more vulnerable members of society—the
#11
�widow, the orphan, the poor. We are commanded to do so by paying the alms tax, the
zakat. Failure to do so will result in grievous punishment in the hereafter.
Prophets have been sent to all peoples and have by and large been ignored. Even
after punishment came upon certain cities that ignored a prophet’s warnings, others did
not heed those examples. God has even sent down two books, the Torah and the Gospel,
to be constant reminders. The people who preserve those books, the “People of the
Book” (i.e., Jews and Christians), continue to bear witness to the one true God, although
even they have altered the true message by corrupting the divine text with human
interpolations. During to these corruptions, Islam, unlike Christianity, does not regard
earlier biblical texts as part of its canon. All the truths of the Torah and Gospel are also to
be found in the Qur'an itself. Muhammad has now been sent as the final prophet, as the
“seal of the prophets,” so this is humanity’s last opportunity to finally get the message.
The message has been essentially the same ever since Abraham, the first monotheist,
brought it to human beings. By submitting his willing to Allah, the one God, Abraham
became the first Muslim, (“one who submits”). The word muslim comes from the same
root as the greeting salām, and is cognate with the Hebrew shalom. According to Islam,
Islam did not begin with Muhammad but rather with Abraham. Muhammad’s importance
lies not in founding Islam, but in restoring it and in being the final prophet. Together
with his son Ishmael, the ancestor of the Arabs, Abraham built and consecrated the
central shrine of Islam, the Kaaba in Mecca.
To receive the message brought first by Abraham, restated by Moses and Jesus, and
finally restored by Muhammad, is to be a believer. To ignore or reject the message is to
#12
�be a non-believer, or infidel. Since the essence of the message is monotheism, infidels
and polytheists are seen as one and the same. Because prophets have been sent to all
peoples, there are no “innocent” polytheists: every people has had an opportunity to
accept the monotheist message. Since there are clear signs everywhere pointing to the
existence of one God, rejecting the oneness of God is taken to indicate not mere
ignorance, but willful ignorance. Polytheists reject God because they want to, not
because they are clueless. Some passages suggest a doctrine of predestination: "God
guides whom he wills and leads astray whom he wills."
The “People of the Book” are not infidels, nor are they believers in the proper sense.
While they have accepted the core of the message—i.e., that God is one—they have
become confused as to other aspects of it. Christians, for example, have mistakenly taken
their prophet Jesus to be not a mere messenger of God, but to be God. Jews have
wrongly rejected Muhammad’s prophetic mission.
Islam asserts a strong dualism of good versus evil and sees them as in constant
struggle with one another. Struggle, or jihād, is a central concept of Islam, although it is
not quite one of the pillars of the faith, at least for Sunnis. Just as in the universe, so too
amongst human beings and in the human soul there is a constant battle between good and
evil, a battle that will last until the Day of Judgment, when all will be resolved by God.
Since God is good, and believers are the ones who have taken God’s side, believers are
inherently on the side of good. This does not mean that believers cannot fall into evil or
err, but it does at least mean that they are on the right side of the cosmic struggle.
Contrariwise, to disbelieve is to go against God, to side with evil against good. Thus
#13
�whatever meritorious action, such as feeding a beggar, disbelievers may do, that action
cannot override the fact that disbelievers have taken the wrong side in the battle of good
versus evil. While they continue in their disbelief, they cannot be saved. Believers, on
the other hand, are not guaranteed salvation, but they will at least receive God’s open ear
and mercy when they ask for forgiveness for their sins.
The struggle against disbelief and evil in oneself and in the world has important
implications for how the Islamic community defines itself in relation to others. During
the Meccan period, when Muslims were a perscuted minority in a largely pagan city, the
message preached sounds something like a message of toleration, as we can see from sura
109:
Say: ‘O unbelievers,
I serve not what you serve
and you are not serving what I serve,
nor am I serving what you have served,
neither are you serving what I serve.’
To you your religion, and to me my religion!’
Now this sura can be taken in more than one way. The weakest reading is that it is a mere
observation that Muslims and polytheists have different religions. But since this is said
directly to polytheists, it is at the very least an act of defiance, for polytheism seeks to
incorporate new gods and cults within itself. It may even, as we can see from Herodotus,
deny the existence of different religions. This sura may be a way of saying, “You may
say that both you and we worship Allah, but in fact we don’t worship the same thing, for
#14
�we worship Allah alone, while you worship him alongside of his supposed daughters and
other false gods.” The last line is thus an assertion of an impassable barrier between
Islam and polytheism.
Another intriguing possibility lies in an ambiguous word in the last line. The
word translated as “religion,” din, can also mean “judgment,” as in the expression,
yawmu d-din, the “Day of Judgment.” Thus we could translate instead, “To you your
judgment, and to me my judgment.” This could be a way of saying, “We fundamentally
disagree, and God will decide between us on Judgment Day.”
Whichever of these possible readings we adopt, something like tolerance is still
being proposed, for in this sura the believer is told to speak the truth to the non-believer,
rather than to attack, oppress, or kill the unbeliever. It does not, however, go against the
idea of a fundamental struggle between good and evil, or between believers and nonbelievers. The Muslim community in Mecca was not in a position to take the offensive
against the Meccan polytheists, so the most that can be expected of them is to maintain
the integrity of their belief by bearing witness to it, i.e., being martyrs for it, in the face of
persecution and oppression.
Once the Muslims migrated to Medina, however, and became powerful enough to
assert themselves against the Meccans, they did so. And the suras from that period reveal
a more aggressive and militant policy against polytheism. Muslims are commanded to
fight the polytheists of Mecca until they cease oppressing Muslims and allow them to
worship in the sacred mosque of Mecca: “Fight them, till there is no persecution and the
#15
�religion is God’s; then if they give over, there shall be no enmity save for
evildoers.” (2.193).
Thus Islam is not a religion that says “Turn the other cheek.” On the other hand,
Muslims are explicity warned not to be the aggressors, “And fight in the way of God with
those who fight with you, but aggress not: God loves not the aggressors.” (2.190) Thus
only defensive warfare is justified, and it is not only justified but even commanded.
Moreover, while Muslims are commanded to spread the word, forced conversion is
explicitly forbidden, “No compulsion is there in religion.” (2.256).
The People of the Book have a special status within Islam. While conflict
between Muslims and polytheists is seen as nearly unavoidable, the People of the Book
should be granted tolerance as fellow, although erring, monotheists. Tolerance in this
context means that Jews and Christians living in a Muslim society are allowed to practice
their own religion under their own laws so long as they recognize Muslim superiority and
pay a tax in exchange for Muslim military protection. While this policy is not explicitly
stated in the Qur’an itself, it did become enshrined in the shari’a or Muslim law. The
Qur’an itself is equivocal on the relations between Muslims and Jews or Christians. To
cite a favorable passage:
Dispute not with the People of the Book
save in the fairer manner, except for
those of them that do wrong; and say,
‘We believe in what has been sent down
to us, and what has been sent down to you;
our God and your God is One, and to Him
we have surrendered.’ (29.46)
#16
�We also read:
Surely they that believe, and those of Jewry
and the Christians, and those Sabaeans,
whoso believes in God and the Last Day, and works
righteousness—their wage awaits them with their Lord,
and no fear shall be on them, neither shall they sorrow. (2.62).
If we turn to the structure of the Islamic society, we find it bound together by religious
and social duties. Although the Qur’an itself does not assign a particular number to these
duties or refer to them as “pillars,” different Islamic sects have enumerated different
“pillars of the faith.” The majority sect, the Sunnis, enumerate five such pillars. Besides
payment of the alms tax, or zakat, that we have already mentioned, we also find the
prescription of five daily prayers, or salat, the pilgrimage to Mecca, or the hajj, as well as
the fast of Ramadan. The remaining duty, the shahada, or testimony of faith, is not
explicitly prescribed as a duty in the Qur’an but may be seen as a precondition for
accepting the Qur’an as a revealed word at all. It goes, “I testify that there is no god but
God, and I testify that Muhammad is the messenger of God.”
What kind of society do these duties promote? First of all, it is one that struggles
against the selfishness of individualism. There is nothing wrong with becoming wealthy
in itself, but there is if one does so at the expense of others, or if one refuses to contribute
to the welfare of those less fortunate. The Qur’an does not seek to abolish or level
existing social hierarchies, whether of rich vs. poor, free vs. slave, or man vs. woman, but
#17
�it does accept the spiritual equality of all before God and insists that all have a duty to
attend not only to the spiritual, but also to the physical, welfare of all others in the
community.
The opposition between the spiritual and the physical, between the spirit and the
“flesh,” so marked in Christianity, is not so strong in Islam. Islamic paradise includes
flowing water, flourishing plants, abundant honey, and beautiful virgins and youths.
Christians have long been scandalised, but that only shows that Muslims do not war
against the flesh as Christians have for so long. Given that God has made both our bodies
and our souls, our flesh and our spirit, to reject the physical is to reject part of God’s
creation. While Islam does believe in a strong opposition between good and evil and
does contrast this current inferior world with the superior world to come, it does not show
a marked contrast between flesh and spirit, nor does it brand the “desires of the flesh” as
inherently evil. There is nothing wrong with desiring and enjoying beautiful things. This
world is inferior to the world to come not because this world is physical and the next
world is spiritual. Even Christians, after all, insist on the resurrection of the body, and
what would a body be good for in a purely spiritual realm? This world is inferior to the
next rather because it is fleeting and filled with injustice and selfishness.
To take one particular example. Islam prohibits the consumption of alcohol not
because it excessively titillates our appetite for gustatory relish, but rather because it
inhibits our ability to act as responsible members of society. Likewise, its sexual
regulations, against adultery and fornication for example, are justified in terms of
mainting a well-regulated society. There is nothing wrong with sexual pleasure per se,
#18
�much less with sexual desire. Modesty in dress is prescribed for both men and women,
although it is more strictly expected of the latter.
Let us take another example. Islam, along with Judaism and Christianity,
prohibits usury on loans to one’s fellow citizens. While economists will rightly point out
that prohibiting usury is both ineffective and inefficient, that criticism misses the point,
for the economists are presupposing a core human selfishness that Islam is striving to
overcome. It is possible to feed the poor to bolster one’s sense of grandeur, or one’s
ranking on some list; it may even work well when all in society simply pursue their
enlightened self-interest. But to do the right thing for the wrong reason is still not to act
morally: one should support charity just because it is the right thing to do.
This is much more that one could say about the Qur’an. I hope the little that I
have said gives you some sense of the context in which it was revealed, of its form and
content, and also of how it conceives of the nature of Islamic society and the relation of
Islam to other religions.
#19
�
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
pdf
Page numeration
Number of pages in the original item.
19 pages
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 Qur'an : an introduction for Johnnies
Description
An account of the resource
Transcript of a lecture given on June 16, 2021 by Ken Wolfe as part of the Graduate Institute Summer Lecture Series. Mr. Wolfe provided this description of the event: "In this introduction to the Qur'an, I will explore the context of its composition within the life of Muhammad and 6th century Arabia, its form and content, its relation to other texts and traditions (the Bible, Judaism, Christianity), and its influence upon certain aspects of the Islamic tradition."
Creator
An entity primarily responsible for making the resource
Wolfe, Kenneth
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
2021-06-16
Rights
Information about rights held in and over the resource
Meem Library has been given permission to make this item available online.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Subject
The topic of the resource
Qur'an
Islam
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
SF_WolfeK_The_Qur'an--An_Introduction_for_Johnnies_2021-06-16
Graduate Institute
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/5f137a68293223dc39a2274aab3c918b.pdf
fd8b0d4719762726a82a488d69fba459
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
paper
Page numeration
Number of pages in the original item.
47 pages
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 question of place in Aristotle's Physics
Description
An account of the resource
Transcript of a lecture given in 1994 by David Bolotin.
Creator
An entity primarily responsible for making the resource
Bolotin, David, 1944-
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
1994
Rights
Information about rights held in and over the resource
Meem Library has been given permission to make this item available online.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Subject
The topic of the resource
Aristotle. Physics.
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
24003315
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/cf1219cc9a669cb161a714337b3350b8.pdf
630c37848fc0a64d1edbe24b5287dca3
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
paper
Page numeration
Number of pages in the original item.
30 pages
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 question of final causality in Aristotle's Physics
Description
An account of the resource
Transcript of a lecture given on April 3, 1992 by David Bolotin as part of the Dean's Lecture and Concert Series.
Creator
An entity primarily responsible for making the resource
Bolotin, David, 1944-
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
1992-04-03
Rights
Information about rights held in and over the resource
Meem Library has been given permission to make this item available online.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Subject
The topic of the resource
Aristotle. Physics.
Causality (Physics)
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
24000038
Friday night lecture
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/4a65bf30753fde968144de5087f5a1a1.pdf
16ab3ce7f64760d83e8ce10568ec5584
PDF Text
Text
The Phenomenology of Blackness
By,
Michael E. Sawyer, PhD
Assistant Professor of Race, Ethnicity, and Migration Studies and
The Department of English
Colorado College
Delivered at
St. John’s College
Santa Fe, New Mexico
Carol J. Worrell Annual Lecture Series on Literature
22 November 2019
�1. Introduction:
My earliest encounter with the academic discipline we generally label as philosophy, occurred
during my sophomore year at the Jesuit prep school I attended in Chicago. My teacher, Brother
McCabe, was a Franciscan Monk who was a Kantian. In addition to endlessly referencing the
categorical imperative, Brother McCabe’s most memorable trope was “always examine your
presuppositions”. I vividly recall him asking me a question and when I began to answer he stopped
me and demanded I account for the presuppositions that would attend my attempt at an answer. My
first inclination was to propose that the central presupposition happened to be that to the extent I
didn’t provide any answer, much less a satisfactory one, I would be in danger of failing his class. That
seemed insufficient so I spent what seemed the better part of 10 minutes examining the assertions and
systems of thinking, understandings, and misunderstandings that marched before the answer I don’t
think I ever managed to provide. Looking back now I realize that the genius of Brother McCabe’s
pedagogical strategy was to insure we left his class first and foremost inclined to examine our
presuppositions. This to me is the foundation of thinking that can best be described as “critical” to
the extent it takes stock of the stuff around the thing you find interesting. Here, I have framed the
title of this talk as a statement: The Phenomenology of Blackness that includes several important
questions and a foundational presupposition. First the questions: What is Phenomenology? We have
an academic answer to that question that we can admit into our archive here as axiomatically the
manner in which the term is commonly employed in the discipline of philosophy. The Stanford
Encyclopedia of Philosophy provides the following definition:
Phenomenology is the study of structures of consciousness as
experienced from the first-person point of view. The central structure
of an experience is its intentionality, its being directed toward
something, as it is an experience of or about some object. An
experience is directed toward an object by virtue of its content or
�meaning (which represents the object) together with appropriate
enabling conditions.
The next term is perhaps more fraught and will be the subject of the talk: What is Blackness?
In referencing that question we identify the central presupposition which feels something like
Blackness is a Thing (das Ding) in the Hegelian sense of the term and it is possible to experience this
thing in the phenomenological sense.
There is quite a bit going on here and we do not have the time to properly account for all of
the moving parts so I will be direct in framing our employment of the elements that are commonly
understood in the field. This primarily revolves around the naming of Blackness as das Ding or the
Thing as the point of inquiry. What I mean here is most productively elaborated in §120 of Hegel’s
Phenomenology of Spirit which reads:
However, the diverse aspects which
consciousness takes upon itself are
determinate in that each is regarded as
existing on its own within the universal
medium. White is only in contrast to black,
etc., and the thing is a “one” precisely in
virtue of its being contrasted with others.
This is not a facile reading of Hegel that extracts this binary opposition of black and white as
philosophically significant without context. Here, Hegel is lingering on the implication of
“Consciousness” and its marginalization by being negatively implicated by sense certainty and
therefore requiring the more robust dialectical process that arrives with his understanding of the
relationship between Lord and Bondsman in the well know passages from §190 as an essential element
of the torturous journey of mediated self-consciousness to the terminus of Absolute Spirit. Here,
Hegel proposes the following:
The master is consciousness existing
for itself. However, the master is no longer
consciousness existing for itself merely as
�the concept of such a consciousness.
Rather, it is consciousness existing for
itself which is mediated with itself through
another consciousness, namely, through an
other whose essence includes its being
synthetically combined with self-sufficient
being, that is, with thinghood itself.
The intellectual water, so to speak, has become very choppy very quickly. This, in many ways
provides the linchpin or hinge to access the preoccupation of this effort: the notion of Blackness as a
discernible way of Being. Here, in the elaboration of consciousness, Hegel proposes that the indicator
of the need for the move to self-consciousness is the failed employment of sense certainty, in his
parlance the apparent separation between black and white, as spectacular. When he arrives at the
dialectic of Lord and Bondsman, the existence of the Thing, the master is only able to relate themselves
as master by the “thinghood” of the bondsman. What this means for my argument is that the
spectacular nature of the color black is the marker of marginal consciousness and with the subjects
that will be the preoccupation of this talk that certainty remains by rendering the dialectic of Lord and
Bondsman, in this instance moribund. The encounter is already over-determined. The dialectic, when
operating properly, does not prefigure the status of the subjects who are its participants. In this
instance, because of the spectacular nature of the color Black, the dialectic is reifying rather than
determinative. The Black subject is locked as the Bondsman because of the overdetermined nature of
the color Black and what we are exposing here as Blackness serves as the prison. I posit that this is
not an error on the part of Hegel. No lesser authority than the Socratic Dialog The Phaedrus establishes
how the color black arrives in the western imaginary as a lack when compared to its opposite, white.
The passage is instructive:
“Let us then liken the soul to the natural union of a team of winged
horses and their charioteer. The gods have horses and charioteers that
are themselves all good and come from good stock besides, while
everyone else has a mixture. To begin with, our driver is in charge of
a pair of horses; second, one of the horses is beautiful and good and
�from stock of the same sort, while the other is the opposite and has
the opposite bloodline. This means that the chariot-driving in our case
is inevitably a painfully difficult business. (246b)”
It is the detail of the description of the horses that concerns our work here in that it speaks to a robust
consideration of black and white as oppositional in quality. Socrates delineates the “goodness of the
good horse and the badness of the bad.” (253d)
“The horse that is on the right, or nobler, side is upright in frame and
well jointed, with a high neck and a regal nose; his coat is white, his
eyes are coal black, and he is a lover of honor with modesty and selfcontrol; companion to true glory, he needs no whip, and is guided by
verbal commands alone. The other horse is a crooked great jumble of
limbs with a short bull-neck, a pug nose, black skin, and bloodshot
white eyes; companion to wild boasts and indecency, he is shaggy
around the ears – deaf as a post – and just barely yields to horsewhip
and goad combined. (253d)”
Here we witness, at the earliest moments of the western philosophical tradition, explicit
reference to the soul as being divided in two with the worst impulses; those that do violence to reason,
represented by physical deformity of which “black skin” is but one manifestation of physical and
metaphysical disability.
Reason and the ability to properly accede to moral authority, in this
formulation, are exemplified by beauty, in conforming to a physical standard, and Whiteness. What
that means, walking with Hegel on our left and Socrates on our right, is that the visuality of black
(Hegel’s sense-certainty and the Socratic “Bad Horse”) arrives at the point of self-consciousness with
an externally imposed system of self-knowledge that we will label here as “Blackness”.
This understanding is most ably exposed by the canonical formulation of what I have labelled
“Tripartite Subaltern Self-Consciousness” by W.E.B. Du Bois in The Souls of Black Folk. In that text,
Du Bois establishes the insufficiency of the Cartesian Cogito for a system of knowing for the subject
we will label here as existing under conditions of externally imposed Blackness and the presupposition
of insufficiency; the figure Du Bois understands as the Negro writing:
�After the Egyptian and Indian, the Greek and Roman, the Teuton and
Mongolian, the Negro is a sort of seventh son, born with a veil, and
gifted with second-sight in this American world,-a world which yields
him no true self-consciousness, but only lets him see himself through
the revelation of the other world. It is a peculiar sensation, this doubleconsciousness, this sense of always looking at one’s self through the
eyes of others, of measuring one’s soul by the tape of a world that
looks on in amused contempt and pity. One ever feels his two-ness,an American, a Negro; two souls, two thoughts, two unreconciled
strivings; two warring ideals in one dark body, whose dogged strength
alone keeps it from being torn asunder. (Du Bois, 6)
Here, Du Bois understands the fractured nature of the relationship of the self to the self on
the part of the figure he understands as the Negro at a discernible remove from being a body that is
understood as Black: “two warring ideals in one dark body”, as the fulfillment of the trace that runs
from the pronouncement of Black as a lack from Socrates through Hegel’s destabilized dialectic. When
I note here that this lack of “true self-consciousness” in the parlance of Du Bois is properly glossed
by studying the result of a self that reaches out to touch itself and receives a negative response. Not
the “I am” that the Western philosophical tradition situates as the normative response to the self
thinking about itself but perhaps an “I am not” in the case of a distorted practice of self-analysis.
Where Du Bois describes a subject that is double-conscious, in that it only knows about itself “through
the eyes of others, of measuring one’s soul by the tape of a world that looks on in amused contempt
and pity” we find a subject out of synch with Descartes. Further, and here Jean-Paul Sartre’s text, The
Transcendence of the Ego, proves useful, he endeavors to explicate what he calls “consciousness in the
first degree” or “unreflected consciousness” as distinct from “consciousness in the second degree” or
“reflected consciousness”. Du Bois’ formulation is productively read alongside the theorizing of
Sartre. What Du Bois seems to mean is that the subject suffering from a lack of true self-consciousness
has, in fact, confused consciousness in the second degree for consciousness in the first degree. Further,
the secondary system of consciousness has contempt for the subject so situated and causes the self
that reflects on itself here to believe that it is, indeed, aberrant when in fact these are externally imposed
�conditions of knowing that fracture the possibility of a way to present the self for recognition by other
subjects.
This question of mutual recognition is the next step along this continuum that, in important
ways, returns to the visual nature of what Hegel understands as consciousness encumbered with sense
certainty and what Socrates describes as the physical manifestation of marginalization as exemplified
by the Black horse. In many ways, and this is, perhaps, an idiosyncratic methodological point, at this
stage of this analysis I believe we reach, what I like to call, Technical Exhaustion. That which has been
exhausted, in my understanding is first the utility of prose or discourse in the form of philosophy or
theory to describe the phenomenon in question. Second, not only does the visual exceed the technical
potentiality of words but Western epistemologies fail here as well. I hope we can discuss this assertion
in the Q&A, it is a statement that is meant to be positively provocative. But on this point, taking the
last assertion first, the manner in which western epistemology finds itself incapable of properly
describing the essence of the subject in question is because this system of knowing in fact is dedicated
to the destruction of the subject in question. One need only note the manner in which language has
already been stacked against the humanity of the subject understood to be Black. Black, the term itself,
is fraught. Our time does not facilitate tracing the tortuous path to resolving this tension but it can
broadly be understood as a component in the requirement to decolonize the canon and the reason the
academy has derived disciplines that range from Africana studies to Feminist and Gender Studies, etc.
One brief methodological point here. I do not believe that one decolonizes the canon through a
project of contraction or excision of the central pillars of the western system of knowing. On the
contrary, in order to exceed the boundaries of that system one must know what the boundaries happen
to be. With that in mind we have employed Socrates, Hegel, Descartes, and at the edges of that
tradition Du Bois to approach the phenomenon of Blackness. Beyond that boundary we will explore
a relationship that I am in the early stages of exploring: that between visual representations of Black
�people being coerced by the police, alternative modes of temporal existence and African-American
literature in the form of fiction. This is something of an intellectual bank shot so to continue the
analogy to a game of billiards, I will just call the shot.
I hypothesize that the bridge forward and backward is the visual. I further hypothesize that
the visual representation of police violence represents a temporal shift that speaks to what I have
called in other spaces the fractured temporality of the subaltern. And finally, I hypothesize that it is
only in fiction that we can grapple effectively with this disorientation. Elements of Roland Barthes’
Camera Lucida speak to this thinking and along with a reference to St. Augustine, will provide the
markers of the boundaries we intend to exceed together. Barthes’ writes:
The Operator is the Photographer. The Spectator is ourselves, all of us
who glance through collections of photographs-in magazines and
newspapers, in books, albums, archives…And the person or thing
photographed is the target, the referent, a kind of little simulacrum,
any eidolon emitted by the object, which I should like to call the Spectrum
of the Photograph, because this word retains, through its root, a
relation to ‘spectacle’ and adds to it that rather terrible thing which is
there in every photograph; the return of the dead. (Barthes, 9)
There is an important note here in that I intend to apply Barthes to analyze video even though
he proposes that the dynamism of moving pictures is discernably different than the immobility of the
photographic image. For our purposes here, we will see that it is the freezing of the subject in space
and perhaps consigning them to death in video representations of police violence that the motion of
the images is in fact only to memorialize what is the substantively coercively imposed immobility or
death.
I have a short bit of video that I would like to play at this point that is intended to further
buttress my argument. I am interested here in bringing into our conversation the way in which the
restriction of mobility memorializes the fractured temporal subjectivity of Blackness. To give some
context here, the mechanical process of walking preoccupied the thinking of Ray Bradbury in the
�writing of the canonical exposition of the dangers of run-away state power in Fahrenheit 451. Bradbury
reveals in the notes preceding his 1951 short story “The Pedestrian” that the arrest of the protagonist
for merely walking down the street near his home was informed by the author’s harassment by the
Los Angeles Police Department for doing the same. It is this notion of walking down a sidewalk that
serves as the opening scene of 451 that is an effect of the causality of police coercive force that restricts
the ability to walk that one might note Kant understands as evidence of maturity and I situate here, in
its restriction, as a technology for the creation and maintenance of marginalized subjectivity. This is
Kant from the essay “What is Enlightenment” for point of reference where he meditates on free
movement.
Here the progress of the subject is alienated from his humanity and therefore his freedom by
the pronouncement of this officer that weaponizes his body and possessions. The shift to spectacle
occurs when the officer asserts that the subject is being audio and video taped.
The restriction of his mobility occurs through the assertion by the officer that he is not free to leave.
There is a way in which that the qualification here, “to leave”. is redundant. The subject is not free.
He cannot walk and his immobility is contrasted by the hyper (in comparison) mobility of the humans
around him.
His possessions are confiscated and inspected.
The officer asserts that she will access the videotape to check for evidence of the attack she witnessed.
In spite of video evidence that is contrary to the claim that the subject had weaponized his golf club
and swung it at the police car he is arrested and charged with a series of crimes. What we have
witnessed is the way in which the fractured temporality of the subaltern body, here understood as
Blackness places this subject immediately in the clutches of death that has arrived with the presence
of the law. Recall the manner in which the officer begins to pronounce his death sentence. One can
detect that it is a recitation that is designed to render whatever coercive force she intends to visit upon
�this body as necessary. There is much to interrogate with this video but our time together requires us
to shift to the next element of my argument, that Western philosophical epistemologies find
themselves technically exhausted here and fiction becomes the most efficacious mode of explicating
what we are witnessing. Prior to addressing our literary reference, we will briefly linger with St.
Augustine as the last stop on the road of the western philosophical canon.
Saint Augustine proposes the following, “si nemo ex me quaerat, scio; si quaerenti explicare
velim, nescio” roughly translated; “if no one asks me what time is I know but if I have to explain it, I
don’t know what time is.” In my reading the Augustine establishes the distinction we need in the
front of our minds between time and temporality. In this instance, I am reading Augustine as actually
commenting on this distinction. The Saint understands time as a force and as a tool for measurement
but when asked to explicate that understanding his relationship to time shifts to the experience of it,
or what I am labelling as temporality or the self-referential understanding of the experience of time by
the subject or subjects in question. So here we are necessarily taking up the challenge posed by St.
Augustine to explicate time and to do so not from the perspective of dealing with it as a force or
measure but as a way of being. In the Augustine we find his explanation of the manner in which we
experience time instructive. Augustine accomplishes this through his careful analysis of reciting a
psalm by memory.
Suppose I am about to recite a psalm which I know. Before I begin,
my expectation is directed towards the whole. But when I have begun,
the verses from it which I take into the past become the object of my
memory. The life of this act of mine is stretched two ways, into my
memory because of the words I have already said and into my
expectation because of those which I am about to say. But my attention
is on what is present: by that the future is transferred to the past. As
the action advances further and further, the shorter the expectation
and the longer the memory, until all expectation is consumed, the
entire action is finished, and it has passed into the memory. (11 28:38)
�Paul Ricoeur, in Volume 1 of his Time and Narrative series writes the following in commenting
on this move by Augustine: “The solution is elegant-but how laborious, how costly, and how fragile!”
The fragility of this proposition is on full display when it is applied to the subject under investigation
here. Memory, or the past, serves as the sine qua non of this understanding of time. Following
Augustine’s analogy, it becomes immediately apparent that there can be no recitation of a psalm from
memory unless there has been a time in which the psalm itself was experienced and remembered in
some past. This does not attend in the case of the subjects who have had their relationship to genealogy
and history fractured via the Middle Passage and its telos, the condition of enslavement. What I mean
is that a set of experiences that serve to destroy, fracture, or confuse a coherent relationship to
memory, and here think of memory as operating as “culture”, also renders normative temporal
existence as established by Augustine, as impossible. Therefore, there can be no past, present, or future
from the perspective of Augustinian Time for the figure harmed by the Middle Passage and its echoes
we can label here as bigotry and or racism. In spite of this fracture with memory it is empirically “true,”
that the subject so harmed has a time in which it exists, it was created, and has an internal time
signature, because it has desire.
We do not have the time here to for me to fully expose the manner in which I am calling on
desire as the foundation of an interior sense of time but let it suffice to say that it is related to the
probative power of Terry Pinkerton’s recent re-translation of Hegel’s Phenomenology of Spirit. In that
essential document, Prof. Pinkerton revised what had been an error in translating the essential passage,
§167 as “self-consciousness is desire itself,” as opposed to the alternative and more traditional “selfconsciousness is desire in general”. This is transformational.
§167…But this opposition between its appearance and its truth has
only the truth for its essence, namely, the unity of self-consciousness
with itself. This unity must become essential to self-consciousness,
which is to say self- consciousness is desire itself. As self-consciousness,
consciousness henceforth has a doubled object: The first, the
�immediate object, the object of sense- certainty and perception, which
however is marked for it with the character of the negative; the second,
namely itself, which is the true essence and which at the outset is on hand
merely in opposition to the first. Self-consciousness exhibits itself
therein as the movement within which, in its own eyes, the
selfsameness of itself with itself comes to be.
It is this movement of the self within the self that I am positing as establishing the internal
time signature necessary for being Human. What this means is that the canonical formulation by
W.E.B. Du Bois in his Souls of Black Folks must be understood in a manner that asserts that what Du
Bois calls a lack of true self-consciousness is a lack of desire which, following this path, amounts to a
lack of a coherent internal time signature. What this means here is that we can understand the shattered
subject that preoccupies Du Bois as frustrated by an imposed understanding of their lack of historical
“situatedness” as an externally imposed subaltern sense of time that we will see Morrison witnesses as
the paralysis of “nows” that cannot recede into “thens” which cannot be supersceded by “whens”. It
is critically important that we carefully attend to what Du Bois has offered here in that what I am
holding up as the nexus of the fracture between the coercive nature of internal time for Blackness and
what Jacques Derrida understands as the “self-calling to the self” when he reflects on elements of
Kant’s Third Critique.
Briefly, I understand the establishment of an internal notion of time, this self-calling or autoaffection, as the foundation of what we understand as being Human, positively and self-referentially
aware of the self as a temporal being that then has an individually established time signature that can
be presented to other similarly situated beings for purposes of mutual recognition and here we should
have Hegel foremost in our minds. Here, operating from an understanding of the Cartesian Cogito, we
understand this self-calling as allowing us to witness the self-thinking about the self as a manner to
establish the subject in time.
What Du Bois describes, the subject with “no true self-consciousness”, is effectively an
externally imposed fracturing of the subject forming and reifying power of self-reflection in terms of
�Descartes and self-authorizing temporality on the part of thinkers like Derrida through Kant. Hence
the paralysis we explore that is exposed by Morrison which, when traced in this way, is what Du Bois
understands as the impossibility of the Black Body being able to productively exist as “Negro” in the
parlance of the day and “American” at the same time.
So here, we should recall the video we observed and witness it as an exemplar of a form of
subjective liminality and recursive ways of Being that render him out of time and beyond the
explanatory capabilities of tools like Heidegger’s understanding of the relationship of Being to
thrownness toward the telos of mortality, death. The subject we observed in the video comes to us as
a “rememory” (and we will attend to this shortly) in the way in which Morrison understands that
phenomenon. We are witness to his treatment in ways that are at no separation from the treatment of
human beings captured and transported against their will through the Middle Passage. His treatment
mirrors that of enslaved bodies who found themselves out of place: think here about those who
became victims to the empowerment of all white people to arrest any Black person with the passage
of the Fugitive Slave Act. His paralysis at the hands of the state is at no remove from that employed
in the aftermath of the Civil War by the Contract Labor System. The same goes for Jim Crow and
here this man finds himself unwittingly a participant in the present day’s carceral system that is fruit
of the poisonous tree of enslavement.
So, let us consider our encounter with the video as a form of memory that must be understood
as placing bodies so coerced and formed with the notion of Blackness as the way of being, as operating
outside of time. Recall here the manner in which I have summoned St. Augustine here. If we view the
video as a “rememory” and insert it into the psyche of a similarly situated being that lacks coherent
relationship to the past, Augustine’s system of temporal existence authorized by recall that I have
noted as fractured through Du Bois’ formulation of the unstable system of consciousness of
Blackness, finds itself exhausted and Morrison takes over. What I mean is that the collective system
�of coercive threat to the subject that functions like racism creates an interwoven system of awareness
that renders it impossible to separate the fate of one Black body from that of another. Further, the
fact that the behavior re-memoried here on video is of the same genus and species as the abuse of
Black bodies since roughly 1619 on these shores, the ability to fix one’s system of memory along a
coherent temporal continuum fails. Morrison, by summoning the ghost of the child Sethe executed to
save her from slavery, allows us to render this experience, this Blackness, legible.
The point of literary reference here is from Toni Morrison’s masterpiece Beloved that, in my
reading, revolves around the fractured temporality of those who are members of a family tree that
touches, even tangentially upon the depravity and howling savagery of the Trans-Atlantic slave trade.
The section in question here opens with the voice of the haunting figure of the murdered child Beloved
pronouncing, “I am Beloved and she is mine.” To read the prose in question is to experience Morrison
at her most sublime. Here she lays out a temporal continuum from the wars in West Africa that
antecede the Middle Passage to the experience of death by a child at the hands of their mother without
pause and without referent to the linear progression of time. What Morrison labels in the same text as
“Rememory” and relates the state of being as “All of it is now it is always now”1
I have posited that Morrison’s preoccupation with the Middle Passage as a subject disforming
catastrophe is a predictable ramification of the destruction of a coherent relationship to history and
culture that is a particularly insidious element of the metaphysical harm done by this transit.
2. Beings Out of Time:
I find Morrison’s Beloved to be an enchanted text that causes me deep, and oftentimes
unresolvable trauma whenever I visit it. But it is necessary and masterful in that capacity. As I proposed
a bit ago, Morrison is about the business of demonstrating that the trauma that is slavery is collective
and that trauma reverberates across the generations. What she calls “ramifications of ramifications”
�in her text Paradise. In Beloved, Morrison’s character Sethe is explaining this temporal confusion to her
daughter Denver. The text reads:
“I was talking about time. It’s so hard for me to believe in it. Some
things go. Pass on. Some things just stay. I used to think it was my
rememory. You know. Some things you forget. Other things you never
do. But it’s not. Places, places are still there. If a house burns down,
it’s gone, but the place-the picture of it-stays, and not just in my
rememory, but out there in the world. What I remember is a picture
floating around out there outside my head…”
“Can other people see it?” asked Denver.
“Oh, yes. Oh, yes, yes, yes. Someday you be walking down the road
and you hear something or see something going on. So clear. And you
think it’s you thinking it up. A thought picture. But no. It’s when you
bump into a rememory that belongs to somebody else.” (Morrison,
43.)
I will make the connection here explicit though I hope it might arrive of its own volition from
the narrative we are weaving together. Morrison is resolving the tension we have located between the
wages of the color black articulated by Plato and Hegel, the imperative of memory on the part of
Augustine, and the notion of fractured or compromised self-consciousness as described by Du Bois.
When we, whatever our subjectivity, encounter pictures of the coercion of Black bodies we are
encountering rememories and our positionality becomes indistinct from that of others similarly
situated across time and space. What I mean is that these pictures, these visual representations of
bodies in pain in the process of being rendered un-free, are what Sethe referred to as “thought”
pictures for two reasons. One might be obvious. They are the disassociated point of view of the bodies
in contact with one another, think Hegel’s Lord and Bondsman here and we as observers occupy a
third place in the room as spectator. However, I wish here to push Hegel a bit and propose that the
relationship between observer and observed is also involved in a dialectical relationship and the Thing
between us, Observer and Observed, in the observation of these videos becomes the time and subject
destabilizing wage of Blackness. The second is that the behavior we are observing is a thought picture
�in that it renders visible the thinking behind systems of white supremacy like Madison’s
pronouncement of fractional humanity in Federalist 54. I’ll quote the text here for clarity and context.
Madison writes:
But we must deny the fact that slaves are considered merely as
property, and in no respect whatever as persons. The true state of the
case is, that they partake of both those qualities; being considered by
our laws; in some respect, as persons, and in other respects, as
property. In being compelled to labor not for himself, but for a master;
in being vendible by one master to another master, and in being subject
at all times to be restrained in his liberty, and chastised in his body, by
the capricious will of another, the slave may appear to be degraded
from the human rank and classed with the irrational animals, which
under the legal domination of property. The Fœderal Constitution
therefore, decides with great propriety on the case of our slaves, when
it views them in the mixt character of persons and property.
Morrison continues and here we are in Book II of the text where she abandons the
conventions of punctuation to express what I am framing here as the rememory induced phantasm of
Blackness.
I AM BELOVED and she is mine. I see her take flowers away from
leaves she puts them in a round basket the leaves are not for her she
fills the basket she opens the grass I would help her but the clouds
are in the way
how can I say things that are pictures I am not
separate from her there is no place where I stop her face is my own
and I want to be there in the place where her face is and to be looking
at it too
a hot thing
All of it is now it is always now there will never be a time
when I am not crouching and watching others who are couching too
I am always crouching and watching others who are crouching too I
am always crouching the man on my face is dead his face is not
mine his mouth smells sweet but his eyes are locked. (Morrison. 248)
These are complex and fragile passages and worthy of our most diligent efforts at close
reading. I have struggled for years with these sections of the text but have found that they yield to my
effort when I address them with the understanding we have traced of Blackness. Blackness as an
externally imposed system of subjective disorientation that mires the subject in the impossible task of
�achieving the form of self-consciousness required to achieve forward progress. Further, I have been
aided in understanding Morrison’s notion of rememory as “thought pictures” by the overwhelming
presence of videos of Black bodies under conditions of coercion. All of this creates the very system
of disorientation that the spectral presence called Beloved experiences that disallows her from being
able to separate the experiences of those in her genealogy who have suffered coercive force.
There is a great deal going on here in these passages which represent the third in a series of
four of this form of narrative where Morrison takes up the challenge that she has embedded in the
passage in question, “how can I say things that are pictures”. In fact, the implicit question here is how
the subject experiencing these visions, these rememories, might process them and situate herself in
time and space and resist the coercion that she is experiencing as a ramification of those ramifications.
The passage opens with a decentering of the notion of internal mind/body separation as well as the
separation between discreet subjects. “I am Beloved and she is mine.” It is important to note here that
this is the only declarative sentence with the employment of a period to eliminate ambiguity. With this
understanding we can read the next passages as if the observer is also the actor. “I see her take flowers
away from leaves she puts them in a round basket the leaves are not for her she fills the basket she
opens the grass” What is important to note here is that Beloved has bumped into rememories of some
other subject in her genealogy. It both is and is not her mother. In this system of perception, it is her
mother and hers and everyone in between, starting with a time before the middle passage. The beauty
of the images she bumps into are necessary as the counterpoint to the depravity of Atlantic World
Slavery. This metaphysical impossibility of separation is experienced by this subject as physical
inseparability. “I am not separate from her there is no place where I stop her face is my own and I
want to be there in the place where her face is and to be looking at it too
a hot thing” If one
explores this text and these sections in particular one will encounter this refrain “a hot thing”, over
and over again. Things indeed become hot and this subjective immersion in the wretched horror of
�the Middle Passage freezes time: “All of it is now
it is always now there will never be a time when
I am not crouching and watching others who are couching too
I am always crouching and watching
others who are crouching too I am always crouching”. Beloved, the spirit come to haunt her mother
for the act of killing her as a technique to emancipate her from slavery, has never picked flowers in
Africa nor directly experienced the Middle Passage but the presentism of these thought pictures is a
result of the terrible power of this regime of coercion. For the specter and for the reader it is now,
always now. The video we experienced a bit ago in the parlance of Morrison., is a hot thing.
3. Conclusion:
The challenge now is to tie this up in a manner that allows us to discuss it and view this
thinking as a point of departure that points in many directions at once and sweeps through and across
multiple systems of knowing and archives past, present, and to come. There are several points of
inflection here that we should mark: the translation of the existence of the visual encounter with the
Other yields to the gloss put on that experience by the Socratic dialog that is absorbed by Hegel whose
thinking then becomes the target of intellectual challenge by Du Bois. This system must again yield to
the somatic or the corporeal encounter with the wages of Blackness as experienced through the
employment of video. That experience confuses and disorients us all independent of subject position,
yields as well and is gathered together, in the parlance of Morrison, by saying things that are pictures.
The real question, the foundational presupposition, is why should we care? I use “we” here
advisedly. I don’t mean the “we” of those who are understood to be living under the experience of
what we have labelled here as “Blackness”. As a practical matter I also do not mean the life of the
mind that privileges this kind of thought experiment as valuable for the sake of the effort. I mean we
as a question of humanity and to be succinct we have to care because we are all participants,
perpetrators, or observers in what we can frame as the wages of the sin of establishing a societal order
that builds its demos on the imperative of exclusion and the notion of freedom as valuable only in the
�presence of the possibility or actuality of its opposite. Blackness, as a master signifier here, can be
understood to, in its abstraction, speak for the plight of all the aggrieved and the maligned. Speaking
for violence against trans bodies in the same way it shouts the despair of children separated from their
guardians and caged because they are seeking safety. The same goes for the mosque, synagogue, or
bar that is attacked for the presence of what are coercively framed as transgressive bodies or systems
of thinking. The same goes for a future that has the potential to erase human existence, in the way we
understand it, based upon the poor stewardship of the earth that is the wretched refuse of rabid
capitalism. I have gestured, perhaps obliquely at the central problematic of how the figure we have
examined here has been called into existence as the unwitting oppositional way of being that allows
something like democracy to exist. This is based upon the notion that the value of freedom is only
discernible and measurable in its dialectical relationship against its opposite way of being: un-freedom.
With that in mind the challenge before us is to imagine and bring into being a type of humanism and
in its aggregation, in the form of a societal order that forms itself outside of the logic of seeking the
middle point between two extremes. The reason for this effort is not to create something like the
debunked and reductive notion of colorblindness. If nothing else remains in our collective minds at
the close of this talk it must be this. Blackness is not a race or a color nor an ethnicity. It is an externally
imposed system of marginalization that renders its victims and purveyors locked in an unnecessary
system of subjective destruction. What this means is that “Blackness” is only related to being a Black
person in that this political epoch has assembled that figure and allowed for it to stand as a master
signifier for the Muslim, the Queer, the Native, the immigrant, the Jew, the Trans, it is endless, which
means that systems of power will always seek a figure to cloak in the subjective disability of Blackness.
In his recent text Necropolitics, Achille Mbembe proposes the following and with its recitation I will
close our time together. Mbembe writes:
The colonial world, as an offspring of democracy, was not the
antithesis of the democratic order. It has always been its double or,
�again, its nocturnal face. No democracy exists without its double,
without its colony – little matter the name and the structure. The
colony is not external to democracy and it is not necessarily located
outside its walls. Democracy bears the colony within it, just as
colonialism bears democracy, often in the guise of a mask…In other
terms, the cost of the mythological logics required for modern
democracies to function and survive is the exteriorization of their
originary violence to third places, to nonplaces, of which the
plantation, the colony, or today, the camp and the prison, are
emblematic figures. (Mbembe)
This quotation, perhaps in some measure, explains the way in which the videos we encounter
serve to memorialize and resist the erasure that might allow us the luxury of believing that these
excesses are either a thing of the past or not meant for us. So long as anyone suffers under the refined
technology of Othering, the result of which we have labelled here as Blackness, we all suffer and more
to the point, are necessarily at risk. The methodological question for all of us that have, in one way or
another, chosen the life of the mind is to focus our attention, across the canon we study and through
the scholarship we create, on requiring that the proper attention be paid to the presence of the tail of
the dragon of hatred that weaves its way through our consciousness. Thank you.
1
Ibid. 248.
�
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/4461b8c6adcb75ab9ba142a9fb596772.pdf
7e23480032f607a5ea7605fb9b37796a
PDF Text
Text
The Phenomenology of Blackness
Presented by,
Michael E. Sawyer, PhD
Colorado College
St. John’s College
Santa Fe, New Mexico
Carol J. Worrell Annual Lecture Series on Literature
22 November 2019
�“However, the diverse aspects which consciousness takes
upon itself are determinate in that each is regarded as
existing on its own within the universal medium. White is
only in contrast to black, etc., and the thing is a “one”
precisely in virtue of its being contrasted with others.”
G.W. F. Hegel Phenomenology of Spirit §120
�“The master is consciousness existing for itself. However,
the master is no longer consciousness existing for itself
merely as the concept of such a consciousness. Rather, it
is consciousness existing for itself which is mediated with
itself through another consciousness, namely, through
another whose essence includes its being synthetically
combined with self-sufficient being, that is, with
thinghood itself.” Ibid. §190
�“‘Let us then liken the soul to the natural union of a team
of winged horses and their charioteer. The gods have
horses and charioteers that are themselves all good and
come from good stock besides, while everyone else has a
mixture. To begin with, our driver is in charge of a pair
of horses; second, one of the horses is beautiful and good
and from stock of the same sort, while the other is the
opposite and has the opposite bloodline. This means that
the chariot-driving in our case is inevitably a painfully
difficult business.’” The Phaedrus (246b)
�“ ‘The horse that is on the right, or nobler, side is upright
in frame and well jointed, with a high neck and a regal
nose; his coat is white, his eyes are coal black, and he is a
lover of honor with modesty and self-control; companion
to true glory, he needs no whip, and is guided by verbal
commands alone. The other horse is a crooked great
jumble of limbs with a short bull-neck, a pug nose, black
skin, and bloodshot white eyes; companion to wild boasts
and indecency, he is shaggy around the ears – deaf as a
post – and just barely yields to horsewhip and goad
combined.’ ” Ibid. (253d)
�“After the Egyptian and Indian, the Greek and Roman, the Teuton
and Mongolian, the Negro is a sort of seventh son, born with a
veil, and gifted with second-sight in this American world, - a world
which yields him no true self-consciousness, but only lets him see
himself through the revelation of the other world. It is a peculiar
sensation, this double-consciousness, this sense of always looking at
one’s self through the eyes of others, of measuring one’s soul by
the tape of a world that looks on in amused contempt and pity. One
ever feels his two-ness,- an American, a Negro; two souls, two
thoughts, two unreconciled strivings; two warring ideals in one dark
body, whose dogged strength alone keeps it from being torn
asunder.” W.E.B. Du Bois The Souls of Black Folk
��““The Operator is the Photographer. The Spectator is ourselves, all of
us who glance through collections of photographs-in magazines
and newspapers, in books, albums, archives…And the person or
thing photographed is the target, the referent, a kind of little
simulacrum, any eidolon emitted by the object, which I should like to
call the Spectrum of the Photograph, because this word retains,
through its root, a relation to ‘spectacle’ and adds to it that rather
terrible thig which is there in every photograph; the return of the
dead.” Roland Barthes Camera Lucida
�������“Suppose I am about to recite a psalm which I know. Before I
begin, my expectation is directed towards the whole. But when I
have begun, the verses from it which I take into the past become
the object of my memory. The life of this act of mine is stretched
two ways, into my memory because of the words I have already said
and into my expectation because of those which I am about to say.
But my attention is on what is present: by that the future is
transferred to the past. As the action advances further and further,
the shorter the expectation and the longer the memory, until all
expectation is consumed, the entire action is finished, and it has
passed into the memory.” St. Augustine Confessions 28:38
�“…But this opposition between its appearance and its truth has
only the truth for its essence, namely, the unity of selfconsciousness with itself. This unity must become essential to selfconsciousness, which is to say self- consciousness is desire itself. As
self-consciousness, consciousness henceforth has a doubled object:
The first, the immediate object, the object of sense- certainty and
perception, which however is marked for it with the character of the
negative; the second, namely itself, which is the true essence and which
at the outset is on hand merely in opposition to the first. Selfconsciousness exhibits itself therein as the movement within which,
in its own eyes, the selfsameness of itself with itself comes to be.”
Hegel §167
�“I was talking about time. It’s so hard for me to believe in
it. Some things go. Pass on. Some things just stay. I used
to think it was my rememory. You know. Some things you
forget. Other things you never do. But it’s not. Places,
places are still there. If a house burns down, it’s gone, but
the place-the picture of it-stays, and not just in my
rememory, but out there in the world. What I remember is
a picture floating around out there outside my head…”
�“Can other people see it?” asked Denver.
“Oh, yes. Oh, yes, yes, yes. Someday you be walking down
the road and you hear something or see something going
on. So clear. And you think it’s you thinking it up. A
thought picture. But no. It’s when you bump into a
rememory that belongs to somebody else.” Toni Morrison
Beloved
�“But we must deny the fact that slaves are considered merely as
property, and in no respect whatever as persons. The true state of
the case is, that they partake of both those qualities; being
considered by our laws; in some respect, as persons, and in other
respects, as property. In being compelled to labor not for himself,
but for a master; in being vendible by one master to another master,
and in being subject at all times to be restrained in his liberty, and
chastised in his body, by the capricious will of another, the slave
may appear to be degraded from the human rank and classed with
the irrational animals, which under the legal domination of
property. The Fœderal Constitution therefore, decides with great
propriety on the case of our slaves, when it views them in the mixt
character of persons and property.” James Madison Federalist 54
�I AM BELOVED and she is mine. I see her take flowers away from
leaves she puts them in a round basket the leaves are not for her
she fills the basket she opens the grass I would help her but the
clouds are in the way how can I say things that are pictures I am
not separate from her there is no place where I stop her face is my
own and I want to be there in the place where her face is and to be
looking at it too
a hot thing
All of it is now it is always now there will never be a time when
I am not crouching and watching others who are couching too I
am always crouching and watching others who are crouching too I
am always crouching the man on my face is dead his face is not
mine his mouth smells sweet but his eyes are locked. Morrison
Beloved
�
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
pdf
Page numeration
Number of pages in the original item.
20 pages
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 phenomenology of blackness
Description
An account of the resource
Transcript of a lecture given on November 22, 2019 by Michael Sawyer as part of the Dean's Lecture and Concert Series.
Creator
An entity primarily responsible for making the resource
Sawyer, Michael E.
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
2019-11-22
Rights
Information about rights held in and over the resource
Meem Library has been given permission to make this item available online.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Relation
A related resource
<p>The following video is referenced in this lecture:</p>
<p><a href="https://youtu.be/n92zfXkRdnk">https://youtu.be/n92zfXkRdnk</a></p>
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
SF_SawyerM_The_Phenomenology_of_Blackness_2019-11-22
Friday night lecture
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/c243d0e09b752feef01b9ac5cc9990b8.pdf
40ff74d84f2a2b7d84d208491d3d62f3
PDF Text
Text
The Pentagon and the Heptagon
Recapitulation
Two weeks ago, we saw how Rafael Bombelli began to suspect that imaginary numbers might be
meaningful as he worked on the cubic equation
x 3 - 15 x - 4 = 0
Using the formula Cardano stole from Tartaglia, got
x=
3
2 + 11
-1
+
3
2 - 11
-1
which he was then able to solve by intuiting that
2 + 11
-1 = 2 +
3
-1 .
The second lecture described Caspar Wessel’s graphic presentation of the arithmetic of complex
numbers. On the complex number plane -(i) complex numbers can be expressed in polar coordinates by giving a distance (modulus) and an
angle (argument);
(ii) multiplication of complex numbers amounts to
(a) multiplication of their distances (moduli) and
(b) adding their angles (arguments); and
(iii) the solutions of equations of the form x n - 1 = 0, known as the “Roots of Unity,” appear
graphically as the vertices of an equilateral n-gon in the unit circle on the complex plane.
Last week, we encountered the idea of “Constructible Numbers.” We showed that Euclid’s postulates
allowed construction of lengths that correspond to the field of rational numbers, a collection of numbers that is closed under the operations of addition, subtraction, multiplication and division. In addition, Euclid’s postulates allow the construction of incommensurable magnitudes (which correspond to
irrational numbers). However, Euclid’s postulates do not permit construction of all incommensurable
magnitudes. We can only construct those that correspond to numbers that can be found in towers of
finite quadratic field extensions, that is, field extensions that have a degree of 2n over the rational
numbers. Plenty of numbers are not included. For example, 2 is not constructible; neither are the
non-algebraic (transcendental) numbers, which form an uncountable infinity far greater than the
countable infinity of the algebraic numbers.
3
�2
bers that is closed under the operations of addition, subtraction, multiplication and division. In addition, Euclid’s postulates allow the construction of incommensurable magnitudes (which correspond to
irrational
numbers).
However, Euclid’s postulates do not permit construction of all incommensurable
4 The
Pentagon and
the Heptagon.nb
magnitudes. We can only construct those that correspond to numbers that can be found in towers of
finite quadratic field extensions, that is, field extensions that have a degree of 2n over the rational
numbers. Plenty of numbers are not included. For example, 2 is not constructible; neither are the
non-algebraic (transcendental) numbers, which form an uncountable infinity far greater than the
countable infinity of the algebraic numbers.
3
We now turn to an application of what we have seen so far: construction of a regular pentagon in a
given circle.
The Lesser-Know Impossibility Problem
Ancient geometry knew several classical problems that seemed impossible; the three most famous
were trisecting the angle, doubling the cube and squaring the circle. These were daunting challenges.
No one had found how to accomplish any of them, but the ancients did not know whether they were
really impossible or only difficulties awaiting clever solutions
Only in the late 18th and 19th centuries did we learn that these three problems really are impossible, at
least with the tools of Euclidean geometry.
In addition to these three, lesser-known but equally interesting problem is that of the heptagon, the
regular seven-sided polygon. In book IV of the Elements, Euclid shows how to construct in a given
circle an equilateral triangle (IV.2), a square (IV. 6), a pentagon (IV. 11) and a hexagon (IV. 15), regular
figures with three, four, five and six sides. He then shows how to construct a regular 15-gon (IV. 16).
Then he stops.
The reader might be expected to wonder, why? Why jump from 6 to 15? Euclid, in his customary
laconic way, says nothing. Some of the figures he skips over were easily constructible. The octagon is
easily made by bisecting the angles of the square. The 10-gon can be gotten similarly from the pentagon and the 12-gon, from the hexagon. But the orderliness of Euclid’s sequence really falls apart with
the 7-gon. With what we learned last week, it is easy to see that the 7-gon is impossible to construct. It
is obtained from the polynomial:
x 7 - 1 = ( x - 1) x 6 + x 5 + x 4 + x 3 + x 2 + x 1 + 1 = 0
That sixth-degree polynomial is irreducible and, since its degree over the rationals is not a power of
two, we can see right away that these complex roots are not constructible.
Did Euclid know that the 7-gon was impossible? He probably suspected it. He surely knew that he
couldn’t do it, which is not quite the same thing.
Beyond the Impossible: the Unsuspected Possible
In 1796, at the age of 19, Carl Friedrich Gauss realized the impossibility of constructing the 7-gon; what
is mor, he realized at the same time that there are other polygons that can be constructed. Looking
only at those with a prime number of sides, in his book Disquisitiones Arithmeticae, he not only showed
that the 17-gon is constructible, he showed how to do it. This is remarkable advance beyond what
Euclid knew.
To help us get to Gauss’s result, it will be helpful to begin with a slightly simpler project: the algebraic
�4 The Pentagon and the Heptagon.nb
3
In 1796, at the age of 19, Carl Friedrich Gauss realized the impossibility of constructing the 7-gon; what
is mor, he realized at the same time that there are other polygons that can be constructed. Looking
only at those with a prime number of sides, in his book Disquisitiones Arithmeticae, he not only showed
that the 17-gon is constructible, he showed how to do it. This is remarkable advance beyond what
Euclid knew.
To help us get to Gauss’s result, it will be helpful to begin with a slightly simpler project: the algebraic
construction of the pentagon.
Euclid’s Construction of the Pentagon
Of course, Euclid knew how to construct a regular pentagon in a given circle. To begin, let’s review how
Euclid’s construction works.
First, a Special Triangle
He begins with construction of a very special triangle, one that is isosceles and whose base angles are
both twice as big as its vertex angle.
θ
2θ
2θ
A little reflection shows why this triangle might be important to the construction of a regular pentagon:
the three angles of the triangle total up to 180°, of course, but they also add up to five times the vertex
angle. This triangle creates one angle that is one-fifth of 180°, and two that are one-fifth of 360°. If this
triangle can be made, it will be the key to constructing the pentagon.
However, constructing this triangle is no simple matter.
To make it, Euclid recalls that back in book II, proposition 11, he had shown how to cut a line at a point
so that the square on one portion of the line is equal to the rectangle contained by the whole line and
the remaining portion of the line.
�4
4 The Pentagon and the Heptagon.nb
Digression
This kind of division is known as one into "mean and extreme ratio," sometimes also referred to as the
"Golden Ratio." It has many cool features, including connection to Fibonacci numbers and logarithmic
spirals, but we haven' t time to get into all these things right now.
If we take the whole AB to be “1” and the distance AC to be “x”, then finding this ratio can be understood as analogous to solving the equation:
x 2 = (1 - x )
x2 + x - 1 = 0
or
whose solutions are:
1±
1 - 4 (-1)
2
=
1±
5
2
You may note that these values are not rational, since they contains the square root of five. They are, of
course, constructible, which we know because (a) we are dealing with a quadratic extension of the
rationals and (b) because Euclid in fact constructs one of them. (No surprise there.)
�1±
1 - 4 (-1)
2
=
1±
5
4 The Pentagon and the Heptagon.nb
2
5
You may note that these values are not rational, since they contains the square root of five. They are, of
course, constructible, which we know because (a) we are dealing with a quadratic extension of the
rationals and (b) because Euclid in fact constructs one of them. (No surprise there.)
Returning to the Construction
Euclid takes a line divided in this way and, using one end as a center, draws a circle with the whole line
as a radius:
A
C
B
He then makes a chord in the circle equal to the larger segment of the divided line:
A
C
B
D
He completes the triangle ABD, and joins CD:
�6
4 The Pentagon and the Heptagon.nb
A
C
B
D
Finally, he draws a circle that goes through points A, C and D:
A
C
B
D
Thanks to a proposition from earlier in Book III, he knows that when from a point outside a circle (like
point B) a line cuts a circle (as line BCA), and another line is draw to the circumference of the circle (as
line BD), and when the rectangle on AB, AC is equal to the square on BD, then the line (BD) is tangent to
the circle (ACD).
With that established, another proposition of Book III allows him to say that the angle CDB (angle 1) is
equal to the angle CAD (angle 2):
�4 The Pentagon and the Heptagon.nb
A
C
2
B
4 5
3 1
D
Add angle CDA to both. Thus angles 2 + 3 are equal to angles 1 + 3. But because AB = AD (in the circle
around A), angles 1 + 3 are equal to angle 5 . So:
angle 5 = angles 1 + angle 3 = angle 2 +angle 3
and because of exterior angles in triangle CBD
angle 4 = angle 2 + angle 3
Therefore, triangle ABD is isosceles and line DB = line DC. And line DB = line AC.
Therefore, angle 3 = angle 2 = angle 1.
This, then, is the isosceles triangle with its base angles equal to twice the vertex angle.
The Pentagon
With the isosceles triangle having the base angles equal to the vertex angle now available, the rest is
easy.
7
�8
4 The Pentagon and the Heptagon.nb
Simply bisect the arcs standing on the longer sides, which are each twice the arc on the shorter side.
Now you have five equal sides and your pentagon is complete.
Join the vertices and you have not only a pentagon, but a pentangle (a regular five-pointed star).
�4 The Pentagon and the Heptagon.nb
9
This construction is completely rigorous and very clever. However, it offers no clues at all about how to
pursue construction of other such prime-sided polygons, such as the 7-gon, the 11-gon, the 13-gon, etc.
Preliminary: the Pentagon
The algebraic construction of the pentagon amounts to finding the roots of the fifth degree cyclotomic
polynomial. That is, we begin with the equation:
x5 = 1
or
x 5 - 1 = 0.
The number 1 is evidently a solution to this equation. It is, in fact, the only rational solution. Therefore,
the equation can be factored by removing the factor (x - 1):
x 5 - 1 = ( x - 1) x 4 + x 3 + x 2 + x + 1 = 0
The Fifth Order Cyclotomic Polynomial
The second expression, x 4 + x 3 + x 2 + x + 1, is irreducible “over the rationals”; that is, it can’t be
simplified by showing it to be the product of factors of lower degree among the rationals. We can be
completely sure that this expression is irreducible because we know that the four roots of the polynomial x 4 + x 3 + x 2 + x + 1 = 0 are complex with imaginary components. They are the four non-real fifth
roots of unity.
�10
4 The Pentagon and the Heptagon.nb
ζ1
ζ2
ζ3
ζ4
But being irreducible over the rationals doesn’t mean that this thing can’t be factored in an extended
field. In fact, it has been shown that every polynomial of nth degree can be factored into n linear factors
in the full complex number field. Our challenge is to find which factors need to be appended to the
rationals in order to factor or “split” this fourth degree polynomial.
Complex Conjugates
We haven’t discussed complex conjugates, but this diagram presents the idea nicely. Notice that the
complex roots of this polynomial appear as two pairs of complex numbers, symmetrically arranged
above and below the real number axis. Root ζ1 is paired this way with root ζ4 and root ζ2 is paired
with root ζ3 . Being so arranged, these roots are written in this form:
a+bi
and
a - bi
The expressions are the same except for the positive and negative signs attached to the imaginary
portions.
Complex conjugates have this handy feature: when a pair of complex conjugates are added, their sum
is a real number. Also, when a pair of complex conjugates are multiplied together, their product is a
real number.
This feature is handy because we are often looking for roots of polynomials whose coefficients are
rational (or, in any case, do not involve imaginaries). Of course, you can construct an arbitrary polynomial with a random selection of complex roots:
(x - (2 + 7 i)) (x - (9 - 3 i)) (x - (-15 + 4 i)) = …
But if you multiply this trio out, you will have some imaginary coefficients.
�is a real number. Also, when a pair of complex conjugates are multiplied together, their product is a
real number.
4 The Pentagon and the Heptagon.nb
11
This feature is handy because we are often looking for roots of polynomials whose coefficients are
rational (or, in any case, do not involve imaginaries). Of course, you can construct an arbitrary polynomial with a random selection of complex roots:
(x - (2 + 7 i)) (x - (9 - 3 i)) (x - (-15 + 4 i)) = …
But if you multiply this trio out, you will have some imaginary coefficients.
(x - (2 + 7 i)) (x - (9 - 3 i)) (x - (-15 + 4 i)) =
(813 + 699 ⅈ) - (142 - 41 ⅈ) x + (4 - 8 ⅈ) x2 + x3
In fact, the only way to eliminate the imaginary components from the expanded polynomial is if the
coefficients occur in pairs of complex conjugates. That way, when the conjugates are multiplied, the
imaginary components disappear.
Return to the Problem
To solve our fourth-degree cyclotomic polynomial:
1 + x + x2 + x3 + x4 = 0
We will proceed in the usual, brash algebraic way: we will pretend that we already have the solutions.
Then we’ll work to discover what they are. The Fundamental Theorem of Algebra tells us that this
fourth degree equation has four solutions, which we will designate (as in the picture)
ζ1 , ζ2 , ζ3 and ζ4 . Roots ζ1 and ζ4 are one pair of complex conjugates; ζ2 and ζ3 are another
pair.
Two-Stage Solution
Take the sums of ζ1 , ζ4 and of ζ2 , ζ3 , like this:
η1 = ζ 1 + ζ 4
η2 = ζ 2 + ζ 3
When added together, η1 and η2 sum up to -1 (because all the fifth roots of unity together sum to zero,
and η1 and η2 include all the roots except (+1 + 0 i):
η1 + η2 = ζ 1 + ζ 4 + ζ 2 + ζ 3 = -1
Also, the product of η1 and η2 works out like this:
ζ 1 + ζ 4 ζ 2 + ζ 3 = ζ 3 + ζ 4 + ζ 6 + ζ 7
Restate this result with the exponents taken Modulo 5, because, on the unit circle in the complex
plane, ζ 5 = ζ 0 = 1. Thus, we have
ζ6 = ζ5 ζ1 = ζ1
ζ7 = ζ5 ζ2 = ζ2
Substitute:
�12
Restate
thisand
result
with the exponents
4 The
Pentagon
the Heptagon.nb
5
taken Modulo 5, because, on the unit circle in the complex
0
plane, ζ = ζ = 1. Thus, we have
ζ6 = ζ5 ζ1 = ζ1
ζ7 = ζ5 ζ2 = ζ2
Substitute:
ζ3 + ζ4 + ζ6 + ζ7 = ζ3 + ζ4 + ζ1 + ζ2 = -1
Presto! We have the sum of the four non-real roots of the equations x 5 - 1 = 0. We know that these
sum to -1.
Building a Quadratic Equation for η1, η2
Great! We have two terms, η1 and η2 . We don’t know what they are, but we do know that their sum is
-1 and their product is also -1. Does that sound like a familiar situation? When we know that when we
know the sum and product of two terms, we can construct a quadratic equation that has these terms as
roots. In this case, we have:
x2 + x - 1 = 0
whose roots are given by the quadratic formula:
η1 and η2 =
-1 ±
1+4
2
=
1
2
-1 +
5 and
1
2
-1 -
5 . (The approximate values of these are
0.61803 and -1.61803.)
Behold! Now It Factors!
Remember that we said that the expression x 4 + x 3 + x 2 + x + 1 = 0 could not be factored over the
rationals? Now it can be factored in an extended field when we append 12 -1 +
append
5 -- or even if we just
5 -- to the rationals.
We have:
( x - ζ1 ) ( x - ζ4 ) = x 2 - ζ1 x - ζ4 x + ζ1 ζ4
( x - ζ1 ) ( x - ζ4 ) = x 2 - (ζ1 + ζ4 ) x + ζ1 ζ4 = x 2 - (ζ1 + ζ4 ) x + 1
= η1
This expression, x
2
- (ζ1 + ζ4 ) x + ζ1 ζ4 , has coefficients that are in the extended field. The
= η1
coefficient of x is the sum of the two roots ζ1 + ζ4 ; we don’t know them individually yet, but we know
that they sum to η1 , which is in the extended field. The constant term is ζ1 ζ4 ; we know right away that
the product of these two is 1 (product of their moduli, sum of their arguments).
�4 The Pentagon and the Heptagon.nb
13
The Four Singletons
Now look at the four roots individually:
ζ1 , ζ2 , ζ3 , ζ4
We know how they sum in pairs:
η1 = ζ 1 + ζ 4
η2 = ζ 2 + ζ 3
We also know the products of the same pairs :
ζ1 ζ4 = ζ5 = 1
ζ2 ζ3 = ζ5 = 1
So we can make two more quadratic equations:
w2 - η1 w + 1 = 0
whose roots are ζ1 and ζ4 , which are solved as w =
η1 ±
y 2 - η2 y + 1 = 0
whose roots are ζ2 and ζ3 which are solved as y =
η2 ±
η1 2 - 4
2
η2 2 - 4
2
We now have enough information to solve for the four roots:
ζ
1
ζ
4
ζ
2
ζ
3
=
=
=
=
η1 +
η1 2 - 4
2
η1 -
η1 2 - 4
2
η2 +
η2 2 - 4
2
η2 -
η2 2 - 4
2
1
=
2
2
2
-1+ 5 -
1
-1- 5 +
1
2
-1- 5 -
= -0.809017 + 0.587785 ⅈ
2
2 -1- 5 - 4
2
= 0.309017 - 0.951057 ⅈ
2
2 -1- 5 - 4
1
= 0.309017 + 0.951057 ⅈ
2
2 -1+ 5 - 4
2
1
=
2
2
1
=
1
2 -1+ 5 - 4
2
1
=
-1+ 5 +
= -0.809017 - 0.587785 ⅈ
You can see that these solutions contain radicals of radicals. These expressions are not in the first
extended field, but we can extend that field again (in a finite quadratic algebraic field extension) so that
it includes these four solutions.
�14
4 The Pentagon and the Heptagon.nb
You can see that these solutions contain radicals of radicals. These expressions are not in the first
extended field, but we can extend that field again (in a finite quadratic algebraic field extension) so that
it includes these four solutions.
These can be plotted on the complex plane:
0.31 + 0.95 ⅈ
-0.81 + 0.59 ⅈ
-0.81 - 0.59 ⅈ
0.31 - 0.95 ⅈ
Voila.
More Important Than the Answer
To summarize and review.
More important that getting the answer or than drawing the pentagon is to notice how the field extensions were built. Beginning with the rationals, which are all constructible, we first got the values for η1
and η2 , which were the sums of ζ 1 + ζ 4 and ζ 2 + ζ 3 respectively, the two pairs of complex conjugates. These values were
1
2
-1 ±
5 , and thus required that we move into an extended field:
Q ⟶ Q(η1, 2 )
This is a quadratic extension and is thus constructible. Then, getting the four roots themselves
required another field extension. The four roots are
η1,2 ±
η1,2 2 - 4
2
, and each will require one more
quadratic field extension.
Q ⟶ Q(η1, 2 ) ⟶ Q(η1, 2 ,
η1,2 ±
η1,2 2 - 4
2
)
Sequences of quadratic field extensions are constructible.
Look again at what is happening here. At the outset, we knew that we had a fourth degree equation with all complex roots.
1 + x + x2 + x3 + x4 = (1 - ζ1 ) (1 - ζ2 ) (1 - ζ3 ) (1 - ζ4 )
By segregating out the pairs of complex conjugates, we separated the factors on the right into two pairs
�4 The Pentagon and the Heptagon.nb
15
Look again at what is happening here. At the outset, we knew that we had a fourth degree equation with all complex roots.
1 + x + x2 + x3 + x4 = (1 - ζ1 ) (1 - ζ2 ) (1 - ζ3 ) (1 - ζ4 )
By segregating out the pairs of complex conjugates, we separated the factors on the right into two pairs
:
1 + x + x2 + x3 + x4 = {(x - ζ1 ) (x - ζ4 )} × {(x - ζ2 ) (x - ζ3 )}
1 + x + x2 + x3 + x4 = x2 - (ζ4 + ζ1 ) x + ζ1 ζ4 × x2 - (ζ2 + ζ3 ) x + ζ2 ζ3
1 + x + x2 + x3 + x4 = x2 - η1 x + 1 × x2 - η2 x + 1
Is this interesting? Yes! If we confine ourselves to rational numbers, then our original equation could
not be factored. If we admit η1 and η2 , it could be factored into two factors. If we admit all the complex numbers -- really, we needed go so far; a finite field extension adding
η1,2 ±
η1,2 2 - 4
2
to the mix would
be enough -- then it factors into four factors:
In Q
1 + x + x2 + x3 + x4
In Q(η1, 2 )
"" factors to x2 - η1 x + 1 × x2 - η2 x + 1
In Q(η1, 2 ,
η1,2 ±
η1,2 2 - 4
2
)
“”
is irreducible
factors to (1 - ζ1 ) (1 - ζ2 ) (1 - ζ3 ) (1 - ζ4 )
The procedure we have followed does exactly what is required for specifying constructible figures: it
has made a sequence of finite field extensions, starting with the rationals, Q, and proceeding by
quadratic field extensions until the polynomial with our desired points as roots is completely factored.
This stepwise factorization works for the pentagon because at each step it was possible to subdivide
the roots into two groups, each of which could be shown to be a quadratic expression of the preceding
group. That is not always possible.
Conclusion
We have seen here an application of the technique of algebraic decomposition. The equation we are
trying to solve is broken into simpler and simpler parts as the field in which we operate is expanded
step-by-step until we arrive at a final field, the “splitting field,” in which the polynomial can be completely decomposed into linear factors.
Unlike Euclid’s way of working, this methodical procedure provides a framework for evaluating which
polygons are constructible and which are not.
We will see this method play out on a larger stage next week with the construction of the hep-
�16
We have seen here an application of the technique of algebraic decomposition. The equation we are
trying
to solve
is broken
into simpler and simpler parts as the field in which we operate is expanded
4 The
Pentagon
and the
Heptagon.nb
step-by-step until we arrive at a final field, the “splitting field,” in which the polynomial can be completely decomposed into linear factors.
Unlike Euclid’s way of working, this methodical procedure provides a framework for evaluating which
polygons are constructible and which are not.
We will see this method play out on a larger stage next week with the construction of the heptadecagon.
Thank you.
�
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
pdf
Page numeration
Number of pages in the original item.
16 pages
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 pentagon and the heptagon
Description
An account of the resource
Transcript of a lecture given on October 1, 2019 by Grant Franks.
Creator
An entity primarily responsible for making the resource
Franks, Grant H.
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
2019-10-01
Rights
Information about rights held in and over the resource
Meem Library has been given permission to make this item available online.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Subject
The topic of the resource
Polygons
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
SF_FranksG_The Pentagon_and_the_Heptagon
-
https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/775e191efa4221b0abec5ca661f8c82b.pdf
7828beef34f7068ae3eac93bf9b1b555
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
St. John's College Lecture Transcripts—Santa Fe
Contributor
An entity responsible for making contributions to the resource
St. John's College Meem Library
Coverage
The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant
Santa Fe, NM
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 at the Santa Fe, NM campus.
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.
Original Format
The type of object, such as painting, sculpture, paper, photo, and additional data
paper
Page numeration
Number of pages in the original item.
26 pages
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 nobility of Sophocles' Antigone
Description
An account of the resource
Transcript of a lecture given on October 31, 1986 by Janet Dougherty as part of the Dean's Lecture and Concert Series.
Creator
An entity primarily responsible for making the resource
Dougherty, Janet
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
Santa Fe, NM
Date
A point or period of time associated with an event in the lifecycle of the resource
1986-10-31
Rights
Information about rights held in and over the resource
Meem Library has been given permission to make this item available online.
Type
The nature or genre of the resource
text
Format
The file format, physical medium, or dimensions of the resource
pdf
Subject
The topic of the resource
Sophocles. Antigone.
Language
A language of the resource
English
Identifier
An unambiguous reference to the resource within a given context
24003733
Friday night lecture
Deprecated: Directive 'allow_url_include' is deprecated in Unknown on line 0