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                    <text>2023

COMMENCEMENT
EXERCISES

Sunday | May 14
Two Thousand and Twenty-Three
Annapolis | Maryland

�PROGRAM
For the Two Hundred Thirty-First
Commencement in the
Three Hundred Twenty-Seventh Year
of the College
Prelude			

Carrollton Brass Quintet

Academic Procession
The National Anthem			

Carrollton Brass Quintet

Francis Scott Key, Alumnus, Class of 1796

Welcome

Nora Demleitner
President of the College

Announcement of Prizes and Awards

President Demleitner
Joseph Macfarland
Dean of the College

Address to the Graduating Class

Nathan Dugan
Tutor

Conferring of Undergraduate Degrees of the Class of 2023			
President Demleitner
Dean Macfarland
Conferring of Graduate Degrees

President Demleitner
Emily H. Brooker Langston
Associate Dean for the Graduate Program

Recognition of the Class of 2020

Academic Recession		

President Demleitner
Dean Macfarland
Emily H. Brooker Langston
Carrollton Brass Quintet

�B A C H E LO R O F A R T S
Kasparas Adomaitis

Vilnius, Lithuania

Huckleberry Finn: the Conscience of Mark Twain’s America

Jack Alford

Vienna, Virginia

The Doomed Romantic: Failure and Fate in Lord Jim

Rhett Anderson

Thetford, Vermont

Wild Apes &amp; Savage Beavers: Examining Identity Post-Colonization in Aime
Cesaire’s Discourse on Colonialism, James Joyce’s Dubliners, and
Dr. Frederick Wiseman’s Dawnland: An Autohistory of the Abenaki Nation

Max Anthon

Haworth, New Jersey

Terminal Philosophy A Closer Look at the End of Socrates

Ella Rose Barron

Denver, Colorado

Voltaire’s Zadig ou la Destinée: A World Not Directed by Wisdom

Joseph Bennett

Amherst, New Hampshire

A Linguistic Christ Figure in Phenomenology of Spirit

Ben Berghammer

Evanston, Illinois

The Torments of a Deep Conscience: What it Means for Ivan Fyodorovich’s
Heart to be Overcome by God in The Brothers Karamazov

James Lawrence Bieneman

West Bloomfield, Michigan

What differentiates Newton and Leibniz in development and understanding
of their calculus?

Malaika Biwott

Fairfax, Virginia

The Demise of the Laborer

Emory Richmond Boll

New York, New York

Magical Mystery Tour Searching for Soul, Self, and Solace in Jung’s Red
Book

Dashiel Donald Buyske-Friedberg
God and Lucifer through Dante’s the Divine Comedy

Philadelphia, Pennsylvania

�Meagan Callahan

Sunderland, Maryland

What You Will: Desire and Happiness in Twelfth Night

Stephen Dela Cruz Cariño

Saipan, Northern Mariana Islands

Fiction and Reality: The Imaginations of Huckleberry Finn and Tom Sawyer

Gabriela Carroll

Austin, Texas

The Fruit of Unconscious Activity in Tolstoy’s War and Peace

Sihwan Cha		

Busan, South Korea

A Touch of Intellect, the Flesh of Man

Bibhusan Chapagain*
Hyeokjae Choi

Kathmandu, Nepal

Seoul, South Korea

Wrestling with the Shade of Plato: The Meno and Lucian’s Hermotimus

Joseph Cunningham

Ambleside, United Kingdom

Out from the Dark Wood: The Importance of Dante’s Awakening and First
Steps into Hell

Sam Detwiler

Jenkintown, Pennsylvania

No More. The Text is Foolish: Understanding Shakespeare’s Props in King
Lear

Jonah Donis

South Hamilton, Massachusetts

To See with Eyes Unclouded: A Look at the Myth of Er in Plato’s Republic

Elizabeth SuAnn Dowdy		

Fort Worth, Texas

Sisters by Chance or Sisters by Choice? An Examination of Sisterhood in
Sophocles’ Antigone

Samantha Duckworth
On Book X of Euclid’s Elements

Mira Dunan Emmart

Holland, Pennsylvania

Baltimore, Maryland

Locke on Labor For the Right or For the Good?

Isabel Emond

Ocean City, Maryland

Spellbound: The Risks of Learning in Plato’s Protagoras

Mathew Fisher

Weathersfield, Vermont

Unnumbered Strands Exploration of Separation in Dream of the Red
Chamber
Vienna, Austria

On Speech, or the Same and the Other

Cristopher Barrett Ford
Ardmore, Pennsylvania

Stamping Ground, Kentucky

A Cry for the Recognition of Living Poetry: Exploring Unreason in Don
Quixote

Alexander Ray Fodor

Of Whales and Men

Amanda Jayne Clemens

Eleanor Carroll Elliott

Reston, Virginia

Freeing Minds: A Treatment for Ignorance

Agnes Sands Galvin

Millerton, New York

To Be Mistress of Pemberley

Kreider Chesterfield Grasse

Philadelphia, Pennsylvania

“The Summit of the Organic Scale” Is Man’s Superciliousness Justified by
Darwin’s View Of the Progression of Man?

Emerson Rieman Graves

Charlotte, North Carolina

“All Myth, The One Reality” of Helen in Egypt

Noah Hale

Seaford, Delaware

The Paradoxical Hero and the Progress of Loss in Job

Allegra Danger Hall

Harrisburg, Pennsylvania

Absence Makes the Heart: The Present Christ in Bach’s St. Matthew Passion

Ella Naomi Harel Kirschner

Philadelphia, Pennsylvania

Phaedra the Philosopher Queen: Passionate Thought, Speech, and Deed in
Euripides’ Hippolytus

�Jonathan Harris

Silver Spring, Maryland

Returning Power to the People: An Analysis of Civil Responsibility in the
Commonwealth, as Established in Thomas Hobbes’ Leviathan

Matthew James Hogan

Ava Marie Elisabeth Lehrman

The Curiosity and Conviction of a Humble Man: On Faraday’s
Electromagnetic Phenomena through Inductive Reasoning
St. Helena, California

Lakeland, Florida

Paradox in Paradise

Matthew Jungheim

Highland Park, Illinois

From Aeneas to Zion: An Account of the Essence of Two Nations

John Frederick Kelly

Annapolis, Maryland

Evanston, Illinois

Shaped by Absence: Negation in Jane Austen’s Persuasion

Eyes Wide Open: An Inquiry into the Second and Third Chapters of the Book
of Genesis

Zixiong Lin

Wenling, People’s Republic of China

Daryl Jeffrey Locke

Upper Marlboro, Maryland

Lucie Carroll McCarthy

Baltimore, Maryland

Delay and the Bounded Self in Shakespeare’s Hamlet

Jenny Kim

Chungju, South Korea

Involuntary Memory in Swann’s Way
New Freedom, Pennsylvania

Baudelaire’s Search for the Red Ideal in Shakespeare’s Macbeth
Cincinnati, Ohio

Corruption and the Art of Philosophy Education and Friendship in Plato’s
Phaedrus
Chicago, Illinois

God, Moses, and the Golden Rule: Does Hillel’s Famous Statement Hold
True When Reading Exodus?

The Ubiquitous Experience of Exile in Baudelaire’s Swan

Las Vegas, Nevada

Outcast Child Monarch of the Clouds

Isabella Martin Kiedrowski

Hyojeong Lee

Cedar Park, Texas

“The Laws and Loss of Nature” Reinterpreting Tocqueville’s Observations on
the Principle of Equality For the Modern Day

Passion Nailed Down in America: An Inquiry on the Sovereignty of People

The Quest for Political and Spiritual Moderation

Rivka Koppel

Nothing Beside Remains: The Breakdown of Morality in Heart of Darkness

Sarah Janice Fairchild Lieberman

Chelsea Johnson

Craig Koch

Ellicott City, Maryland

Joshua Levins

The “Dark, Sweet Stench” of Loam in Toni Morrison’s Sula.

Mia Kobylski

Needham, Massachusetts

Law, Principle, Rule: What If Darwin’s Laws Are Taken as Laws of Physics?
Burke, Virginia

Ella Hougie

Jake Lees

Incheon, South Korea

Isabel Lucille McDonald

Gambier, Ohio

The Social Outsider: Society, Adventure, and Freedom in Mark Twain’s
Adventures of Huckleberry Finn

Sylvie McKnight-Milles

Ellicott City, Maryland

In Search of Physical Reality: A Journey through Schrödinger’s
Mathematical Derivation of the Quantum Wave Function

Elizabeth Lynn Meade

Pewaukee, Wisconsin

“You Must Not Be Here:”: Alienation through Law in War and Peace

Ezra Melchor

Catonsville, Maryland

On Nietzsche’s Genealogy of Morals: The Doubt of Faith and the Will to Truth

Megli Micek

Las Vegas, Nevada

Love, the Inevitable End of Everything in Leo Tolstoy’s Anna Karenina

�Jozef Miklasz

Crownsville, Maryland

The Five Rings Freedom through the Way of the Warrior

Grace Frances Miller

Trinity, North Carolina

Santa Fe, New Mexico

Louisville, Kentucky

The Life of Isaac McCaslin From Faulkiner’s Go Down, Moses

Giovanni Morgan

Stuart, Florida

The I That is We: How Community Sets Us Free in Hegel’s Phenomenology
of Spirit

Elizabeth “Birdi” Mueller

Ridgway, Colorado

Saving Orlando: Marriage and Mannage in Shakespeare’s As You Like It

Brooke Anna Murphy-Petri

Darien, Connecticut

To Love One’s Neighhhbor a Voyage into Gulliver’s Travels

Lillian Pearl Lemmert Naill

Baltimore, Maryland

Old Westbury, New York

The Torment and Ecstacy of Spleen and the Ideal

Tom Ni

Pfäffikon, Switzerland

Virtue or Virtù: The Case of Scipio

Matthew Nugent*
Joseph Padgett

Keene, New Hampshire

Memphis, Tennessee

Die Müh Ist Aus or Finding Rest in Bach’s Passion

Lysithia Page
Do We Get What We Want? An Essay on Plato’s Symposium

Gorham, Maine

Francis Proctor

Annapolis, Maryland

A Consideration of Inheritance, Rights, and Liberty

Alayna Kay Louise Raymond

Spanaway, Washington

Finding Her Epos: An Investigation into Dorothea Brooke’s Potential for Epic
Life in George Eliot’s Middlemarch

Elli Megan Rees

Llanelli, Wales

Seduction: A means of Destruction or Reconciliation?

Jennifer Reid

Atlanta, Georgia

Movement in Passivity: A Phenomenological Approach to the Development
of Passivity in the Girl in Simone De Beauvoir’s The Second Sex

Caroline Grace Ritter

Arlington, Virginia

On Personal Freedom and Historical Inevitability in Tolstoy’s War and Peace

The Four Elements in the Epic of Gilgamesh: An Exploration of Duality within
Nature

Noah Negri

Silas Nathanael Pillsbury
On Attention and Identity: A Study of Virginia Woolf’s To the Lighthouse

Passion and Gospel: Bach’s Double Scoring of Matthew

Caroline Kohl Moeller

Chicago, Illinois

The Dialectic of Violence: Frantz Fanon’s Wretched of the Earth

Economy, Passion, Love Exploring What It Means to Marry Well in Jane
Austen’s Pride and Prejudice

Ruby Mariah Miller

Angel Pantoja Jr.

Garrick Buchanan Rodman
The Identity and Ideals of a Man Betrayed

Jacob Richard Rowley

Bend, Oregon

A Platform of Stability: Eternal Moment in To The Lighthouse

Kostiantyn Rymar

Kyiv, Ukraine

Sin, Death, Suffering, and Baudelaire: How Evil Penetrates l’esprit and
l’corps to Create the Deplorable Nature of the Human Condition

Apurva Sharma

Kathmandu, Nepal

“A Shadow Here and a Light There”: An Exploration of the Self and the Other
in Virginia Woolf’s To the Lighthouse

Lirian Selene Spolaore
Burlingame, California

Glennville, Pennsylvania

Washington, District of Columbia

A Journey through Memory: Experiencing Time with Marcel Proust

�Sachin Cyril Stanislaus

Templeton, California

On Legitimate Authority in Jean-Jacques Rousseau’s On the Social Contract:
An Inquiry into the Possibility of Justified Social Relations

Honor Dubarry Stanton

Toronto, Canada

The Degradation of Soul and Regime in Plato’s Republic

Wyatt Sweeney

Cleveland, Ohio

Bona Fide: The Anatomy and Apotheosis of Identity through Montaigne’s
Essays

Nathaniel Talbutt

Providence, Rhode Island

Colossus! Interpreting Real and Symbolic Power in Shakespeare’s Tragedy
of Julius Caesar

Nicolas Turner

Arlington, Virginia

What Wants to Be Told: Self-knowledge through the Fables of Aesop and
Jean de La Fontaine

Nya Uberman-Gillon

Olympia, Washington

The Human Frequency: A Marriage of Opposites and Acceptable Paradox

James “Cooper” Ussery

Naples, Florida

A Standpoint Beyond: On The German Ideology

Paul William Vosteen Jr.

Monterey, Virginia

The Frenzied Pied Piper of Idolatry

Libby Watson

Houston, Texas

Thinking of Thinking: Why the Phenomenology of Spirit is a Story

Leonard Wells

Washington, District of Columbia

The Odyssey of L’russe Besuhof in Tolstoy’s War and Peace

Tessa Wild

St. Louis, Missouri

“Unnatural Troubles”: An Exploration of Madness in Macbeth

Tong “Cynthia” Wu

Suzhou, People’s Republic of China

Man as a Beautiful Animal: The Evolution of Human Civilization through
Beauty in Charles Darwin’s The Descent of Man

Jingyi Zhang

Fuzhou, People’s Republic of China

Mal of Les Fleurs du Mal

Rose Meien Zhang

Beijing, People’s Republic of China

Something New Under the Sun in Kepler’s Astronomia Nova

Yingxin Zhang

Wenling, People’s Republic of China

How One Should Live a Life: Moderation in Plato’s Republic

�MASTER OF ARTS IN LIBERAL ARTS
Mark Acebo

Perth, Australia

Sarah Pendleton Alexander
Alexander Phillip Andreosatos
Jeffrey Robert Boatwright†
Adam Burkhart

§

Annapolis, Maryland
§

Quincy, Massachusetts
Los Angeles, California
Pittsburgh, Pennsylvania

§

David Somerset Christian

Cypress, California

Jesse Clagett

Dahlonega, Georgia

Ari Spencer Coleman†

South Jordan, Utah

Benjamin Crocker

Sydney, Australia

Folke Egerstrom

Santa Ana, Costa Rica

§

Matthew Ely

Des Moines, Iowa

Derek Foret

Washington, District of Columbia
§

Fort Washington, Maryland

†

Jay Alexander Gold

Madison, Wisconsin

Ansley Starr Green

Alachua, Florida

Tanya Malita Fuller

Steven Hayes§

Lawrenceville, New Jersey

Jordan Loch Hill

Perth, Australia
†

Joshua C Laperche
§

Connor Morrison

Benjamin Stewart Pershall†
Siobhán Autumn Petersen
†

David R. Pierce
Ethan Pyle

§

Exeter, Rhode Island
Philadelphia, Pennsylvania
Severna Park, Maryland
Shady Side, Maryland
Santa Teresa, New Mexico
Hartland, Wisconsin

Walker Beau Rogalsky

Winnipeg, Canada

Kasey M Siciliano

Cincinnati, Ohio

Abdullah Wadood

Bethesda, Maryland

Chase Waller

§

Judith Jankowski Wrenn§
*

Upon completion of requirements

†

Degree conferred January 18, 2023

§

Degree conferred October 26, 2022

Chattanooga, Tennessee
Fountain Inn, South Carolina

�Welcoming back our Class of 2020
B A C H E LO R O F A R T S
Tbel Abuseridze		

Batumi, Georgia

On Voyaging

Zainab Noor Ahmad

Lucknow, India

With Feminine Love &amp; Beauty – A Close Look at Prince Andrei Bolkonsky

Angelika Nora Alberstadt

New York, New York

Who Painted the Lion? An Analysis of the Wife of Bath in Chaucer’s
Canterbury Tales

Alexander Thompson Albert

Towson, Maryland

The Influence of War and Peace on Individuals in War and Peace

Emmet Allen

Lexington, Kentucky

Discovery and Concealment – An Examination of Meaning and
Sedimentation in Husserl’s Crisis of European Sciences

August Baganz

Annapolis, Maryland

And Repudiating Immolation: The Motion Towards Understanding in
Faulkner’s Go Down, Moses

Thomas Balding

Long Island, New York

Free Will and the Individual in War and Peace – An Investigation of Man’s
Ability to Act in a Historical World

Morgan Robert Edward Ballard-Wheeler

Tucson, Arizona

Love’s Language in Tristan und Isolde – An Exploration of Musical Meaning

Darcy Marie Bohlin

Santa Fe, New Mexico

On the Amorality of Forgiveness

Anne Lindsay Brong
The Cumulative Action of Creation – A Calculus Under Darwin

Wyndmoor, Pennsylvania

�Martha Campbell

Annapolis, Maryland

The Perfect Words in The Perfect Order – The Power of the Poetic in Soren
Kierkegaard’s Fear and Trembling and Whether it Succeeds in Saying The
Unspeakable

Charles John Carpenter

Farmington, Connecticut

The Natural Social Man – Finding Morality Free of Prejudice in Rousseau’s
Emile

Cole Caudle

Sulphur, Oklahoma

Dynamic Language – An essay concerning how our conception of language
shapes how we engage with the world

Yu Chieh Chang

Taichung, Taiwan

Between Love and Suffering – Confessional Language in Dostoevsky’s Crime
and Punishment

Parker James Chlovechok

Cambridge, Ohio

Hindsight is Twenty-Twenty - An Analysis of the Different Forms of Memory
and their Applications to Life and the Creative Process

Raeann Clement

Port of Spain, Trinidad &amp; Tobago

A Pursuit of Virtue
Adelaide, Australia

Time, Nature, &amp; Empathy: A Portrait of Humanity in Goethe’s Faust
Los Angeles, California

Kathmandu, Nepal

Zarathustra’s Solitude – A Path to Self-Overcoming

Collin Doherty

Richmond, Virginia

A Moral Bath- Pierre’s Pursuit of Happiness in War and Peace

“…And the Battlefield is the Heart of Man”: An Inquiry into Love, Judgement,
and Suffering in The Brothers Karamazov

Sierra Engdahl

Estero, Florida

How to Build a Good European: Nietzsche’s Pursuit of a Reformed Science

Noah Estes

Mosier, Oregon

On Storytelling: The Art of Experience, Perception, and Symbolism in Mark
Twain’s Adventures of Huckleberry Finn

Sadie Bromberg Funk

Minneapolis, Minnesota

Inseparably Interconnected Infinitesimal Elements: An Exploration of
Tolstoy’s Science of History and its Mathematical Possibilities

Tianqi (Simon) Gao

Beijing, People’s Republic of China

The Idea of “Conflicts” in Federalist Papers and Its Misuse in American
Politics
Pasadena, Maryland

Pain: The Price for Greatness
Carlsbad, California

Death for Philosophers –A Guide to Dispelling the Fear of Death Through
Lucretius’ On the Nature of Things

Beebs Hartzell

Houston, Texas

Maddie Nell Fortier Jane

Charlottesville, Virginia

What Giants? An Exploration of Laughter in Cervantes’ Don Quixote
Annapolis, Maryland

On the Isolation Present in the Modern Poetry of Charles Baudelaire

Kesi Emily Dremel

Pittsburgh, Pennsylvania

Antigone: Oedipus’s Complex Daughter

Gorgias: The Rhetoric of Shame

Mary Dray

Ryan Andrew Eberlein

Mitchell Harris

How Ghosts and Art Catch Consciences

Nishan Dhungel

Washington, District of Columbia

Marx, Engels, Ricardo, &amp; Smith- Social labor: Marx’s Critique of Vulgar
Economy

Jodi-Ann Griffin

Jack McGregor Condie
Sam Cooper

Samuel Dreyer

Annapolis, Maryland

Kaila Nicole Johnson

Hagerstown, Maryland

On Violence and Virtue: Hamlet and the Nature of Satisfaction in Revenge
Stories

�Anton Olegovich Kalmysh

Yekaterinburg, Russia

The Limitations of Prospero’s Project and Power

Sila Karabiber

Sakarya, Turkey

Why Two Geometries: The Story of Parallel Lines

Beimnet Kebede

Addis Ababa, Ethiopia

Irregularity: The Pseudo-Perfect Garden- A Deliberation of Perfection and
its Place Within God’s Creative Process

Alexander Kocinsky

Trumbull, Connecticut

Cairo, Egypt

Alone on the Island of Man’s Knowledge – An Examination of the Social and
Asocial in Hayy ibn Yaqzan

Jameson DeWeese Marshall

Fairfax, Virginia

Loving others, by loving ourselves – An exploration of happiness in
Rousseau’s Emile

Tyler Michael Mazur

Cambridge, Ohio

The Fruits That We Bear – An exploration of character and action in
Aristotle’s On Poetics and what the art of fiction tells us about our
perception of reality

Scott Stanaway McCrae

Silver Spring, Maryland

The Tension Within the Leaf: How Conflicting Ends Manifest in the Growing
Plant
Portland, Oregon

A Quantum of Causality – Kantian Metaphysics and Quantum Mechanics
Joined
Vienna, Virginia

Achilles’ Wager – On Fate in the Iliad
West Chester, Pennsylvania

Fear of Change: A Look at Time in the Charles Baudelaire’s Les Fleurs du Mal

Mackenzie Richard–Daffner

Williamstown, Massachusetts

Sitting in an English garden waiting for the sun – I Am the Walrus – The
Beatles

Erica Rochelle Root

Ludington, Michigan

Redemption and Damnation of Humanity: A Story Portrayed by the Lovers of
Gounod’s Faust

James Edward Rubidge

Westport, Connecticut

well maybe I’ve been wrong, maybe my intentions are irrelevant, but
honestly, it’s not just for me. An Analysis of Maturity Through Henry IV Part I
Sterling, Virginia

Change We Can Believe In – Knowledge, Necessity, and History in
Philosophical Fragments
Hanover, Pennsylvania

Finding Satisfaction in a Material World - An Analysis of Lucretius’ On the
Nature of Things

Family and Leadership in Shakepeare’s The Tempest

Elinatan Nelson

Zachary Radford

Kassem Ayad Mansour

John Schuyler Monday

To Whom Do I Owe This Alienation? An Inquiry into Words and Meaning in
Hegel’s Phenomenology of Spirit

John Pell
Sacramento, California

Hegel’s Critique of Pure Morality – The Use and Abuse of Conscience in
Hegel’s Elements of the Philosophy of Right

Sean Miller

Stockholm, Sweden

Edan Moses Otto

“Man Errs as Long as he Will Strive”

Zachary Leveroni

Elisabeth Morell

Thousand Oaks, California

Gabriela Sanchez

Wellington, Florida

Miraculous Artworks – On Hannah Arendt’s The Human Condition

Anna Seban

Davis, California

Swann in Search of Love and Art: The Role of Art, Aesthetics, and
Appreciation in Swann’s Way

Eireamhán Mitha Semple

Dublin, Ireland &amp; Islamabad, Pakistan

The Generalisation of Economic Thought – An essay on The General Theory
by John Maynard Keynes

�Aishwarya Sharma

Kathmandu, Nepal

MASTER OF ARTS IN LIBERAL ARTS

Who Is Antigone? Living a Life Opposed to Life

Glen “Quinn” LeRoy Shivel V

Kennebunkport, Maine

What Sort of Things Are we? Understanding our behavior through the
teachings of Epictetus.

Jakob Stief

St. Louis, Missouri

The Leap of Faith: Resolving the Dialectic between the Esthetic and the
Ethical in Kierkegaard’s Either/Or

Zuowen Tang
How Kant Saved Me from the Void

Mary Elizabeth Turner

Cedar City, Utah

Descartes’ Approach
Muntinlúpa City, Philippines

The Traveler’s Mind, from Both Sides of Jonathan Swift’s Fourth Wall

Grace Villmow

Brookings, South Dakota

Judged, Sentenced, And Condemned - An Exploration of the Truth Behind
Misogyny in Christine de Pizan’s The Book of the City of Ladies

Logan Jefferey Zimmerman

Andrew Dorchester

Washington, District of Columbia

Gregory LaMontagne

Jordan Timothy Poyner
Jordana Leah Stern Rozenman

Quang Ngai, Vietnam

Eloping in Adventures of Huckleberry Finn

Alberto Clemente Antonio Valenzuela III

Chestnut Hill, Massachusetts

Jaime Marquez

Shenzhen, People’s Republic of China

Hung “Jay” Tram

Lee David Cranberg

Wray, Colorado

Confronting the Mask. An Investigation of the Prelude to a Philosophy of the
Future.

Ryan Shinkel

Annapolis, Maryland
Arlington, Virginia
Winona Lake, Indiana
Washington, District of Columbia
Annapolis, Maryland

�PRIZES AND AWARDS

To the Graduate Institute student who has written a distinguished preceptorial
essay in the year 2022.
Offered by the Alumni Association

How Might a Machine Intend?

To the member of the Senior Class who has written the best senior essay.

Siobhán Petersen, Co-Winner

Offered in memory of Susan Irene Roberts, of the Class of 1966

In Search of Physical Reality: A Journey Through Schrödinger’s Mathematical
Derivation of the Quantum Wave Function

The Three Peaks of Aristotle’s Ethics:
A Critique of Magnanimity through Friendship and Wisdom

Sylvie McKnight-Milles

Quinn Rifkin, Co-Winner

Honorable Mention
On Personal Freedom and Historical Inevitability in Tolstoy’s War and Peace
Caroline Ritter

To the member of the Junior Class who has written the best annual essay.

Offered by Mrs. Leslie Clark Stevens in memory of her daughter-in-law, Kathryn Mylroie
Stevens

To a member of the Senior Class, for excellence in speaking.
Offered in memory of Senator Millard E. Tydings of Maryland

Isabel Emond

To the student who submits the best English version of a Greek text.
Offered in memory of John S. Kieffer, President Emeritus

Stephen Carino

Dorothea Brooke’s Death and Resurrection through Art

Dolan Polglaze

Honorable Mention
Yucen Han

To the member of the Sophomore Class who has written the best annual essay.
Offered under the will of the late Judge Walter I. Dawkins, of the Class of 1880

To the student who submits the best English version of a French poem.

Let Him Who Has Ears Listen – Parable, Christian Privacy,
and the Mechanics of Transformation

Finn Yepkle

To the member of the Freshman Class who has written the best annual essay.
Saving Glaucon’s Soul – An Examination of Socrates’ Answer
in the Republic in Relation to Glaucon’s Character

Offered by the Board of Visitors and Governors

LE CORBEAU ET LE RENARD FABLE

Ava Lehrman

To the student who submits a fine original English poem.
Offered by Dr. George Austin in memory of his brother, Henry

Untitled
Lucie McCarthy

Victoria Horsman
Honorable Mention
A Noble Risk and a Noble Lie: The Doubt and Deception of Socrates
Kaleth Cushman

Honorable Mention
When Rajani poured her love
Bibhusan Chapagain

Honorable Mention
Character in Nicomachean Ethics: Who is to Blame?

Honorable Mention
A Reservation
Michael Engstrom

Chelsea Simons

To the Graduate Institute student who has written a distinguished tutorial essay in the
2021-2022 academic year.
Offered in Memory of Laurence Berns by a donation from Joe Coelho

Neither Evening nor Dawn: A Comparison of Μεγαλοψυχία and Σοφία in the
‘Nicomachean Ethics’

Jules Spiese

To the student who submits a fine original musical composition.

Offered in memory of Mary Joy Belknap, President of the Caritas Society 1987-1991
Alma redemptoris mater

Tarik Mahmud

�To the student who submits a fine essay on a piece of music.

Offered in memory of Mary Joy Belknap, President of the Caritas Society 1987-1991
Elli Rees

To the student who submits the most elegant solution of a synthetic problem.

Offered by the Class of 1986 in memory of Bryce Jacobsen, of the Class of 1942, Tutor and
Director of Athletics Emeritus
Paolo Medelius

To the student who submits the most elegant solution of an analytical problem.
Offered in memory of James R. McClintock, of the Class of 1965

Paolo Medelius

To the student who carries out a fine laboratory project.
Offered in memory of Curtis Wilson by the Class of 1963

Not awarded

To the members of the Senior class, through participation, leadership, and
sportsmanship, have contributed most to the SJC athletic program, a special blazer.
Jacob Lees
Ava Lehrman

To the member of the Senior Class who has contributed outstanding service to the
Greater Annapolis Community.
Offered by the Caritas Society of St. John’s College

Ezra Melchor

To the member of the Senior Class who has demonstrated the greatest care for and
service to youth who reside in the City of Annapolis.
Offered by the friends and family of Marvin B. Cooper, of the Class of 1969
Alayna Raymond

To the member of the Senior Class who submits the best work of visual art to the
Community Art Exhibition, the Charles Vernon Moran Prize.
Daryl Locke

To the Senior who has demonstrated excellence in the arts, literature, or sciences,
the Walter S. Baird prize.
World within Hjelmslev Disk

Rose Zhang

�CONGRATULATIONS, ALUMNI.
Stay in touch. Get involved. Give back.
sjc.edu
410-626-2531

�</text>
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&lt;ul&gt;
&lt;li&gt;1796&lt;/li&gt;
&lt;li&gt;1835-1836&lt;/li&gt;
&lt;li&gt;1842&lt;/li&gt;
&lt;li&gt;1852&lt;/li&gt;
&lt;li&gt;1856-1857&lt;/li&gt;
&lt;li&gt;1870&lt;/li&gt;
&lt;li&gt;1878&lt;/li&gt;
&lt;li&gt;1880&lt;/li&gt;
&lt;li&gt;1890&lt;/li&gt;
&lt;li&gt;1893&lt;/li&gt;
&lt;li&gt;1895&lt;/li&gt;
&lt;li&gt;1897&lt;/li&gt;
&lt;li&gt;1907&lt;/li&gt;
&lt;li&gt;1910-1918&lt;/li&gt;
&lt;li&gt;1920-1924&lt;/li&gt;
&lt;li&gt;1928-1929&lt;/li&gt;
&lt;li&gt;1932&lt;/li&gt;
&lt;li&gt;1936-1937&lt;/li&gt;
&lt;li&gt;1939-1945&lt;/li&gt;
&lt;li&gt;1947-present  &lt;/li&gt;
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                    <text>2024

COMMENCEMENT
EXERCISES

Sunday | May 12
Two Thousand and Twenty-Four
Annapolis | Maryland

��PROGRAM
For the Two Hundred Thirty-Second
Commencement in the
Three Hundred Twenty-Eighth Year
of the College
Prelude			

Carrollton Brass Quintet

Academic Procession
The National Anthem			

Carrollton Brass Quintet

Francis Scott Key, Alumnus, Class of 1796

Welcome

President Demleitner

Announcement of Prizes and Awards

President Demleitner
Dean Paalman

Address to the Graduating Class

Mark Roosevelt

Collegewide and Santa Fe Campus President

Conferring of Undergraduate Degrees

President Demleitner
Dean Paalman

Conferring of Graduate Degrees

President Demleitner
Associate Dean Boyle

Academic Recession		

Carrollton Brass Quintet

��B A C H E LO R O F A R T S

Erin Allen		

Coatesville, Pennsylvania

On Physical Analogies, Quantum Physics, and Determinism

Logan Arendt

Levittown, Pennsylvania

Beatrice or Ulysses: On Dante’s Sins in the Divine Comedy

Parth Sarthi Bajaj

Katni, India

The Allure of War: An Examination of the Individual lives and the historical
thoughts in Tolstoy’s War and Peace

Emma Barron

Monroe, Louisiana

The Women of the Aeneid: An Exploration of Vergil’s Female Characters

Lucinda-Cecilia Bellamy

Washington, District of Columbia

The True Vision of the Uncertain Self - (An Investigation of the Mystical
Relationship Between Memory and Conversion)

Peter Boersema

Shoreline, Washington

What are Political Economists For?

Ana Bojaxhi

Camarillo, California

A Mental Discourse on the Jealousy, Greed, and Glory of Man in Hobbes’
Leviathan

Luke Briner
The Alone to the Alone: The One and the Good in Plotinus

Northport, New York

�Allyson Brink

St. Louis, Missouri

Reading the Oresteia with Cassandra in Mind

Carter Brown

Fairfax, Virginia

Moving Without Ark: On Conversation and Compassion in the Book of Job

Acacia Burnham

Santa Fe, New Mexico

Edgar I Nothing Am: Paradox &amp; Performance in Shakespeare’s King Lear

Leul Campbell

Newburgh, New York

The Inequality of Man: The Leap from Property to Slavery in Rousseau’s
Second Discourse

Seneca Case

Grand Rapids, Michigan

Eros and the Divine: An Exploration of the Role of Beatrice and the Feminine
Form in the Divine Comedy

Carolynn Ceci

Bel Air, Maryland

The Wind and the Wise: An Inquiry into a human’s application of Divine
Wisdom

Hannah Chavers

Farmington, New Mexico

The Lullaby of Philosophy and the Importance of Storytelling: A Musical
Endeavor Propounded by Fear and an Effort to Uncover What Is

Yiyang Cheng

Suzhou, People’s Republic of China

Home: the Stability, the Power, the Source of Struggle: The Power Distortion
in the Family as a Cause of the Tragedy in Jean Racine’s Phèdre

Paige Chesney

Washington Grove, Maryland

On the Sun, Equinox, and Celestial Recurrence in Dante Alighieri’s Paradiso

�Caleb Clark

Hollis, New Hampshire

Perfectibilité and the Fall of Man in Rousseau’s Discourse on Inequality

Marshall Cooperman

Washington, District of Columbia

The Role of Justice in Thucydides’ History of the Peloponnesian War

Elisabeth Crosby

New Haven, Connecticut

Nietzsche’s Meditation on How to Survive History: A Return to Being Human

Charles Dang

Pohnpei, Federated States of Micronesia

A Tale of Two Cities: An Exploration of the Atlantis Myth in Plato’s Timaeus
and Critias

Sydney Davis

Arnold, Maryland

Knowing Clytemnestra: On Fully Understanding Character, Past Initial
Impression

James Dormer

Pleasanton, California

A Web Sovereignty: On People and Power in Democracy in America

Frederic Duch Clerici

Barcelona, Spain

The Fundamental Law of Natural Philosophy: An Investigation into the
Scientific Mind

Michael Engstrom

Kenosha, Wisconsin

Over Better Waters: Guidance and Innate Freedom in Dante’s Purgatorio

Tallis Estevez

Silver Spring, Maryland

Purely to Look On: Consciousness’s Self-Examination in The Phenomenology
of Spirit

�Hannah Fodor

Vienna, Austria

What Does it Take to Believe a Story? A Look into the Imaginative Space in
Don Quixote

Audrey Fox

Swarthmore, Pennsylvania

The Drama of Justice: Thrasymachus and Glaucon’s Performance in Plato’s
Republic

Adam Francavilla

Traverse City, Michigan

Democracy in America: The Dangers of Democratic Revolution

Tait Geijer

San Clemente, California

Things Change: Heraclitus’ Cosmology of Πῦρ

Hannah Glick

Warminster, Pennsylvania

The Psychology of a City

Ian Glueckert

Columbus, New Jersey

The Three Deaths of Prince Andrei: On Inevitability, Freedom, and God in
War and Peace

Daniela Gonzalez

Lynn, Massachusetts

What is invisibility?

Oscar Gonzalez

Silver Spring, Maryland

The Hermit’s Sojourn: A Study of the Inner and Outer in Either/Or

Stephanie Harris

Happy Valley, Oregon

Navigating Life’s Deluge: The Metaphor of Movement in Rimbaud’s
Illuminations

�Nelli Harutyunyan

Yerevan, Armenia

What is Philosophy Good For?

Karina Havener

Washington, District of Columbia

Memory in Swann’s Way

Troy Herner-Brown

Alameda, California

The Eye of the Beholder: Ptolemy, Copernicus, Kepler, and the Nature of
Scientific Progress

Gabiel Hess

Stanardsville, Virginia

An Essay on the Role of Reason in Dante’s Salvation and Ours

Adelina Hill

Bridgeport, West Virginia

The Hidden Philosophy of Psychology and the Mystery of Man

Henry Hills

Easton, Maryland

On the Mythical Riddles Concerning Paternal Succession in Hesiod’s
Theogony

Veronique Hornquist

Miami Beach, Florida

An Exploration into the Notion of Beauty in Charles Baudelaire’s Les Fleur du
mal

Ian Howell

Blue Hill, Maine

“The Love of the Grotesque”: Nostalgia in Baudelaire

Genevieve Hoyah

Laurel, Maryland

On Those Who Dwell in the Hollow of the Earth: A Hegelian Critique of the
Human Body as Presented in Plato’s Phaedo

�Zeinip Ibragim Kyzy

Bishkek, Kyrgyzstan

Love’s Alchemy: Exploring Faust’s Transformative Journey and Its
Implications in Goethe’s Faust

Annelise Invanescu

Dearborn Heights, Michigan

Against Birth: An Analysis of the Concept of Death in Sophocles’ Antigone

Alexandra Jacobs

Austin, Texas

Language and Signs: An Inquiry into the Embodied “Word” in the Gospel of
John

Robert Jacobsen

Orient, Ohio

Broken Crockery: The Greatness of General Mikhail Kutuzov

Songeun Grace Jang

Seoul, Republic of Korea

An Attempt to Know the World: Exploration into Aristotle’s Physics

Francis Kane

Easton, Maryland

The Machiavellian Notion of Virtue: (An Investigation of the use of Virtue in
The Prince by Niccolo Machiavelli)

Ranger Kasdorf

Waterford, Virginia

Milton’s Elements

Katherine Kennedy

Charlottesville, Virginia

When the Lights of Health Go Down: Virginia Woolf on Illness

JinHyung Kim

Haeundae, Republic of Korea

On Self-Love and Perfect Liberty in Adam Smith’s The Wealth of Nations

�Sahn Kim

Seongnamsi Bundang-Gu, Republic of Korea

A Meaningful Life: In Thus Spoke Zarathustra

Ranna Kisswani

Las Vegas, Nevada

From Soil We Were to Dust We Shall Be: Likeness &amp; Createdness’ Role in
the Fall

Daniella Kolen*

Baltimore, Maryland

Dagny Kulkarni

Redlands, California

The Resilience of the Wild Rose-Bush: On the Interconnectedness of
Human Nature and the Natural World in Nathaniel Hawthorne’s The Scarlet
Letter

Benjamin Kushnir

Tokyo, Japan

Concerning Spinoza’s Idea of Human Nature

Valentine Lamb

Falls Church, Virginia

Criminal Blood, Justified Blades: Debt, Duty, Desire, and Death in Virgil’s
Aeneid

Ye Seul Lim*
Alexander Macaulay

Seoul, Republic of Korea
Charlestown, Massachusetts

The Moral Failure of Civilization in Gulliver’s Travels

Bridget Mace
Power for the Wife of Bath in The Canterbury Tales

Chappaqua, New York

�Benjamin Maier

Oakland, California

The Drowning of Hippasus: How Incommensurable Magnitudes Became the
Real Numbers

Nathaniel Martin

Fort Worth, Texas

Anomalistic Growth as Proof of Nature’s Goal: Metamorphosis’
transformation of understanding in Goethe’s Metamorphosis of Plants

Allayah Mason-Koehler

Valparaiso, Indiana

Sinning and Sighing in Sign: Lies and Their Repair in a Paradise Lost

Max Mersmann-Jones

Stockton, California

The Power of Two: In Search of a Fundamental Theory of Subjection

Madelyn Minor

Rochester, New York

Repeat After Me: On the Relationships Between Images and Their Origins in
the Tale of Echo and Narcissus

Tuyět-Nhi Nghiêm

Cerritos, California

The Good in Bad Friends: Reaching for The Good Beyond Justice by
Examining Friendship in the Nichomachean Ethics

Charlotte Nicholas

Cowbridge, Wales

Natasha’s Final Harmony: On Music in War and Peace

George Augustus Pananas

Staunton, Virginia

According to Nature: The Application of the Stoic System to Human Life

Mila Passoff

New York, New York

One is Not a Lonely Number: Dedekind’s “Free Creation” and Scientific
Knowledge

�Dolan Polglaze

Walkersville, Maryland

Aristotle’s Biologist: Is Our Contemplation the Animals’ Salvation?

Abigail Poppleton

Storrs Manfield, Connecticut

Songs of the Underworld: An Essay on Homer’s Odyssey

Adam Powers

New York, New York

American Jeremiah: Providential Fact in Alexis de Tocqueville’s Democracy
in America

Daniel Redish

Bloomington, Minnesota

A Polity by Any Other Name: An Analysis of Naming in Thucydides’ History
of the Peloponnesian War

Catherine Row

Glen Allen, Virginia

Lilacs in a Thunderstorm – or – The Necessity of Art in the Face of Nature: A
Reflection on Swann’s Way by Marcel Proust

Sori Ryu

La Mirada, California

A Study of Geneology on the Royal Line of David

Ahan Sagreiya

Jabalpur, India

Understanding Nature and the Nature of Understanding: On the Natural
Path to Understanding and Science in Aristotle’s Physics

Antonina Schlussman

New York, New York

Free Me from the Bonds of Passion! An Exploration of Moral Virtue in
St. Thomas Aquinas’ Summa Theologiæ

Bennett Scott

Cherry Hulls Village, Colorado

An Ethics of Eternity: Deciding on Infinity in the Iliad and Pascal

�Andrew Selway

Baltimore, Maryland

Finding Freedom in the Woods: Isaac’s Path to Liberty in The Bear

Aaron Shelsby

Washington, District of Columbia

The Origin of Deliberative Speech and Its Consequences for the Minds of
Milton’s Fallen Angels

Samuel Soderland

King George, Virginia

Converging on Conversion: Why Should One Follow God? According to
Augustine

Dohyun Song

Jeonju, Republic of Korea

Man’s Perpetual Journey of Sensing and Conversing in Gulliver’s Travels by
Jonathan Swift

Zhiheng Song

Hangzhou, People’s Republic of China

On Knowing: Boundaries of the Recollectible from Meno

Chloe Steer

Catonsville, Maryland

But His Flaw’d Heart Burst Smilingly: Despair, Trifling, and Cures in King Lear

Emma Strauss

Pasadena, Maryland

Heaven or Hell: Which One Will You Choose? An Exploration into
Nonconformity and Making a Way for Oneself

Joshua Tague

Annapolis, Maryland

An End of Grief

Kimberly Taylor
The Elements of Government: A Mathematical Examination of the U.S.
Founding Documents

McLean, Virginia

�John Teague

Woodlands, Texas

Beyond Experience: The Faculties of Consciousness and Rationality in Leo
Tolstoy’s War and Peace

Christopher Thomas

Okemos, Michigan

Gathering Time: Memory and Wholeness in Augustine’s Confessions

Stephaun Turner

Bridgeton, Missouri

Threading the Weaver’s Loom: Plotinus and Hegel Illuminate Creation’s
Metaphysical Depths in Genesis

Christopher Turney

Atlanta, Georgia

The Teachable Yahoo: An Exploration of Identity in Gulliver’s Travels

Naina Wagh

Pune, India

Zarathustra’s Descent into Mankind: Looking at Zarathustra’s Prologue as a
Tragedy

Micah Warren

Alexandria, Virginia

The Perfection of the State in Machiavelli’s Discourses on Livy

Isabella Watmough

Damariscotta, Maine

The “special providence in the fall of a sparrow” in Shakespeare’s Hamlet:
A Study of the Noble Conscience in the Midst of Tragedy (with Help from
Hegel)

Senrui Yang

Chongqing, People’s Republic of China

A Reading of Les Fleurs du Mal: Is Life Worth Living? Reflections on the Dark
Side of the Contemplative Life and Modern Man’s Search for Positive Beliefs
and Meaning in Life

�Nicholas Yurasko

Medford, Massachusetts

On the Psychodynamic Theory of Dreams &amp; Neurotic Symptomatology

Henry Zheng

Washington, District of Columbia

Look Back at Education’s Past, Present, and Future: The History of
Education from Europe to America and Anticipatory Predictions of What
Happens to Education After Alexis De Tocqueville’s Democracy in America

�MASTER OF ARTS IN LIBERAL ARTS

Sylvie Bernhardt

Annapolis, Maryland

Michael Connors

Brisbane, Australia

Lyndsey Cornell§

Palm Beach Gardens, Florida

Jose Coronado Flores
Nicholas Dacosta
Brian Doyle§
Dominic Escorpeso
William Galloway
Grant Gombert†
Elizabeth Hartley
Sonya Holbrook§
Theodorus Johan Jeursen†
Stuart Kenny
Yonas Ketsela

Frederick, Maryland
Washington, District of Columbia
Bethesda, Maryland
Waldorf, Maryland
Camden, South Carolina
Boerne, Texas
The Dalles, Oregon
Cornelius, North Carolina
‘s-Gravenzande, Netherlands
Lincoln, Nebraska
Ethiopia

Jordan Klein

Annapolis, Maryland

Viktoriya Kudryavtseva§

Baltimore, Maryland

Melissa Moore§
Benjamin Morgan
Luis Parrales

Wolfboro, New Hampshire
Washington, District of Columbia
Arlington, Virginia

�Bethany Pautrat§

Laurel, Maryland

Noah Perrin§

Richmond, Virginia

Quinn Rifkin

Ellicottville, New York

Gavin Robillard
Armando Rodriguez§

Fairfax, Virginia
Puebla, Mexico

Anthony Roncalli

New York, New York

Brandon Rothenberg§

Rockville, Maryland

Jack Ryan
Luca Stephen-Sujdovic
Houston Taylor

Annapolis, Maryland
Sydney, Australia
Chattanooga, Tennessee

Kevin Telford

Oceanside, New York

Noah Vancina†

Fort Wayne, Indiana

Ray Whitney

Millsboro, Delaware

*

Upon completion of requirements
Degree conferred January 24, 2024
§
Degree conferred October 21, 2023
†

�PRIZES AND AWARDS

To the member of the Senior Class who has written the best senior essay.
Offered in memory of Susan Irene Roberts, of the Class of 1966

Aristotle’s Biologist: Is Our Contemplation the Animals’ Salvation?

Dolan Polglaze
Honorable Mentions
Edgar I Nothing Am: Paradox &amp; Performance in Shakespeare’s King Lear
Acacia Burnham
Sinning and Sighing in Sign—Lies and Their Repair in a Paradise Lost

Allayah Mason-Koehler

To the member of the Junior Class who has written the best annual essay.

Offered by Mrs. Leslie Clark Stevens in memory of her daughter-in-law, Kathryn Mylroie
Stevens
Honorable Mentions
The Everlasting Covenant or Immortalizing Leviathan—
The Maintenance of the Commonwealth
Helen Felback
On le forcera d’etre libre: An Essay on Rousseau

Benjamin Kay

To the member of the Sophomore Class who has written the best annual essay.
Offered under the will of the late Judge Walter I. Dawkins, of the Class of 1880

Reaching Up to the Heavens–A Study of How We Know
About God Through Aquinas

Victoria Horsman

To the member of the Freshman Class who has written the best annual essay.
Death in Athens

Henry Haggard

ἡ Θεωρία τῶν Θεῶν: The Activity and Forms of Contemplation in
Aristotle’s Nicomachean Ethics

Albert Shapiro

�To the Graduate Institute student who has written a distinguished tutorial essay in the
2023-2024 academic year.
Offered in Memory of Laurence Berns by a donation from Joe Coelho

The Last Man: An Introduction to Nietzsche’s on the Advantages
and Disadvantages of History for Life

Armando Rodriguez

To the Graduate Institute student who has written a distinguished preceptorial
essay in the year 2023.
Offered by the Alumni Association

The Constitution of Opinion: On David Hume’s First Principles of Government

Benjamin Crocker

To a member of the Senior Class, for excellence in speaking.
Offered in memory of Senator Millard E. Tydings of Maryland

Michael Engstrom

To the student who submits the best English version of a Greek text.
Offered in memory of John S. Kieffer, President Emeritus

Max Dreyfuss

To the student who submits a fine original English poem.
Offered by Dr. George Austin in memory of his brother, Henry

Astraea’s Rhapsody Or, Innocence Departed

Luke Briner
Honorable Mention
The Midwife
Mia Guill

To the student who submits a fine original musical composition.

Offered in memory of Mary Joy Belknap, President of the Caritas Society 1987-1991
Ave verum corpus

Tarik Mahmud

To the student who submits a fine essay on a piece of music.

Offered in memory of Mary Joy Belknap, President of the Caritas Society 1987-1991
Honorable Mention
An End Beyond Endings: Word-Tone Relationships in the Seikilos Epitaph
Helen Wagner

�To the student who submits the most elegant solution of a synthetic problem. Offered
by the Class of 1986 in memory of Bryce Jacobsen, of the Class of 1942, Tutor and Director of
Athletics Emeritus
Peter Boersema
Honorable Mention
Theodora Montague

To the student who submits the most elegant solution of an analytical problem.
Offered in memory of James R. McClintock, of the Class of 1965

Troy Herner-Brown

To the student who carries out a fine laboratory project.
Offered in memory of Curtis Wilson by the Class of 1963

Honorable Mention
The Barber-pole Effect
Ahan Sagreiya

To the members of the Senior class, through participation, leadership, and
sportsmanship, have contributed most to the SJC athletic program, a special blazer.
Charlotte Nicholas
Frederic Duch Clerici

To the member of the Senior Class who has contributed outstanding service to the
Greater Annapolis Community.
Offered by the Caritas Society of St. John’s College

Madelyn Minor

To the member of the Senior Class who has demonstrated the greatest care for and
service to youth who reside in the City of Annapolis.
Offered by the friends and family of Marvin B. Cooper, of the Class of 1969
Luke Briner

To the Senior who has demonstrated excellence in the arts, literature, or sciences,
the Walter S. Baird prize.
Two Mechanical Voices, Fluent as a Dance

Ranger Kasdorf
Thinking Outside the Boox A Mini-Portfolio of Poetry, Inspired by the Liberal Arts

Tuyět-Nhi Nghiêm

���CONGRATULATIONS, ALUMNI.
Stay in touch. Get involved. Give back.
sjc.edu
410-626-2531

�</text>
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&lt;ul&gt;
&lt;li&gt;1796&lt;/li&gt;
&lt;li&gt;1835-1836&lt;/li&gt;
&lt;li&gt;1842&lt;/li&gt;
&lt;li&gt;1852&lt;/li&gt;
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&lt;li&gt;1870&lt;/li&gt;
&lt;li&gt;1878&lt;/li&gt;
&lt;li&gt;1880&lt;/li&gt;
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&lt;li&gt;1893&lt;/li&gt;
&lt;li&gt;1895&lt;/li&gt;
&lt;li&gt;1897&lt;/li&gt;
&lt;li&gt;1907&lt;/li&gt;
&lt;li&gt;1910-1918&lt;/li&gt;
&lt;li&gt;1920-1924&lt;/li&gt;
&lt;li&gt;1928-1929&lt;/li&gt;
&lt;li&gt;1932&lt;/li&gt;
&lt;li&gt;1936-1937&lt;/li&gt;
&lt;li&gt;1939-1945&lt;/li&gt;
&lt;li&gt;1947-present  &lt;/li&gt;
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To learn more about the Greenfield Library's archives use policy or to obtain high-resolution images, please see the &lt;strong&gt;&lt;a title="Image Collection Use Policy" href="http://www.sjc.edu/academic-programs/libraries/greenfield-library/policies/#photographicarchivepolicy" target="_blank" rel="noreferrer noopener"&gt;Library’s Image Collection Use Policy&lt;/a&gt;&lt;/strong&gt;.&lt;br /&gt;&lt;br /&gt;Click on &lt;strong&gt;&lt;a title="Photographic Archives" href="http://digitalarchives.sjc.edu/items/browse?collection=7"&gt;Items in the Image Collection—Annapolis Collection&lt;/a&gt;&lt;/strong&gt; to view and sort all items in the collection.</text>
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                  <text>&lt;p&gt;The Greenfield Library Image Collection houses over 5,000 photographs, postcards, drawings, and other graphic materials.  The images in the collection document the history, academic, and community life of St. John’s College. &lt;/p&gt;
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                    <text>Constructible Numbers
Grant Franks
June 3, 2019, revised September 21, 2019

Introduction
Recapitulation
Two weeks ago, we saw that Rafael Bombelli confronted the possibility that the square root of negative
one might have some positive significance when he solved the cubic equation
x 3 - 15 x - 4 = 0
and, 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  .

Last week, we followed Caspar Wessel in developing a graphic understanding of the arithmetic of
complex numbers. In particular, we saw that
(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.
So far, so good.

Today’s Agenda

�2 ���

3 Constructible Numbers.nb

Today, we’re going in diﬀerent direction. Today, we will talk about constructing numbers and the
emptiness of the Euclidean plane.

The Geometrical Plane
The Awkward Question
“Are there holes in the geometric plane?” This is a question that rarely gets asked when doing geometry. There is no reason to raise such a question until some awkward questions get asked, and there is
no way to answer the question without being able to step back from intuitively given geometrical space
by thinking algebraically.

Construction
In Euclid’s geometry, one begins with five postulates, three of which authorize constructions:
Postulate One: to draw a line between any point and any other point;
Postulate Two: to continue a line indefinitely; and
Postulate Three: to draw a circle with any center and radius.
The references to “any point,” “any center” and “any radius” give the impression that the Euclidean
plane contains all possible points. But that isn’t quite true or, to be more precises, there are many
points in a plane to which Euclidean geometry provides no access.
For instance, suppose you wanted to “square the circle.” This is a classical problem that amounts to
drawing a rectangle with an area equal to that of a given circle. Archimedes shows that such a rectangle would have a height equal to the radius of the circle and base equal to half its circumference. If you
can draw a circle with any center and radius, could you draw one whose radius is equal to the semicircumference of the circle you are trying to square? If you could do that just by saying it, your problem
would be solved.
That would not satisfy a mathematician. She or he would want to know how to get that radius length,
that is, how to construct it. Beginning with the one length that is given -- the radius of the circle to be
squared -- and using only permissible manipulations, how can we construct the desired straight line
with a length equal to the semi-circumference of the given circle?

The Meno Problem
This problem sounds a little like the geometrical problem of the Meno: Socrates asks the slave boy, “If
you are given this square with these sides, can you show the side of a square with double the area?”
The slave boy is stumped. Some modern students think they have a better answer than the slave boy
and say, “That’s easy! It’s the square root of two!” They don’t realize that they are not giving an

�3 Constructible Numbers.nb

���

3

answer at all. The phrase “square root of two” is just a slightly shortened version of “the magnitude
which, when multiplied by itself, gives two.” So, to Socrates’ question, “What is the magnitude which,
when multiplied by itself, yields two?” they have answered, “The magnitude which, when multiplied by
itself, gives two.” True, no doubt, as any tautology is true, but it doesn’t really advance our knowledge
of, well, anything.
Socrates provides what is really needed: a geometrical construction for finding the square root of two
by looking at the diagonal of a square with sides equal to one. (Really, he gives the construction for 2
2 because his original square had sides equal to two, but that is minor detail.)
For the squaring of the circle, we want a construction for a straight line whose length is equal to the
semidiameter of the given circle. Alas, not all that humankind desires does it obtain! There is no
Euclidean construction for that length. The story that leads to that result culminates in the late nineteenth century. It involves Ferdinand Lindemann’s demonstration of that π is not just an irrational
magnitude -- that had been known for over a century (Johann Lambert 1761) -- but that it is a particular
kind of irrational magnitude. Getting to that result, if anyone here is interested in it, would require
work that goes beyond what this series of talks will cover. If it is any consolation, however, the ideas
we will cover tonight are necessary preliminaries to that work.

The Delian Problem
Once upon a time, a long time ago, a great plague aﬀlicted the city of Delos. The citizens consulted
Apollo’s oracle at Delphi who told them that the god was dissatisfied with the altar of his temple. The
altar was made in the shape of a cube, and the oracle said that the god wanted an altar twice as big.
The citizens, eager to be rid of the plague, got a great piece of marble and built an altar twice as long,
twice as deep and twice as tall as the one that was there. The plague continued. The priest of the
oracle corrected the people saying the god wanted an altar with twice the volume of the present one.
The newly built altar had eight times the volume, and was not what was wanted. The people were
understandably annoyed, but everyone in the ancient world knew that gods love to mess with people
by issuing weirdly misleading oracular pronouncements. Gods are cruel, that’s all there is to it.
Geometers realized immediately that what was needed was, in eﬀect, an altar whose side was 2
times bigger than the present one. When they set out to design it, however, they found that the god
had been even crueler than expected. They couldn’t figure out how to construct the altar. Finding a
construction for 2 was easy, but finding one for the 2 was surprisingly diﬀicult.
3

3

To get to the bottom of the problem that they faced, we have to sketch out a new kind of algebraic
operation, finite field extensions. Ordinarily this would be a semester-long study, but since this is St.
John’s College, I will try to compress it into about fi�een minutes.

�4 ���

3 Constructible Numbers.nb

Rational Operations
What is a Field?
To begin, we need to define “a field.”
For our purposes, a “field” is a collection of objects -- we’re going to be talking about numbers -- that
are closed under the operations of addition (and its inverse, subtraction) and multiplication (and its
inverse, division).
Consider first, then, the whole numbers: 1, 2, 3, …. These are closed under addition, that is, if you add
any two whole numbers you get a whole number. But they are not closed under subtraction. Although
you can subtract 5 from 7 to get 2, you cannot subtract 7 from 5.
So set aside the whole numbers and take up the integers: … -3, -2, -1, 0, +1, +2, +3, …. Now you have
closure under addition and subtraction. You also have closure under multiplication but not division. 10
divided by 5 is 2, but 10 divided by 3 is not among the integers.
So set aside the integers and take up the rational numbers: …

-1 -1 -2
, 5 , 11 ,
3

…0…

1
, 2, 5
10 7 3

…. That is,

all the numbers made up by ratios of integers with one another (forbidding division by zero). Now we
have it: this is a field. It is closed under addition and subtraction, it is closed under multiplication and
division.
For convenience’ sake, we will give the rational numbers a symbolic name, Q. (Why Q and not R?
Because R is reserved for the real numbers. Alas.)

Euclid’s Operations Allow Us to Form a Field
The operations of Euclid’s geometry allow us to construct lengths on a line that correspond to the field
of rational numbers. If we begin with a given length that we will call the “unit,” we can with straightedge and compass easily make a double length, a triple length, etc. If we define “negative” to be mean
motion in one direction from an arbitrary starting point and “positive” to mean going in the opposite
direction, we can construct lengths corresponding to all integers. By an easy construction, we can also
divide our given unit length into equal parts corresponding to any whole number. Thus we can make
lengths corresponding to any positive proper fraction; by multiplying these we can make any proper or
improper fraction, and by directing them toward the negative side of our arbitrary zero point, we can
identify places corresponding to any rational number. The lengths from zero to these points can be
added, subtracted, multiplied and divided at will and the result will always be another rational length.
We have a field.
If we erect two such lines at right angles to one another, we can locate and label any point on a plane

�3 Constructible Numbers.nb

���

5

that corresponds to (a, b), where a and b are rational numbers. As my grandfather used to say, “Now
we’re cookin’ with gas!”

Other Lengths
All points with rational coordinates is a lot of points, but we know that there are other lengths that can
be found in Euclidean geometry. There is, for example, 2 , which is the diagonal of the square with
sides of unit length. In fact, we can construct lengths equal to the square roots of any lengths we can
find through other means.
D

A

C

B

If you want to find a length equal to r , draw line AB in length equal to r + 1. Here, let Ac = r and let CB
= 1. Erect a semi-circle on line AB and a perpendicular at C meeting the semicircle at D. Join AD and DB.
Triangle ADC is similar to triangle DCB and to the combined triangle ADB. Therefore:
AC : CD :: CD : CB
AC × CB = CD2
But AC = r and CB = 1; therefore:
r × 1 = 2 = CD2
r = CD.

Combinations
A little examination will show that these are all the operations that are available to us. We have addition, subtraction, multiplication, division -- these are suﬀicient to find any rational lengths. In addition
to this, we can take the square root of any length that we can find. Not only that: we can do so as many
times as we please. So, can construct

2 , or

5 , or

17
3

, or of any rational length. And that’s not

�6 ���

3 Constructible Numbers.nb

all! We can construct

2 . Or

2+

2 . Or

17
3

+

2+

7
3

… or any sequence or combina-

tion of the rational operations and repeated extraction of square roots.
With these techniques in hand, someone might easily jump to the conclusion that these Euclidean
operations can construct any length whatsoever. That’s probably what I would have said if anyone had
asked me back when I was a Johnnie Freshman more years ago than I care to think about. But I would
have been wrong.

Jumping to Conclusions
When I learned about the Pythagorean theorem and irrational numbers (or their equivalents, incommensurable lengths), I didn’t take time to think about these new numbers as carefully as, in retrospect,
I should have. Looking back, I think my understanding ran something like this:
“We had the whole numbers, but they weren’t enough to do subtraction so we added the negative
numbers and got the integers. But the integers weren’t enough to do division, so we added the fractions and got the rational numbers. But even the rationals weren’t enough to account for all the
lengths we could find in geometry -- the 2 is irrational! (Hey! I was just as surprised by this as the
Greeks were!) -- so we added the irrational numbers to the rational numbers and now we have all the
real numbers, which is all that there are!”
That understanding didn’t get challenged for decades until I began working on a preceptorial on Galois
Theory and Professor Charles Hadlock, author of Field Theory and Its Classical Problems, introduced me
to finite field extensions. It was here that I learned the humbling lesson that not all irrational numbers
are the same. Some numbers are more irrational than others, and lumping them all together blurred
together distinctions that are best kept separate.

Baby Steps
Let’s start over. Go back to when we had just the rational lengths and could find any point with rational
coordinates. Now, we read the Meno and find out about 2 . Instead of pretending that we are now
able to generate all possible irrational lengths, look carefully at what we have. We can make any
rational length, and we can make the square root of two. If we continue now to use just with our
rational operations (+, -, x, ÷) on the two lengths we have at hand, 1 and 2 , we can make any number
that looks like this:
a+b

2

where a and b are rational numbers. Notice something important about these numbers: we can add
them, subtract them, multiply them, and divide them any way we please and we always get other

�3 Constructible Numbers.nb

���

7

numbers of this same kind. So, if we have:
3 + 5

2  + -1 + 7

2  = 2 + 12

3 + 5

2  × -1 + 7

2  = -3 + 21

2.

or
2 -5

2

2 + 35  2  = 67 + 16

2

Division is a bit more complicated, but it works as well. The upshot is this: the numbers a + b 2 form
a field of their own. This new field is called an extension field. Because we added a finite number of
elements to form it (in this case, just one), it is called a finite field extension. And because the element
we added was a solution of a polynomial with elements of the original field as coeﬀicients -- in this
case, x 2 - 2 = 0 -- it is called a finite algebraic field extension. And because it was made by adding an
element that is the square-root of a member of the original field, it is called a quadratic finite algebraic
field extension. Let’s call it F1 and write F = Q( 2 ) to signify that F was formed by appending 2 to Q
and making all the numbers of the form a + b 2 where a and b are elements of Q.
F is big. It’s bigger than Q, the rational numbers. It includes Q as a subset, so we write:
F1 ⊃ Q.
But F does not include all the numbers (lengths) that we can construct because we can adjoin other
elements if we wish. We can even take the square root of some squirrely element of F1 that already has
a square root of two, say:
3+7

2

We can append this element to F1 and form the numbers:
c+d

3+7

2

where c and d are elements of F. This is a quadratic finite algebraic field extension of F1 . Let’s call it F2
and write F2 = F1 (

3+7

2 ) to signify that F2 was formed by appending

ing all the numbers of the form c + d

3+7

3+7

2 to F1 and mak-

2 where c and d are elements of F1 .

F2 is big. It’s bigger than F1 and much bigger than Q. It includes F1 as a subset, so we write:
F2 ⊃ F1 ⊃ Q.
Do you see where this is going? We can continue this process as long as we wish.

�8 ���

3 Constructible Numbers.nb

… F5 ⊃ F 4 ⊃ F 3 ⊃ F 2 ⊃ F 1 ⊃ Q
When I put them all together, I have a tower of finite quadratic field extensions. Every number that
corresponds to every possible constructible length is somewhere in that tower. Altogether, they are
called the constructible numbers. The set of constructible numbers is very big.
But it’s not everything.

There are Non-Constructible Numbers
There are, as it turns out, non-constructible numbers, as can be shown in several ways. For instance,
2 (the real cube root of two) is not a constructible number. It does not belong to any tower of
quadratic field extensions over the rationals.
3

For, proceeding in the time honored way of reductio ad absurdum, suppose that 2 were constructible. Also remember that, since in the real numbers y = x 3 is strictly increasing, there is only one
real cube root of two. Now, if 2 were constructible, it would belong to some quadratic field extension
of a field that was itself part of a tower of quadratic field extensions leading back to the rationals.
3

3

2 ∈ Fn

3

That means that
include 2 .

3

where

2 =a+b

Fn ⊃ Fn-1 ⊃ Fn-2 ⊃ Fn-3 ⊃ … F1 ⊃ Q

c , where a, b and c are all parts of Fn-1 , but where Fn-1 does not itself

3

Cube both sides of this equation.
3

 2  = a + b
3

3

c  = a3 + 3 a 2 b

2 = (a3 + 3 a b2 c ) + ( 3 a2 b + b3 c )

c + 3 a b 2 c + b3 c

c

c

The number 2 is a part of Fn - 1 ; we know this because it is a member of Q. Therefore, it has no component multiplied by c , which means that ( 3 a2 b + b3 c ) = 0.
Next consider (a3 + 3 a b2 c ) - ( 3 a2 b + b3 c ) c (notice the minus sign). Since 3 a2 b + b3 c = 0, this
3
has the same value as (a3 + 3 a b2 c ) + ( 3 a2 b + b3 c ) c . But it unpacks into a - b c  . So we
have two cube roots of 2:
a+b

c

a-b

c

But there is only one value ; therefore b = 0. That means that a + b

c is really just a, and

3

2 is in

�3 Constructible Numbers.nb

���

9

Fn - 1 . By the same reasoning, it is a member of Fn - 2 and Fn - 3 and so on until we discover it is a member
of Q, that is, that it is rational.
But

3

2 is NOT rational. (Those who doubt this can see Appendix 1.)

Thus, 2 is NOT in ANY tower of quadratic field extensions beginning with the rationals. It is not
constructible.
3

The Delian Problem is Thus Solved
At this point, the architects and engineers at Delos should despair: if 2 is not constructible, then
they cannot double the size of the altar of Apollo with Euclidean mathematics. The gods are cruel, but
they are are mathematically well-informed.
3

There are Lots of Non-Constructible Numbers.
The demonstration that the 2 is not constructible is all well and good, but it is rather ad hoc. It
doesn’t immediately produce any broad conclusions about constructible vs. non-constructible numbers.
3

A slightly more detailed investigation of field theory allows broader conclusions. It is possible to
characterize the size or “degree” of one algebraic field extension over another. Compare, for example,
the quadratic extension that results from adjoining 2 to the rationals with what would be called the
“cubic” extension that occurs when you adjoin 2 . In the first case, we can express any number in the
extended field by an expression that looks like this:
3

2.

a+b

These numbers form a field. You can add, subtract, multiply and divide to your heart’s content and
never leave the field. If you try this with 2 , however, a problem arises. Form a number like:
3

a+b

3

2.

You can add and subtract alright, but as soon as you start multiplying, you’ll find yourself running into:
3

2 ×

3

2 =

3

4

The cube root of four is not the same as the cube root of two. It can’t be expressed by combinations of
rational numbers and the cube root of two. It is outside of the (purported) field. This problem did not
arise with 2 because
2 ×

2 =2

�10 ���

3 Constructible Numbers.nb

which is within the field defined by a + b 2 . With the cube root, it is not enough to add one term; you
need to add two. To get a field that includes 2 , you need numbers of this form:
3

a+b

3

2 +c

3

4.

A little experimentation will persuade you that these numbers do form a field.
Notice that when you formed a extended field with 2 , your new numbers had two terms, a + b 2 .
With 2 of two, your new numbers have three terms. The quadratic extension is of “degree two,”
while the cubic extension is of “degree three.” Field extensions can be compounded: an extension of
degree two followed by an extension of degree three will yield an extended field of degree six over the
original field. Field extensions can get remarkably complex, but for our purposes it will be enough to
focus on relatively simple extensions of relatively small degrees.
3

3

The degree of an extension is measured by complexity of the minimal polynomial needed to produce
the new elements whose addition to the original field leads to the extension. Determining whether a
polynomial is “minimal” poses some problems, but this approach can produce sweeping knowledge
about whole classes of extensions. So, for instance, every quadratic field extension over the one before
it, so that a tower of quadratic field extensions -- that is, the collection leading to any constructible
numbers -- will have powers 2, 4, 8, 16 … 2n over the rationals. At the same time, it can be shown that
for equations of the form:
xn - 2 = 0
the number n gives the degree of the extension resulting from appending one of the solutions of the
equation to a field. This result allows is to know that
Solutions of

Appended to Q are of degree

And thus are generally

2

2

Constructible

2

3

Not Constructible

- 2

4

2

4

Constructible

- 2

5

2

5

Not Constructible

x6 - 2

6

2

6

Not Constructible

x7

7

2

7

Not Constructible

x2

I.e.

- 2

2

x3 - 2

3

x4
x5

- 2

Also, it can be show that the solutions of the equations for the nth roots of unity, a�er factoring out (x 1), are minimal when n is prime. So
Roots of
structible
Roots of
structible

x3 - 1 = 0

appended to Q are of degree

2

over the rationals, so con-

x5 - 1 = 0

appended to Q are of degree

4

over the rationals, so con-

�3 Constructible Numbers.nb

Roots of
structible
Roots of
structible
Roots of
structible
Roots of
structible
Roots of
structible
etc.

���

x7 - 1 = 0

appended to Q are of degree

6

over the rationals, so NOT con-

x 11 - 1 = 0

appended to Q are of degree

10

over the rationals, so NOT con-

x 13 - 1 = 0

appended to Q are of degree

12

over the rationals, so NOT con-

x 17 - 1 = 0

appended to Q are of degree

16

over the rationals, so con-

x 19 - 1 = 0

appended to Q are of degree

18

over the rationals, so NOT con-

11

Many -- indeed, most -- of these create extensions whose degrees are not powers of two over the
rationals. Thus they create field extensions filled with numbers that are not constructible. Literally
infinite fields of non-constructible numbers emerge.

Hierarchy of Irrationals
Viewing all the irrationals as an undiﬀerentiated mob is a mistake. We can distinguish between those
irrationals that are constructible and those that are not. The constructible numbers are built by successive quadratic field extensions starting from the rationals.

Non-Constructibles

Constructibles

Rationals

The distinction between the constructible and non-constructible numbers is interesting enough, but
situation is even stranger than that.
As we have seen, the Constructibles are made from towers of field extensions all of degree two. We

�12 ���

3 Constructible Numbers.nb

have just seen that many polynomials have solutions which, when appended to a field, give an extension of degree other than two; the equation x 3 - 2 = 0, for instance, has solutions of degree 3, which
takes us away from the powers-of-two towers of constructible numbers.
Suppose we toss away that restriction. Suppose we consider all algebraic field extensions of any
degree. What if we allow ourselves to build towers in which each step can be of any degree -- that is, to
append to a field the solutions of a polynomial of any degree. In this way, we could make towers of
fields that include the solutions of any finite polynomial equations. That would include all the constructible numbers and much, much more. This immense collection is known as the algebraic numbers. It includes the rational numbers and the constructible numbers and much, much more.
It does not, however, include everything. There are numbers that are not included among the algebraic
numbers. You know a few: π is not an algebraic number. Neither is e, the base of the natural logarithm
system. Leibniz and later Euler called these non-algebraic numbers “transcendental numbers,” a
wonderfully mystical “woo-woo” name that stuck and is in common use today.

Transcendentals

Algebraics
Constructibles

Rationals

There are LOTS of Transcendental Numbers
When I name π and e as transcendental numbers, you may be misled into thinking that there are only a
few such numbers and that each of them is a precious rarity, much treasured by mathematicians like
these two specimens.
Au contraire! Far from being scarce, the transcendental numbers not only surpass all other numbers in

�3 Constructible Numbers.nb

���

13

quantity, they do so by an infinite amount. Of course, to characterize one infinity as greater or lesser
than another is a controversial project first pioneered by nineteenth century mathematician Georg
Cantor. According to Cantor, the smallest sort of infinity is like that of the natural numbers which can
be ordered in such a way that one can count oﬀ the members of an infinite set sequentially and be sure
eventually to encounter every member. The natural numbers are obviously countable in this way, as
are the integers if we number them like this:
0
+1 -1 +2 -2 +3 -3 …
1st 2nd 3rd 4th 5th 6th 7th …
It takes a little more work to see that the rationals can be placed in countable order (they can), and a
bit more still to figure out that the algebraic numbers can also be ordered and counted. But they can.
Cantor designates this infinity by the symbol ℵ0 .
The complete collection of real numbers -- and also the collection of all complex number a + b i where a
and b can be any real number -- cannot be so ordered. These numbers, according to Canto, form a
higher degree of infinity, the infinity of the continuum, ℵ1 which is widely take to be equivalent to 2 ℵ0 ,
a quantity distinctly diﬀerent, and distinctly bigger than ℵ0 -- insofar as “bigger” is a concept applicable to infinities.
From this perspective, the relation of the algebraic numbers to the entire set of complex numbers is
pretty much what Ptolemy would call “the ratio of a point to a line.” The whole realm of all our manipulations, geometric and algebraic, occur in a vanishingly small subset of the totality of real numbers. Yet
though we speak of infinities, do not imagine that these transcendentals are far away. They are not far
away in heaven, so that you have to ask, “Who will ascend into heaven to get them?” Nor are they
beyond the sea, so that you have to ask, “Who will cross the sea to get them?” No, they are very near to
you always on every side, crowding about with incredible density. And the net of constructible numbers seems now to spread across the Euclidean plane like ever-thinning gossamer network of barely
perceptible points, each separated from the next by gulfs teeming full with inaccessible points.
While we all contemplate the miserable smallness of all our endeavors, let us take a brief break and
then there will be time for questions.

Appendix 1: Irrationality of 2
3

This result follows from Euclid, Book X, proposition 9. It can also be shown from arithmetic principles
as follows:
Suppose that

3

2 is rational. Then it can be expressed as a fraction

numbers. We may assume also that the fraction

a
b

a
b

where a and b are finite whole

is expressed in lowest terms, so that a and b have no

�14 ���

3 Constructible Numbers.nb

factors in common.
We have:
3

a
b

2 =

2=

a3
b3

2 b3 = a 3
That means that a3 is even, which means that a is even; thus a3 is divisible by 8. Let it be 8 c3 .
2 b3 = 8 c 3
b3 = 4 c 3
That means that b3 is divisible by 4, which means that b is even (and that b3 is in fact divisible by 64).
But we began with the hypothesis ab was a fraction where a and b have no factors in common.
Therefore,

3

2 cannot be expressed as a fraction ab .

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