The Declaration of Independence Is Almost a Euclidean Proof

Jefferson borrowed Euclid's structure, axioms and all, to argue for independence.

Every fall, I stand at the board with a new group of students and teach them the same six-part skeleton. Enunciation. Exposition. Specification. Justification. Construction. Conclusion. I tell them Euclid used it almost 500 times without variation, and that once they see the pattern, the wall of text stops being a wall.

I never expected to find it again in a document about a king.

Read the Declaration of Independence slowly, the way you’d read Proposition I.1, and the parts show through. A general principle, stated in the abstract. A grounding in the specific case. A restatement of what’s about to be shown. A long list of justifications, cited one after another like margin notes. An act, not just an argument. A conclusion.

This isn’t an accident, and it isn’t just a nice metaphor for a summer newsletter. Jefferson’s own draft shows him reaching for the word. He first wrote that these truths were “sacred and undeniable.” He crossed it out. He wrote “self-evident” instead, the word Euclid used for an axiom, a thing you don’t prove but grant, and build everything else on top of.

Jefferson studied Euclid at William and Mary under a tutor, William Small, who historians believe drilled him in exactly this kind of reasoning: definitions, postulates, axioms, and the long chain of justified steps that follows once you accept them. Lincoln did the same thing decades later with a saddlebag copy of the Elements, teaching himself what Jefferson had been taught.

Today, everyone else is writing about 1776. I want to write about geometry, because Jefferson did too.

Euclid begins with his definitions, postulates, and common notions. On these, he builds mathematical arguments that reach certain conclusions, because mathematical objects are simple and abstract. A geometric proof can grow enormously complex and its conclusion will still be certain, so long as the logic holds.

Jefferson is not proving a geometric fact. He’s making a political argument, using the closest thing to Euclid’s certainty that he can build: a king’s actions standing in for a circle’s radii. But a king’s actions aren’t radii. They have to be weighed and judged, not measured. Jefferson compensates for that harder, messier subject matter the only way he can: he keeps the logical structure as tight and simple as possible, so that once a reader grants the axioms, the conclusion is difficult to escape.

John Trumbull's painting, Declaration of Independence

Euclidean Structure

The entire Declaration of Independence can be read through the parts of a Euclidean proposition.

The Introduction Explains the Necessity

Euclid never explains why he’s writing. He simply begins. Jefferson can’t afford that luxury. The opening sentences of the preamble to the Declaration of Independence are not part of the argument. They explain why the document is necessary.

When in the Course of human events, it becomes necessary for one people to dissolve the political bands which have connected them with another, and to assume among the powers of the earth, the separate and equal station to which the Laws of Nature and of Nature’s God entitle them, a decent respect to the opinions of mankind requires that they should declare the causes which impel them to the separation.

The Axioms: Self-Evident Assumptions

Before Euclid proves anything, he asks you to grant his definitions and postulates. Jefferson does the same thing, and does it in one sentence, before he ever mentions the King.

We hold these truths to be self-evident,

1. That all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.

2. That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed,

3. That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it, and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness.

Strip away the king, the grievances, even Euclid, and a single conditional is doing all the work: if a government becomes destructive of these rights, the people may abolish it. Everything from here to the end of the document exists to establish one thing, and one thing only: that the antecedent is true in this case. Prove that, and the conclusion follows whether anyone likes it or not.

Here the analogy briefly strains. Euclid's postulates are genuinely generic: "draw a line between any two points" has nothing to do with triangles yet. Jefferson's third axiom already names the exact principle his whole document will run on, that a destructive government may be abolished. His Enunciation, one sentence later, restates that same principle almost word for word. Where Euclid moves from the fully general to the fully specific in two clean steps, Jefferson circles the same claim twice before he ever names a king. Perhaps this is not a flaw so much as emphasis.

He wanted this axiom, above the other two, to be unmistakable before the grievances began.

The Enunciation Declares the General Principle

Prudence, indeed, will dictate that Governments long established should not be changed for light and transient causes; and accordingly all experience hath shewn, that mankind are more disposed to suffer, while evils are sufferable, than to right themselves by abolishing the forms to which they are accustomed. But when a long train of abuses and usurpations, pursuing invariably the same Object evinces a design to reduce them under absolute Despotism, it is their right, it is their duty, to throw off such Government, and to provide new Guards for their future security.

Only now does a name enter the argument.

The Exposition and Specification Name the Parties and State the Charge

Euclid often folds these two moves into a single sentence: Let AB be the given line... I say that an equilateral triangle can be constructed upon it. Jefferson does something similar.

Such has been the patient sufferance of these Colonies; and such is now the necessity which constrains them to alter their former Systems of Government. The history of the present King of Great Britain is a history of repeated injuries and usurpations, all having in direct object the establishment of an absolute Tyranny over these States.

The general principle now has its two parties: a King, and a people who have suffered patiently long enough. What remains is to show it.

The Construction Builds the Argument, Indicting King George III

Later editors added an extra part, the Justifications, which are marginal notes. Jefferson doesn’t include them. But you can supply the citation yourself for any grievance on the list, just trace it back to one of the three self-evident truths above.

To prove this, let Facts be submitted to a candid world.

1. He has refused his Assent to Laws, the most wholesome and necessary for the public good.

2. He has forbidden his Governors to pass Laws of immediate and pressing importance, unless suspended in their operation till his Assent should be obtained; and when so suspended, he has utterly neglected to attend to them.

3. He has refused to pass other Laws for the accommodation of large districts of people, unless those people would relinquish the right of Representation in the Legislature, a right inestimable to them and formidable to tyrants only.

4. He has called together legislative bodies at places unusual, uncomfortable, and distant from the depository of their public Records, for the sole purpose of fatiguing them into compliance with his measures.

5. He has dissolved Representative Houses repeatedly, for opposing with manly firmness his invasions on the rights of the people.

6. He has refused for a long time, after such dissolutions, to cause others to be elected; whereby the Legislative powers, incapable of Annihilation, have returned to the People at large for their exercise; the State remaining in the mean time exposed to all the dangers of invasion from without, and convulsions within.

7. He has endeavoured to prevent the population of these States; for that purpose obstructing the Laws for Naturalization of Foreigners; refusing to pass others to encourage their migrations hither, and raising the conditions of new Appropriations of Lands.

8. He has obstructed the Administration of Justice, by refusing his Assent to Laws for establishing Judiciary powers.

9. He has made Judges dependent on his Will alone, for the tenure of their offices, and the amount and payment of their salaries.

10. He has erected a multitude of New Offices, and sent hither swarms of Officers to harrass our people, and eat out their substance.

11. He has kept among us, in times of peace, Standing Armies without the Consent of our legislatures.

12. He has affected to render the Military independent of and superior to the Civil power.

13. He has combined with others to subject us to a jurisdiction foreign to our constitution, and unacknowledged by our laws; giving his Assent to their Acts of pretended Legislation:

14. For Quartering large bodies of armed troops among us:

15. For protecting them, by a mock Trial, from punishment for any Murders which they should commit on the Inhabitants of these States:

16. For cutting off our Trade with all parts of the world:

17. For imposing Taxes on us without our Consent:

18. For depriving us in many cases, of the benefits of Trial by Jury:

19. For transporting us beyond Seas to be tried for pretended offences:

20. For abolishing the free System of English Laws in a neighbouring Province, establishing therein an Arbitrary government, and enlarging its Boundaries so as to render it at once an example and fit instrument for introducing the same absolute rule into these Colonies:

21. For taking away our Charters, abolishing our most valuable Laws, and altering fundamentally the Forms of our Governments:

22. For suspending our own Legislatures, and declaring themselves invested with power to legislate for us in all cases whatsoever.

23. He has abdicated Government here, by declaring us out of his Protection and waging War against us.

24. He has plundered our seas, ravaged our Coasts, burnt our towns, and destroyed the lives of our people.

25. He is at this time transporting large Armies of foreign Mercenaries to compleat the works of death, desolation and tyranny, already begun with circumstances of Cruelty & perfidy scarcely paralleled in the most barbarous ages, and totally unworthy the Head of a civilized nation.

26. He has constrained our fellow Citizens taken Captive on the high Seas to bear Arms against their Country, to become the executioners of their friends and Brethren, or to fall themselves by their Hands.

27. He has excited domestic insurrections amongst us, and has endeavoured to bring on the inhabitants of our frontiers, the merciless Indian Savages, whose known rule of warfare, is an undistinguished destruction of all ages, sexes and conditions.

Twenty-seven grievances, and not one of them is the argument itself. They are evidence, the raw material a proof still has to reason over.

The Proof Demonstrates Legitimacy of Independence

In every stage of these Oppressions We have Petitioned for Redress in the most humble terms: Our repeated Petitions have been answered only by repeated injury.

A Prince, whose character is thus marked by every act which may define a Tyrant, is unfit to be the ruler of a free people.

Nor have We been wanting in attentions to our British brethren. We have warned them from time to time of attempts by their legislature to extend an unwarrantable jurisdiction over us. We have reminded them of the circumstances of our emigration and settlement here. We have appealed to their native justice and magnanimity, and we have conjured them by the ties of our common kindred to disavow these usurpations, which, would inevitably interrupt our connections and correspondence. They too have been deaf to the voice of justice and of consanguinity. We must, therefore, acquiesce in the necessity, which denounces our Separation, and hold them, as we hold the rest of mankind, Enemies in War, in Peace Friends.

The reasoning is done. What’s left is to say, formally, what has now been shown.

The Conclusion Restates the General Principle in the Specific Case as Emphatically Proven: The Colonies Demand Independence from Great Britain

We, therefore, the Representatives of the united States of America, in General Congress, Assembled, appealing to the Supreme Judge of the world for the rectitude of our intentions, do, in the Name, and by Authority of the good People of these Colonies, solemnly publish and declare, That these United Colonies are, and of Right ought to be Free and Independent States; that they are Absolved from all Allegiance to the British Crown, and that all political connection between them and the State of Great Britain, is and ought to be totally dissolved; and that as Free and Independent States, they have full Power to levy War, conclude Peace, contract Alliances, establish Commerce, and to do all other Acts and Things which Independent States may of right do.

A Euclidean geometric proposition ends here with a Q.E.D. or Q.E.F., at the moment of proof, because the thing is now certain.

Jefferson’s cannot.

Jefferson ends the Declaration with a pledge, because the thing is not certain yet. Fifty-six men signed their names, risking everything on a future they could only promise, not prove.

And for the support of this Declaration, with a firm reliance on the protection of divine Providence, we mutually pledge to each other our Lives, our Fortunes and our sacred Honor.

Image of the Declaration of Independence

Euclid’s proposition ends this way because a mathematical proposition, once proved, is certain forever. Jefferson’s document was never a theorem, and never a construction either. It was something Euclid never had to write: a pledge on a future that hadn’t happened yet.

We are that future.

Two hundred and fifty years of a nation still standing, still free, still governed by consent, is not a geometric proof. But it is a kind of confirmation Euclid would have recognized.

The intended thing has been done.

Quod erat faciendum.

Today, we get to say it.