Reform College · original work VI

Euclid as the logic of consent

A companion to a mathematics course that still teaches the Elements — not a copy of anyone’s lectures. For the student who will copy propositions in the Lincoln method, then ask what a people has actually granted.

Euclid does not begin with a useful formula. He begins with what must be granted if anything is to follow: points, lines, common notions, postulates. A student who will not grant the common notions cannot be argued into a triangle. A people who will not grant a form cannot be argued into a Union as if they had. That is the whole of this essay. Hillsdale’s mathematics course exists to put Euclid back in the liberal arts. This page exists to say why that restoration is political, not merely pedagogical — without taking a sentence from their films.

I. Common notions

The common notions are not theorems. They are what a mind already has if it is a mind: things equal to the same are equal to each other; the whole is greater than the part. You do not vote them in. You recognize them, or you have left geometry for rhetoric.

The Declaration’s self-evident truths occupy that place in politics. They are not statutes. They are what must be granted if a compact is to be a compact and not a conquest. “Created equal” is not a policy preference of 1776. It is a common notion: no man is another’s by nature. A later bureau that redraws “the people” by age or by a paid debt is denying a common notion while keeping the stationery of a demonstration. Reese and Cockerham are, in that sense, Euclidean: they ask what was granted, not what would be convenient.

II. Postulates

A postulate is a grant you make so that construction may begin. Let it be granted that a straight line may be drawn between two points. If you will not grant it, Book I stops. You are not refuted. You have refused the game.

Ratification is the political postulate. Nine states granted the Philadelphia plan among themselves. North Carolina, at Hillsborough, did not. That is not a failure of geometry. That is a student who will not grant the fifth postulate until it is written as a bill of rights. Sixteen months outside the Union is the time a people may spend refusing a postulate. The amendments are the rewritten grant. A college that teaches the Constitution as if the postulate had always been universal has already erased the only interesting line in the Elements of 1788.

Lincoln’s method is the right one here: copy the postulates until you can recite them, then try to see which later “as if” is a new postulate smuggled in as a corollary. August 6, 1861 — acts “legalized… as if they had been issued under the previous express authority of Congress” — is not a corollary. It is a new postulate, written after the construction, to save a figure that was drawn without it. Euclid would tear the sheet.

III. Proposition and Q.E.D.

A proposition is a claim that follows. The proof may be a construction, or a reduction to absurdity. In the second, you grant the opponent’s figure and show that it destroys a common notion or a prior proposition. Brutus did that to necessary-and-proper. Taney in Merryman, whatever else is said of him, did it to a presidential suspension sitting in Article I: if the President is the judge of the clause, the writ is not a law but a favor. The full Court did not reach the live body. That is a demonstration interrupted, not a demonstration answered.

Bruen is the Court’s instruction to do Euclidean work: the text is the common notion (“the people,” “keep and bear”); the analog is the prior proposition (what the Founding actually did); interest-balancing is a different science, closer to a merchant’s bargain than to a proof. A student trained only in policy will call this backward. A student who has copied Book I will recognize it as the only method that does not let the desired figure dictate the grant.

IV. What the Lincoln class copies

In the accelerated track, Euclid I–IV is six weeks of hand work. No film until the proposition is proved on a blank page. That is not hostility to lecturers. It is the order of the arts. A lecture heard before the copy becomes a formula sheet. Lincoln worked the Elements as a grown man in Congress for the same reason a surveyor does: so the line he draws is the line that follows, not the line he wishes.

The political use of that habit is this hall’s: when a magistrate, a bureau, or a court offers a construction, ask — is this a common notion, a postulate we granted, a proposition that follows, or a figure drawn first and justified after? If the last, withhold. North Carolina already showed how. The Bill of Rights is what a withheld compact looks like when it finally grants.

Work VI of the dossier. Read the Elements (public domain) with the hand, not with this page as a substitute. Hillsdale’s Mathematics and Logic may be enrolled after the copy: their course, their lectures. This essay is PXD2’s and does not reprint them. Companion to Work I and LYC 02.