Elements

Overview

The Elements has no prose in it. There is no preface, no motivation, no worked example, no word about who Euclid was or why any of this matters. It opens with twenty-three definitions and then begins proving things, and it does not stop for five hundred pages. Nothing else on the shelf is written this way.

The unit is the proposition, and every one has the same six parts: the general statement, the setting out with lettered points, the specification of what is to be shown, the construction, the proof, and the conclusion, which restates the general claim and closes with the formula that later became QED. Once you have the rhythm you can read them quickly. What you cannot do is read them passively. A proposition is a set of instructions for building something in your own head, and if you are not drawing you are not reading.

Two things to know about authorship. Euclid almost certainly did not discover most of this. Book V is Eudoxus’ theory of proportion, Book X owes much to Theaetetus, and the Pythagorean material is older still. What he did was select, order and connect, which is an intellectual achievement of a different kind and possibly a rarer one. And the text we have has been edited, glossed and occasionally corrected for twenty-three centuries, so the received version includes definitions Euclid probably did not write.

The argument

Book I is the one every course reads. Twenty-three definitions, five postulates, five common notions, forty-eight propositions. The postulates are strikingly modest: you may draw a line between two points, extend a line, draw a circle with any centre and radius, and all right angles are equal. Then the fifth, which takes a whole paragraph to state and is not obvious at all. Euclid clearly knew it was different, and he avoids using it for twenty-eight propositions.

The first proposition constructs an equilateral triangle on a given line. Proposition 4 proves triangles congruent by two sides and the included angle, by a method of superposition that has troubled readers since antiquity. Proposition 5, that the base angles of an isosceles triangle are equal, acquired the name pons asinorum, the bridge of asses, because its figure looks like a trestle and because medieval students stalled there. From Proposition 29 the parallel postulate comes in, and everything about parallels, angle sums and parallelograms follows fast. The book ends at Propositions 47 and 48 with the Pythagorean theorem and its converse, which is a piece of composition: the whole of Book I has been built to arrive there.

Books II to IV continue in the plane. Book II does in geometry what we would now do in algebra, proving identities about rectangles and squares. Book III is circles, tangents and inscribed angles. Book IV inscribes and circumscribes regular figures, including the pentagon.

Books V and VI are the hard turn. Eudoxus’ theory of proportion in Book V is designed to handle magnitudes that have no common measure, and its fifth definition, about what it means for equal multiples to exceed alike, is a genuinely difficult piece of thinking that anticipates the nineteenth-century definition of a real number. Book VI applies it to similar figures.

Books VII to IX leave geometry for number. Here a number is a multitude of units, drawn as a line segment. Book VII gives what we call the Euclidean algorithm for a greatest common measure. Book IX contains the most famous single proof in the collection, Proposition 20, that there are more primes than any given collection of them, argued by constructing a new one from the product of the old.

Book X, the longest and least read, classifies incommensurable magnitudes into an elaborate taxonomy. Books XI to XIII move into three dimensions, with Book XII using the method of exhaustion to compare cones, cylinders and spheres, and Book XIII constructing the five regular solids and proving, in the last proposition of the work, that there can be no others.

How to read it

Get a pencil, paper and a straightedge, and do not read a proposition without drawing its figure yourself. This is not a suggestion. The lettered diagram is the argument, and following it on the page while someone else’s picture sits beside it teaches you almost nothing.

The traditional way, and the reason programs assign it, is to go to a board and reconstruct a proposition from memory in front of other people. If you are reading alone, do the equivalent: read it, close the book, and rebuild it.

Do not attempt all thirteen books. Book I entire is the standard assignment and is enough to change how you read. If you want more, add Book V definitions 4 and 5, Book VI on similarity, and Book IX Proposition 20. Book X is for specialists.

Watch the postulates. Every time a proof does something, ask which postulate lets it. And watch for the places where Euclid does something his postulates do not authorise: Proposition 1 assumes that two circles cross, which nothing in the list guarantees. Hilbert wrote a whole book in 1899 filling these gaps, and finding one yourself is more instructive than being told.

The common misreading is to treat this as a book of results. You know that the angles of a triangle sum to two right angles. The point is to see how little Euclid needs to assume before he is entitled to say so.

Questions it raises

• Euclid proves everything and assumes ten things. Why these ten? What would make a starting point self-evident rather than merely convenient?
• The fifth postulate is stated as if it were an observation. Is it one? Two thousand years of attempts to derive it from the others all failed. Why?
• Numbers in Books VII to IX are drawn as lines. What does the geometry give the arithmetic, and what does it cost?
• Proposition 4 proves congruence by picking up one triangle and laying it on another. Is motion a legitimate move in a proof about figures?
• If the five regular solids are exactly five and no more, is that a fact about space, about our imagination, or about the definitions?
• Spinoza wrote ethics and Newton wrote physics in this form. Does the geometrical style carry its certainty with it into other subjects, or only its appearance?