Euler's Eyes
2026-10-14
The Elements: definitions, postulates and common notions
A point is that which has no part. A line is breadthless length.
— Euclid, Elements, Book I, Definitions 1 and 2
Imagine you want to convince someone, beyond any possible doubt, that the angles of a triangle always add up to 180°. Measuring is not enough: the protractor makes errors, and you can only measure a few triangles. You need an argument. But every argument starts from something. Where should you start? If every claim must be justified by another, and that one by another, the chain never ends.
Euclid’s solution was to declare, on the very first page, exactly where he starts from. Everything else would have to be proved.
The three pieces
Book I of the Elements begins with three lists.
- Definitions (23 in Book I): they say what is being talked about. “A point is that which has no part.” “A circle is a plane figure bounded by a line such that all the straight lines falling on it from one point inside it are equal.”
- Postulates (5): things the reader is asked to accept about what can be done in geometry.
- Common notions (5): general truths, valid for any magnitude and not only in geometry.
The five postulates are, in a free translation:
- A straight line can be drawn from any point to any point.
- A finite straight line can be extended continuously in a straight line.
- A circle can be described with any centre and any radius.
- All right angles are equal to one another.
- If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side.
The first three describe what can be done with an unmarked straightedge and a compass: drawing lines, extending them, drawing circles. Every construction in the Elements uses only these two instruments. The fourth guarantees that the right angle is a standard measure. The fifth, longer and less obvious than the others, is the famous parallel postulate.
The five common notions are:
- Things equal to the same thing are equal to one another.
- If equals are added to equals, the wholes are equal.
- If equals are subtracted from equals, the remainders are equal.
- Things that coincide with one another are equal to one another.
- The whole is greater than the part.
The first is what we now write as: if $a=c$ and $b=c$, then $a=b$. It appears already in the first proposition.
The map of the thirteen books
| Books | Subject |
|---|---|
| I | Triangles, parallels and Pythagoras’s theorem |
| II | Geometric algebra: areas of rectangles and squares |
| III and IV | Circles and inscribed polygons |
| V | Eudoxus’s theory of proportions |
| VI | Similar figures |
| VII to IX | Number theory: greatest common divisor, primes, perfect numbers |
| X | Incommensurable magnitudes, such as $\sqrt2$ |
| XI to XIII | Solid geometry and the five regular polyhedra |
The book ends by proving that there are exactly five regular polyhedra: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. Plato associated them with the elements of nature, and some think the whole book was written to reach them.
The circle and the method of exhaustion
Book XII answers a very old question: what is the area of a circle? The Egyptians had a recipe. In the Rhind papyrus, from around 1550 BC, the area of a round field of diameter 9 is that of a square of side 8, which amounts to using $\pi\approx\frac{256}{81}\approx3.16$. In the Bible, the great bronze basin of Solomon’s temple was “ten cubits from the one brim to the other” and “a line of thirty cubits did compass it round about” (1 Kings 7:23): there, $\pi=3$.
Euclid gives no number at all. In Proposition XII.2 he proves that the areas of two circles are to one another as the squares on their diameters. In today’s language, the area is $kd^2$, with the same constant $k$ for every circle. The proof, which goes back to Eudoxus, inscribes in the circle a square, then an octagon, then a 16-sided polygon, and so on. The square already fills more than half of the circle, and each time the number of sides is doubled the polygon covers more than half of what was missing. What remains becomes smaller than any given quantity. This is the method of exhaustion: exhausting the figure with polygons, without ever speaking of infinity.
Shortly afterwards, Archimedes used the same method, with 96-sided polygons inside and outside the circle, to show that $3\frac{10}{71}<\pi<3\frac17$, that is, $3.1408<\pi<3.1429$. The letter $\pi$ only appeared in 1706 and spread through Euler.
A chain of dependencies
What makes the Elements special is the order. Proposition I.1 uses only definitions, postulates and common notions. I.2 also uses I.1. I.47, Pythagoras’s theorem, uses I.4, I.14, I.41 and I.46, which in turn use others. From the last theorem of Book XIII one can follow a chain of propositions, each resting on another, all the way back to the five postulates.
Exercise: which common notion justifies the conclusion: if AB = CD and CD = EF, then AB = EF?
The first: things equal to the same thing are equal to one another. Here $AB$ and $EF$ are both equal to $CD$. Today we call this property of equality transitivity.
Exercise: with straightedge and compass, how do you find the midpoint of a segment AB?
Draw two circles of radius $AB$, one centred at $A$ and the other at $B$. They cross at two points, $C$ and $D$, one on each side of $AB$. The line $CD$ cuts $AB$ at the midpoint $M$. Why? The four sides of $ACBD$ are equal to $AB$, so it is a rhombus, and the diagonals of a rhombus bisect each other. This is Proposition I.10, done with postulates 1 and 3.
Sources: Euclid’s Elements, Book I, D. E. Joyce’s edition · Thomas L. Heath, The Thirteen Books of Euclid’s Elements (Dover, 1956) · For the terms and the first proofs, see also this site’s section on the foundations of the Elements.
All topics on Euclid · ← Euclid of Alexandria: the author without a face · The first proposition: the equilateral triangle →