Math 409/409G History of Mathematics Books X – XIII of the Elements

After investigating number theory in Books VII – IX of the Elements, Euclid moved on to a discussion of incommensurable numbers in Book X. If you recall from a previous lesson, the Pythagoreans believed that all segments were commensurable in the sense that the ratio of the measures of any two segments is a rational number.

Then one day the Pythagorean Hippasus discovered that the diagonal of a square and its side are not commensurable! In particular, he discovered that the square root of 2 is not a rational number. Since this discovery totally ruined the Pythagoreans’ development of similar triangles, the Pythagoreans took Hippasus far out into the Mediterranean and threw him overboard!

This took place in the 5 th century B.C.E. Euclid wrote the Elements in the 3 rd century B.C.E. Between these centuries, many mathematicians developed the theory of incommensurable numbers and this is what Euclid presented in Book X of the Elements. Today, the theory of incommensurable numbers is called the theory of the square roots of numbers.

The last three Books, XI – XIII, mainly covered three-dimensional geometry. For example, Proposition 10 in Book XII, showed that the volume of a right circular cone is 1/3 the volume of the cylinder having the same circular base and height. That is, V cone  (πr 2 h)/3.

In Proposition 2 of Book XII, Euclid stated that: Circles are to each other as the squares of their diameters. As a consequence of this we have that for any circle, the ratio of its area to the square of its diameter is constant. But Euclid failed to give a numerical estimate of this rather important and useful constant. Today we know this constant as π/4.

In a similar vein, the last proposition of Book XII established that the ratio of the volume of a sphere to the cube of its diameter is a constant. But again, Euclid made no attempt to approximate this constant. The value of these constants were finally given in 225 B.C.E. by Archimedes.

The last Book of the Elements, Book XIII, contained 18 propositions about regular solids. By Euclid’s day, 5 such solids were known: 1. Tetrahedron (a pyramid with equilateral triangles as each of its 4 faces.) 2. Cube

3. Octahedron (with equilateral triangles as each of its 8 faces.) 4.Dodecahedron (with regular pentagons as each of its 12 faces.) 5.Icosahedron (a 20-faced solid with equilateral triangles as faces.)

As the 465 th and last proposition of the Elements, Euclid proved that these five solids are the only possible regular solids. How did he do this? Well he had already established two important facts:  The sum of the plane angles of the faces of the solid meeting at a vertex must be less than 360 o. (Proposition 21 of Book XI.)  A solid angle must be formed by the intersection of 3 or more faces meeting at a vertex. (Intuitive understanding.)

Putting these two facts together we see that we must have the minimal condition that: 3  (plane  of a face) < 360 o. A regular solid is made up of regular polygons. The plane angles of the regular polygons are: Regular Polygon 3  Plane Angle Equilateral Triangle 3  60 o  180 o Square 3  90 o  270 o Pentagon 3  108 o  324 o Hexagon or larger  3  120 o  360 o

This shows us that the only possible faces for a regular solid are the equilateral triangle, the square, and the pentagon. Using Proposition 21 from Book XI, The sum of the plane angles of the facesof the solid meeting at a vertex must be less than 360 o. we get the following results.

The sum of the plane angles of the faces of the solid meeting at a vertex must be less than 360 o. If the faces of the regular solid are equilateral triangles, then the solid must have less than six faces meeting at a vertex since 6  60 o is not less than 360 o. So the solid has 3 faces meeting at a vertex (the tetrahedron), 4 faces meeting at a vertex (the octahedron), or 5 faces meeting at a vertex (the icosahedron).

The regular solids having equilateral triangles as faces: Tetrahedron 4 faces, 3 meeting at a vertex Octahedron 8 faces, 4 meeting at a vertex Icosahedron 20 faces, 5 meeting at a vertex

The sum of the plane angles of the faces of the solid meeting at a vertex must be less than 360 o. If the faces of the regular solid are squares, then the solid must have exactly 3 faces intersecting a vertex since four or more faces meeting at a vertex would result in the sum of the plane angles being greater than or equal to 4  90 o  360 o. So the cube is the only regular solid having square faces.

The sum of the plane angles of the faces of the solid meeting at a vertex must be less than 360 o. If the faces of the regular solid are pentagons, then the solid must have exactly 3 faces intersecting a vertex since four or more faces meeting at a vertex would result in the sum of the plane angles being greater than or equal to 4  108 o  432 o. So the dodecahedron is the only regular solid having pentagonal faces.

So we have established, as Euclid did, that there are only five regular solids. Since these regular solids were featured in the writings of Plato before Euclid wrote the Elements, they are today referred to as the “Platonic solids.”

This ends the lesson on Books X – XIII of the Elements