A History of Mathematics From Mesopotamia to Modernity

58 A History ofMathematics

H

D F

E

C

B

A

K

Fig. 1Menaechmus’s construction (from Fauvel and Gray p. 86). A and E are given, and it is required to construct B and C mean

proportionals. The parabola is given by the rule (square on DK=rectangle DF by A); the hyperbola by the rule (rectangle KD by DF=

rectangle A by E). DF equals C, and DK equals B.

A good example of the interaction of theory and practice is provided, surprisingly, by the classical

problem of the duplication of the cube. The problem seems to have surfaced some time before

400 bcein the form:

Given a cube C, to construct a cube D whose volume shall be twice the volume of C.

Using an argument similar to the one I have given in Chapter 2 (for Plato’s ‘Meno’), it is easy to

see that, if the side of C isafeet, that of D must bea^3

√

2 feet; and to generalize from doubling to
increasing in any proportion. This was done quite early. The earliest solution, by Menaechmus is
said to have involved the invention of the curves which we call conic sections. In modern notation,
we would take a parabola whose equation isy=x^2 and a hyperbola whose equation isxy= 2 a^3.
They meet atx=a^3

√

2,y=x^2 (see Fig. 1).

So far so good—see Exercise 1 for a check that this solves the problem. We have not said how

Menaechmus defined the curves, and the sources do not either. In later times, they were defined in

the first place as sections of a cone (e.g. the shadows which a lampshade casts on the walls or the

floor); the information which is encoded in the equations I have given had to be proved, and was

given its definitive form in Apollonius’s difficult (late third centurybce)Conics, one of the major

works in the ‘classical’ Greek tradition. However, there are doubts about whether such arguments

were available to Menaechmus. The record which we have (which itself is late, see Fauvel and

Gray 2.F.4) is related to the problem of two mean proportionals as in Exercise 2, and looks more

like the coordinate definition which we would use. Knorr (1986, p. 62) gives a ‘reconstruction’ of

how Menaechmus might have thought of it, with criticisms of earlier reconstructions by Heath

and others.

We have a substantial amount of information on the cube-duplication problem, even if some of

it is hard to interpret. One particularly interesting source is a supposed document by Eratosthenes

(third centurybce), called the ‘Platonicus’, which only survives in quotation.^1 Eratosthenes gives

1. In Eutocius’ (sixth centuryce) commentary on Archimedes’Sphere and Cylinder.