A History of Mathematics From Mesopotamia to Modernity

Greeks,Practical andTheoretical 59

S

k:2 a:2

TK L

AEMNZ H

DP

G

O

R

B

S

k:2 a:2

TKL

AEMNZ H

DP

O G

R

B

Fig. 2The ‘mesolabe’ (Shown for the duplication problem). The triangular plate AET is fixed, while MZK and NHL move. To find two

mean proportionals between AS and GL slide the two plates so that the meeting points R and O are in a straight line with A and G.

(Try this at home.)

a folklore account of the origin of the problem (doubling the size of the altar at Delos). Though not

reliable history, this at least has a practical appearance. He dismisses previous solutions, without

describing them in detail, as ‘impractical’ or ‘unwieldy’; and presents his own solution, by means

of a machine called a ‘mesolabe’, or ‘mean-taker’—which can construct any number of mean

proportionals between A and B, if properly calibrated (Fig. 2).

This ‘mechanical’ solution in itself goes against the standard image of Greek geometry as purely

abstract; and this is still further contradicted by Eratosthenes’s claims that the method can be used

in all sorts of ways:

We shall be able, furthermore, to convert our liquid and dry measures, the metretes and the medimnus, into a cube,

and from the size of this cube to measure the capacity of other vessels in terms of these measures. My method will also

be useful for those who wish to increase the size of catapults and ballistas. For, if the throw is to be increased, all the

elements of these engines, the thicknesses, lengths, and the sizes of the openings, wheel casings and cables must be

increased in proportion. (Eratosthenes quoted by Eutocius, in Fauvel and Gray 2.F.3.)

There seems to be some much stronger link between theory and practice here, though Knorr warns:

Eratosthenes was chiefly a man of letters, and one suspects that his vision of the practicality of a sensitive special-

purpose instrument like the ‘mesolabe’ was rather overstated. Still, the ideology behind its invention seems genuine.

(Knorr 1986, p. 212)

Knorr may underestimate the extent to which at least some Greek geometers thought of the problem

as applied—the same motive for the construction appears in the undoubtedly military work of

Eratosthenes’s near contemporary Philo of Byzantium. Here then, without straying outside the

boundaries of ‘classical’ geometry, we find a mechanical solution to a problem which is at least

being promoted for its practical uses. Clearly—and this is the point which we shall investigate in

this chapter—the nature of Greek mathematics is more complex than one might have thought. We

shall look at some examples of the later tradition which do not entirely fit into the Euclidean mould,