A History of Mathematics From Mesopotamia to Modernity

TheCalculus 181

–0.5 0.5

2

4

6

8

2 –1.5 –1 1 1.5

A

C

B

T

W

T 0

c

Fig. 6The catenaryy=acosh(x/a)=(a/ 2 )(ex/a+e−x/a).

[Jakob Bernoulli] wrote that one shouldat leastgive a construction by quadrature of an algebraic curve. It was

betterto give a construction by rectification of an algebraic curve, or a ‘pointwise construction’...Thebestway to

represent a curve, however, was a construction by curves ‘given in nature’ (as theElasticaor, e.g., theCatenary. (Bos

1991, p. 34).

We seem to have a vicious circle here; if the catenary is so natural that it is best to represent other

curves by means of it, then what point is there in trying to find another way of describing it? The

point is illustrated, in a slightly surreal way, in Leibniz’s paper which ‘solves’ the problem. It should

be noted that the solution (given in Gerhardt 1962, vol. V, pp. 258–63, in French for easy reading)

is a purely geometrical construction of points on the curve, with no proof, either by old-fashioned

geometry or by calculus; and that the same is true of the slightly different versions of the other

competitors.

Having shown that the catenary was related to the logarithm (it involves the functionex), Leibniz

proposed that ships at sea should have a chain suitably suspended among their instruments, so that,

if they lost their invaluable tables of logarithms, they could work them out from measurements on

the curve.

Question.Given the equation of the catenary, how would you use measurements on a chain to

work out logz?

Question.Is this an entirely stupid and impractical idea, or is it on the contrary ingenious and

practical?

The catenary was only one of a host of practical and pseudo-practical problems for which the

calculus proved to be uniquely well-adapted, partly because it dealt with rates of change. So we

find it applied to the flight of cannon-balls in a resisting medium; to the vibrations of stretched

strings; to the shape of sails, and so on. These mundane problems took their place with the grander

questions of the movement of the planets and the shape of the earth which Newton had discussed

in thePrincipia.