A History of Mathematics- From Mesopotamia to Modernity

182 A History ofMathematics

And it is now time to return to a version—if possible updated—of Boris Hessen’s thesis: that the

whole concern of thePrincipiawas with precisely those questions whose solutions were urgently

required by the British bourgeoisie.

At that time the most progressive class, it demanded the most progressive science...The necessity arose of not merely

empirically resolving isolated problems, but of synthetically surveying and laying a stable theoretical basis for the

solution by general methods of all the aggregate of physical problems set for immediate solution by the development

of the new technique.

And since (as we have already demonstrated) the basic complex of problems was that of mechanics this encyclopaedic

survey of the physical problems was equivalent to the creation of a harmonious structure of theoretical mechanics

which would supply general methods of resolving the tasks of the mechanics of earth and sky. (Hessen 1971,

pp. 170–1)

If for ‘Principia’ we read ‘calculus’ and for ‘British’ we read ‘European’, is there any way of sustaining

this thesis? The problem is worth the attention of the next generation of historians.

Exercise7.Using the equilibrium of a section of the chain,AB(where, say,Ais the lowest point), derive

the equation of the catenary given above.

10. Afterword

Boyer’s history of the calculus (1949) ends tidily with the nineteenth-century reformulation of

analysis—usually ascribed to Weierstrass in the 1860s—which banished infinitesimals and made

everything rigorous (in principle, after the model of Greek geometry) again. Whether this vindicated

Berkeley or the hopeful analysts who expected that their methods would sooner or later (in this

case much later) be justified, is unclear. However, it is worth mentioning that any impression that

mathematicians stopped using such ill-founded methods once they had been shown the correct ones

is very oversimplified, since mathematics is a large subject—by 1860 it was already out of hand—

and not all areas are going to come under the same régime. In particular, the rigorous justification

of the calculus is, on the whole, too hard to be taught outside universities, while its results are so

useful that they need to be available much earlier. Hence, the language of infinitesimals lingered

on. Here are two examples from personal experience.

1. At school, at the age of 17, we were taught to find areas of curves, in particular those given
   by ‘polar equations’ involving the coordinatesr,θas in Fig. 7. The heart-shaped curve shown is
   appropriately called a ‘cardioid’, and its equation is:

r=a( 1 +cosθ)

To find the area, we used the ‘element of area’; the infinitely small triangle of heightrand angledθ.
As a triangle, it has area^12 r^2 ·dθ—this may not be immediately obvious, but it follows by neglecting
second-order infinitesimals, and we were told that it was true. Sinceθruns from 0 to 2π, the area
inside the curve is
∫ 2 π

0

1

2

a^2 ( 1 +cosθ)^2 ·dθ