The History of Mathematics: A Brief Course

1. COMPLEX ANALYSIS 491

for rectification and an equation

fy 1 1 fx 1

/ , dt=- „ dt,

Jo %/T^t^2 Wo

which can be written in differential form as

dx ndy

for the division of an arc.
Trigonometry helps to solve this last equation. Instead of the arccosine function
that the integral actually represents, it makes more sense to look at the inverse of
it, the cosine function. This function provides an algebraic equation through its
addition formula,

aoy - 022/ +a 3 y = x,

relating the abscissas of the end of the given arc (x) and the end of the nth part of it
(y). The algebraic nature of this equation determines whether the division problem
can be solved with ruler and compass. In particular, for ç = 3 and a 60-degree
arc (x = 1/2), for which the equation is Ay^3 — 3y — 1/2, such a solution does not
exist. Thus the problems of computing arc length on a circle and equal division
of its arcs lead to an interesting combination of algebra, geometry, and calculus.
Moreover, the periodicity of the inverse function makes this equation easy to solve
(see Problem 17.1).
The division problem was fated to play an important role in study of integrals
of algebraic functions. The Italian nobleman Fagnano (1682-1766) studied the
problem of rectifying the lemniscate, whose polar equation is r^2 = 2cos(20). Its
element of arc is Ë/2(1 — 2sin^2 0)_1//2dt?, and the substitution u = tanf? turns
this expression into \/2(l - ui)~ll^2 du. Thus, the rectification problem involves
evaluating the integral

/

Jo

: du ,

VI -u^4

while the division problem involves solving the differential equation

dz ndu

Fagnano gave the solution for ç = 2 as the algebraic relation

Euler observed the analogy between these integrals and the circular integrals just
discussed, and suggested that it would be reasonable to study the inverse function.
But Euler lived at a time when the familiar functions were still the elementary ones.
He found a large number of integrals that could be expressed in terms of algebraic,
logarithmic, and trigonometric functions and showed that there were others that
could not be so expressed.