2 Ordinary Differential Equations
which is equivalent to finding tangents to the curves; he also introduces into
mathematics t he custom of using letters which appear first in the alp habet for
constants and letters which appear last for variables; and he formulates our
present system of exponents in which x^2 denotes x · x , in which x^3 denotes
x · x · x, etc. Pierre de Fermat's claim to priority in the invention of a nalytic
geometry is b ased on a letter he wrote to Gilles Roberval in Septemb er 1636,
in which he claims that the ideas he is advancing are seven years old. Fermat's
method for finding a tangent to a curve was devised in conjunction with his
procedure for determining maxima and minima. Thus, Fermat was the first
mathematician to develop the central idea of differential calculus- t h e notion
of a derivative. Fermat also had great success in the t heory of integration. By
1636 or earlier, Fermat had discovered and proved by geometrical means the
power formula for positive integer exponents- that is, Fermat had proved for
positive integers n
xndx= --.
l
a an+l
0 n+l
Later, Fermat generali zed this result to rational exponents n-=/=--1. In many
respects, t he work of Descartes and Fermat were antipodal. Generally speak-
ing, Descartes started with a locus and then derived its equation, whereas
Fermat began with an equation and t hen found the locus. Their combined
efforts illustrate the two fundamental and inverse properties of analytic ge-
ometry.
Until approximately the middle of the seventeenth century, integral and
differential calculus appeared to be two distinct branches of mathematics.
Integration, in the case of calculating the area under a curve, consisted of
finding the limit approached by the sum of a very large number of extremely
thin rectangles as the number of rectangles increased indefinitely and the
widt h of each rectangle approached zero. Different iation, on the other hand,
consisted of finding the limit of a difference quot ient. About 1646, Evangelista
Torricelli (1608-1647), a student of Galileo and inventor of the barometer
(1643), showed that integration and differentiation were inverse operations
for equations of the form y = xn where n is a positive integer. That is,
Torricelli showed for n a posit ive integer
- d 1x tndt--d ( --xn+l)
dx o dx n+l
Isaac Barrow (1630-1677) is usually given credit for being t he first mathemati-
cia n to recognize in its fullest generality that differentiation and integration
are inverse operations. In his Lectiones, Barrow essentially stated and proved
geometrically the fundamental theorem of calculus- that is, if f(x) is a
continuous function on the interval [a, b] and if x is in the interval [ a , b], then
dx d 1x a f(t) dt = f(x).