Chapter 16. The Calculus
The infinite occurs in three forms in calculus: the derivative, the integral, and the
power series. Integration, in the form of finding areas and volumes, was developed as
a particular theory before the other two subjects came into general use. As we have
seen, infinitesimal methods were used in geometry by the Chinese and Japanese,
and the latter also used infinite series to solve geometric problems. In India also,
mathematicians used infinite series to solve geometric problems via trigonometry.
According to Rajagopal (1993), the mathematician Nilakanta, who lived in South
India and whose dates are given as 1444-1543, gave a general proof of the formula
for the sum of a geometric series. The most advanced of these results is attributed to
Madhava (1340-1425), but is definitively stated in the work of Jyeshtadeva (1530-
ca. 1608):
The product of the given Sine and the radius divided by the Co-
sine is the first result. From the first,...etc., results obtain...a
sequence of results by taking repeatedly the square of the Sine as
the multiplier and the square of the Cosine as the divisor. Divide
... in order by the odd numbers one, three, etc... From the sum of
the odd terms, subtract the sum of the even terms. [The result]
becomes the arc. [Rajagopal, 1993, p. 98]These instructions give in words an algorithm that we would write as the follow-
ing formula, remembering that the Sine and Cosine used in earlier times correspond
to our r sin è and r cos È, where r is the radius of the circle:Ë r(^2) sinf? r (^4) sin (^3) 0 r (^6) sin (^50)
rf? = 1
r cos è 3r^3 cos^3 è 5r^5 cos^5 è
The bulk of calculus was developed in Europe during the seventeenth century,
and it is on that development that the rest of this chapter is focused.
Since analytic geometry was discussed in Section 1 of Chapter 12, we take
up the story at the point where infinitesimal methods begin to be used in finding
tangents and areas. The crucial step is the realization of the mutually inverse
nature of these two processes and their consolidation as a set of algebraic and limit
operations that can be applied to any function. At the center of the entire process
lies the very concept of a function, which was a seventeenth-century innovation.- Prelude to the calculus
In his comprehensive history of the calculus (1949), Boyer described "a century of
anticipation" during which the application of algebra to geometric problems began
to incorporate some of the less systematic parts of ancient geometry, especially the
infinitesimal ideas contained in what was called the method of indivisibles. Let us
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