The History of Mathematics: A Brief Course

468 16. THE CALCULUS

published a work on geometry in 1668 in which he stated the equivalent of the

formula given earlier (unbeknown to Gregory, of course) by Jyeshtadeva:

J*3 ^,5 ^.7

arctan ÷ — ÷ 1 \- • • •.

3 5 7

Similarly, infinite product expansions were known by this time for the number ð.

One, due to Wallis, is

2 _ 1 ·3-3·5·5·7··

ð ~ 2·2·4·4·6·6··· '

The binomial series. It was the binomial series that really established the use of
infinite series in analysis. The expansion of a power of a binomial leads to finite
series when the exponent is a nonnegative integer and to an infinite series otherwise.
This series, which we now write in the form

(1+Ir=,+ir<r-\>-t+v

was discovered by Isaac Newton (1642-1727) around 1665, although, of course, he
expressed it in a different language, as a recursive procedure for finding the terms.
In a 1676 letter to Henry Oldenburg (1615-1677), the Secretary of the Royal Society,
Newton wrote this expansion as

P + PQ\— = P\—-r —AQ + ——BQ + — CQ + —- DQ + etc.

(^1) η (^1) ç ç 2n 3n An
"where Ñ + PQ stands for a quantity whose root or power or whose root of a power
is to be found, Ñ being the first term of that quantity, Q being the remaining terms
divided by the first term and m/n the numerical index of the powers of Ñ + PQ...
A stands for the first term P\^, Â for the second term ^AQ, and so on... ."
Newton's explanation of the meaning of the terms A, B, C,..., means that the
kth term is obtained from its predecessor via multiplication by { [(m/n) — fc]/(fc +
1)}Q. He said that m/n could be any fraction, positive or negative.

2. Newton and Leibniz

The results we have just examined show that parts of the calculus were recognized
by the mid-seventeenth century, like the pieces of a jigsaw puzzle lying loose on
a table. What was needed was someone to see the pattern and fit all the pieces
together. The unifying principle was the concept of a derivative, and that concept
came to Newton and Leibniz independently and in slightly differing forms.

2.1. Isaac Newton. Isaac Newton discovered the binomial theorem, the general
use of infinite series, and what he called the method of fluxions during the mid-

16608. His early notes on the subject were not published until after his death, but
       a revised version of the method was expounded in his Principia.