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NEWTON, ISAAC (b. Woolsthorpe, England,
25 December 1642; d. London, England, 20 March
1727), mathematics, dynamics, celestial mechanics,
astronomy, optics, natural philosophy.

Mathematics. Any summary of Newton's contributions
to mathematics must take account not only
of his fundamental work in the calculus and other
aspects of analysis--including infinite series (and most
notably the general binomial expansion)--but also his
activity in algebra and number theory, classical and
analytic geometry, finite differences, the classification
of curves, methods of computation and approximation,
and even probability.

reducing any dignity [power] of any Binomial into
such a series....”23 He further stated that:

In the winter between the years 1664 & 1665 upon
reading Dr Wallis's Arithmetica Infinitorum & trying
to interpole his progressions for squaring the circle
[that is, finding the area or evaluating 0?1 (1 -
x2)1/2 dx],
I found out another infinite series for squaring the
circle & then another infinite series for squaring the
Hyperbola....24

On 13 June 1676, Newton sent Oldenburg the
“Epistola prior” for transmission to Leibniz. In this
communication he wrote that fractions “are reduced
to infinite series by division; and radical quantities
by extraction of roots,” the latter

where P + PQ signifies the quantity whose root or
even any power, or the root of a power, is to be found;
P signifies the first term of that quantity, Q the remaining
terms divided by the first, and m/n the numerical
index of the power of P + PQ, whether that power is
integral or (so to speak) fractional, whether positive
or negative.25

A sample given by Newton is the expansion

?(c2 + x2) or (c2 +
x2)1/2 = c + x2/2c -
x4/8c3
+ x6/16c5 - 5x8/128c7 +
7x10/256c9 + etc.

where

P = c2, Q = x2/c2, m = 1, n = 2,
and
A = Pm/n = (c2)1/2 = c, B = (m/n) AQ =
x2/2c,

What is perhaps the most important general statement
made by Newton in this letter is that in dealing with
infinite series all operations are carried out “in the
symbols just as they are commonly carried out in
decimal numbers.”

Wallis had obtained the quadratures of certain
curves (that is, the areas under the curves), by a
technique of indivisibles yielding 0?1 (1 -
x2)n dx
for certain positive integral values of n (0, 1, 2, 3);
in attempting to find the quadrature of a circle
of unit radius, he had sought to evaluate the
integral 0?1 (1 -
x2)1/2 dx by interpolation. He showed
that

Newton read Wallis and was stimulated to go considerably
further, freeing the upper bound and then
deriving the infinite series expressing the area of a
quadrant of a circle of radius x:

In so freeing the upper bound, he was led to recognize
that the terms, identified by their powers of x, displayed
the binomial coefficients. Thus, the factors
1/2, 1/8, 1/16, 5/128, ... stand out plainly as (?),
(?), (?),
(?),..., in the special case q = 1/2 in the
generalization

In this way, according to D. T. Whiteside, Newton
could begin with the indefinite integral and, “by
differentiation in a Wallisian manner,” proceed to
a straightforward derivation of the “series-expansion
of the binomial (1 - xp)q ... virtually in its
modern
form,” with “| xp | implicitly less than unity
for convergence.”
As a check on the validity of this general
series expansion, he “compared its particular expansions
with the results of algebraic division and squareroot
extraction (q = 1/2).” This work, which was
done in the winter of 1664-1665, was later presented
in modified form at the beginning of Newton's De
analysi.

He correctly summarized the stages of development
of his method in the “Epistola posterior” of
24 October 1676, which--as before--he wrote for
Oldenburg to transmit to Leibniz: