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nydus/A History of MathematicsPublic

This text examines the transition from the Middle Ages to the Modern era, highlighting how the fall of Constantinople and the invention of the printing press catalyzed a revival of classical learning. It traces the shift toward scientific inquiry through the rise of pure mathematics and astronomy, detailing the intellectual struggle against established scholastic and ecclesiastical authority.

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Table of Contents

Descartes to Newton

When x=1, these values become respectively 1, 23, 815, 48105, etc. Now since the ordinate of the circle is y=(1−x2)12, the exponent of which is 12 or the mean value between 0 and 1, the question of this quadrature reduced itself to this: If 0, 1, 2, 3, etc., operated upon by a certain law, give 1, 23, 815, 48105, what will 12 give, when operated upon by the same law? He attempted to solve this by interpolation, a method first brought

into prominence by him, and arrived by a highly complicated

and difficult analysis at the following very remarkable expression: π2=2·2·4·4·6·6·8·8⋯1·3·3·5·5·7·7·9⋯

He did not succeed in making the interpolation itself, because he did not employ literal or general exponents, and could not conceive a series with more than one term and less than two, which it seemed to him the interpolated series must have. The consideration of this difficulty led Newton to the

discovery of the Binomial Theorem. This is the best place to

speak of that discovery. Newton virtually assumed that the same conditions which underlie the general expressions for the areas given above must also hold for the expression to be interpolated. In the first place, he observed that in each expression the first term is x, that x increases in odd powers, that the signs alternate + and −, and that the second terms 03x3, 13x3, 23x3, 33x3, are in arithmetical progression. Hence the first two terms of the interpolated series must be x−12x33. He next considered that the denominators 1, 3, 5, 7, etc., are in arithmetical progression, and that the coefficients in the numerators in each expression are the digits of some power of the number 11; namely, for the first expression, 110 or 1; for the second, 111 or 1, 1; for the third, 112 or 1, 2, 1; for the fourth, 113 or 1, 3, 3, 1; etc. He then discovered that, having given the second digit (call it m), the remaining digits can be found by continual multiplication of the terms of the series m−01·m−12·m−23·m−34· etc. Thus, if m=4, then 4·m−12 gives 6; 6·m−23 gives 4; 4·m−34 gives 1. Applying this rule to the required series, since the second term is 12x3\Fstr3, we have m=12, and then get for the succeeding coefficients

in the numerators respectively −18\Fstr, +116, −5128, etc.; hence the required area for the circular segment is x−12x3\Fstr3−18x55−116x77− etc. Thus he found the interpolated expression to be an infinite series, instead of one having more than one term and less than two, as Wallis believed it must be. This interpolation suggested to Newton a mode of expanding (1−x2)12, or, more generally, (1−x2)m, into a series. He observed that he had only to omit from the expression just found the denominators 1, 3, 5, 7, etc., and to lower each power of x by unity, and he had the desired expression. In a letter to Oldenburg (June 13, 1676), Newton states the theorem as follows: The extraction of roots is much shortened by the theorem

(P+PQ)mn=Pmn+mnAQ+m−n2nBQ+m−2n3nCQ+etc., where A means the first term, Pmn, B the second term, C the third term, etc. He verified it by actual multiplication, but gave no regular proof of it. He gave it for any exponent whatever, but made no distinction between the case when the exponent is positive and integral, and the others.

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