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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

Theory of Numbers

printed in the Philosophical Transactions for 1861 and 1867. They treat of linear indeterminate equations and congruences, and of the orders and genera of ternary quadratic forms. He established the principles on which the extension to the general case of n indeterminates of quadratic forms depends. He contributed also two memoirs to the Proceedings of the Royal Society of 1864 and 1868, in the second of which he remarks that the theorems of Jacobi, Eisenstein, and Liouville,

relating to the representation of numbers by 4, 6, 8 squares, and other simple quadratic forms are deducible by a uniform method from the principles indicated in his paper. Theorems relating to the case of 5 squares were given by Eisenstein, but Smith completed the enunciation of them, and

added the corresponding theorems for 7 squares. The solution of the cases of 2, 4, 6 squares may be obtained by elliptic functions, but when the number of squares is odd, it involves processes peculiar to the theory of numbers. This class of theorems is limited to 8 squares, and Smith completed the group. In ignorance of Smith's investigations, the French Academy offered a prize for the demonstration and completion of Eisenstein's theorems for 5 squares. This Smith had accomplished fifteen years earlier. He sent in a dissertation in 1882, and next year, a month after his death, the prize was awarded to him, another prize being also awarded to H. Minkowsky

of Bonn. The theory of numbers led Smith to the study of elliptic functions. He wrote also on modern geometry.

His successor at Oxford was J. J. Sylvester.

Ernst Eduard Kummer (1810–1893), professor in the University

of Berlin, is closely identified with the theory of numbers. Dirichlet's work on complex numbers of the form a+ib, introduced by Gauss, was extended by him, by Eisenstein, and Dedekind. Instead of the equation x4−1=0, the roots of which yield Gauss' units, Eisenstein used the equation

x3−1=0 and complex numbers a+bρ (ρ being a cube root of unity), the theory of which resembles that of Gauss' numbers. Kummer passed to the general case xn−1=0 and got

complex numbers of the form α=a1A1+a2A2+a3A3+⋯, where ai are whole real numbers, and Ai roots of the above equation.59 Euclid's theory of the greatest common divisor is not applicable to such complex numbers, and their prime factors cannot be defined in the same way as prime factors of common integers are defined. In the effort to overcome this difficulty, Kummer was led to introduce the conception of "ideal numbers." These ideal numbers have been applied by

G. Zolotareff of St. Petersburg to the solution of a problem

of the integral calculus, left unfinished by Abel (Liouville's

Journal, Second Series, 1864, Vol. IX.). Julius Wilhelm Richard Dedekind of Braunschweig (born 1831) has given in the second

edition of Dirichlet's Vorlesungen über Zahlentheorie a new

theory of complex numbers, in which he to some extent deviates from the course of Kummer, and avoids the use of ideal numbers. Dedekind has taken the roots of any irreducible equation with integral coefficients as

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