Beginning of Modern Math

In considering the history of modern mathematics two questions at

once arise: (1) what limitations shall be placed upon the term

Mathematics; (2) what force shall be assigned to the word Modern? In

other words, how shall Modern Mathematics be defined?

In these pages the term Mathematics will be limited to the domain of

pure science. Questions of the applications of the various branches

will be considered only incidentally. Such great contributions as

those of Newton in the realm of mathematical physics, of Laplace in

celestial mechanics, of Lagrange and Cauchy in the wave theory, and

of Poisson, Fourier, and Bessel in the theory of heat, belong rather

to the field of applications.

In particular, in the domain of numbers reference will be made to

certain of the contributions to the general theory, to the men who

have placed the study of irrational and transcendent numbers upon a

scientific foundation, and to those who have developed the modern

theory of complex numbers and its elaboration in the field of

quaternions and Ausdehnungslehre. In the theory of equations the

names of some of the leading investigators will be mentioned,

together with a brief statement of the results which they

secured. The impossibility of solving the quintic will lead to a

consideration of the names of the founders of the group theory and

of the doctrine of determinants. This phase of higher algebra will

be followed by the theory of forms, or quantics. The later

development of the calculus, leading to differential equations and

the theory of functions, will complete the algebraic side, save for

a brief reference to the theory of probabilities. In the domain of

geometry some of the contributors to the later development of the

analytic and synthetic fields will be mentioned, together with the

most noteworthy results of their labors. Had the author’s space not

been so strictly limited he would have given lists of those who have

worked in other important lines, but the topics considered have been

thought to have the best right to prominent place under any

reasonable definition of Mathematics.

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