266 V. MATHEMATICS A A LANGUAGE
schemes is quite elaborate and ingenious, and impossible to analyse here more fully; so that only one example can be given.
The solution of mathematical equations is perhaps to be considered as the central problem of mathematics. The word 'equation' is derived from Latin aequare, to equalize, and is a statement of the symmetrical relation of equality expressed as An
equation expresses the relation between quantities, some of which are known, some unknown and to be found. By the solution of an equation, we mean the finding of values for the unknowns which will satisfy the equation.
Linear equations of the typenecessitated the introduction of
fractions. Linear equations with several variables led to the theory of determinants and matrices., which underwent, later, a tremendous independent development; yet they originated in the attempt to simplify the solution of these equations.
involve the problems of quadratic and cubic equations. When we consider equations of a degree higher than the fourth, we find that we cannot solve them by former methods; and mathematicians have had to invent theories of substitutions, groups, different special functions and similar devices. The solution of differential equations introduced further difficulties, allied with the theory of function.
The linear transformations of algebraic polynomials with two or more variables in connection with the theory of determinants, symmetrical functions, differential operations., necessitated the development of an extensive theory of algebraic forms which, at present, is far from being complete.
In the above analysis, I have refrained from giving details, most of which would be of no value to the layman, and unnecessary for the mathematician; but it must be emphasized that the theory of function and the theory of groups, with their very extensive developments,