the units for his complex numbers. Attracted by Kummer's investigations, his pupil, Leopold Kronecker (1823–1891) made researches which he applied to algebraic equations.
On the other hand, efforts have been made to utilise in the theory of numbers the results of the modern higher algebra. Following up researches of Hermite, Paul Bachmann of Münster
investigated the arithmetical formula which gives the automorphics of a ternary quadratic form.89 The problem of the equivalence of two positive or definite ternary quadratic forms was solved by L. Seeber; and that of the arithmetical automorphics
of such forms, by Eisenstein. The more difficult problem
of the equivalence for indefinite ternary forms has been investigated by Edward Selling of Würzburg. On quadratic
forms of four or more indeterminates little has yet been done. Hermite showed that the number of non-equivalent classes of
quadratic forms having integral coefficients and a given discriminant is finite, while Zolotareff and A. N. Korkine, both
of St. Petersburg, investigated the minima of positive quadratic forms. In connection with binary quadratic forms, Smith
established the theorem that if the joint invariant of two properly primitive forms vanishes, the determinant of either of them is represented primitively by the duplicate of the other.
The interchange of theorems between arithmetic and algebra is displayed in the recent researches of J. W. L. Glaisher
of Trinity College (born 1848) and Sylvester. Sylvester gave a Constructive Theory of Partitions, which received additions from his pupils, F. Franklin and G. S. Ely.
The conception of "number" has been much extended in our time. With the Greeks it included only the ordinary positive whole numbers; Diophantus added rational fractions
to the domain of numbers. Later negative numbers and imaginaries came gradually to be recognised. Descartes fully grasped the notion of the negative; Gauss, that of the imaginary.
With Euclid, a ratio, whether rational or irrational, was not a number. The recognition of ratios and irrationals as
numbers took place in the sixteenth century, and found expression with Newton. By the ratio method, the continuity of the
real number system has been based on the continuity of space, but in recent time three theories of irrationals have been advanced by Weierstrass, J. W. R. Dedekind, G. Cantor, and
Heine, which prove the continuity of numbers without borrowing