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 it from space. They are based on the definition of numbers by regular sequences, the use of series and limits, and some new mathematical conceptions.
