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.

APPLIED MATHEMATICS.

From the

Notwithstanding the beautiful developments of celestial mechanics reached by Laplace at the close of the eighteenth century, there was made a discovery on the first day of the present century which presented a problem seemingly beyond the power of that analysis. We refer to the discovery of Ceres by Piazzi in Italy, which became known in Germany just after the philosopher Hegel had published a dissertation proving a priori that such a discovery could not be made. positions of the planet observed by Piazzi its orbit could not be satisfactorily calculated by the old methods, and it remained for the genius of Gauss to devise a method of calculating elliptic orbits which was free from the assumption of a small eccentricity and inclination. Gauss' method was developed further in his Theoria Motus. The new planet was re-discovered with aid of Gauss' data by Olbers, an astronomer who promoted science not only by his own astronomical studies, but also by discerning and directing towards astronomical pursuits the genius of Bessel.

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Friedrich Wilhelm Bessel (1784-1846) was a native of Minden in Westphalia. Fondness for figures, and a distaste for Latin grammar led him to the choice of a mercantile career. In his fifteenth year he became an apprenticed clerk in Bremen, and for nearly seven years he devoted his days to mastering the details of his business, and part of his nights to study. Hoping some day to become a supercargo on trading expeditions, he became interested in observations at sea. With a sextant constructed by him and an ordinary clock he determined the latitude of Bremen. His success in this inspired him for astronomical study. One work after another was mastered by him, unaided, during the hours snatched from

sleep. From old observations he calculated the orbit of Halley's comet. Bessel introduced himself to Olbers, and submitted to him the calculation, which Olbers immediately sent for publication. Encouraged by Olbers, Bessel turned his back to the prospect of affluence, chose poverty and the stars, and became assistant in J. H. Schröter's observatory at Lilienthal. Four years later he was chosen to superintend the construction of the new observatory at Königsberg.92 In the absence of an adequate mathematical teaching force, Bessel was obliged to lecture on mathematics to prepare students for astronomy. He was relieved of this work in 1825 by the arrival of Jacobi. We shall not recount the labours by which Bessel earned the title of founder of modern practical astronomy and geodesy. As an observer he towered far above Gauss, but as a mathematician he reverently bowed before the genius of his great contemporary. Of Bessel's papers, the one of greatest mathematical interest is an "Untersuchung des Theils der planetarischen Störungen, welcher aus der Bewegung der Sonne ensteht" (1824), in which he introduces a class of transcendental functions, J(x), much used in applied mathematics, and known as "Bessel's functions." He gave their principal properties, and constructed tables for their evaluation. Recently it has been observed that Bessel's functions appear much earlier in mathematical literature.98 Such functions of the zero order occur in papers of Daniel Bernoulli (1732) and Euler on vibration of heavy strings suspended from one end. All of Bessel's functions of the first kind and of integral orders occur in a paper by Euler (1764) on the vibration of a stretched elastic membrane. In 1878 Lord Rayleigh proved that Bessel's functions are merely particular cases of Laplace's functions. J. W. L. Glaisher illustrates by Bessel's functions his assertion that mathematical branches growing out of physical inquiries as a rule "lack the easy flow

or homogeneity of form which is characteristic of a mathematical theory properly so called." These functions have been studied by C. Th. Anger of Danzig, O. Schlömilch of Dresden, R. Lipschitz of Bonn (born 1832), Carl Neumann of Leipzig (born 1832), Eugen Lommel of Leipzig, I. Todhunter of St. John's College, Cambridge.

Prominent among the successors of Laplace are the following: Siméon Denis Poisson (1781-1840), who wrote in 1808 a classic Mémoire sur les inégalités séculaires des moyens mouvements des planètes. Giovanni Antonio Amadeo Plana (17811864) of Turin, a nephew of Lagrange, who published in 1811 a Memoria sulla teoria dell' attrazione degli sferoidi ellitici, and contributed to the theory of the moon. Peter Andreas Hansen (1795-1874) of Gotha, at one time a clockmaker in Tondern, then Schumacher's assistant at Altona, and finally director of the observatory at Gotha, wrote on various astronomical subjects, but mainly on the lunar theory, which he elaborated in his work Fundamenta nova investigationes orbitæ veræ quam Luna perlustrat (1838), and in subsequent investigations embracing extensive lunar tables. George Biddel Airy (18011892), royal astronomer at Greenwich, published in 1826 his Mathematical Tracts on the Lunar and Planetary Theories. These researches have since been greatly extended by him. August Ferdinand Möbius (1790-1868) of Leipzig wrote, in 1842, Elemente der Mechanik des Himmels. Urbain Jean Joseph Le Verrier (1811-1877) of Paris wrote, the Recherches Astronomiques, constituting in part a new elaboration of celestial mechanics, and is famous for his theoretical discovery of Neptune. John Couch Adams (1819-1892) of Cambridge divided with Le Verrier the honour of the mathematical discovery of Neptune, and pointed out in 1853 that Laplace's explanation of the secular acceleration of the moon's mean motion accounted for only half the observed acceleration,

Charles Eugène Delaunay (born 1816, and drowned off Cherbourg in 1872), professor of mechanics at the Sorbonne in Paris, explained most of the remaining acceleration of the moon, unaccounted for by Laplace's theory as corrected by Adams, by tracing the effect of tidal friction, a theory previously suggested independently by Kant, Robert Mayer, and William Ferrel of Kentucky. George Howard Darwin of Cambridge (born 1845) made some very remarkable investigations in 1879 on tidal friction, which trace with great certainty the history of the moon from its origin. He has since studied also the effects of tidal friction upon other bodies in the solar system. Criticisms on some parts of his researches have been made by James Nolan of Victoria. Simon Newcomb (born 1835), superintendent of the Nautical Almanac at Washington, and professor of mathematics at the Johns Hopkins University, investigated the errors in Hansen's tables of the moon. For the last twelve years the main work of the U. S. Nautical Almanac office has been to collect and discuss data for new tables of the planets which will supplant the tables of Le Verrier. G. W. Hill of that office has contributed an elegant paper on certain possible abbreviations in the computation of the long-period of the moon's motion due to the direct action of the planets, and has made the most elaborate determination yet undertaken of the inequalities of the moon's motion due to the figure of the earth. He has also computed certain lunar inequalities due to the action of Jupiter.

The mathematical discussion of Saturn's rings was taken up first by Laplace, who demonstrated that a homogeneous solid ring could not be in equilibrium, and in 1851 by B. Peirce, who proved their non-solidity by showing that even an irregular solid ring could not be in equilibrium about Saturn. The mechanism of these rings was investigated by James Clerk Maxwell in an essay to which the Adams prize was awarded.