JAMES JOSEPH SYLVESTER *

We have come together to do honor to the memory of the great man whose work in initiating and for seven years conducting the mathematical department of this institution will always remain one of the proudest traditions of the Johns Hopkins University. To me, as one who was long his pupil, and who owes so much to his inspiration, has been assigned the task of saying something about the work and the genius of Sylvester, and especially about the influence which he exerted, while in Baltimore, upon the study of mathematics here and upon the advancement of mathematical research in America.

Since his death there has appeared in the English journal Nature, and has been reprinted in the Johns Hopkins University Circulars, a review of his life. and work by Major MacMahon; and in 1889, when that work was well-nigh ended, Sylvester's great compeer and friend, Professor Cayley, contributed to the columns of the same journal a sketch of Sylvester's labors. One of his Baltimore pupils, too, Professor Halsted, of the University of Texas, has given in the columns of Science an account of his life and achievements. It is therefore the less necessary to undertake here to give anything in the nature of an enumeration of even his most signal contributions to mathematics.

His influence upon the development of mathe

* Memorial address delivered at the Johns Hopkins University, May 2, 1897.

matical science rests chiefly, of course, upon his work in the Theory of Invariants. Apart from Sir William Rowan Hamilton's invention and development of Quaternions, this theory is the one great contribution made by British thought to the progress of pure mathematics in the present century, or indeed since the days of the contemporaries of Newton. From about the middle of the eighteenth century until near the middle of the nineteenth, English mathematics was in a condition of something like torpor. The second half of the eighteenth century was one of the most brilliant periods in the history of mathematics; but the magnificent achievements of Euler, Lagrange, Laplace, awakened no response on the other side of the narrow seas. It seems almost incredible that the complacent conservatism of Cambridge went so far that even the notation of mathematical analysis as used on the Continent was untaught there until about 1820. Babbage tells us, in his "Passages from the Life of a Philosopher," how he, together with Herschel, Peacock, and a few others, founded in 1812 the "Analytical Society" for promoting (as Babbage humorously expressed it) "the principles of D-ism in opposition to the Dot-age of the University." It is from the translation by these three men (in 1816) of Lacroix's Treatise on the Differential and Integral Calculus, together with the publication by them four years later of two volumes of illustrative examples, that the first impulse toward a revival of mathematics in England is usually dated. Nothing could show more thoroughly the insular and retrograde condition of English mathe

case that in the magnificent extension of of mathematics which was effected by nental mathematicians during the first f of the present century, England had n is almost literally correct to say that th mathematics for about a hundred year written without serious defect with Eng matics left entirely out of account.

That the like statement cannot be ma to the past fifty years is due pre-emin genius and labors of three men: Hamil and Sylvester. Hamilton was a high genius, who constructed and developed great mathematical method. Great as wa it lay so entirely apart from the gen research that it did not, in his own ti awaken widespread activity on the par either at home or abroad. On the othe Theory of Invariants had a history of be called the normal type. Its origin is in Boole's discoveries of isolated instanc ance; these led Cayley to institute a investigation of this remarkable and sig nomenon; and Cayley's researches aw ardent interest of Sylvester. Under th

these two great masters, a new and important province was rapidly added to the domain of algebra. Not only did other English mathematicians join in the work, but Hermite in France, Aronhold and Clebsch in Germany, Brioschi in Italy, and other Continental mathematicians seized upon the new ideas, and the theory of invariants was for three decades one of the leading objects of mathematical research throughout Europe. It is impossible to apportion between Cayley and Sylvester the honor of the series of brilliant discoveries which marked the early years of the theory of invariants. Their names are linked together as the creators of a new and beautiful development of algebra, the ideas of which have profoundly influenced the progress also of geometry and of analysis generally. "The theory of invariants," says MacMahon, "sprang into existence under the strong hand of Cayley, but that it emerged finally a complete work of art, for the admiration of future generations of mathematicians, was largely owing to the flashes of inspiration with which Sylvester's intellect illuminated it." It is pleasant to know that the triumphs of neither were marred by any dispute as to personal claims or by anything even approaching jealousy. On the contrary, these two men of genius, antipodes of each other in temperament and habits of work, were alike in the constancy of their mutual friendship, regard, and admiration.

I have dwelt thus long on Sylvester's connection with the creation of the Theory of Invariants, because it is by that chiefly that he left his trace upon the history of mathematics in its large outlines. But

his genius is quite as strikingly shown in researches of a more isolated character. Ten years before the date of his work in invariants, he wrote in quick succession several remarkable memoirs on algebraic subjects, especially on Sturm's functions and on elimination. His researches in the Theory of Partitions of Numbers are among the most original and remarkable of his works. In the Theory of Numbers he was especially interested in Ternary Cubic Forms. The question of the distribution of prime numbers had a great fascination for him; and he succeeded while in Baltimore in making an impression upon this recondite problem in that he contracted the limits found by Tchebycheff for the number of primes contained within a given range. His work seldom touched on geometry, but his "theory of residuation" in connection with cubic curves is a beautiful structure, to which he made some remarkable additions while in Baltimore. I am not, however, attempting to give a survey of his work; suffice it to add that, in adidtion to the subjects named, he made contributions to astronomy, to dynamics, and to the theory of link-motion, besides other special subjects.

One of the most striking of Sylvester's achievements was his demonstration and extension of Newton's unproved rule concerning the number of the imaginary roots of an algebraic equation. Newton had left no trace of the process of thought by which he had arrived at his rule, nor had he given any indication of the basis on which it rests. All attempts of later mathematicians to establish it had proved futile. It was characteristic of Sylvester to set himself the task of filling up this lacuna in mathematics. The