Polars in Curves and Surfaces of the Second Order," and "Development and Dissection of Riemann's Surfaces." Should it be claimed that these theses are the work of immature students, then we may answer that for candidates for the bachelor's degree they are nevertheless creditable. The writer of the first thesis (L. M. Hoskins) is now doing excellent work as instructor in engineering at the university. The writer of the second thesis (L. S. Hulburt) is now professor of mathematics at the univer sity of Dakota. M. Updegraff, of the class of '84, wrote a thesis on "Resultants." He holds now a responsible position at the National Observatory at Cordoba, Argentine Republic, South America. Titles of later "special-honor theses" are, "Approximation to the Roots of Numerical Equations," "Maxima and Minima," "On the Equation sin my. cos nysin mx. cos nx," "Different Systems of Co-ordinates." These theses are certainly indicative of a healthful activity in the under-graduate mathematical department.

For special studies pursued after graduation and the presentation of an acceptable thesis, the degree of Master is conferred. The following are the titles of two theses written to secure the degree of "master of science in mathematics: " "The Hodograph," "On a Quadratic Form" (in the theory of numbers).

The present courses in mathematics offered at the University of Wisconsin (catalogue 1887-88) are as follows:

Subcourse I, Algebra. Five exercises a week during the fall term. (Professor Van Velzer and Mr. Slichter.)

Required of Freshmen in all courses.

Subcourse II, Theory of Equations, including the elements of determinants, and graphic algebra. Five exercises a week during the winter term. (Professor Van Völzer and Mr. Slichter.)

Required of Freshmen in the Modern Classical, English, General Science, and Engineering Courses.

Subcourse III, Solid Geometry. Five exercises a week during the winter term. (Professor Van Velzer or Mr. Slichter.)

Required of Freshmen in the Ancient Classical Course.

Subcourse IV, Trigonometry. Five exercises a week during the spring term. (Professor Van Velzer and Mr. Slichter.)

Required of Freshmen in all courses.

Subcourse V, Descriptive Geometry. The topics taught embrace the projection of lines, planes, surfaces, and solids, the intersection of each of these with any one of the others, tangent lines to curves and surfaces and tangent planes to surfaces, problems in shades and shadows, of lines and surfaces, linear perspective and isometric projection. The class-room exercises are accompanied by work in the draughting room. The text-book used is Church's Descriptive Geometry. Full study during the spring term, Freshman year, and three-fifths study during the fall term, Sophomore year. (Professor Bull.)

Required of Freshmen in Civil and Mechanical Engineering. Elective for other students. Subcourse VI, Analytic Geometry. Five exercises a week during the fall term. (Professor Van Velzer.)

Required of engineering Sophomores and scientific Sophomores who pursue mathematical, physical, or astronomical studies. Elective for other students.

Subcourse VII, Differential Calculus. Five exercises a week during the winter term. (Professor Van Velzer.)

Required of engineering Sophomores and scientific Sophomores who pursue mathematical, physical, or astronomical studies. Elective for other students.

Subcourse VIII, Integral Calculus. Five exercises a week during the spring term. (Professor Van Velzer.)

Required of engineering Sophomores and scientific Sophomores who pursue mathematical, physical, or astronomical studies. Elective for other students.

Subcourse XIX, Method of Least Squares. This is a course in the theory of probabilities as applied to the adjustment of errors of observation. It will be first given in 1889. Must be preceded by subcourses VI, VII, and VIII, three-fifths study during the winter term. (Mr. Hoskins.)

Required of Seniors in Civil Engineering.

Subcourses IX to XVIII, special advanced electives. Courses varying from year to year are offered in the following subjects: IX, Modern Analytic Geometry; X, Higher Plane Curves; XI, Geometry of Three Dimensions; XII, Differential Equations; XIII, Spherical Harmonics; XIV, Elliptic Functions; XV, Theory of Functions; XVI, Theory of Numbers; XVII, Quantics; and XVIII, Quaternions.

Very good work has been done, at times, by students in the department of mathematical physics. Prof. John E. Davies, the professor of physics, takes a living interest in pure as well as applied mathematics. His reading in pure mathematics has, indeed, been very extensive. Mathematical reading is a recreation to him. He would not unfrequently take with him some mathematical work-as, for instance, Tait's Quaternions-to faculty meetings, that he might pass pleasantly the otherwise tedious sessions of that august assembly. Many years ago he made, for his own use, a complete translation of Koenigsberger's work on Elliptic Functions.

The university offers excellent facilities for the study of astronomy. The Washburn Observatory has a large equatorial for use in original work, and also a smaller one for the use of students. After the death of Professor Watson, Professor Holden became director of the Observatory. He held this position until his appointment as director of the Lick Observatory. Prof. George C. Comstock is now professor of astronomy and associate director of the Washburn Observatory. Professor Comstock is a pupil of Watson, and came to Wisconsin from Ann Arbor with Watson. Before assuming the duties of his present position he was for two or three years professor of mathematics and astronomy at the University of Ohio.

The instruction in analytical mechanics is in charge of Mr. L. M. Hoskins, a young man of very marked mathematical talent. He graduated in 1883 at the head of a class of sixty-five, and was afterward appointed fellow in mathematics in Harvard University. Through his influence, the study of analytical mechanics had been made much more prominent in the engineering courses than it had been formerly. Two terms are now devoted to it instead of only one. Bowser's Elements of Analytical Mechanics is the text-book used.

Mr. Hoskins has contributed to the Annals of Mathematics, the Mathematical Magazine, and Van Nostrand's Engineering, Magazine.

JOHNS HOPKINS UNIVERSITY,

President Daniel C. Gilman once said to the trustees of the Johns Hopkins University, when the question of "How to begin a university " was upon their minds, "Enlist a great mathematician and a distinguished Grecian; your problem will be solved. Such men can teach in a dwelling-house as well as in a palace. Part of the apparatus they will bring, part we will furnish. Other teachers will follow them."* So it came to pass that, before there were any buildings for classes, a professor of mathematics and a professor of Greek were secured for the new university.

When President Gilman was engaged in the all-important work of selecting men for the above positions, he may have been actuated in his choice by thoughts similar to those of Prof. G. Chrystal, who, before a learned body of English scientists, once expressed himself as follows:† "Science can not live among the people, and scientific education can not be more than a wordy rehearsal of dead text-books, unless we have living contact with the working minds of living men. It takes the hand of God to make a great mind, but contact with a great mind will make a little mind greater. The most valuable instruction in any art or science is to sit at the feet of a master, and the next best, to have contact of another who has himself been so instructed."

Is there a student among us who has studied with Sylvester and who will deny the truth of the above? Is there a mathematician, who has sat as a pupil at the feet of Benjamin Peirce, who will deny it? It is a fortunate circumstance for the progress of the exact sciences in this country that, at a time when the "Father of American Mathematics" was approaching his grave, there came among us another master who gave the study of mathematics aresh and powerful impulse. Professor Sylvester is a mathematical genius, who has no superior in England, except, perhaps, Professor Cayley.

James Joseph Sylvester was born in London in 1814, and was educated at the University of Cambridge. He came to this country to fill the professorship at the University of Virginia when he was a very young man, but his stay among us then was very short. He became a member of the Royal Society at the age of twenty-five. For some time he was professor of natural philosophy in University College, London. In 1855 he became professor of mathematics in the Royal Military Academy at Woolwich, and in 1876 was elected for the position at the Johns Hopkins University.

Sylvester's activity has been wonderful. Prior to 1863 he published 112 scientific memoirs, which are recorded in the Royal Society's Index of Scientific Papers. A most important paper, printed in the Philo

* Annual report of the president of the Johns Hopkins University, 1888, p. 29. tNature, September 10, 1885, Section A of Brit. Assocation, opening address by Prof. G. Chrystal, president of the section.

sophical Transactions of 1864, is Sylvester's Theorem on Newton's Rule for discovering the number of real and imaginary roots of an equation. Of this Todhunter says: "If we consider the intrinsic beauty of the theorem, the interest which belongs to the rule associated with the great name of Newton, and the long lapse of years during which the reason and extent of that rule remained undiscovered by mathematicians, among whom Maclaurin, Waring, and Euler are explicitly included, we must regard Professor Sylvester's investigations made to the theory of equations in modern times justly to be ranked with those of Fourier, Sturm, and Cauchy." A few of his numerous other investigations, made before coming to Baltimore, are on the Rotation of a Rigid Body; on the Analytical Development of Fresnel's Optical Theory of Crystals; on Reversion of Series; on the Involution of Six Lines in Space, " culminating in the result that if these six lines represent forces in equilibrium they must lie on a ruled cubic surface;" on a general theorem by which, for instance, the quintic can be expressed as the sum of three fifth powers. In 1859 he gave a course of lectures at King's College, London, on the subject of The Partitions of Numbers and the Solution of Simultaneous Equations in Integers, in which it fell to his lot "to show how the difficulties might be overcome which had previously baffled the efforts of mathematicians, and especially of one bearing no less venerable a name than that of Leonard Euler," and also laid the basis of a method which has since been carried out to a much greater extent in his "Constructive Theory of Partitions," published in the American Journal of Mathematics, in writing which he "received much valuable co-operation and material contributions" from his "pupils in the Johns Hopkins University." +

Professor Sylvester's most celebrated work has been in modern higher algebra. A very large portion of the theory of determinants is due to him, and the epoch-making theory of invariants owes its origin and early development almost exclusively to his genius and that of Professor Cayley.

The Johns Hopkins University offered to Professor Sylvester every facility for original work that could be desired. By the system of "fellowships" a number of talented young men were drawn to Balti more, who were capable not only of understanding the teachings of their great master, but, in many cases, also of aiding him in his researches. The university, moreover, started the American Journal of Mathematics, in which all investigations in mathematics could be pub. lished and thereby brought before the mathematical public. Professor Sylvester's time was not taken up by the usual routine work in school, but was almost wholly given to the pursuit of his favorite subjects. He lectured, perhaps, two or three times per week, but these lectures generally disclosed some new discovery in algebra.

*Theory of Equations, page 250.

+ Inaugural Lecture delivered by Professor Sylvester before the University of Oxford, December 12, 1885, published in Nature, January 7, 1886.

Though he had passed his sixtieth year before he came to the Johns Hopkins University, his mind seemed to be as strong and active as ever. The group of students he had gathered about him were almost constantly made to feel the glow of new ideas or of old ones in a new form. From 1877 to 1882, Professor Sylvester contributed thirty articles and notes to the American Journal of Mathematics; twenty-two to the Comptes Rendus de l'Académie des Sciences de l'Institut de France; one paper to the Proceedings of the Royal Society, "On the Limits to the Order and Degree of the Fundamental Invariants of Binary Quantics" (1878); four to the Messenger of Mathematics; four to the Lon. don, Edinburgh, and Dublin Philosophical Magazine; six to the Journal für reine und angewandte Mathematik, Berlin.* If this list be complete, the number of original papers published by him while at the Johns Hopkins University was sixty-seven. Special mention may be made here of a proof by Professor Sylvester, printed in the Philosophical Magazine for 1878, of a theorem on the number of linearly independent differentiants, which had been awaiting proof for over a quarter of a century. He was led to undertake the investigation of this subject by a question put to him by one of his students in connection with a footnote given at one place in Faà de Bruno's Théorie des Formes Binaires. Since his return to England, Sylvester has been developing a new subject, which he calls the "Method of Reciprocants." The lectures which he delivered on this subject at the University of Oxford have been reported by Mr. Hammond and published in the American Journal of Mathematics.

Sylvester has manufactured a large number of technical terms in mathematics. He himself speaks on this point as follows: "Perhaps I may, without immodesty, lay claim to the appellation of the mathematical Adam, as I believe that I have given more names (passed into general circulation) to the creatures of the mathematical reason than all the other mathematicians of the age combined."t

In his writings, Professor Sylvester is often very eloquent. His style is peculiarly flowery, and indicative of very powerful imagination. His articles are frequently interspersed with short pieces of poetry, either quoted or of his own composition. Thus, in his article in Nature, January, 1886, is given a short poem, "On a Missing Member of a Family Group of Terms in an Algebraical Formula;" followed by this sentence: "Having now refreshed ourselves and bathed the tips of our fingers in the Pierian spring, let us turn back for a few brief moments to a light banquet of the reason."

Since the beginning of the Johns Hopkins University, twenty fellowships have been open annually to competition, each yielding five hundred dollars and exempting the holder from all charges for tuition. This system was instituted for the purpose of affording to young men * U. S. Bureau of Education, Circular of Information No. 1, 1888, p. 220. Nature, Dec. 15, 1887, p. 152, note.