of their complicated motion is already in the printer's hands. The two outer planets, Uranus and Neptune, have not yet been begun." Of Professor Newcomb's labors Professor Cayley has said: "Professor Newcomb's writings exhibit, all of them, a combination on the one hand of mathematical skill and power, and on the other of good hard work, devoted to the furtherance of astronomical science." His book on Popular Astronomy (1877) is well known. It has been republished in England and translated into German. The treatise on Astronomy by Newcomb and Holden, and their "Shorter Course" on Astronomy, are works which have been introduced as text-books into our colleges almost universally. Professor Newcomb's scientific work has not been confined to astronomy. He has carried on investigations on subjects purely mathematical. One of the most important is his article on "Elementary Theorems Relating to the Geometry of a Space of Three Dimensions and of Uniform Positive Curvature in the Fourth Dimension," published in Borchardt's Journal, Bd. 83, Berlin, 1877. Full extracts of this very important contribution to non-Euclidian geometry are given in the Enclyclopædia Britannica, article "Measurement." It is gratifying to know that through Professor Newcomb America has done something toward developing the far-reaching generalizations of non-Euclidian geometry and hyper-space. In Volume I of the American Journal of Mathematics he has a note "On a Class of Transformations which Surfaces may Undergo in Space of more than Three Dimensions," in which he shows, for instance, that if a fourth dimension were added to space, a closed material surface (or shell) could be turned inside out by simple flexure without either stretching or tearing. Later articles have been on the theory of errors in observations. In former years he also contributed to the Mathematical Monthly and the Analyst. Professor Newcomb has written a series of college text-books on mathematics. In 1881 appeared his Algebra for Colleges and his Elements of Geometry; in 1882 his Trigonometry and Logarithms, and School Algebra; in 1884 his Analytical Geometry and Essentials of Trigonometry; in 1887 his Differential and Integral Calculus. These works have been favorably reviewed by the press, and are everywhere highly respected. Professor Newcomb's fundamental idea has been to lead up to new and strange conceptions by slow and gradual steps. "All mathematical conceptions require time to become engrafted upon the mind, and the more time the greater their abstruseness." The stu dent is gradually made familiar in these books with the conceptions of variables, functions, increments, infinitesimals, and limits, long before he takes up the calculus, so in the study of the calculus he is not confronted, at the outset and all at once, by a host of new and strange ideas, but possesses already a considerable degree of familiarity with them. With the publication of Newcomb's Algebra has begun a con. siderable "shaking" of the "dry bones" in this science, and we now possess works on this subject that are of considerable merit, Professor Newcomb studies political economy as a recreation, and every now and then there is a commotion in the camp of political economists, caused by a bomb thrown into their midst by Professor Newcomb, in the form of some magazine article or bo ok. In 1884 Professor Newcomb added to his duties as superintendent of the Nautical Almanac that of professor of mathematics and astronomy at the Johns Hopkins University. He generally delivers at that institution two lectures per week. The effect of his connection with the mathematical department has been that the mathematical course is more thoroughly systematized and more carefully graded than formerly, and that the attention of students is drawn also to higher astronomy, theoretical and practical. An observatory for instruction is now provided by the university. Besides a telescope of 9 inches aperture there is a meridian circle with collimators, mercury-basin, and other appliances. Professor Newcomb entered upon his duties at the Johns Hopkins University in 1884 by giving a course of lectures on celestial mechanics. Among other things it embraced his own paper on the "Development of the Perturbative Function and its Derivative in Sines and Co-sines of the Eccentric Anomaly and in Powers of the Eccentricities and Inclinations." The lectures were well attended by the graduate students. At the blackboard Professor Newcomb does not manipulate the crayon with so great dexterity as do his associates, who have been in the lecture-room all their lives, but his lectures are clear, instructive, original, and popular among the students. Since the departure of Professor Sylvester the following courses of lectures have been given to graduate students: Courses of Instruction, Hours per Week, and Attendance, 1884-'88. Analytical and Celestial Mechanics: Prof. Newcomb, 1884-'85, 2 hrs. (11). Theory of Special Perturbations: Prof. Newcomb, 1887-'88, 1st half-year, 2 hrs. Computation of Orbits: Prof. Newcomb, 1887-'88, May, 2 hrs. Theory of Numbers: Dr. Story, 1884-'85, 1st half-year, 2 hrs. (9). Modern Synthetic Geometry: Dr. Story, 1884-'85, 1st half-year, 3 hrs. (8). Introductory Course for Graduates: Dr. Story, 1884-'85, 5 hrs. (10); 1885-'86, 5 hrs. (7); 1886-'87, 5 hrs. (10); 1887-'88, 5 hrs. Modern Algebra: Dr. Story, 1884-'85, 2d half-year, 2 hrs. (9). Quaternions: Dr. Story, 1884-'85, 2d half-year, 3 hrs. (8); 1886-'87. 3 hrs. (5); 1887-88, 3 hrs. Finite Differences and Interpolation: Dr. Story, 1885–86, 1st half-year, 2 hrs. (5). Advanced Analytic Geometry: Dr. Story, 1885-'86, 3 hrs. (4); 1886-'87, 2 hrs. (8); 1887-'88, 2 hrs. Theory of Probabilities: Dr. Story, 1885-'86, 2d half-year, 2 hrs. (5). Calculus of Variations: Dr. Craig, 1884-'85, 1st half-year, 2 hrs. (5). Theory of Functions: Dr. Craig, 1884-'85, 3 hrs. (5); 1885-'86, 1st half-year, 3 hrs. (4); 1886-'87, 3 hrs. (6); 1887–288, 1st half-year, 3 hrs. Hydrodynamics: Dr. Craig, 1884-'85, 1st half-year, 3 hrs. (6); 1885-'86, 1st half-year, 3 hrs. (4); 1886-'87, 1st half-year, 3 hrs. (4); 1887-88, 1st half-year, 3 hrs. Linear Differential Equations: Dr. Craig, 1884-'85, 2d half-year, 3 hrs. (3); 1885–86, 2 hrs. (4); 1887-'88, 2d half-year, 2 hrs. Theoretical Dynamics: Dr. Craig, 1887-'88, 2d half-year, 2 hrs. Differential Equations: Dr. Craig, 1887-'88, 2 hrs. Mathematical Theory of Elasticity: Dr. Craig, 1885-'86, 2d half-year, 3 hrs. (4). Elliptic and Abelian Functions: Dr. Craig, 1885-'86, 2d half-year, 3 hrs. (4); 1886-'87, 1st half-year, 2 hrs. (6). Abelian Functions: Dr. Craig, 1887-88, 2 hrs. Problems in Mechanics: Dr. Franklin, 1884-'85, 2 hrs. (5); 1885-'86, 2 hrs. (6); 1886'87, 2 hrs. (8); 1887-88, 2 hrs. Since the fall of 1884 Dr. Story has been giving every year an Introductory Course to graduate students, which consists of short courses of lectures on the leading branches of higher mathematics. They are intended to give the student a general view of the whole field, which afterward he is to enter upon and study in its details. The Johns Hopkins University went into operation primarily as a University, giving instruction to students who had graduated from college. A regular college course was, however, organized, and it has been growing rapidly from year to year. In the college the student has the choice between several parallel curricula, which are assumed to be equally honorable, liberal, and difficult, and which therefore lead to the same degree of bachelor of arts. Seven groups have been arranged. Some of them embrace no mathematics at all; but, in those courses where it does enter, the instruction is very thorough. Take, for instance, Dr. Story's lectures on conic sections; the method of treatment is entirely modern, and presupposes a knowledge of determinants. A syllabus has been prepared for the use of the students. The lectures resemble the course given in the work of Clebsch. The student who may have studied such books as Loomis's Analytical Geometry, and who may labor with the conceit that he has mastered analytical geometry and conic sections, will soon discover that he has learned only the A B C, and that he is wholly ignorant of the more elegant methods of modern times. Connected with the mathematical department of the university has always been a mathematical seminary, which during the time of Syl vester constituted in fact the mathematical society of the university. The meetings were held monthly. In it the instructors and more advanced students would present and discuss their original researches. Care was taken to eliminate papers of little or no value by immature students. Professor Sylvester generally presided. "If you were fortunate," says Dr. E. W. Davis, "you had your paper first on the program. Short it must be and to the point. Sylvester would be pleased. Then came his paper, or two of them. After him came the rest, but no show did theystand; Sylvester was dreaming of his own higher flights and where they would yet carry him." Since the time of Newcomb this mathematical seminary has been called the Mathematical Society. It is carried on in the same way as before. Three mathematical seminaries proper have since existed, one conducted by Professor Newcomb, another by Dr. Story, and the third by Dr. Craig. The meetings are held in the evening, and weekly. Each instructor selects for his seminary topics from his special studies; Newcomb, astronomical subjects; Story, geometrical subjects or quaternions; Craig, theory of functions or differential equations. Professor Newcomb's seminary work is closely connected with his lectures. The student elaborates some particular points of the lectures or makes practical application of the principles involved. In one case the computation of the orbit of a comet was taken up. Dr. Story, in the year 1885-86, took up the subject of plane curves for his seminary, and dwelt considerably on quartic and quintic curves, giving matter from Möbius and Zeuthen, and the result of his own study on quintics. The stu dent was expected, if possible, to begin where he had left off and carry on investigations along lines pointed out by him. Dr. Story's talk on this subject in this seminary suggested to one of the students a subject of a thesis for the doctor's degree. In the fall of 1888 Dr. Story began his seminary work with the seventeenth example, p. 103, in Tait's Quaternions. Dr. Craig's seminary has generally been upon subjects in continuation and extension of those upon which he is lecturing at the time. If, for instance, he is lecturing on functions, following the "Cours de M. Hermite,” he may in his seminary bring up matter from Briot and Bouquet. At other times he has introduced work into his seminary intended to be preparatory to certain advanced courses which he expected to offer. MATHEMATICAL JOURNALS. The mathematical journals which we are about to discuss were of a much higher grade than those of preceding years. First in order of time is the Mathematical Miscellany, a semi-annual publication, edited by Charles Gill. He was teacher of mathematics at the St. Paul's Collegiate Institute at Flushing, Long Island. Eight numbers were published; the first in February, 1836, and the last in November, 1839. Like many other journals of this kind, it had a Junior and Senior department-the former for young students, the latter for those more advanced. The first number was entirely the work of the editor, excepting two or three new problems. Mr. Gill was much interested in Diophantine analysis. In 1848 he published a little book on the Application of Angular Analysis to the Solution of Indeterminate Problems of the Second Degree, which contains some of his investigations on this subject. Another enthusiastic worker in the field of Diophantine analysis, and a frequent contributor to Gill's journal, was William Lenhart, a favorite pupil of Robert Adrain. Having been afflicted for twenty-eight years with a spasmodic affection of the limbs, occasioned by a fall in early life, which confined him in a measure to his room, he had devoted a considerable portion of his time to Diophantine analysis. To him is attributed the solution of the problem, to divide unity into six parts such that, if unity be added to each, the sums will be cubes. The evident defect in Lenhart's processes was their tentative character. In fact, this criticism applies to all work done in Diophantine analysis by American computers, down to the present time. It is true even of old Diophantus himself. To this ancient Alexandrian algebraist, who is the author of the earliest treatise on algebra extant, as well as to his American followers of modern times, general methods were quite unknown. Each problem has its own distinct method, which is often useless for the most closely related problems. It has been remarked by H. Hankel that, after having studied one hundred solutions of Diophantus, it is difficult to solve the one hundred and first. It is to be regretted that American students should have wasted so much time over Diophantine analysis, instead of falling in line with European workers in the theory of numbers as developed by Gauss and others. Previous to the publication of the American Journal of Mathematics, our journals contained no contributions whatever on the theory of numbers, excepting the Mathematical Miscellany, which had some few articles by Benjamin Peirce and Theodore Strong, which involved Gaussian methods. Among the contributors to the Mathematical Miscellany were Theodore Strong, Benjamin Peirce, Charles Avery, Marcus Catlin of Hamilton College, George R. Perkins, O. Root, William Lenhart, Lyman Abbott, jr., B. Docharty, and others. The next mathematical periodical was the Cambridge Miscellany of Mathematics, Physics, and Astronomy, edited by Benjamin Peirce and Joseph Lovering, of Harvard, and published quarterly. The last prob. lems proposed in Gill's journal were solved here. Four numbers only were published, the first in 1842. The list of contributors to this jour nal was about the same as to the preceding. The most valuable arti. cles were those written by the editors. During the next fifteen years America was without a mathematical journal; but in 1858, J. D. Runkle, of the Nautical Almanac office in Boston, started the Mathematical Monthly. He has since held the dis tinguished position of professor of mathematics at (and, for a time, president of) the Massachusetts Institute of Technology, where he has been especially interested in developing the department of manual training. As will be seen presently, the time for beginning the publi cation of a long-lived mathematical journal was not opportune. Three volumes only appeared. On a fly-leaf the editor announced to his subscribers that over one-third of the subscribers to Volume I discontinued their subscriptions at the close. "I have supposed," he says, "that those who continued their subscription to the second volume would not be so likely to discontinue it to the third volume, and I have made my arrangements accordingly. If, however, any considerable number should discontinue now, it will be subject to very serious loss. * I ask as a favor for all to continue to Volume III, and notify me during |