At the University of Michigan his teaching was marked by great thoroughness. He was a rather slow man, and took great pains with the poorer students. He had the happy faculty of inducing all students to perform faithful work. It is related that the son of a certain prominent Congressman once labored under the conceit that his father's reputation would exempt him from the necessity of studying whenever he felt disinclined to do so. Once, when being called upon to recite, he answered, "not prepared." Professor Olney assured him that the lesson was easy, asked him to rise from his seat, and then proceeded, much to the amusement of the rest of the class, to develop with him the entire lesson of the day by asking him questions. In that way was spent the whole hour. The class was made to assist him in some of the more difficult points. The Congressman's son concluded, on that occasion, that it was, after all, more agreeable to his feelings to prepare his mathematics carefully in his own room than to expose his ignorance before the whole class by being kept reciting for a whole hour. At times Professor Olney enjoyed joking at the expense of those who would not be injured by it. The result of his teaching was a high average standing among students. The first important step toward reaching good results consisted in a strict adherence to the requirements laid down for admission. If a student failed in his entrance examination, then Professor Olney took much pains to see that the deficiencies would be made up under a competent private teacher who was personally known to him. The rigid requirements for admission gave the mathematical department great leverage.

Professor Olney was an active promoter of various humanitarian enterprises, and was much interested in the educational work of the Baptist denomination, of which he was a member. He was interested in the progress of Kalamazoo College (Baptist) quite as much as in that of Michigan University. His library is now the property of that college. At the time of his death he was engaged in the revision of his series of text-books to meet the increased demands of the times.

In 1860, before Olney was connected with the university, the terms for admission were-to the classical course, arithmetic, and algebra through simple equations; to the scientific course, arithmetic, algebra through quadratic equations and radicals, and the first and third books of Davies' Legendre. In 1864 quadratic equations were added to the classical course, and to the scientific course the fourth book of Legendre. In 1867 the requirements for the classical course were raised so as to equal those in the scientific course, but in the following year quadratic equations were temporarily withdrawn. The fifth book of Legendre was added in the scientific course in 1869. In 1870 all of Legendre was required, and five books in the classical course. In the next year arithmetic, Olney's Complete Algebra, and Parts I and II of Olney's Geometry (including plane, solid, and spherical geometry), were the requirements in both courses. No changes have been made since.

The college curriculum in 1854 was, for both courses, algebra, geom. etry, trigonometry, analytical geometry, and calculus. The next year calculus was withdrawn from the classical course, but was re-instated in 1864, and in 1868 was made elective. In 1878 all courses except those for the degree of B. L. (English) embraced calculus. In 1881 the B. L. course included trigonometry. Since then calculus has been elective in all courses except the scientific. Analytical geometry has been added to the B. L. course.

During the last eight or ten years the "university system" has been growing rapidly at Ann Arbor. Mathematical studies of university grade have been 'offered. Determinants, quaternions, and modern analytical geometry were first announced in 1878; higher algebra in 1879; synthetic geometry and elliptic functions in 1885; theory of functions in 1886; differential equations (advanced) in 1887. The calculus of variations (probably as much as is contained in Church's or Courtenay's Calculus) was announced first in 1866.

The text-books which have been used at the University of Michigan, at different periods, are as follows:

Algebra.-Davies' Bourdon, Ray's—Part II, Olney's University Algebra, Newcomb's College Algebra, Chas. Smith's Treatise on Algebra, Salmon's Higher Algebra, Burnside and Panton.

Determinants.-Muir, Scott, Dostor, Peck.
Geometry.-Davies' Legendre, Olney, Ray.

Trigonometry.-Davies' Legendre, Loomis, Olney.

Synthetic Geometry.-Reye, Steiner.

Analytic Geometry.-Davies, Loomis, Church, Olney, Peirce's Curves, Functions, and Forces, Chas. Smith, Salmon, Frost, Aldis, Whitworth, Ciebsch.

Calculus.-Davies, Church, Loomis, Courtenay, Olney, Price, Todhunter, Williamson, Jordan.

Differential Equations.-Boole, Forsyth.

Calculus of Variations.-Todhunter, Carll.
Quaternions.-Kelland & Tait, Hardy, Tait.
Elliptic Functions.-Durége, Bobek, Jordan.

Prof. G. C. Comstock, of the Washburn Observatory, gives the following reminiscences of the mathematical instruction at Ann Arbor:* "I entered the University of Michigan in the fall of 1873, with a preparation in mathematics consisting of arithmetic, elementary algebra through quadratic equations and including a very hurried view of logarithms, and plane, solid, and spherical geometry. The mathemati cal course given in the university at that time comprised, in the Freshman year, Olney's University Algebra, inventive geometry (consisting of an assignment of theorems for which the student was expected to find demonstrations), and plane and spherical trigonometry. In the Sophomore year, general geometry and differential and integral calcu lus. Descriptive geometry was required of engineering students, and was occasionally taught to others.

* Letter to the writer, November 6, 1888.

"The Freshmen were taught by instructors, usually young men of not much experience in teaching, but once a week they (the students) went up to Professor Olney for a review of the week's work, and these occasions were the trials of a Freshman's life. Olney's stern and rigid discipline had won for him among students the sobriquet "Old Toughy." He was not, however, a harsh man, and although the students stood in awe of him I think that he was generally liked by them. One feature of the weekly reviews may serve to illustrate his discipline and his power of enforcing it. He insisted upon the attention of each student being given to the demonstrations and explanations which the person reciting was engaged upon, and given so closely that the latter might be stopped at any point and any other student required to take up the demonstration at that point and carry it on without duplicating anything which had already been given.

"The University Algebra given the Freshman class contained an elementary view of infinitesimals, extending to the differentiation of algebraic functions and the use of Taylor's formula; and also a presentation of loci of equations, by which the student became familiar with the geometrical representation of an equation. The Sophomore thus came to this study of general geometry and calculus with some preliminary notions of these subjects. The study of the calculus was elective, but every Sophomore was required to take an elementary course in general geometry, and to make use here of the principles of the calculus which he had learned as a Freshman.

"Professor Olney's tastes were decidedly geometrical in character, and he constantly sought to translate analytical expressions into their geometrical equivalents, and much of his success as a teacher is probably due to this.

"Professor Bemàn, on the other hand, is an analyst, a 'lightning mathematician' in the student vernacular, and, in my day, the facility with which he handled mathematical expressions dazed and discouraged the student, who usually felt that he did not get much from Professor Beman.

"The criticism which I should now make upon the mathematical teaching which I received, is that little or no attempt was made to point out the applications of mathematics, and to encourage the student to apply it to those numerous problems of physical science, of engineering, and of navigation, which serve as powerful stimulants to the interest. The student was taught how to solve a spherical triangle, and how to look out logarithms from a table, but was never required to solve such a triangle and obtain numerical results.

"The text-books in use were those written by Professor Olney, none other being employed even for reference. There were no mathematical clubs or seminaries, and no facilities offered for the study of mathematics beyond the prescribed curriculum."

Professor Olney is the author of a complete set of mathematical text-books, which have displaced the works of Davies, Loomis, and Robinson in many schools, both in the East and in the West. His works are quite distinctive in the arrangement of subjects, and mark a decided advance over the other books just named. In the explanatory notes added here and there, in the tabular views at the end of chapters, in the judicious selection of examples, we see the fruits of long experi ence in the class-room. His books exhibit him in the light of a great teacher rather than a great mathematician. He was greatly aided in his work by Professor Beman, who prepared all the "keys" to the mathematical books, and did a great deal of critical work. It has been stated that Professor Olney could never get his publishers to print the books in the form which seemed the most perfect to him. He considered the traditional classification of mathematical subjects very defective, and wished to write a System of Mathematics in which he could embody his own ideals on this point. He thought, for example, that a considerable part of algebra should be taught before taking up the advanced parts of arithmetic, such as percentage and its applications, and that plane geometry should precede mensuration in arithmetic. By discarding the usual division of mathematics into separate volumes on arithmetic, algebra, geometry, etc., and by writing a system of mathematics he hoped to introduce great improvements. The publishers, on the other hand, preferred the traditional classification, as the books would then meet with larger sale. Professor Olney was thus hampered, to some extent, in the execution of his ideal scheme.

In his published works, the science of geometry is brought under two great heads,' Special or Elementary Geometry, and General Geometry. The former consists of four parts: The First Part is an empirical geometry, designed as an introduction, in which the fundamental facts are illustrated but not demonstrated. The Second Part contains the elements of demonstrative geometry, designed for schools of lower grade. The Third Part was written to meet the special needs at the University of Michigan. It was studied in the Freshman class by students who had mastered the Second Part. The effort is made here to encourage original research. This part contains also applications of algebra to geometry, and an introduction to modern geometry. The Fourth Part consists of plane and spherical trigonometry, treated geometrically. The old "line-system" is still retained here.

General Geometry was intended to be developed by him in two separate volumes, but only the first was published. The first treats of plane loci, the second was intended for loci in space. This first vol ume may be very roughly described as covering the field generally oc cupied by analytical geometry and calculus. Olney favored the in finitesimal method, which he used also in his Elementary Geometry, where he permits the number of sides of a regular polygon circumscribed about a circle to become "infinite," and to coincide with the

circle. We are glad that this method is at the present time being more and more eliminated from elementary text-books. It is worthy of note that in his calculus Olney gives the elegant method, discovered by Prof. James C. Watson, of demonstrating the rule for differentiating a logarithm without the use of series.

In some courses the subjects have been taught exclusively by lectures, but the present tendency is to use the best text-book available, and supplement it with lectures as may be found advisable. Of late years a good deal of attention has been given to the careful and critical reading of such works as Salmon's Conic Sections, Higher Algebra, Geometry of Three Dimensions, Frost's Solid Geometry, Jordan's Cours d'Analyse, Forsyth's Differential Equations, Price's Calculus, Carll's Calculus of Variations, Burnside and Panton's Theory of Equations, Reye's Geometrie der Lage, Steiner's Vorlesungen über synthetische Geometrie, Clebsch's Vorlesungen über Geometrie der Ebene. It is thought that better results have been secured in this way than when the student's attention is largely given to the taking of notes.

Since the death of Professor Olney, Professor Beman has been filling the professorship of mathematics. He graduated at the University of Michigan in 1870. Excepting the first year after graduation (when he was instructor in Greek at another institution), he has been teaching continually at his alma mater—from 1871 to 1874 as instructor in mathematics, then as assistant professor and as associate professor of mathematics, and, finally, as full professor. He has done much toward introducing the "university system" in his department, and has been a contributor to our mathematical journals, particularly to the Analyst and the Annals of Mathematics.

For several years Charles N. Jones has been professor of applied mathematics. He has been a very successful teacher of mechanics. Professor Beman has two or three assistants in the department of pure mathematics.

A mathematical club was organized in 1887. It is under the control of the students, but an active interest is continually shown by the various instructors. Papers of some length are presented, problems discussed, etc.

UNIVERSITY OF WISCONSIN.

The University of Wisconsin was organized in 1848, and formally opened in 1850. A preparatory department was established in 1849, and it was not till 1851 that regular college classes were formed. Like most other State universities, the University of Wisconsin had a hard struggle for existence during its early years. Our State Legislatures did not always pursue a wise course toward their higher institutions of learning. The lands which were granted to the States by the General Government for the support of higher education were disposed of in a manner intended to "encourage immigration," rather than to foster a great