The University of Michigan, an encyclopedic survey ... Wilfred B. Shaw, editor.

THE DEPARTMENT OF MATHEMATICS

The History of the Department

WHEN the seven young men who constituted the first student body came to Ann Arbor in the fall of 1841 to enter the newly organized University of Michigan, one of the two professors who welcomed them was the Reverend George Palmer Williams (Vermont '25, LL.D. Kenyon '49), Professor of Mathematics. To this year may therefore be assigned the birth of the Department of Mathematics.

Professor Williams, who was thirty-nine years old at that time, had had a varied teaching experience as principal of the preparatory school at Kenyon College, Ohio, for four years, and as a teacher of ancient languages in Western University of Pennsylvania and at Kenyon College for six years. He came to Ann Arbor from Pontiac, where he had been principal of the University's Pontiac branch since 1837 (see Part I: Branches) . When the first class graduated in 1845 Professor Williams, as President of the Faculty, gave the members their diplomas.

The history of the Department of Mathematics may be divided into four periods, the first extending from the entrance of the first class in 1841 to the appointment of Edward Olney in 1863, the second to the death of Professor Olney in 1887, the third to the death of Professor Beman in 1922, and the fourth to the present time.

The first period, 1841-63. — For a number of years after 1843 George P. Williams was listed in the annual Catalogue as Professor of Natural Philosophy and Mathematics. The total enrollment in the 1840's was so small that he was able to conduct all of the classes in both of these subjects, but as the University attracted more students it became necessary that his burden of teaching should be divided. Two professorships were formed from his chair in 1854, and his title was changed to Professor of Mathematics. In 1854-55 there were sixty-three freshmen — almost as many students as were in the other three classes combined. The following year Williams was assisted in mathematics by Alexander Winchell, then Professor of Natural History, and William G. Peck, Professor of Physics and Civil Engineering. In May, 1856, William Petit Trowbridge (U. S. Mil. Acad. '48, Ph.D. hon. Princeton '79, LL.D. Michigan '87) was also appointed Professor of Mathematics, but he served for only one year, and later had a distinguished career in engineering. One of the University's own graduates, John Emory Clark ('56, A.M. '59), was Assistant Professor of Mathematics for the following two years. During the remaining four years of the period Williams was assisted by James Craig Watson ('57, Ph.D. Leipzig '70, LL.D. Columbia '77). Watson, a gifted mathematician, was only twenty-one years old when he became Instructor in Mathematics and temporary Professor of Astronomy in 1859. He later brought fame to Michigan as Professor of Astronomy and Director of the Detroit Observatory (see Part III: Department of Astronomy) .

In this early period the work in mathematics included algebra and geometry in the freshman year, and plane and spherical trigonometry and some analytical geometry and calculus in the sophomore year. The books used, Davies' translations of Bourdon's Algebra and Legendre's Geometry, Davies and Loomis' Trigonometry, Loomis' Analytical Geometry, and Loomis' Differential and Integral Calculus, give some idea of the ground covered in the early courses. The

new University soon showed its broadminded attitude toward education by providing a scientific course parallel with the classical course. As early as the year 1854-55 the number of freshmen in the scientific course exceeded the number pursuing the classical course, this year marking the beginning of the popularity of the new course. The number of upperclassmen who had chosen it was only eleven — less than one-sixth of the enrollment in the junior and senior classes. The curriculum for students enrolled in the scientific course included an additional term of calculus and such applied subjects as surveying, navigation, descriptive geometry, and drawing and architecture.

Even in these very early days extensive plans were in mind for a more advanced development:

university course

This Course … for those who have taken the degree of Bachelor of Arts or … Bachelor of Science … when completely furnished with able professors and the material of learning, will correspond to that pursued in the Universities of France and Germany.

Cat.

A skeleton outline of the proposed course lists twenty subjects of study, in which higher mathematics occupies the seventh place. The first step toward a realization of this more advanced program was signalized by the appearance, in the Catalogue of 1856-57, of a list of books for reference. In the higher mathematics mention was made of Church's Analytical Geometry, Church's Differential and Integral Calculus, Courtenay's Calculus, and Peirce's Curves and Functions.

In the Catalogue of 1858-59 appeared the announcement of a "Programme of Studies for the Degrees of A.M. and M.S." Professor Williams offered further work in calculus, and Assistant Professor Clark offered courses in the same program — Higher Algebra; Calculus, which proposed to give a general view of definite integrals, differential equations, including the theory of singular solutions, and partial differential equations; and Method of Variations. It is interesting to speculate what the results might have been if Assistant Professor Clark had remained to perfect this ambitious program.

With a staff consisting of Professor Williams and Instructor Watson it was inevitable that the emphasis during the remaining four years of this first period should be on applied mathematics. Watson, in addition to his instructorship in mathematics, was Professor of Astronomy in 1859-60 and then for three years was Professor of Physics. He offered as Physics in the graduate program analytical mechanics and the mathematical theory of heat, light, and sound.

The second period, 1863-87. — The fall of 1863, with the departure of Brünnow as Professor of Astronomy and the appointment of Edward Olney (A.M. hon. Madison University '53, LL.D. Kalamazoo '73) as Professor of Mathematics, marks the beginning of the second period. Watson became Professor of Astronomy and Director of the Detroit Observatory, and Williams, Professor of Physics, a position which he held until 1875. From that year until his death in 1881 he was Professor Emeritus.

At the opening of the second period Professor Olney and one instructor constituted the staff. Olney was thirty-five years old when he came to the University to take charge of the work in mathematics. He was not a college graduate, but had educated himself while working on a farm. He had served five years as principal of the Perrysburg Union School in Perrysburg, Ohio, and for ten years had held the chair of mathematics in Kalamazoo College. During his first two years at the University of Michigan Olney

was assisted by Allen Jeremiah Curtis (Kalamazoo '60, A.M. Michigan '61), who had been his colleague on the Kalamazoo College faculty. At the University Curtis was an instructor in both mathematics and rhetoric. He was succeeded in 1865 by William Butler Morgan (Haverford '53, A.M. ibid. '57, Michigan '63e), who remained only one year and taught civil engineering as well as mathematics. Then for five years George Benjamin Merriman (Ohio Wesleyan '63, A.M. Michigan '64), Assistant Professor of Mathematics, divided the classes with Professor Olney. Merriman had studied law and had been admitted to the bar. In 1871 he left the department, but continued in the University, in the Department of Physics, through 1874-75. During a part of the time that Merriman taught in the Department of Mathematics Mark Walrod Harrington ('68, A.M. '71, LL.D. '94), an instructor, gave a part of his time to the subject, though his interests were rather in the natural and physical sciences. He definitely entered the fields of geology and biology in 1872 and was later a noted astronomer and meteorologist (see Part III: Meteorology and Department of Astronomy ).

In 1871-72 Merriman was replaced by three instructors — Wooster Woodruff Beman ('70, A.M. '73, LL.D. Kalamazoo '08), Edward Laurens Mark ('71, Ph.D. Leipzig '76, LL.D. Michigan '96), and Marcus Baker ('70, A.M. '76, LL.B. Columbian University '96). Of these, Beman was the only one who remained, Mark leaving after one year and Baker after two.

The staff took on greater permanence during 1874-75, for Olney and Beman, who now became an assistant professor, were joined by Charles Nelson Jones (Oberlin '71), and these three, with the addition of an occasional assistant or instructor, constituted the staff of the department for the remaining thirteen years of the period.

The undergraduate curriculum in mathematics at the beginning of this second period was much like that which had obtained from the earliest days. In the classical course there were plane geometry and trigonometry in the first semester of the freshman year, algebra in the second; plane analytical geometry and differential calculus in the first semester of the sophomore year, and solid geometry and integral calculus in the second; in the junior year there was physics, supplemented by spherical geometry and spherical trigonometry. The curriculum in the scientific course devoted more time to calculus and included descriptive geometry and analytical mechanics. Ray's Plane and Solid Geometry and Ray's Algebra, Part II, replaced some of the earlier texts, and soon some of Olney's own books came into use.

The Catalogue for the year 1867-68 contains the following description of the work in mathematics (pp. 63-64):

The course pursued in Pure Mathematics has reference both to intellectual training and the acquisition of practical knowledge. Ray's text-books in Elementary Geometry and Algebra were used in the Freshman year. During this year the student is kept pretty close to the methods of the author; but much stress is laid upon the solution of problems and the performance of all practical exercises tending to promote thoroughness and independent thought. In the Sophomore year, while text-books are used, a part of the course is given by lectures in order to give greater breadth of view, and to develop in the pupil the power of investigation, by following out suggestions made by the lecturer. For General Geometry and Calculus Loomis' class-book is used. In the Scientific courses Davies' Analytical Geometry and Davies' Descriptive Geometry and Warren's Perspective Drawing are used on their several subjects.

Ample provision is made for those who wish to pursue a more extended course of

Mathematical studies, by allowing them to substitute mathematical for other studies, according to the preceding synopsis of the Courses of Instruction. In this way the Geometry, Infinitesimal Calculus, Calculus of Variations, and the Calculus of Finite Differences may be pursued as far as students may desire.

The Catalogue for the following year gives a detailed outline (p. 51), listing with great detail the topics included in the various courses:

VI. — Pure Mathematics Classical Course

Freshmen. — Geometry, Problems in construction, Review of Properties of Triangles, Polygons, Plane Areas, Solid and Spherical Geometry; Algebra, Quadratics, Ratio, Proportion, Progressions, Theory of Indeterminate Coefficients, Binomial Theorem and its application to the development of Functions, Theory and Use of Logarithms, Indeterminate Analysis, and the Elements of the Theory of Equations.

Sophomores. — Trigonometry, Plane and Spherical; General Geometry, Construction of Equations, Production of Equations of Plane Loci, transformation of Coördinates, and the Properties of the Conic Sections; Elective; Calculus, Differential, including Differentiation of Functions of a Single Variable, McLaurin's and Taylor's Theorems with Binomial Theorem and Theory of Logarithms deduced, Maxima and Minima of Functions of a Single Variable, Radius of Curvature, and the Elementary Principles of Tracing Curves; Integral, Elementary Forms, Binomial Differentials, Rectification and Quadrature of Plane Curves, and Surfaces and Volumes of Solids of Revolution.

For the students who took the scientific course there were added topics in algebra: resolution of cubic equations and a more complete view of the theory of equations, including Sturm's theorem and Horner's method of resolving numerical higher equations.

In the civil engineering course there was a further requirement:

Juniors. — General Geometry, Polar Coördinates, Lines, Planes and surfaces in Space including Surfaces of the 2d Order; Calculus, Differential including Functions of Several Variables with a fuller view of the Theory of Curves, and Integral including Functions of two Variables and Special Processes.

When these descriptions were rewritten for the Catalogue of 1872-73, it was possible to mention Olney's Treatise on Special or Elementary Geometry, Part III, and Olney's Algebra, Part III, which had recently appeared, as indicating the ground covered. These texts were fairly complete, beginning with first principles. The Part III in each case embraced the topics especially fitted for university work. The ideal that Olney had in mind for college work in mathematics and for the appropriate high-school preparation is clearly set forth in the following paragraph from the Preface of his Geometry:

Part III, which is contained only in the University Edition, has been written with special reference to the needs of students in the University of Michigan. Our admirable system of public High Schools, of which schools there is now one in almost every considerable village, promises ere long to become to us something near what the German Gymnasia are to their Universities. In order to promote the legitimate development of these schools, it is necessary that the University resign to them the work of instruction in the elements of the various branches, as fast and as far as they are prepared in sufficient numbers to undertake it. It is thought that these schools should now give instruction in Elementary Geometry, which has hitherto been given in our ordinary college course. The first two parts of this volume furnish this amount of instruction, and students are expected to pass an examination upon it on their entrance into the University. This amount of preparation enables students to extend their knowledge of Geometry, during the Freshman year in the University, considerably beyond what has hitherto been practicable. As a text-book for

such students, Part III has been written. At this stage of his progress, the student is prepared to learn to investigate for himself. Hence he is here furnished with a large collection of well classified theorems and problems, which afford a review of all that has gone before, extend his knowledge of geometrical truth, and give him the needed discipline in original demonstration. To develop the power of independent thought, is the most difficult, while it is the most important part of the teacher's work. Great pains have therefore been taken, in this part of the work, to render such aid, and only such, as a student ought to require in advancing from the stage in which he has been following the processes of others, to that of independent reasoning. In the second place, this part contains what is usually styled Applications of Algebra to Geometry, with an extended and carefully selected range of examples in this important subject. A third purpose has been to present in this part an introduction to what is often spoken of as the Modern Geometry, by which is meant the results of modern thought in developing geometrical truth upon the direct method. While, as a system of geometrical reasoning, this Geometry is not philosophically different from that with which the student of Euclid is familiar, and which is properly distinguished as the special or direct method, the character of the facts developed is quite novel. So much so, indeed, that the student who has no knowledge of Geometry but that which our common text-books furnish, knows absolutely nothing of the domain into which most of the brilliant advances of the present century have been made. He knows not even the terms in which the ideas of such writers as Poncelet, Chasles, and Salmon, are expressed, and he is quite as much a stranger to the thought. In this part are presented the fundamental ideas concerning Loci, Symmetry, Maxima and Minima, Isoperimetry, the Theory of Transversals, Anharmonic Ratio, Polars, Radical Axes, and other modern views concerning the circle.

Olney was a master of clear exposition, and the influence of his books was widespread among the high schools of the region.

The Calendar of 1878-79 shows a considerable advance in the direction of our present practice. For the first time, all courses were listed by number and the number of recitations per week was stated. The list (p. 31) follows:

• first semester
  • 1. Advanced Algebra. Four times a week.
  • 2. General Geometry and Calculus. Four times a week.
  • 3. Advanced General Geometry and Calculus. Five times a week.
  • 4. Determinants. Once a week.
• second semester
  • 5. Advanced Geometry; Plane and Spherical Trigonometry. Four times a week.
  • 6. General Geometry and Calculus. Four times a week.
  • 7. Modern Geometry and Trilinear Co-ordinates. Twice a week.
  • 8. Calculus of Variations. Twice a week, first half of semester.
  • 9. Quaternions. Twice a week, second half of semester.

In 1879-80 a two-hour course in trigonometry and a four-hour course in analytical mechanics were added, and the following statement appeared (p. 33):

It is proposed to add a two years' course of Mathematical reading as soon as there shall be a sufficient demand for it, in such standard works as Salmon's Higher Algebra, Frost's Solid Geometry, Doctor's Determinants, Todhunter's or Price's Integral Calculus and Taite's Quaternions.

Two years later this reading course was listed as Course 17, and Scott's Determinants and Routh's Rigid Dynamics were mentioned among the suggested books. The course in analytical mechanics was extended to two semesters, the second being devoted to dynamics. Two-hour courses in advanced algebra and geometry were also added, but these represented rearrangements of work already given with but little new subject matter. In 1885-86 the course in geometry was extended to two semesters, and included

synthetic projective geometry. It was, however, a course intended for freshmen. A two-hour course in the elements of the theory of functions, including elliptic functions, was added, completing the list of courses offered in 1886-87, the end of the second period.

Professor Olney was well aware that the department, to do effective work, must be provided with library facilities and other equipment. The inadequate library appropriations permitted few additions to the small group of mathematical books included in the original 3,700 bought in Europe by Asa Gray. The records show that in 1881 Olney addressed the Board of Regents on the needs of the Department of Mathematics, recommending the purchase of additional textbooks. Crelle's Journal was mentioned as being particularly needed. An appropriation of $500 was made for the purchase of the complete set of this Journal, but the appropriation was not used, for Mr. E. C. Hegeler, of La Salle, Illinois, placed the same amount at Professor Olney's disposal for this purpose. On two earlier occasions Olney had asked for funds to procure means of illustration in his department and had been granted $50 in 1864 and $70 in 1865. Some of the models included in the department's collection were doubtless part of these early purchases.

Two circumstances point to the conclusion that the work in mathematics was exacting in those early times: Professor Olney was commonly known among the students by the nickname of "Toughy," and in November, 1880, the Board of Regents found it desirable to pass a resolution directing an investigation of the subject, as follows:

Resolved, That the Literary Faculty be requested carefully to examine as to the amount of time required by the students to prepare their mathematical recitations, and to see that these studies do not interfere with others equally important and necessary to a course of liberal education.

R.P.

The third period, 1887-1922. — Upon the death of Professor Olney, Associate Professor Beman was promoted to a professorship in mathematics and Assistant Professor Jones was named Professor of Applied Mathematics. With these changes the department entered the third period. Jones held his position only during 1887-88.

By the following year, 1888-89, of the three who had constituted the staff for the last twelve years Beman alone remained; with him were associated as instructors Alexander Ziwet (C.E. Karlsruhe Polytechnic School '80), Charles Puryear (Richmond '81, B.S.[C.E.] Virginia '85, LL.D. Daniel Baker Coll. '14), and Frank Nelson Cole (Harvard '82, Ph.D. ibid. '86), the first doctor of philosophy to have a place on the staff. Both Cole and Ziwet took an active part in the New York Mathematical Society, their names appearing as authors of reviews in the first volume of its Bulletin. Some of the instructors appointed in these years remained only a short time, but the staff was gradually acquiring a more stable character. The appointments that proved more permanent were those of Joseph Lybrand Markley (Haverford '85, Ph.D. Harvard '89) and Elmer Adelbert Lyman ('86, LL.D. Berea College '18) in 1890, Arthur Graham Hall ('87, Ph.D. Leipzig '02) in 1891, James Waterman Glover ('92, Ph.D. Harvard '95) and Edwin Charles Goddard ('89, '99l) in 1895, and William Henry Butts ('78, Ph.D. Zurich '07) in 1898. Cole left in 1895 to accept a professorship in Columbia University. In 1898 Lyman went to Michigan State Normal College.

The year 1899-1900 closed with the staff made up of Professor Beman, Junior Professor Ziwet, Assistant Professor Markley, and four instructors — Hall,

Glover, Goddard, and Butts. Goddard had been studying law at the same time that he was Instructor in Mathematics.

The next few years witnessed a remarkable growth in the staff; there were so many changes that it seems desirable to record by name only those instructors who were later promoted. Walter Burton Ford (Harvard '97, Ph.D. ibid. '05) came as Instructor in 1900-1901. In 1901-2 Junior Professor Ziwet was placed in charge of mathematics for engineering students, a step which initiated the gradual separation of the work in mathematics into two distinct departments (see Part VII: Department of Engineering Mathematics) . The three instructors added in 1903-4 who remained more than the one year were: Archie Burton Pierce (California '90, Ph.D. Zurich '03), Theodore Rudolph Running (Wisconsin '92, Ph.D. ibid. '97), and Peter Field (Minnesota '96, Ph.D. Cornell '02). Louis Charles Karpinski (Cornell '01, Ph.D. Strassburg '03) and John William Bradshaw ('00, Ph.D. Strassburg '04) were appointed instructors in 1904-5, and in the same year Ziwet was made a professor. Of those who later advanced in the department, one was added as an instructor in each of the next four years — Clyde Elton Love ('05, Ph.D. '13) in 1905-6, Louis Allen Hopkins (Butler College '05, Ph.D. Chicago '14) in 1906-7, Vincent Collins Poor (Kansas '01, Ph.D. ibid. '15) in 1907-8, and Frank Howard Stevens (Chicago '08) in 1908-9. Markley became a full professor in 1907. Butts, though continuing to teach mathematics in the Literary Department, became Assistant Dean in the Department of Engineering in May, 1908. Pierce was transferred to the civil engineering staff in the autumn. At the same time Bradshaw dropped his administrative work (see Part II: Office of the Registrar) , and Arthur G. Hall, who had been a member of the mathematics faculty for several years before 1903, returned to the University as Professor of Mathematics, Registrar of the Department of Literature, Science, and the Arts, and Editor of University Publications. Several promotions having been made, the staff in 1908-9 consisted of four professors, two junior professors, five assistant professors, and nine instructors.

Theophil Henry Hildebrandt (Illinois '05, Ph.D. Chicago '10) came as Instructor in 1909-10, and Carl Jenness Coe ('09, Ph.D. Harvard '29) in 1910-11. In 1911 Glover was promoted from an associate professorship to a full professorship of mathematics and insurance.

The following five men joined the staff in the next five years: Louis Joseph Rouse (Princeton '08, Ph.D. Michigan '18) and Tomlinson Fort (Georgia '06, Ph.D. Harvard '12) in 1913-14, Alfred Lewis Nelson (Midland College '11, Ph.D. Chicago '15) in 1915-16, Harry Clyde Carver ('15) in 1916-17, and Rainard Benton Robbins (Indiana '09, Ph.D. Harvard '14) in 1917-18. Meanwhile, in 1914-15, the title of junior professor was changed to that of associate professor. In 1915-16 Stevens was transferred to the Department of Engineering Mechanics, and Hopkins accepted, in addition to his teaching duties, the position of Secretary of the Colleges of Engineering and Architecture. In 1917-18 Ford was named Professor of Mathematics, and Tomlinson Fort accepted a position at the University of Alabama. William Wells Denton ('07, Ph.D. Illinois '12) came as Instructor in the fall of 1918, and between 1919 and 1921 Field, Karpinski, Butts, and Running were promoted to professorships. Norman Herbert Anning (Queen's University '05, A.M. ibid. '06) came as Instructor in 1920-21. The following year, there were ten professors, four associate professors, seven assistant professors, and fifteen instructors.

Because of the practice of introducing new courses, often with but slightly modified content, and of giving them new numbers, the list of courses in 1886-87 presents a very confused picture: Courses 10, 12, and 15, which together covered much the same ground as Course 5, were for freshmen, but Courses 2 and 3 were more advanced. In 1887-88 a partial renumbering somewhat improved the situation.

Beyond a good two-year program in trigonometry, analytical geometry, and elementary calculus, only four courses were offered besides the reading course. These were the five-hour course Analytical Mechanics, a two-hour course called Modern Algebra, Differential Equations, a two-hour course, and the three-hour course designated Quaternions. There were eight freshman sections handled by the two instructors; Professors Beman and Jones took care of the other courses. With the coming of Dr. Cole in 1888-89 two new two-hour courses were offered, Mathematical Theory of Elasticity and Elements of the Theory of Functions.

Changes in textbooks and in the names of courses were also made soon after Professor Beman took charge of the department; the term "general geometry," introduced by Olney, disappeared. Olney had a strong antipathy to the name "analytical geometry," as he maintained that its method was no more analytical than that of the so-called "synthetic geometry." He distinguished "special" or "elementary" geometry, which deals with the properties of a particular curve, especially the circle, from "general" geometry, which treats of the common properties of curves and develops methods applicable to the investigation of all curves. The latter is aptly illustrated by the solution of the problem, to find the tangent at a given point on any curve.

Beman did not care to retain this terminology. He accepted the use of the prevalent name of "analytic geometry," and soon replaced Olney's texts, especially with those of English authors. C. Smith's Treatise on Algebra, his Conic Sections and Solid Geometry, Loney's Trigonometry, Williamson's Calculus, and Edwards' Integral Calculus for Beginners were in use in the nineties. A little later, Beman, in collaboration with David Eugene Smith, published a series of texts, but these were for secondary schools — no books of college grade were included.

Two years were sufficient to demonstrate that the new numbering of courses was not satisfactory, and a complete revision took place in 1889-90. Courses in trigonometry, algebra, analytic geometry, calculus, and analytical mechanics were numbered consecutively from one to six, with modifications indicated by an appended "a"; elective courses bore the numbers from seven to thirteen inclusive. Among these we find but one new course, one of two hours in modern geometry. The changes of the next two years merely lengthened courses already included in the list. The University had been empowered to grant the teacher's certificate in 1891, but long before that some of the departments had been giving teachers' courses, and as early as 1880 the teacher's diploma was awarded in certain subjects.

In 1892-93 the department, recognizing its responsibility for the preparation of teachers for the secondary schools, introduced two courses for teachers — Teachers' Seminar in Algebra and Teachers' Seminar in Geometry. There were 128 Michigan high schools on the accredited list of the University in 1893. These teachers' seminars included a review of the content of high-school courses with occasional reference to more advanced points of view, together with a few lectures on the history of mathematics. Each student was expected to write a criticism of some text selected from Professor

Beman's large collection. To one who was willing to undertake the task, a text in a foreign language was assigned.

New courses offered in the years that followed included Fourier Series and Spherical Harmonics, Theory of Substitutions, Partial Differential Equations, Theory of Numbers, Theory of Invariants, and Theory of the Potential.

The present statistical and actuarial work in the department began in 1902-3, when Glover offered three courses in the theory of annuities (see p. 654) .

Before the close of the third period, January, 1922, there were other developments in the curriculum. Professor Ziwet, Professor Field, and others worked out a series of courses in applied mathematics — Vector Analysis, Hydrodynamics, and Theory of Elasticity. Professor Ford offered a course in infinite series and products, and another in topics in the theory of divergent series. The teachers' courses were elaborated, and separate courses were introduced in the history of mathematics, graphical methods, and celestial mechanics.

The physical equipment of the department was greatly increased during Professor Beman's administration. During his first year he addressed a letter to the Board of Regents, asking for an appropriation of $500 to buy models and supporting his request by reference to Professor Olney's generosity in turning back into the general fund the appropriation of an equal amount for the purchase of Crelle's Journal. This request was granted, and the purchases made at that time constitute most of the collection of models now in the possession of the department.

The fourth period, since 1922. — Professor Beman began the year 1921-22 in apparently good health, but an attack of arthritis compelled him to turn over his classes to others and was the cause of his death on January 18, 1922. He had completed fifty years of active service as a member of the faculty. Professor Markley was appointed Chairman of the Department of Mathematics in 1922. He directed its affairs until 1926, when, because of failing health, he resigned the chairmanship, but continued teaching for another year before his retirement.

Several important changes in the staff occurred during the chairmanship of Professor Markley. In 1922-23 two instructors came, Ruel V. Churchill (Chicago '22, Ph.D. Michigan '29) and Cecil Calvert Craig (Indiana '20, Ph.D. Michigan '27). Bradshaw and Hildebrandt became professors in 1923-24, and Assistant Professor Robbins left to take a position with the New York State Insurance Department. James Alexander Shohat (Magister of Pure Mathematics, Petrograd '22) came as Instructor in 1924-25. After a long and painful illness, Professor Hall died on January 10, 1925 (see Part II: Office of the Registrar) . Assistant Professor Nelson resigned in 1925 to accept a professorship in the College of the City of Detroit. Ben Dushnik ('24, Ph.D. '31) and Walter Otto Menge ('25, Ph.D. '31) began as instructors in 1925-26, the year in which Alexander Ziwet was made Professor Emeritus, Love was promoted to a professorship, and Field was appointed Chairman of the Department of Mathematics in the College of Engineering.

In the spring of 1926 James W. Glover became Chairman, and immediately set himself to the task of revivifying the department. No new major appointment had been made since the death of Professor Beman, and at first the University attempted to secure a man of national reputation as professor. The attempt having proven unsuccessful, it was decided to add several promising younger men to the staff. In the year 1926-27 three assistant professors were

appointed — James Andrew Nyswander (California '13, Ph.D. Chicago '24), George Yuri Rainich (Magister of Pure Mathematics, Kazan '13), and Raymond Louis Wilder (Brown '20, Ph.D. Texas '23). At the close of that year Markley became Professor Emeritus.

By resolution of the Regents in 1928 the Department of Mathematics in the College of Literature, Science, and the Arts and the Department of Mathematics in the College of Engineering were reunited under the chairmanship of Professor Glover. William Dowell Baten (Baylor '14, Ph.D. Michigan '29) was added as an instructor in the fall of 1928. Alexander Ziwet died on November 18, 1928, and Joseph L. Markley a little more than a year later, April 19, 1930. In 1929-30 two new assistant professors were appointed — William Leake Ayres (Southwestern '23, Ph.D. Pennsylvania '27) and Arthur Herbert Copeland (Amherst '21, Ph.D. Harvard '26). Also in that year Assistant Professor Shohat resigned.

Professor Glover was absent on leave during the two years 1930-32 in order to devote his time to the Teachers Insurance and Annuity Association, of which he was president, and Field was Acting Chairman of the department. Glover, upon his retirement from the chairmanship in the fall of 1934, was named Edward Olney Professor of Mathematics, a distinction conferred in recognition of his services to the University and to the department. T. H. Hildebrandt was designated as his successor. At the beginning of the year the department lost Assistant Professor Denton.

During the five years 1935-40 the following changes in the staff took place: Glover and Ford retired and were named professors emeritus, and Associate Professor Menge and Assistant Professor Baten left the University; Carver, Wilder, and Rainich were appointed to full professorships; Copeland and Churchill became associate professors; and Edwin Wilkinson Miller ('26, Ph.D. '30), Sumner Byron Myers (Harvard '29, Ph.D. ibid. '32), and Paul Sumner Dwyer (Allegheny '21, Ph.D. Michigan '36) were appointed assistant professors. At the close of the academic year 1939-40 the Department of Mathematics had a staff of nine professors, seven associate professors, seven assistant professors, and eleven instructors.

When the most recent period in the history of the department began in 1922, all the main branches of mathematics were represented by courses. The additions to the staff made possible extension in the fields of topology, differential geometry and relativity, modern algebraic theory, and probability. The number and variety of graduate courses has been increased, and the use of the seminar method has been extended. The most striking characteristic of this period has been the emphasis placed upon research and graduate work. This is reflected in the number of doctor's theses written. Although only eleven doctor's degrees had been conferred up to 1922, seventy-four were conferred in the eighteen succeeding years. Increased interest and activity in mathematical research on the part of members of the staff have naturally accompanied this growth. Other activities of the department have not been neglected. Not only is there adequate provision for those whose interest is along mathematical lines, but the increased need of mathematics in other fields has called for expansion in courses of interest primarily to engineering students and in courses in actuarial science and in mathematical statistics and their applications. The Department of Mathematics is growing not only by furthering its own interests but also by serving the the needs of other departments.

John W. Bradshaw

Courses in Actuarial Mathematics

In this country university courses in the theory of probability, up to the beginning of this century, were largely confined to the solution of questions of a priori probability, that is, throwing of dice, drawing of cards, tossing of coins, and employment of combinations, permutations, substitutions, and the like, to find the numerator and denominator of the fraction expressing the required chance. This approach to the subject was the natural outcome of adherence to English texts and acceptance of the practices of the British school of mathematicians.

The first volume of Biometrika appeared in 1901-2, but it was a long time before the English and Scotch actuaries knew much about the new methods of approach used by Karl Pearson and his followers. The application of the methods of empirical probability to important practical problems, largely social in character — one of which was life insurance — was hardly known to our college and university mathematicians, and little study had been given by any of them to this unlimited field of useful and interesting material awaiting refined mathematical treatment.

It might fairly be said that students of the natural sciences recognized this situation before the mathematicians did. With mathematical equipment unequal to the task, they were trying as best they could to solve problems which they knew could be solved but with which they were unprepared to deal except by methods of elementary mathematical approach. It was this situation which first decided the writer to introduce in the University of Michigan courses in mathematics involving primarily the study of empirical probability.

One of the most important applications of this theory was actuarial mathematics. Early in the present century a number of foreign universities had developed actuarial courses in their departments of mathematics. This was, of course, to a considerable extent, due to national insurance and pension plans already under way. Although the total insurance in force in the United States was more than that of all the rest of the companies in the world put together, no training of technical actuarial content was available in this country. It was, therefore, under most favorable conditions that such courses were started at the University of Michigan.

Personal conversations and conferences made it apparent that the life insurance companies were favorably disposed toward this new plan. Accordingly, in the fall of 1902, the first course in this field was offered in the University of Michigan. It appeared in the University Calendar for 1902-3 as Mathematics 45, "Theory of Annuities and Insurance (II), two hours, Dr. Glover." It was elected by eleven men and one woman, and was given on Tuesday and Thursday in Room 17, University Hall, North Wing (now known as Mason Hall). Oliver Winfred Perrin ('01, A.M. '04), Associate Actuary of the Penn Mutual Life Insurance Company, was a member of this class and the first graduate of the University to enter and remain in the actuarial profession.

These courses were first announced both in the Department of Mathematics and in the Department of Political Economy (see Part III: Department of Economics) . The faculty of the latter department was cordially disposed and was most helpful in the organization of the courses; it co-operated by encouraging election of these courses by students of political economy who had sufficient preparation in mathematics to undertake them with profit. The plan was to supplement the technical courses in actuarial mathematics by courses in

political economy which would develop more fully the social aspects of insurance.

It should be acknowledged here that from the beginning to the present time busy executives and officers of insurance companies throughout the country have given valuable and cordial support to this work and have frequently taken the trouble to come to Ann Arbor and lecture to our classes on various phases of their business. This interest from outside the University has stimulated the students and has undoubtedly contributed much to the success of the venture. The companies also have recognized the training received here by sending their officers year after year to select University of Michigan actuarial graduates for technical positions in various departments of the home offices. Many students from this department have advanced from modest actuarial positions to become secretaries, vice-presidents, presidents, and directors in the important life and casualty insurance companies of this country. University of Michigan students from China, Japan, Mexico, the Philippine Islands, and other countries are now holding responsible positions in such companies organized in their native lands. In a number of cases University of Michigan graduates in actuarial mathematics have organized successful companies which they now head.

Up to the present time about four hundred students have taken all the actuarial courses and most of them are now actively engaged in executive positions of high rank. Not a few of them hold official positions in government insurance offices in this and other countries. Among them are about fifty women graduates, of whom one-third have married and have retired from active business life.

Although the number of professional actuarial graduates is relatively small — about four hundred — the elementary courses necessary to prepare students for advanced actuarial theory led to a new development for students not planning to become actuaries. They wanted an elementary course in financial mathematics, in which the mathematical content included simple and compound interest, annuities, sinking funds, valuation of securities, and depreciation. When such a course was offered, it attracted many students who were interested in the above subjects as a matter of general information and as a preparation for one of the many lines of modern business. This course injected into the classroom work a certain practical interest not ordinarily found in elementary mathematics courses. The result was a steadily increasing demand for Mathematics 51, (now Mathematics 47), for which in some years as many as three hundred students each semester were enrolled. And, since the elementary course in financial mathematics was begun at the University of Michigan in 1902, a similar course has been introduced into the mathematics department of almost every college and university in this country.

These courses at the University were organized and given at first by the writer, but additions to the teaching force were soon required because of the increasing number of students. Most of the work of instruction in actuarial science has been carried on of late by H. C. Carver, C. C. Craig, W. O. Menge, J. A. Nyswander, T. E. Raiford, and R. B. Robbins.

The curriculum in actuarial science has, in effect, developed a small professional field within the Department of Mathematics and a new group of elementary courses in finance, insurance, and statistics which have a strong appeal for many students who do not plan to enter the actuarial profession.

James W. Glover

[Died July 15, 1941.]

Courses in Mathematical Statistics

The first work in mathematical statistics which was offered by this University was presented in a two-hour combined course, Mathematics of Insurance and Statistics, listed in the 1902-3 Announcement of the College of Literature, Science, and the Arts by the Department of Political Economy and Sociology. This course was initiated and taught by James W. Glover, and was listed among the mathematical courses with the note: "For a detailed description of the same, consult this Announcement under Political Economy and Sociology."

Although the description of this two-hour course indicated that an important place was given to statistical theory, subsequent Announcements reveal that the course was developed in the direction of insurance, rather than of statistics. Thus, in 1906-7 and the succeeding years the description was as follows: "This course includes an elementary treatment of the following subjects: Interest, investment securities, averages, mortality tables, annuities, computations of life insurance premiums, and reserves."

The first course that was devoted exclusively to statistical theory was offered in 1912 by Professor Glover, and this date should be regarded as marking the birth of our curriculum in statistics. The following year the course was taught by Edward Brind Escott ('95, M.S. Chicago '96) and in the next two years by Chester Hume Forsyth (Butler '06, Ph.D. Michigan '15). It was described (Cal., 1912-13, p. 213) as follows: "The subjects treated in this course are averages, graphical representation of statistics, frequency curves, correlation, smoothing of statistics; with applications to statistical problems in economics, biology, insurance, and physics." It was given for two hours credit and was continued through the second semester as Course 50. Elderton's Frequency Curves and Correlation, recommended by the Actuarial Society of America, was used as a text, and this work was supplemented by lectures on interpolation and mechanical quadrature.

During recent years research workers in nearly all fields have recognized the necessity of utilizing statistical methods in measuring the validity of results derived from observational data, and consequently a number of courses in statistical methodology are now being offered in an effort to serve the particular needs of the various departments and schools. The Department of Mathematics offers a special course in mathematics and statistics designed to meet the needs of students in the School of Forestry and Conservation, another course for students of sociology, the basic Courses 49 and 50 (now listed as Mathematics 43 and Mathematics 44), for which one year's work in freshman mathematics is a prerequisite, an intermediate course requiring a knowledge of calculus, and an advanced course designed for students working for higher degrees and specializing in the more theoretical aspects of probability and statistics. Until the end of June, 1940, nineteen students had received their doctor's degrees in mathematical statistics, the first doctorate in this field having been conferred in 1915.

The members of the mathematical-statistical staff are constantly being consulted on matters concerning statistical research from all corners of the campus. They also offer informal courses in statistics for staff members of the University who use statistical methods in their researches but who cannot afford the time required to master the mathematical background so necessary for a complete understanding of the statistical methodology which they employ.

The unusual success of this University in teaching and utilizing statistical methods

has been achieved largely through the use of excellent mechanical equipment provided by the administration. Since a computing machine is available for each student in all recitation periods, it is possible for the student to work out numerical exercises simultaneously with the presentation of new topics. The University utilizes two complete Hollerith installations, one in the Rackham Building and the other in the University Hospital. The statistical laboratory possesses a very complete set of instruments such as adding machines, integraphs, and harmonic analyzers.

In addition to the personnel of its mathematical-statistical staff, two other factors have contributed largely to the leadership of the University of Michigan in statistical research. The Department of Mathematics is providing an essential mathematical background through courses in probability, finite differences, and other branches of pure mathematics which are of great value in developments of theoretical statistics. Also, the Annals of Mathematical Statistics — the only journal of its kind in the country and the official publication of the Institute of Mathematical Statistics, with a worldwide circulation — was founded within the University's Department of Mathematics in 1930 and was edited here until 1938, when its editorial office was transferred to Princeton University.

Harry C. Carver

SELECTED BIBLIOGRAPHY

Calendar Catalogue … Catalogue and Register General Register Issue History of the University of Michigan. President's Report Proceedings of the Board of Regents … University of Michigan Regents' Proceedings …, 1837-1864.