- Chapter 7: Graduate Education in Mathematics in 1894
Among the scientists important in the early development of the University of Florida which we discussed in the last four chapters, many still have a kind of presence on campus, even though it might not be noticed by most of the current undergraduates.
Where Grinter Hall now stands, next to Walker Hall, Kermit Sigmon recalls that in the 1960's stood a very dilapidated building named Benton Hall which was torn down for the construction of Grinter Hall in 1969.
But then a building in the present engineering complex, was named John R. Benton Hall. This building currently serves as a laboratory building for Electrical Engineering, fittingly enough. Among other facilities in this building, are the Florida Solid State Electronics Laboratory, the Micro-electronics Laboratory (which has a commemorative plaque of its own as the Rosser Laboratory), the Electronics Laboratory, the University of Florida Power Affiliates, the Power Systems Simulations Laboratory, and the Noise Research Laboratory. We also still have today on campus the Benton Engineering Council, the Engineering College Student Council, which has undoubtedly evolved from the Benton Engineering Society, found in the Seminole yearbook as early as 1915. There is no plaque marking the date of construction of the newer Benton Hall, with the Governor of Florida and Board of Control at that time grandly inscribed, as for the older campus buildings like Walker Hall and even Grinter Hall, but we find the following memorial plaque to Dean Benton in one of the corridors near an entrance to the new Benton Hall:
In Memoriam John Robert Benton, Ph.D. First Dean of the College of Engineering 1910--1930 University of Florida He being dead yet speaketh
Dean Thomas Simpson has a less grand presence on campus in the form of Simpson Hall, a dormitory in the Graham Residence Area. Nestled between a Coca-cola machine and a snack machine on an outer wall of this building, we find the following memorial plaque to Dr. Simpson:
THOMAS MARSHALL SIMPSON EMINENT SCHOLAR, TEACHER AND AUTHOR THOMAS MARSHALL SIMPSON SECOND DEAN OF THE GRADUATE SCHOOL AND FOR MANY YEARS HEAD OF THE DEPARTMENT OF MATHEMATICS CONTRIBUTED SIGNIFICANTLY TO THE EDUCATION OF STUDENTS AT THE UNIVERSITY OF FLORIDA. HIS SCHOLASTIC INTEGRITY STANDS AS A BEACON FOR THEM WHO SEEK KNOWLEDGE.
On this marker, unlike that of Dean Benton, no dates are given. Also, Simpson's second wife has left materials with the Center for Florida Studies, which is a second evidence of Simpson's presence on campus.
While no building on campus is named after Dr. Franklin Kokomoor, just during the spring of 1994, the University of Florida Chapter of the Phi Kappa Phi honorary society established seven scholarships which were to be named after distinguished past presidents of the Florida Chapter of Phi Kappa Phi. One of these scholarships so established is the Franklin W. Kokomoor Scholarship.
The first Professor of Mathematics and Astronomy, Dr. Karl Schmidt has a rather less public presence on campus than the three scientists just discussed. The main sources attesting to Schmidt's presence are the information found in the 1905--1906 Record as reported in Chapter 3; the recruitment letter from President Sledd, reported in Chapter 2 and further correspondence at the Center for Florida Studies; and Schmidt's book From Science to God in Smathers Library. Unfortunately, the Seminole yearbook was not issued until 1910, so we have seen no photograph of Professor Schmidt in any campus sources.
No building on campus or honorary society has been named in honor of our second Head Professor, Dr. Herbert Keppel. But he survives in the Seminole yearbooks from 1910--1918 and also from Mrs. Benton's reminiscences, as recorded in Chapter 5. Even before I had discovered Mrs. Benton's account of Herbert Keppel, one afternoon when I was working in the Library Archives, the gentleman Carl Van Ness, Assistant University Architect, who had first steered me to the 1911 Catalogue and Herbert Keppel as being the Professor of Mathematics and Astronomy during my first time ever to set foot inside Library East, told me that he recalled that some diaries of Herbert Keppel were contained in storage in the University Stadium and that these things could be retrieved for me, if I wished. Naturally, I jumped at the chance, eagerly looking forward to reading Keppel's comments about the teaching load and student body at the University of Florida between 1908 and 1918.
Unfortunately, it turned out that the material in the Archives is not a diary describing events at the University of Florida at all, but rather two volumes of notes Keppel took at Clark University while a graduate student in mathematics. Indeed, as mentioned in Chapter 5, the first page of this material in Keppel's own handwriting in the first portion (as packaged by the Archives) is inscribed with the following title:
Mathematics and Pedagogy Notes taken while reading Clark University, April 20, 1894and this dating is consistent with the chronology given in the 1909 Record for Professor Keppel's educational experience prior to joining the University of Florida. Especially, this notebook reveals that Keppel devoted a good portion of his studies to high school and university curricular matters and teaching methods.
The second portion of this material is untitled. An unsolved puzzle is also presented by this notebook. For in the first part of this material, Keppel only wrote on every other side of the pages. Someone else who obtained possession of Keppel's notebook has written out quizzes for various mathematics classes, including trigonometry and mathematics of finance.
There is even placed in the middle of this notebook a yellow flyer, which turns out to be The Orange and Blue Florida Daily Bulletin, Vol. III, No. 7 for Tuesday, September 18, 1928. On the back of this flyer, are found more drafts of quizzes and tests.
Since I am confident from evidence provided by the 1955 American Men of Science, that Dr. Simpson did indeed arrive at the University of Florida in 1918, and Dr. Keppel is no longer to be found in the yearbook, I also have confidence in Dr. Sam Proctor's statement that Dr. Keppel died in October, 1918.
Unfortunately, this second person who was writing down all these examinations in Keppel's note book a decade or so after Keppel's death, did not once identify himself or herself by name. In any event, if anyone wants to know what tests were like in the 1920's at the University of Florida, this material awaits them in the University Archives. The Orange and Blue Florida Daily Bulletin which accidentally thus came to light is itself interesting because it gives an example of the Chapel Programs that were held in those times.
The student body was directed to be seated in the University Auditorium by classes, the faculty was to be in place on stage, and the following program was promised:
Organ Prelude | |
Hymn | |
Scripture and Prayer. | Dr. T. V. McCaul |
Introduction of President Tigert | by Dr. Farr |
Address by President Tigert | |
University Songs, | led by University Cheer Leader, Keezell. |
Let us return to Keppel's notebook and see what sort of curricular matters were of concern to our profession in the late nineteenth century. As mentioned in Chapter 5, the first portion of this notebook consists of Keppel copying apparently in its entirety, a study on curricular matters called the Report of the Committee of Ten. Professor John Kenelly, Professor Charles Nelson, and Dean Bob Burton Brown all told me that such names were common for national educational reports in those days.
Kenelly told me that educational concerns of that era included setting national standards for high schools so that the colleges and universities would not have to test individually all applicants for admission to their institutions. Indeed, in my browsing through the University of Florida Records of the 1910's, I found that students from approved senior high schools of Florida could gain admission by certificate; the principal had to fill out a standardized form available from the University.
A later catalogue contained a list of certified high schools. If a student had not attended a certified high school, then he was obliged to take entrance examinations. The Committee of Ten according to Keppel's notes contained university faculty, and high school and preparatory school principals. The Committee was chaired by Professor Newcomb and included Professor Byerly of Harvard (Harvard has a Byerly Hall), Professor Cajori of the University of Colorado, Professor Fine of Princeton University, Professor Olds of Amherst College, and Professor Safford of Williams College.
The Table of Contents of this report as copied by Keppel was as follows:
- General Statement and Conclusion
- Special report on teaching arithmetic
- Special report on teaching concrete geometry
- Special report on teaching of algebra
- Special report on teaching formal geometry
- A radical change in teaching arithmetic. The course should be (a) abridged and (b) enriched.
- Abridged by omitting entirely those subjects which perplex and exhaust the pupil without affording any really mental discipline.
- Enriched by a (1) greater number of exercises in simple calculation and (2) in the solution of concrete problems
- Subject matter to be curtailed or omitted:
- compound proportion
- cube root
- abstract mensuration
- obsolete denominate quantities
- the greater part of commercial arithmetic
- Percentage should be reduced to needs of life
- In profit and loss, simple and compound interest examples not easily intelligible to the pupil should be omitted
- Fractional periods of time in compound interest are useless and undesirable
Here are three further paragraphs from this document:
``Colleges should supplement their written admission examination in geometry by oral ones; and a substantial part of the examination, whether written or oral, should be devoted to testing the ability of the candidate to construct original demonstrations.'' ``The binomial formula for fractional and negative exponents had better be reserved for college; for positive integer exponents, the pupil should verify it by actual multiplication.'' ``Also either in school or college synthetic or projective geometry should be taught.''
Following up on the Report of the Committee of Ten, Keppel made notes on admission requirements of various universities and colleges, often with recommended texts given. Thus the following is found for M.I.T.:
``In June, 1894 and thereafter, applicants will be required to pass an examination in Solid Geometry or in Advanced Algebra. It is the intention of the Faculty to require both of these subjects at no distant date.''
A goodly number of colleges on Keppel's list required Wentworth's Elements as the required geometry text. It is then interesting that elsewhere in Keppel's notebook, he comments on his experiences while teaching a course in geometry at the Academy of Northwestern University during the time period September to June 1895--96 using Wentworth's Plane and Solid Geometry. [This entry points up a difficulty with possibly inaccuracies in the University of Florida Record, for this source places Keppel at Northwestern as an Instructor only at a slightly later date beginning in 1896.] Keppel notes which theorems or constructions gave the students difficulty, improvements to proofs, and corrections to the text.
As mentioned in Chapter 2, after Keppel studied the material on the Report of the Committee of Ten, he copied or translated material concerning educational matters from various journals. He translated first from the German an article by Dr. Rethwisch, Deutschlands hoheres Schulmesser in neunzehnten Jahrhundert. Then he took notes on an article by Professor T. H. Safford published in the October, 1893 Bulletin of the New York Mathematics Society on ``Instruction in mathematics in the United States.'' Keppel copied an article comparing curricula in American, French, and German high schools. Then he took notes on an article by Professor Alexander Zimet published in the October, 1891 Bulletin of the New York Mathematical Society on ``The Teaching of Elementary Geometry in German Schools.'' This article contains the comment that an inferior geometry text by Kambly was in the 74th edition and used in 217 German high schools. This article recommends, rather, the use of a much less popular book Jacob Falke, Propadeutik der Geometrie, published in 1866 in Leipzig by Quandt and Haindel. This entry is also interesting, because we shall later see that in Graduate Research Professors A. D. Wallace's address to the Florida chapter of the Phi Beta Kappa in 1971, one of the fancy (by present day standards) vocabulary words Wallace employed is this same propaedeutic.
The next portion of Keppel's notebook consists of a listing of various mathematics departments throughout the entire world, together with their faculties, at differing dates. Keppel's own heading for this material is Professors of Mathematics as given in Minerva 1894--95. From this source we learn that in the late nineteenth century, it was common for educational institutions to have a position of Professor of Mathematics and Astronomy, even if the department contained more than one Professor. We also find that in 1837, the Alexander Zimet referenced above was an Assistant Professor at the University of Michigan which even then had a faculty of 2 Professors, 2 Assistant Professors, and 4 Instructors. There, however, the titles of the two professors were Professor of Mathematics, and Professor of Descriptive Geometry and Drawing. The Department of Mathematics at Johns Hopkins had a Professor of Pure Mathematics, a Professor of Mathematics, and a Professor of Mathematics and Astronomy.
It is also interesting to see in Keppel's notes, that at the time the Faculte de Bordeaux was surveyed for this listing in the Minerva, that Hadamard was a Dozent in Astronomy and Mechanics. Harvard could be proud of the following four Professors: James Mills Peirce, Professor of Astronomy and Mathematics; Charles Joyce White, Professor of Mathematics; William Elwood Byerly, Professor of Mathematics; and Benjamin Osgood Peirce, Professor of Mathematics and Natural Philosophy.
The next portion of the notebook changes character. In retrospect, after reading this second portion of Keppel's notebook completely, we may deduce that the Mathematics Seminar at Clark University met on Saturdays and what Keppel was doing in this later portion of his notebook was first, taking notes at the lectures in the Seminar, and secondly, preparing materials for his own presentation. Also, dates in this section indicate that parts of this material were written down in 1893, before the first portion on curricular matters which I have described above. This second notebook begins with several lectures by Dr. Henry Taber, delivered starting at 4 pm on Friday, April 14, 1893 on the topic of Symbolic Logic. [Taber was perhaps William Story's best Ph.D. student at John Hopkins, taking the degree in June, 1888. When Story was selected as the founding Professor of Mathematics at the new Clark University, Taber joined him as the docent, Oskar Bolza as the associate, cf. [2].] The first page contains a reference to a research article of De Morgan, ``On the structure and syllogism,'' published in the Cambridge Philosophical Society in Volume 8. Also an article of Peirce in the American Journal of Mathematics, Volume 7, and Peirce's Studies in Logic are cited. This part of the notebook contains beautifully drawn Venn diagrams. In Dr. Taber's estimation,
``Charles Peirce has written ably on Boolean calculus,'' as recorded by Keppel.
The next portion of the notebook deals with questions in analytical and constructive plane geometry. The first lecture, delivered by Dr. Taber on February 18, 1893 is entitled To express mathematical properties graphically. References include Clifford's Mathematical Papers, Salomon's Conic Sections, and work of Poncelet. Taber gave the following definition:
`` Metrical Theorems: these are theorems which require the measurement of lines or angles to be taken into consideration.''
Projective geometry and something called the harmonic conjugate involving distances between points plays a role in this mathematics. Keppel records the following conclusion to this course of lectures:
``We thus see that any angle may be expressed graphically. The property of an angle which is not destroyed by projection is then---that if from the vertex of the angle lines be drawn to the two circular points at infinity, these lines will form an harmonic range or ratio, which divided by -2 sqrt{-1} gives the value of the angle.
Dr. Taber said in conclusion,
`I believe this is one of the most wonderful theorems in Geometry. I think it is due to Laguere; the whole subject is due to Poncelet.' ''
These lecture notes are followed by reading notes on Dowling's Notes on Analytical Metrics, then by a presentation of Mr. Hill on Saturday, March 4, 1893, in which Hill discussed the problem:
``express graphically the theorem that the three angles of a triangle are equal to two right angles.''
Again the anharmonic ratio and complex variable arithmetic was used in this demonstration. During this same seminar, Mr. Nichol's also spoke about the problem to
``Express graphically the statement that the tangent is perpendicular to the radius at the point of contact in a circle.''The demonstration employed the harmonic conjugates.
Next Keppel took reading notes from a book entitled Carr's Synopsis. [you information superhighway junkies, I pulled down the LUIS to track the date of this reference and found 755 entries under CARR; too much trouble.] Here are three examples of the sort of thing discussed:
- Section 4717.
- All circles pass through the same two imaginary points at infinity and through two real or imaginary finite points.
- Section 4718.
- Concentric circles touch in four imaginary points at infinity.
- Section 4722.
- Any two lines including an angle theta form, with the lines drawn from the two circular points at infinity to their point of intersection, a pencil of which the anharmonic ratio is exp(i(pi - 2 theta)).
Next Keppel took reading notes in French from the book M. Chasles, Apercu des Methodes de Geometries, Paris, 1875.
Then, apparently also in preparation for his own Seminar lecture, Keppel starts consulting the German equivalent of the current Mathematical Reviews, the Jahrbuch uber die Fortschritte der Mathematik, published by the Prussian Academy of Science in Berlin, starting first in 1868, thus even predating the Zentralblatt, which only started publishing in the 1930's. With the Jahrbuch material, sometimes Keppel copied the German text, sometimes he translated it into English. He surveyed the works of the following authors: J. Frischauf, Die Geometrischen Constructionen von Mascheroni und Steiner; De Coatfront, Sur la geometrie de la regle, Nouvelle correspondance de mathematiques III, 204--208; F. Kessler, Beitrage zur Geometrie des Zirkels; F. Bessel, Grundzuge der Geometrie des Cirkels, Archiv der Math. und Physik von R. Hoppe, LXVII, 44--63; and finally, J. S. Mackay, Solution of Euclid's problem with a rule and one fixed aperture of the compass by the Italian geometers of the sixteenth century, Edinb. Math. Soc. Proc. V, 2--22. The next portion of Keppel's notebook contains material translated from the German taken from the collected works of Jacob Steiner, specifically from Die geometrischen Constructionen ausgefuhrt mittelst der geraden Linie und eines festen Kreises, als Lehrgegenstand auf hoheren Unterricht-Austalten und zur practischen Benutzung. The following amusing statement, as translated by Keppel, is to be found on the applicability question in those days:
``If the constructions of Mascheroni are of great benefit to the mechanic and the manufacturer of astronomical instruments, then it is reasonable to believe that the present work will be of benefit to engineers and surveyors.''
Now we come to the final portion of Keppel's notes in this volume, which I have copied in their entirety, for this is Keppel's preparation for his own lecture in the Saturday seminar after all this studying of these French and German sources.
Restricted Conditions in Geometrical Constructions (A synopsis of a report made to the Math Sem at Clark University in 1894)
Preliminary
With a ruler we can- Draw a straight line when two points of it are given.
- Find the point of intersection of two lines
- When one of the lines is given entire and the other only by two of its points
- When both lines are only given by points which determine them
- When a circle is in the plane to find the intersection of it with a line of which only two points are given
- Draw a circle or arc when size and position are given
- Find the points of intersection of two circles
- When one is given entire in the plane and the other has only its size and position given
- When both have only their size and position given
- When a line is given in the plane to find the intersection of a circle with it when the position and size of the latter are given
- To lay off agiven length on a given line.
- Compass with fixed opening--Pappus, etc.
- Ruler alone--Brianchon, and Schroten (16?? - 1659)
- Ruler and compass with fixed aperture---Italians of the 16th century
- Compass alone---Mascheroni
- Ruler and fixed circle in the plane--Steiner
The ancients considered a problem in geometry as capable of solution when its construction required no other apparatus than the ruler and compass. They seem however to have known that for some problems at least the conditions could be further reduced.
Cantor (Vorlesung uber Geschichte I, p. 383) says that passages in the works of Pappas of Alexandria show plainly that the Greeks were acquainted with a geometry which admitted of a single opening of the compass.
Several mathematicians have also since then tried to solve geometrical problems by assuming fewer conditions than those required by the ancients, or by somewhat changing the conditions. The several efforts may be summed up under the following six heads:
- Using one fixed aperture of the compass
Besides Pappus and the Grecian geometers, we know that Abul Wafa Muhammud (940--998 A.D.) a Persian astronomer and geometer solved problems by this means. Ref: Cantor, Geschitchte I, p. 638
- Using the ruler alone
The first to adopt the ruler alone seems to have been Franciscus von Schroten (1615--1660) Professor at the University of Leyden Ref: Cantor, Geschichte II, pp. 606--628 Chasles, Apercu..., Paris, 1875, pp. 98--99 Many of the problems were of practical application to surveying and in them besides using the ruler he considers that accessible lines can be measured. Brianchon (1785--18--) and Servois ... have also solved many problems by means of the right line alone: they gave the name Geometrie de la regle to this method (Ref: Chasles, Apercu du math., Paris, 1875, pp. 98--99, pp. 213--215). Other problems of the same kind are found in Recreations mathematiques d'Azaman (?) (edition of 1778). A work on surveying by Mascheroni also adopts this method.
- Using a ruler and a compass with fixed aperture
This seems to have been a favorite problem with the Italian geometers of the 16th century. Among them who used this method was Scipione del Ferro (--1525), Hieronimo Cardano (1501--76), Nicolo Tartaglia (1506--1559), and Luigi Ferrari (1522--1565). The first named above is also known to have discovered the solution of the cubic equation having a special form. This discovery was not published, but was communicated to his friend Fiore, who proposed the problem to Tartaglia in a contest. The later solved the problem and generalized it, but he also refrained from publishing his results and communicated them in confidence to Cardan. He however betrayed the trust and published the results some years later. Ferrari, a devoted pupil of Cardano took up the defense of his master when Tartaglia began his severe attacks. He challenged Tartaglia to a public contest and in one of these the latter proposes a number of problems to be solved by means of a compass with one fixed aperture (Ref: Cantor II, p. 453)
Some of Tartaglia's problems follow.
- In any triangle to construct a square of which one side shall
lie one outside of the triangle
``Let bc be the longest side of the triangle abc ...''
With that incomplete phrase, nothing more is written in the notebook, and so ends our knowledge about Herbert Keppel's preparation for his lecture in the Mathematics Seminar at Clark University during the late nineteenth century.
We were interested to find in the October, 1994 issue of the American Mathematical Monthly, a short article [1] by Professor Norbert Hungerbuhler of the ETH-Zurich entitled A Short Elementary Proof of the Mohr-Mascheroni Theorem. A sentence in the introduction to this article helps put some of Keppel's studies in perspective:
``In 1879 Lorenzo Mascheroni surprised the mathematical world with the theorem that every geometric construction that can be carried out by compasses and ruler may be done without ruler .... It turned out that Georg Mohr had proved this theorem in 1672 already.... The proofs given by Mohr and Mascheroni are quite complicated. Later easier proofs have been developed.''
References:
- Hungerbuhler, N., ``A Short Elementary Proof of the Mohr-Mascheroni Theorem'', American Mathematical Monthly 101 (1994), pp. 784--787.
- Cooke, Roger and Rickey, V. Fredrick, ``W.E. Story of Hopkins and Clark' in American Mathematical Society History of Mathematics, Vol. 3 (1989), A Century of Mathematics in America, Part III, ed. P. Duren, pp. 29--76.