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statement as to the specific abilities which should be developed (such as the ability to understand and use a formula, the ability to interpret an algebraic result in a concrete setting, the ability to analyze and solve concrete problems, etc.).

The following principles have accordingly been tentatively adopted to govern our proposed reconsideration of college entrance requirements in mathematics: 1. The scholarship implied in the present customary requirement of 211⁄2 units should not be lowered.

2. Drill in algebraic manipulation should be limited to those processes and to the degree of complexity necessary for a thorough understanding of principles and required by the probable applications either in common life or in subsequent mathematics.

3. More emphasis should be given to such immediately useful elementary topics as:

(a) The understanding and use of the formula.

(b) The interpretation of graphic representation and the use of graphic methods.

4. More emphasis should be placed on the acquisition of insight and power and less time devoted to acquiring mere facility in the solution of formal exercises. 5. While specific minimum requirements in separate subjects like algebra, geometry, etc., are still necessary, adequate provision should be made, perhaps through the so-called comprehensive examination, for the pupils in those schools who are developing "general courses" in mathematics where these subjects are not taught in separate courses.

6. College entrance requirements should be stated not only in terms of subjects and topics but also in terms of specific mathematical abilities to be developed.

7. Means should be found to introduce a proper and desirable amount of flexibility into the requirements in order to encourage progress in secondary education through the experimental introduction of new topics and methods. On the basis of these principles we hope to formulate our final report for subm'ssion to the Council not later than its next summer meeting.

Respectfully submitted, for the Committee,

J. W. YOUNG,
Chairman.

(7) The subject of mathematical needs of students in physics, as presented in Dr. Page's paper naturally divides itself into two parts: first, the mathematical needs of the undergraduate student in physics, and second, those of the graduate.

The elements of algebra and trigonometry are essential prerequisites for an undergraduate course in physics. The parts of algebra of importance are the solution of equations of the first and second degree, simultaneous equations of the first degree, logarithms and exponentials, and the binomial theorem; Dr. Page pleaded for more thorough preparation in these topics and greater practice in

their application, even at the expense of those parts of the subject which the undergraduate is likely to have less occasion to use. At the very beginning of his course in physics the student takes up composition and resolution of vectors, and in order to obtain a clear idea of the significance of these directed quantities he must have a good working knowledge of the elements of trigonometry. It is more important that he should be able to distinguish readily sine from cosine when neither side of the angle is horizontal, than that he should be acquainted with a large number of trigonometrical relations which he may never have occasion to use.

Very early in his course in physics, the student takes up derivations which can be explained satisfactorily only by the use of the calculus. It seems unfortunate to the teacher of physics that the college course in general physics is usually given before the student is able to use the calculus; the student's comprehension of the calculus suffers when he is deprived of the excellent opportunity to apply it which is afforded by the study of physics. The tendency to bring into freshman courses in mathematics at least the elements of differentiation and integration was looked upon with favor, and hope was expressed that this movement will permeate down to the last year of high school.

The mathematical preparation of the man who intends to pursue graduate study and engage in research in physics cannot be too thorough. Its value to him is of two kinds: first, it provides him with tools which are essential for his work, and second, it tra'ns him in those habits of logical thought which are exemplified preeminently in mathematical reasoning.

The first requirement of the graduate student in physics is a good course in advanced calculus and ordinary differential equations. Soon after must come courses in partial differential equations, and the study of Legendre's, Bessel's and Laplace's functions, in order to enable him to solve problems in electrostatics and allied subjects. The historical importance of Fourier's series involved in the solution of problems on the flow of heat has been greatly augmented by applications of this form of analysis to the theory of radiation. Some knowledge of the theory of probabilities is essential before courses in statistical mechanics and the kinetic theory of gases are undertaken.

Above all, emphasis was laid on the importance of vector analysis to the student of physics. With few exceptions, all the important quantities with which physics deals are either vectors or the scalar products of vectors. As mathematics is employed by the physicist as a tool in carrying out reasoning in physics, it is important that the notation employed should emphasize the physical rather than the mathematical significance of the logical processes involved. Hence the superiority of vector over scalar analysis. Of the current notations, none is comparable in simplicity and utility to that of Gibbs. One who has been brought up on this form of vector analysis and has made constant use of it cannot help feeling astonished that it has not been universally adopted by physicists.

In discussing this paper Professor Ransom held that while the speakers on

the program maintained that students expecting to go into scientific work need a wide acquaintance with mathematics, it is still to be said that many students do not know early enough what courses they need or even that there are mathematical subjects which serve their special purposes. Here is the opportunity for a survey course of a half year, where the instructor advisedly does the greater part of the work and suggests what will be of later use, a course which is profitable for sophomores, certainly for juniors and seniors. Such a course would help in furnishing a better knowledge of the language of mathematics. Professor Page replied that the student would doubtless not know of the requirements for, say, electrodynamics and the electromagnetic theory of light, where some such course as Gibbs's vector analysis is essential; he will, if wise, ascertain from his instructor in physics or from some other well informed person what the necessary preparation is. Professor Coolidge described a survey course at Harvard characterized by the students as "seeing mathematics"; the course was elected for the most part by those who had no adequate preparation in earlier mathematics and proved a disillusionment for those who had hopefully inaugurated the plan. Professor Hurwitz instanced a similar course at Cornell in which the instructor refused to make it an easy course, the result being that its enrollment has been very small. He added that uneducated laymen can today pick up a popular science journal and can from this acquire the essential principles of new methods in non-technical language; nevertheless only the exterior results can thus be transmitted, and it is our distinct duty to contribute as far as may be through mathematics also to the development of a well-informed public.

Professor Hawkes made the very important remark that much of what is sought through survey courses can be done by mathematical clubs. We need merely to refer to the past few volumes of the MONTHLY to see how much of this desirable work has actually been accomplished in the live universities and colleges of America. Dr. Helen Owens said that the junior mathematical club at Cornell considers application of mathematics to other subjects, thus enabling students to get preliminary notions of the mathematical courses needed in other branches. The experimental plan, which thus far has included applications to physics, statistics, actuarial work and chemistry, has proved very interesting.

A good natured controversy between Professors Bowden and Page as to whether Professor Gibbs was a pure mathematician or a member of the department of physics, turned the discussion to the question of a coördination between the two departments. Professor Page said that the pure mathematician does not keep closely enough in touch with the experimentalist and his work is apt to suffer so far as its immediate usefulness to the physicist is concerned.

Professor Woods emphasized the portion of the paper that called for those parts of differential geometry and other advanced mathematics which in recent years have become of the utmost importance to the physicist and chemist, e.g., the Einstein theory. He also thinks it unfortunate to teach physics without calculus. Under a new plan at the Massachusetts Institute of Technology,

trigonometry is an entrance requirement, the freshman begins with calculus, even before he studies analytic geometry; this course is to be followed by a course in the beginnings of analytic geometry for one term, then further work in calculus.

MEETING OF THE COUNCIL OF THE ASSOCIATION.

The following seventy-two persons and two institutions, on applications duly certified, were elected to membership:

To individual membership.

W. C. BARTOL, A.M. (Bucknell); Ph.D. (Adrian). Prof. of math. and astr., Bucknell Univ., Lewisburg, Pa.

ANNETTE BENNETT, A.M. (Columbia). Head of dept. of math., Eureka Coll., Eureka, Ill.

W. N. BIRCHBY, A.M. (Colo. College). Pasadena, Cal.

L. I. BONNEY, A.B. (Bates College). Asst. prof., Middlebury Coll., Middlebury,
Vt.

F. E. BRASCH. Asst. reference librarian, John Crerar Library, Chicago, Ill.
W. M. BRODIE, M.E. (Va. Polytech. Inst.); A.M. (Columbia). Prof., Va.
Polytech. Inst., Blacksburg, Va.

J. S. BROWN, A.M. (Texas). Prof., Southwest Texas St. Normal College, San
Marcos, Tex.

MARGARET BUCHANAN, A.B. (West Va. Univ.). Grad. stud., Bryn Mawr Coll., Bryn Mawr, Pa.

S. E. CROWE, A.B. (Ohio State). Asst. prof., Mich. Agric. Coll., East Lansing, Mich.

W. L. CRUM, Ph.D. (Yale). Instr., Yale Coll., New Haven, Conn.

R. D. DOUGLASS, A.M. (Maine).
W. F. DOWNEY, A.B. (Amherst).
Annex, Boston, Mass.

Instr., Mass. Inst. of Tech., Cambridge, Mass.
In charge Collins Bldg., English High School

G. C. EVANS, Ph.D. (Harvard). Prof., Rice Inst., Houston, Tex.

A. J. FLEISIG, A. B. (St. Procopius). Instr., St. Procopius Coll., Lisle, Ill.
C. D. GARLOUGH, A.M. (Illinois). Prof., Wheaton Coll., Wheaton, Ill.
H. H. GAVER, A.M. (Virginia). Instr., U. S. Naval Acad., Annapolis, Md.
R. E. GILMAN, Ph.D. (Princeton). Asst. prof., Brown Univ., Providence, R. I.
W. W. GORSLINE, B.S. (Chicago). Teacher of collegiate math. and surv.,
Crane Jun. Coll., Chicago, Ill.

W. C. GRAUSTEIN, Ph.D. (Bonn). Asst. prof., Harvard Univ., Cambridge, Mass.
T. H. GRONWALL, Ph.D. (Upsala); C.E. (Berlin). Math. and dynamics expert,
Technical Staff, U. S. Ord., Washington, D. C.

V. G. GROVE, A.M. (Kentucky). Asst. prof., Mich. Ag. Coll., E. Lansing, Mich. H. E. GUDHEIM, M.E. (Royal Univ. of Tech., Stockholm). Assoc. prof. of graphics, Va. Polytech. Inst., Blacksburg, Va.

HOWARD HARDING, B.M.E. (Michigan). With Rochester Railway and Light Co., Rochester, N. Y.

W. L. HART, Ph.D. (Chicago). Asst. prof., Univ. of Minnesota, Minneapolis, Minn.

J. R. HITT, B.S. (Miss. Coll.). Asst. prof., Mississippi Coll., Clinton, Miss. YUN HUANG HO, A.B. (Cornell). Grad. stud., Cornell University, Ithaca, N. Y. Asso. prof., Univ. of South Carolina, Colum

J. B. JACKSON, A.M. (Columbia). bia, S. C.

J. S. W. JONES, D.Sc. (Washington Coll.). Prof., Washington Coll., Chestertown, Md.

G. B. KING, A.B. (Ark. Cumberland College). Prof., Cumberland Coll., Clarksville, Ark.

J. R. KLINE, Ph.D., (Pennsylvania). Asso., Univ. of Illinois, Urbana, Ill.

L. C. KNIGHT, Ph.B. (Wooster). Asst. prof., College of Wooster, Wooster, Ohio. ELMER LATSHAW, Grad. preparatory engineering course (Drexel Inst.) Mech. designing, W. Philadelphia, Pa.

LENA R. LEWIS, A.M. (Texas). Head of dept. of math., Thorp Spr. Chr. Coll., Thorp Springs, Tex.

H. M. LUFKIN, S.T.B. (Phila. Div. Sch.). Asst., Cornell Univ., Ithaca, N. Y. R. M. MCDILL, A.M. (Indiana). Prof., Hastings Coll., Hastings, Neb.

MARTHA P. MCGAVOCK, A.M. (Chicago). Head of dept. of math., Sullins Coll., Bristol, Va.

J. B. MEYER, M.S. (Purdue). Head of dept. of math., St. Normal School,
Valley City, N. D.

NORMAN MILLER, Ph.D. (Harvard). Lecturer, Queens Univ., Kingston, Can.
E. B. MODE, B.S. (Boston Univ.). Instr., Boston Univ., Boston, Mass.
H. M. MORSE, Ph.D. (Harvard). Benjamin Peirce Instr., Harvard Univ., Cam-
bridge, Mass.

G. W. MULLINS, Ph.D. (Columbia).
M. A. NORDGAARD, A.M. (Maine).
YEISUKE ONO. First Higher School,
HELEN B. OWENS, Ph.D. (Cornell).
S. F. PARSON. Head of dept. of math., No. Ill. St. Normal School, De Kalb, Ill.
W. H. PEARCE, A.M. (Michigan). Head of dept. of math., Central St. Normal
School, Mount Pleasant, Mich.

Asst. prof., Barnard Coll., New York, N. Y.
Asst. prof., Grinnell Coll., Grinnell, Iowa.
Tokio, Japan.

Instr., Cornell Univ., Ithaca, N. Y.

H. P. PETTIT, A.M. (Kentucky). Asst., Univ. of Illinois, Urbana, Ill.

E. C. PHILLIPS, Ph.D. (Johns Hopkins). Prof. of math. and astr., Woodstock Coll., Woodstock, Md.

HILLEL PORITSKY. Asst. in math. and phys., Cornell Univ., Ithaca, N. Y.

C. LOIS REA, A.B. (Allegheny). Prof. of math. and sc., Cedarville Coll.,. Cedarville, Ohio.

F. W. REED, Ph.D. (Virginia). Instr., Cornell Univ., Ithaca, N. Y.

J. N. RICE, Ph.D. (Catholic Univ.). Instr., Catholic Univ., Washington, D. C.
J. M. ROBB, A.M. (Michigan). Head of math. dept., high school, Everett, Wash.
G. M. ROBISON, Ph.D. (Cornell). Instr., Cornell Univ., Ithaca, N. Y.
L. V. ROMIG, A.M., M.S. (Michigan). Prof., Augustana Coll., Rock Island, Ill.

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