PI MU EPSILON JOURNAL TEE OFFICIAL PUBLICATIONOF THE HONORARY MATHEMATICAL FRATERNITY

VOLUME 1 NUMBER 6

CONTENTS

Page Euclid Alone Has Looked on Beauty Bare ...... Edna St. Vincent Millay 215 Archimedes and the Theory of Numbers . . . . E. T. Bell 216 Problem Department ...... 222 Problems for Solution ...... 222 Solutions...... 225 Pi Mu Epsilon Journal Staff ...... 229 General Officers of the Fraternity (Continued from No. 5) . 232 Reports of the Chapters ...... 233 Errata...... 240 Medals, Prizes and Scholarships...... 241 An Open Letter to the Members of the Pi Mu Epsilon Fraternity ...... J. S. Frame 242 News and Notices...... 244 Directory...... 247 Initiates, Academic Year 1950-1951 (Continued from No. 5) . 252 Initiates, Academic Year 1951-1952 ...... 257

APRIL 1952 PI MU EPSILON JOURNAL PI MU EPSILON JOURNAL THE OFFICIAL PUBLICATIONOF THE OFFICIAL PUBLICATION THE HONORARY MATHEMATICAL FRATERNITY OF THE HONORARY MATHEMATICAL F

RUTH W. STOKES, Editor VOLUME 1 NUMBER 6 ASSOCIATE EDITORS J, 8. FRAME, Michigan State College, East Lansing, Michigan N. H. McCOY, Smith College, Northampton, Massachusetts LEO MOSER, Univorsity of Alberta, Edmonton, Alberta, Canada CONTENTS R. J. WALKER, Cornoll University, Ithaca, New York HOWARD C. BENNETT, Business Manager Page Euclid Alone Has Looked on Beauty Bare ...... GENERAL OFFICERS OF THE FRATERNITY Edna St. Vincent Millay 215 Director Got~oral:C. C, MacDuffee, University of Wisconsin Archimedes and the Theory of Numbers .... E. T. Bell 216 VICO-DtroctorGonoral: W. M. Whyburn, University of N. C. Problem Department ...... 222 Socrotnry-Troa8uror General: J. S. Frame, Michigan State College Problems for Solution ...... 222 Councillore Gonoral: Solutions...... 225 8. 5, Cairns, Unlvor~ityof Illinois, Urbana, Illinois Pi Mu Epsilon Journal Staff ...... 229 Tomllnaon Fort, University of Georgia, Athens, Georgia Soplilo McDonald, University of California, Berkeley, calif. General Officers of the Fraternity (Continued from No. 5) . 232 Ruth W. Stokes, Syracuse University, Syracuse, New York Reports of the Chapters ...... 233 EX-offlcioMombor of tho Council: Howard C. Bennett Errata...... 240 Medals, Prizes and Scholarships...... 241 MANUSCRIPTS and EDITORIAL CORRESPONDENCE, including CHAPTER REPORTS, NEWS ITEMS, etc., should be addressed An Open Letter to the Members to the Editor, Ruth W. Stokes, 15 Smith College, Syracuse Uni- of the Pi Mu Epsilon Fraternity ...... J. S. Frame 242 voralty, Syracuse 10, New York. News and Notices...... 244 PI MU UPSILON JOURNAL is published semi-annually at Syra- Directory...... 247 cuiio Univorsity. Initiates, Academic Year 1950-1951 (Continued from No. 5). 252 SUBSCRIPTION PRICE: To Individual Members, $1.50 for 2 years; to Non-Members and Libraries, $2.00 for 2 years. Initiates, Academic Year 1951-1952 ...... 257 Subscriptions, orders for back numbers and correspondence concorning subscriptions and advertising should be addressed to Howard C. Bennett, 15 Smith College, Syracuse University, APRIL Syracuse 10, Now York. Copyright 1952 by Pi Mu Epsilon Fraternity, Inc. "It" is a dodecahedr ometric "solids"

EUCLID ALONE HAS LOOKED ON BEAUTY BARE* Edna St. Vincent Millay

Euclid alone has looked on Beauty Bare. Let all who prate of Beauty hold their peace, And lay them prone upon the earth and cease To ponder on themselves, the while they stare At nothing, intricately drawn nowhere In shapes of shifting lineage; let geese heroes seek release into luminous air.

0 blinding hour, 0 holy, terrible day, When first the shaft into his vision shone Since the time of Pythagoras, men have recognized that the chniques of many creative arts - architecture, sculpture, paint- Of light anatomized! Euclid alone ing, music, and poetry - draw upon the principles which mathe- Has looked on Beauty bare. Fortunate they maticians have discovered. Less frequently, however - and least frequently in poetry - has a mathematical subject been transmuted Who, though once only and then but far away, into creative art. - Of poems treating mathematical subjects, probably the best Have heard her massive sandal set on stone. known is Millay's sonnet on Euclid. The poem does not need the myth that has grown up about its creation: that, instead of study- *"~uclidAlone Has Looked on Beauty Bare" ing for a geometry examination, Millay was so engrossed in from the HARPWEAVER AND OTHER POEMS achieving the artistic synthesis of her appreciation of Euclid that she failed the examination. Flawless in form, it is a poetic rec- published by Harper & Brothers ognition of the pure beauty of mathematical form and mathemati- copyright 1920 by Edna St. Vincent Millay cal reasoning, and of mathematics as the essence of systematic thinking. -- Raven I. McDavid, Jr. 1952 ARCHIMEDES 217

"Compute, 0 Friend, the host of the oxen of the Sun, giv- ing thy mind thereto; if thou hast a share of wisdom, com- pute the number which once grazed upon the Sicilian isle Thrinacia, and which were divided according to color into ARCHIMEDES AND THE THEORY OF NUMBERS~ four herds, one milk white, one black, one yellow, and one E. T. Bell, California Institute of Technology dappled. The number of bulls formed the majority of the animals and the relations between them were as follows."- As we would continue today, letW,w be the respective num- Tradition credits Archimedes with only one contribu- bers of white bulls and white cows, and (X,x), (Y,y), (Z,z) tion to the theory of numbers. This is the famous Cattle the numbers of bulls and cows in the black, yellow, and dap- Problem, which probably will never be fully disposed of in pled herds respectively. The first part of the problem is to the sense of producing the complete numerical answer.' solve (in positive integers) the following seven equations The circumstances under which the problem came to light between the eight unknowns: in modern times are of interest. The German critic and dramatist, G. E. Lessing (1729-1781), spent most of the last decade of his laborious life dusting and sifting the treasures of the Wolfenbutel library (North Germany), where Leibniz had slaved out his last unrewarded years. Lessing was chief librarian. He made many discoveries of at least scholarly interest for their time and published them in a series of learned volumes. The first (1773) contained the Cattle Problem, a Greek 'epigram' in 44 lines of verse, headed "A problem which Archimedes found among (some) epigrams and sent, in his letter to Eratosthenes of Cyrene, to be solved by those in Alexandria who occupy themselves with such matters." The problem may have been suggested by a passage in the twelfth book of Homer's Odyssey: "Next, you will reach the island of Thrinacia (or Thrinakia, Sicily) where in great numbers graze kine and the sturdy flocks of the Sun." The complete problem is in two parts. The first is easy today, the second is not, even today. Somewhat short- ened the statement is as follows.

excerpt from "The Last problem." Notice that the fractions are expressed in 'unit fractions' ^M~account is indebted to the article by R. C. Archibald, as an Egyptian of the time of Ahmes might have done, thus American Mathematical Monthly, vol. 25, 1918, pp. 411 -414, 1/2 + 1/3 instead of 5/6. These equations are (today) an which contains numerous references to the extensive liter- easy indeterminate system there are only seven equations ature on the problem, - for the eight unknowns. The solutions, where n is any pos- itive integer, are PI MU EPSILON JOURNAL April 1952 ARCHIMEDES

The last equation is of the general type

where D is a positive integer having no square divisor ex- ceeding 1. By a historical mistake it is named after an ec- centric character but mediocre mathematician, John Pel1 Even for n = 1, giving the smallest numbers satisfying (1610-1685), and is called a Pellian equation. It is too late the equations, the herd must have stood several deep on the now to right the mistake; the equation should have been island of Sicily. But this is only a paltry beginning. In named after Format. The method for solving it completely passing, the calculations for this much of the problem, sim- is now well known, and is the important algorithm of con- ple to us, might have taxed a skilled computer having at his tinued fractions in the current theory of numbers. (The command only the Greek alphabetical system of writing older English school algebras used to include the algo- numbers or even the Babylonian sexagesimal. Archimedes rithm.) It is by no means obvious that an equation of this may have suspected as much, for he says, "If thou canst type for any given Dof the prescribed form necessarily has give, 0 Friend,, the number of bulls and cows in each herd a solution T, U, and indeed an infinity of solutions. The thou art not unknowing or unskilled in numbers, but still not existence of a solution was proved only in 1776, when La- yet to be counted among the wise." He then stiffens the grange (1736-1813) succeeded after many attempts which, problem very considerably by imposing two further condi- he said, had cost him more thought than any of his other tions which the numbers of bulls must satisfy. great successes and perhaps more than they may have been 'Consider, however, the following additional relations worth. between the numbers of bulls of the Sun: The least solution T,U for Archimedes' D = 4729494 was computed in 1880 by A. Amthor. It is W + X = a square number,

Z + Y = a triangular number.

When thou hast computed the totals of the herds, 0 Friend, U = 50,549,485,234,315,033,074,477,819,735,540,408,986,340. go forth as a conqueror, and rest assured that thou art proved most skilled in the science of number." I willingly leave to the reader the pleasure of verifying that hat was rather rubbing it in. With the values of W, X, these T,U actually satisfy the equation. Z,Y as above satisfying the first part, the problem now de- If - which seems most unlikely -Archimedes knew that mands the solution of his final equation must have a solution he was about twenty centuries ahead of his time. In fact he would have had to know as much as Lagrange unless, of course, he succeeded in calculating the required numbers T,U - an extremely la- where u,v are unknown numbers, and this in turn, as easily borious process - and proceeded thence in an attempt to shown, requires numbers T,U to satisfy compute the total number of cattle in the herd. If he suc- ceeded in the last he surpassed any calculating machine yet invented or ever likely to be. For Amthor proved that W is PI MU EPSILON JOURNAL April ARCHIMEDES 221 a number of 206545 digits, and the total number in the herd The evidence is not mathematical but human: "The unmis- also requires this many. To give some idea of the magni- takable vein of satire in the opening words of the epigram, tude of these numbers, Amthor saidUItis easy to show that and in the transition from the first to the second part, and a sphere having the diameter of the Milky Way, across in the last lines, was a shaft directed toward Apollonius." which light (speeding at 186000 miles a second) takes ten This is the only time that Apollonius, "the Great Geom- thousand years to travel, could contain only a part of this eter'' as he was justly called by the Greeks, the Moslems, great number of animals even if the size of each is that of and their successors, figures as a contributor to the theory the smallest bacterium." Amthor gives another illustration. of numbers. A geometer who could inspire anything as To print all eight numbers of the solution, with 2500 digits hellish as the Cattle Problem must have been an arithmeti- to the page, would require a volume of over 660 pages. Af- cian, even if only an involuntary one, of no small order. If ter all this it seems improbable that Gauss solved the Cat- Archimedes ever sent his close and respected friend, Era- tle Problem completely, as one of his uncritical admirers tosthenes, anything for himself nearly so provocative as the asserted that he had. There must have been a misunder- Thrinacian bulls and cows, it has not survived. standing somewhere. The occasion for Archimedes' 'shaft' was the problem A cynic has remarked that as long as there is an un- of approximating to TT, the ratio of the circumference of any solved problem, some fool will try to solve it, especially if circle to its diameter. To 7 places TT = 3.1415926, which is the problem is unsolvable. Perhaps A. H. Bell (no relation considerably farther than any of the Greeks got. (An Eniac to the present writer) and his two collaborators, constituting calculating machine in about 70 hours performed the super- the Hillsboro Mathematical Club of Hillsboro, Illinois, in human feat of computing TI to 2035 places.) Archimedes had 1 10 the 1890'~~did not fully qualify for the cynic's remark, but shown that TT is less than 3 - and greater than 3 -. Apol- they came close. After nearly four years of unpaid hard 7 71 labor the members of the Club computed 30 or 31 of the lonius claimed a closer approximation, and is said to have first digits and 12 of the last for each of the eight unknown boasted about his superiority over Archimedes. He also bull and cow numbers and for the total number in the herd. exploited a system for writing and manipulating large num- The results disagree with the more modest computations by bers that he claimed was better than the similar venture of Amthor. Does anybody wish to check the Club's computa- Archimedes in his "sand reckonern -which need not be de- tions? Or Amthor's? The field is open and wider than the scribed as it has nothing to do with the theory of numbers; plains of Thrinacia. it was a scheme, ingenious for its time, of constructing There is the inevitable question, was it really Archi- large numbers. Archimedes drove Apollonius to cover by medes who proposed the problem? If it was not, who else hurling the Cattle Problem at him from ambush. The huge had brains enough to imagine such a horror? The histori- numbers required to solve the problem were beyond human cal experts agree that Archimedes probably was responsi- computation by any means Apollonius could possibly have ble for the first part. Some, but not all, credit him also known. They might even cause an electronic calculator to with inventing the second part. Again, if it was not Archi- blow a battery of tubes. medes, what unknown genius constructed an elaborate solv- -able problem leading to indeterminate equations of the sec- ond degree? Equations of this kind written down at random are far more likely than not to have no solution. The final verdict according to some scholars is that it was Archi- medes after all who successfully set the hard second part. PROBLEM DEPARTMENT 223

37. Proposed by Victor Thebault, Tennie Sarthe, France

Find all pairs of three digit numbers M and Nsuch that M*N = P and M'-N' = P', where M'-N' and P' are the num- bers M*N and P written backwards. For example, PROBLEM DEPARTMENT Edited by Leo Moser, University of Alberta and

This department welcomes problems believed to be new and, as a rule, demanding no greater ability in problem 38. Proposed by C. W. Trigg, Los Angeles City College solving than that of the average member of the Fraternity, but occasionally we shall publish problems that should chal- In the triangle ABC, AA' is a median. Prove that if lenge the ability of the advanced undergraduate and/or can- dictate for the Master's degree. Solutions of these problems -AM =l.? should be submitted on separate, signed sheets within five MA' q months after publication. Address all communications con- cerning problems to Leo Moser, Mathematics Department, then CM extended divides AB in the ratio University of Alberta, Edmonton, Alberta, Canada.

PROBLEMS FOR SOLUTION 39. Proposed by Pedro Piza, San Juan, Puerto Rico

35. Proposedby N. S. Mendelsohn, University of Manitoba Find digits m, a, b, c, d, e, £ such that A point moves in a straight line starting from rest and

-.9m + abcdef = fedcba finishing at rest, and covers unit distance in unit time. 9m Prove that at some point its acceleration has a magnitude of at least 4 units. 40. Proposed by J. H. Butchart, Arizona State College

Proposed by Joan Sherley, Syracuse University Three infinitely long parallel wires carry electrostatic charges el, ez, e, per unit length, where 2e1 = 2ez = -e3. A man wishes to plant an orchard with n trees in ten Show that if the cross section of the wires is an equilateral straight rows, five in a row. What is the smallest value n triangle then the circumscribed circle to this triangle is a can have? line of force. 224 PI MU EPSILON JOURNAL April 1952 PROBLEM DEPARTMENT 225

41. Proposed by Chester McMaster, New York City originated them. They appeared elsewhere in mathematical literature, without proof, but we believe they will be of suf- There are more chess masters in New York City than ficient interest to our readers to warrant reprinting. in the rest of the U. S. combined. A chess tournament is planned, in which all American masters are expected to at- tend. In determining the site of the tournament it is agreed that the primary consideration should be the minimization SOLUTIONS of the total inter-city distance covered by all participants. The New York masters claim that by this criterion the site chosen should be their city. The west coast players and 5. Proposed by P. M. Anselone, College of Puget Sound some others claim that a city at or near the center of grav- M ity or centroid of the players would be better. Prove that (M+N - i)! the New Yorkers are right. If F(M,N) = M! (N + 1) E (M i)! i=o -

42. Proposed by Me1 Stover, Winnipeg, Manitoba Show that F(M,N) = F(N,M)

Prove that the volume of a tetrahedron determined by Solution by William Moser, University of Toronto two line segments lying on two skew lines is unaltered by sliding the segments along their lines (but leaving their We use induction over M to prove F(M,N) = (M + N + 1) ! lengths unaltered). which clearly implies the required result. For M = 1, F(1,N) = (N + 2)! is easily checked directly. Assuming now that F(M,N) = (M + N + 1)! we have 43. Proposed by Paul W. Gilbert, Syracuse University M+l (M + 1+ N - i) Four solid spheres lie on top of a table. Each sphere F(M + 1,N) = (M + l)!(N + 1) JIZi is tangent to the other three. If three of the spheres have i=O (M+ 1 - i)! the same radius R, what is the radius of the fourth sphere?

44. Proposed by Paul W. Gilbert, Syracuse University

Assuming that c is a positive constant, solve the fol- lowing equation for x:

= (M+N+ l)!(N+ 1 +M+1) = (M+ N+ 2)!

-Editorial-- Note. Professor Gilbert, as Faculty Adviser which completes the proof. for the New York Alpha Chapter, used problems 43 and 44 as "Problems of the Month." He does not claim to have Also solved by C. W. Trigg. 226 PI MU EPSILON JOURNAL April 1952 PROBLEM DEPARTMENT 227

' 6. Proposed by C. W. Trigg, Los Angeles City College of (0,l) and (p,q) as diameter. Show that if the roots are complex, the real part is the abscissa of the center and the Starting with a straight edge, closed compasses, and coefficient of i is the tangent from (p/2, 0) to this circle. two straight line segments a and b, construct the harmonic mean of a and b in the least number of operations. Chang- Solution by Roy Sinclair, University of Alberta ing the opening of the compasses, drawing a circle or the arc of a circle, and drawing a straight line are each con- The coordinates of the center of the circle are ($, sidered an operation. (q+l)/2). The square of its radius is (3+ ((q+1)/2 - if = Partial Solution by the Proposer (8 + (q-l)2)/4. The distance between (p/2,0) and the center In the following discussion the number in parentheses is (qtl)/2. Hence if L is the tangent length then1 L' = (q-l)V4 z z is the ordered number of the operation. - (p t (q-1)')/4 = (4q - p ) /4 or L = (4q - pz)i/2. Sincethe1 Construction: Open the compasses to a radius a (I), complex roots of 2 - px t q = 0 are p/2 ? i(4q - P'): /2 the and with an arbitrary point A as center describe a circle (2). With B an arbitrary point on the circle as center, de- theorem is proved. scribe a circle cutting the first circle at C and D (3). Draw Also solved by C. W. Trigg. AB extended (4) and AD extended (5). Change the opening of the compasses to equal b (6). With A as center describe a circle cutting AD at E (7). Draw CE cutting AB at F (8). 33. Proposed by C. W. Trigg, Los Angeles City College Change the opening of the compasses to equal AF (9). With F as center describe a circle cutting AB at A and G (10). It is well known that the elements of the fourth row (or AG = c is the harmonic mean of a and b. column) of the Pascal triangle are tetrahedral numbers. Proof: Draw CA and draw FH parallel to CA cutting Establish the following properties of the fourth row. d AD ~~LDAB= 60 = LCAB = LAFH, so triangle AFH is 1. The difference of two consecutive elements is a tri- equilateral and FH =AH = AF = c/2. It follows that (b = c/2): angular number. c/2::b:a, so c = 2ab/(a + b). 2. The difference of two alternate elements is a square. 3. The difference of the (n + 2)nd and the nth elements Editorial Note: Though the above constructions seem increased by the (n + 1)st element of the third row is a pen- quite efficient, it is not proved that ten is the least number tagonal number. of operations possible. A proof that this is indeed the case, 4. Six times the nth element added to the (n + 1)st ele- or a construction involving fewer operations would be wel- ment of the second row is a cube. come. 5. The nth element is equal to the sum of the first n elements of the third row.

30. Proposed by J. H. Butchart, Arizona State College Solution by the proposer

A well known construction for the roots of 2-px + q The nth element of the rth row of the Pascal triangle = 0 is to find the x intercepts of the circle having the join is C(n+r-2, r-1). 228 PI MU EPSILON JOURNAL April PI MU EPSILON JOURNAL STAFF 1. C(n+3,3) - C(n+2,3) = (n+2)(n+l)/2 = C(n+2,2), the (n+l)st element of the third row. 2. C(n+4,3) - C(n+2,3) = (n+4)(n+3)(n+2)= (n+2)(n+l)n/6 = (n+2)', the square of the (n+2)ndelement of the second row. 3. C(n+4,3)-C(n+2,3) + C(n+2,2) = (n+2)' + (n+2)(n+l)/2 = (n+2)(3 n+ 5) /2. 4. 6C(n+2,3) + C(n+l,l) = (n+2)(n+l)n+ (n+l) = (n+l)=. 5. C(n+r-2,r-1) = C(n+r-3,r-2) + C(n+r-3,r-1) = C(n+r-3,r-2) + C(n+r-4,r-2) + C(n+r-4,r-1) --. . .= C(n+r-3,r-2) + C(n+r-4,r-2) +. . .+ C(r-1,r-1). Now C(r-1,r-1) = C(r-2,r-2), so the nth element of the rth row is equal to the sum of the first n elements of the (r-1)st row. That is, this property is not restricted to the fourth row alone.

34. Proposed by J. S. Frame, Michigan State College For what values of k are the following twelve points the vertices of a regular icosahedron? (0, tk, tl),(21, 0, tk), (tk, tl, O)?

Solution by the proposer RUTH WYCKLIFFE STOKES

If k satisfies the condition, so do -k, 1/k, and -1/k. We EDITOR-IN-CHIE F therefore look first for all solutions between 0 and 1 and RUTH WYCKLIFFE STOKES, Assistant Professor of Mathematics, Syra- find the rest from these. Five vertices are adjacent to and cuse University. Native of Greenville, S. C. A.B, Winthrop Col; M.A, Van- equally distant from the vertex (O,k,l), and the five vertices derbilt; Ph.D. (math), Duke. Teacher,high sch, S.C; asst. prof. math, Win- opposite these are adjacent to (0 -k,-1). From (O,k,l) to throp Col; instr, Duke; asso. prof, N. Texas State Col; prof. and dept. head, i Mitchell Col; prof. and dept. head (astron. and math), Winthrop Col, 36-46; (0,-k,l) the squared distance is 4k , whereas from (O,k,l) to visiting prof, Brown Univ, summer, 42; asst. prof. math, Syracuse, 46--. each of the other adjacent vertices the squared distance is Math. Soc; Math. Asn. (v-chr, 41-46; chr, 46-47) So. E. Sec; Nat. Council 1 + k% (1-k)'. Hence Teachers Math. (director, 45-47); Astron. Soc. Pacific; fellow, AAAS; Sigma Delta Epsilon (Secy, 49-50); Sigma Xi; Pi Mu Epsilon; founder and ed. 'Pi Mu Epsilon Journal (49--). A geometric theory of solution of linear inequalities; symbolic quadratic forms; construction of models illustrating almost all From the one solution between 0 and 1 we obtain the four branches of mathematics. solutions: ASSOCIATE EDITORS JAMES SUTHERLAND FRAME, Professor of Mathematics and Chair- man of Department, Michigan State College. (Under the title 'SECRETARY- TREASURER GENERAL,' Professor Frame's photograph and biographical It is clear that for these values of k the given points have sketch appeared in the November 1951 issue, this Journal, along with the the required property. other general officers.) 230 PI MU EPSILON JOURNAL April 1952 PI MU EPSILON JOURNAL STAFF 231

ASSOCIATE EDITOR ASSOCIATE EDITOR

NEAL HENRY McCOY, Profes- ROBERT JOHN WALKER, Pro- sor of Mathematics and Depart- fessor of Mathematics and De- ment Chairman. Smith College. partment Head, Cornell Univer- Native of ~aukokis,Okla. A~B, sity, Ithaca, N.Y. Native of Pit- Baylor; M.S. Ph.D, Iowa. Teach- tsburg, Pa. B.S, Carnegie Inst. er, highsch, Texas; nat. research Tech; fellow, Ph.D (math), instr, fellow, Princeton; asst. prof. Princeton; asst. prof, assoc. prof, math, assoc. prof, prof, Smith prof (48--), dept. head (SO--), Col, 42--. Math. Soc; Math.Asn. Cornell. Lecturer, Princeton, (2nd v-pres, 49-50; assoc. ed. 40-41; mathematician, Aberdeen 'Math. Monthly'). Author of Carus Proving Ground, Md, 43-45. As- Monograph, "Rings and Ideals." soc. ed, 'Duke Math. Jour.' Math. Editor of Duke Math. Journal, Soc; Math. Asn. (assoc. ed, 'Math. 1951--. Algebra and quantum Monthly'). Singularities of alge- mechanics; theory of matrices; braic manifolds; magic squares; modern algebra; theory of rings. artillery rockets.

NEAL HENRY McCOY ROBERT JOHN WALKER

BUSINESS MANAGER

HOWARD CURTIS BENNETT, ASSOCIATE EDITOR Mathematics Instructor, Syra- cuse University. Native of Green- LEO MOSER, Lecturer in Math- wich, N.Y. Ph.B, Syracuse; S.T.B, ematics, University of Alberta. Boston; M.A, Columbia. Instruc- B.Sc, Manitoba; M.A, Toronto; tor and Registrar, Nanchang Ph.D, University of North Caro- Academy, Nanchang, China; Cler- lina. Lecturer, University of gyman, Troy Conference of Meth- Manitoba, 1947-49; asso. prof, odist Church; Instr, Hobart Col; Texas Technological College, Math. Instr., Syracuse, 1946~; 1950-51. Member American Business Mgr. 'Pi Mu Epsilon Mathematical Society; Mathe- Journal' (49--). Member AMS matical Association of America; and Phi Delta Kappa. Member of Canadian Mathematical Con- the Mathematical Club, of Syra- gress; Sigma Xi; Pi Mu Epsilon. cuse University, which became Number theory; geometry; math- the first chapter of the Pi Mu ematical recreations. Epsilon Fraternity; also, student of Professor Edward Drake Roe, Jr., founder of the Fraternity.

LEO MOSER HOWARD CURTIS BENNETT GENERAL OFFICERS OF THE FRATERNITY REPORTS OF THE CHAPTERS (Continued from Volume 1, Number 5) (Send reports to Ruth W. Stokes, 15 Smith College, Syracuse Uni- versity, Syracuse 10, New York.)

COUNCILOR GENERAL EDITOR'S NOTE. According to Article VI, Section 3 of the Con- stitution: "The Secretary shall keep account of all meetings and TOMLINSON FORT, Professor of Mathematics and transactions of the chapter and, before the close of the academic Head of the Department, University of Georgia. Native of year, shall send to the Secretary-General and to the Director- Albany, Georgia. A.B, A.M. U. of Georgia; A.M. Ph.D, General, an annual report of the chapter activities including pro- grams of meetings, results of elections, etc." The Secretary- Harvard; also, Kirkland traveling fellow from Harvard, General now suggests that an additional copy of the annual report Europe. Instructor in mathematics, Georgia, Harvard, of each chapter be sent to the editor of the Pi Mu Epsilon Journal. Michigan; Asst. Prof, Michigan; Professor, Alabama, Hun- Besides the information listed above we are especially interested ter College; Professor and Dean, Graduate School, Lehigh; in learning what the chapters are doing by way of competitive ex- Prof. Mathematics, Georgia, 1945--. A.A; Math. Soc. aminations, medals, prizes and scholarships. These annual re- (Asso. sec'y); Math. Asn. (v. pres.), Director General of Pi ports will be published in the chronological order in which they Mu Epsilon Fraternity; Editor, Bulletin of American Math- are received. ematical Society. Author: Infinite Series 1930; Analytic Geometry (with others) 1932; Difference Equations 1948; Alpha of Iowa, Iowa State College Calculus 1951. Research: Infinite series; finite difference; linear difference equations. During the school year 1950-1951, the following papers were presented at meetings of Iowa Alpha chapter: "Paradoxes simple and complex" by Dr. Joe Foote "The use of matrix notation for transfer functions of physical COUNCILOR GENERAL systems" by Mr. A. J. Bradt "Some equations for absorption of gases" by Dr. Robert S. RUTH WYCKLIFFE STOKES (See 'PI MU EPSILON Hansen. JOURNAL STAFF' for photograph and biographical sketch.) We held eight meetings during the year, including two initia- tion banquets held immediately after fall and spring initiation in honor of the initiates. We initiated 58 persons this year. Officers for 1950-1951 were: Director, Eleanor Johns; Vice- Director, Richard E. Johnson; Secretary, Ella Mae Whatley; EX OFFICIO MEMBER OF THE COUNCIL Treasurer, Leslie Gates; Faculty Adviser, Ralph M. Robinson. Officers elected for 1951-1952 are: Director, Richard E. HOWARD CURTIS BENNETT (See'PI MU EPSILON Johnson; Vice-Director, John Druyor; Secretary, Caroline Iverson; JOURNAL STAFF' for photograph and biographical sketch.) Treasurer, Donald Klippenstein; Faculty Adviser, Ralph M. Robin- son. 1952 REPORTS OF THE CHAPTERS 235 PI MU EPSILON JOURNAL April Alpha of Alabama, University of Alabama Alpha of Michigan, Michigan State College

The Alabama Alpha chapter of Pi Mu Epsilon held four pro- The chapter held thirteen meetings during the year 1950- gram meetings during the academic year 1950-1951, in addition 1951, including business meetings, program meetings, initiations, to two combined business meetings and initiation ceremonies. the annual winter banquet and the spring picnic. Topics at pro- Two social functions, a Christmas party and a spring picnic, were gram meetings were as follows: held. "Dissection problems on squares" by Dr. Stewart The following papers were presented: "Applications of mechanics to geometry" by Dr. L. M. Kelly "Methods of reducing curves to unit curves in integration" by "Soap film problem" by James Powell Herbert Diekhans "Mathematics as a career," a panel discussion led by Doctors "Statistics, common sense and epistemology" by Professor Stewart, Powell, Wells, Katz and Herzog 0. L. Lacey "The problem of the queens" by Dr. Herzog "The origin of mathematics" by Hassel Palmer "Education in Puerto Rico" by Dr. Grove "Theorem of Helly" by B. Vanderburg. "The modern theory of dimensions" by Dr. Nordhaus. The chapter initiated into membership twenty-four students At the winter banquet, Professor George Piranian, of the Uni- and faculty members. versity of Michigan, was the guest speaker. The subject of his The following officers were elected for the academic year talk was "Series and Sequences." 1951-1952: Director, Donald A. Swenson; Vice-Director, Ying Officers for the year 1951-1952 are: President, Paul Buben; Victor Wu; Secretary, Betty Campbell; Treasurer, Louis Jaffe; Vice-President,Douglas Behr; Secretary, Henry Leonard; Treas- Librarian, Dr. J. D. Mancill; Social Chairman, Elizabeth Cathey. urer, Charles Parker; Permanent Secretary, James Powell; Fac- ulty Advisers, Dr. L. M. Kelly and Dr. J. E. Powell.

Delta of New York, New York University Gamma of Ohio, University of Toledo At the beginning of the academic year 1950-1951, Elaine Weiss was Director; Alvin Saperstein, Vice-Director; Mordecai The activities of the Ohio Gamma chapter for the year 1950- Schwartz, Secretary; and Dr. Gottfried E. Noether, Permanent 1951 included business and social meetings, the annual banquet Secretary-Treasurer. Due to the mid year graduation of Director and a "pot luck" supper. Topics for the talks given at the meet- Weiss and Secretary Schwartz, Vice-Director Saperstein was ings were as follows: asked to assume the duties of both director and secretary for the "The mathematical method and problem solving" by Profes- second semester. sor E. H. C. Hildebrandt The chapter held five meetings during the year and the fol- "One engineer's mathematics" by Clarence M. McDowell lowing papers were presented: "The Golden Section" by Dr. Wayne Dancer, banquet speaker "The principle of the computing machine" by Dr. Gerald "The study of the cardioid by inversion" by Miss Violet Davis Goertzel "Mathematical engineering" by Ben Clymer (at the "pot luck" "Mathematics between the Scylla of physics and the Charyb- supper). dis of logic" by Dr. John Van Heijenoort Officers for the year 1950-1951 were: Director, Miss Grace "Probability theory and games of chance" by Dr. Gottfried E. Cutler; Vice-Director, Martha Goodwin; Secretary, Anna Tom; Noether Treasurer, Norman Krohn; Corresponding Secretary, Dr. Wayne "The drunkard's path or the gambler's ruin" by Dr. Bernard Dancer. Friedman Officers for the year 1951-1952 are: Director, Dr. C. E. "The foundations of physics" by Dr. Henry Margenau. Amos; Vice-Director, William McCord; Secretary, Jacqueline Officers elected for 1951-1952 are: Director, Willard L. McLain; Treasurer, James F. Machen; Corresponding Secretary, Maranker; Vice-Director, Walter Koppelman; Secretary, William Dr. Wayne Dancer. C. Zoellner. PI MU EPSILON JOURNAL April 1952 REPORTS OF THE CHAPTERS 237

Beta of California, University of California, Berkeley "Rational decimic curves invariant under D-14" by James McKay During the year 1950-1951, the California Beta chapter held "Arithmetic of transfinite numbers" by Arthur Moskin seven regular meetings exclusive of the fall initiation, banquet "Cantor's ternary set" by Douglas Newton and spring initiation picnic. The pattern usually followed at meet- "Job opportunities for the mathematician" by Dr. Roy M. ings was this: a business meeting, the speaker for the evening, Winger. and adjournment for refreshments. Talks given were: The annual Spring Picnic for members, prospective mem- 'Large scale high-speed computers" by Professor D. H. bers, faculty and friends; a "bridge-party"; and several short Lehmer ucoffee-sessions" made up the social program for the year. "The inner and outer development of mathematics" by Pro- There were seven new members elected and initiated into the fessor Abraham Seidenberg Fraternity during the year. In addition, there are 37 "associates" "Algebraic identities involving sums of squares" by Profes- elected and whose talks will be heard at future meetings, where- sor Raphael M. Robinson upon they will be elected to full membership. "Codification of geometry" by Dr. Ting-Kwan Pan "Post algebra" by Mr. Ralph Willoughby "Gambling and mathematics" by Professor Michel Loeve. Alpha of Virginia, University of Richmond At the semi-annual initiation banquet twenty-four new mem- bers joined the chapter. The Virginia Alpha chapter held nine meetings during the ac- Swami Stoughton Bell baffled and delighted all present (at the ademic year 1950-1951, including business meetings, program April meeting) by his feats of magic and prestidigitation and his meetings, fall initiation meeting and the annual banquet in May. accompanying mathematical commentary. Topics discussed at program meetings were the following: The semi-annual initiation picnic was held at Live Oak Park "Mathematical side of astronomy" by Mr. Ogburn in Berkeley. Twenty-seven new members joined the chapter. "Topology" by Marion Jeffries In the spring semester, Stoughton Bell I1 was elected to the "Opportunities in the field of mathematics after graduation position of treasurer left vacant by Leonard Stimpson's leaving from college", a panel discussion led by Dr. C. H. Wheeler in, the University to accept a position. The officers for the year Samuel Smith and Barbara McGeehee 1951-52 are: Director, William Noh; Vice-Director, Vivian Kob- "A glimpse of time" by Miss Isabel Harris. lick; Secretary, Susan Chakmakjian; Treasurer, NewmanH. Fisher, On the average, there were from twelve to twenty-five per- Jr.; Librarian, Sarah Hallum. sons present at each meeting; and there were seven new mem- bers initiated during the year, bringing the total membership in the chapter since the charter was granted to 60. Beta of Washington, University of Washington Officers for 1950-1951 were: Director, Dr. C. H. Wheeler in; Vice-Director, Samuel L. Smith III; Secretary, Ann Jones; Treas- The first meeting for the 1950-1951 session was a business urer, James D. Sutherland. meeting at which the following officers were elected: Director, Officers for 1951-1952 are: Director, Dr. C. H. Wheeler III; Richard Woollett; Vice - Director, Robert Wisner; Secretary- Vice-Director, Willard E. Meador, Jr.; Secretary, James L. Jud- Treasurer, Miss Mary Ann Deggeller. son; Treasurer, William F. Herget. The following talks were presented during the academic year 1950-51: "Cesaro, on one type of divergent integral" by Gerald Rogers Alpha of Kentucky, University of Kentucky "Formal power series systems" by Dr. Richard W. Ball "Waring's problem" by John Selfridge During the academic year 1950-1951, the Kentucky Alpha "Tensor analysis" by Ted R. Jenkins If chapter held six regular meetings in addition to the initiation "Two-tuple rings" by Richard Byrne banquet held in January. The following papers were presented: 238 PI MU EPSILON JOURNAL April REPORTS OF THE CHAPTERS 239 "Report on the International Congress of Mathematicians" by Alpha of Nebraska, University of Nebraska Dr. V. F. Cowling "An elementary geometry problem" by Dr. A. W. Goodman "Calculating machines" by Dr. M. S. Davis Four program meetings were held by the Nebraska ~l~h~ "On a problem of S. Bernstein" by Dr. V. F. Cowling chapter during 1950-1951, in addition to the social meetings The program consisted of the following talks: "A problem in linear algebras" by Mr. A. E. Foster "Elasticity" by Dr. Tadeusz Leser. "An application of the solution of a special recursive At the initiation banquet Dr. K. 0. Lang gave an entertaining by Dr. H. B. Ribeiro talk on "Rain Making." "A method of quality production control" by J. w. Adams Officers for 1950-1951 were: Director, Wimberly C. Royster; "Biology and mathematics" by Professor B. H. Burma Vice-Director, Donald C. Rose; Librarian, Cordell B. Moore, Sec- "Movement of pressure systems" by Professor L. K. Jackson. retary, A. E. Foster; Treasurer, Sherman B. Vanaman. The new officers for 1951-1952 are: Director, George cobel; Officers for 1951-1952 are: Director, Donald C. Rose; Vice- Vice-Director, Bruce Emmons; Secretary, Donna Grueber; T~~~~- Director, Ruric E. Wheeler; Librarian, Wilson M. Zaring; Secre- urer, Winfred Zacharias. tary, William Swift; Treasurer, Ralph C. Brown. The number of active members at the beginning of the year was 23 and the number of members initiated during the year was Beta of Pennsylvania, Bucknell University 16. The total membership in the chapter since the charter was granted is 256. The following papers were presented to the Pennsylvania Beta chapter during the year 1950-1951: Alpha of the District of Columbia, Howard University "The background of necessary and sufficient" by Emil Polak "Diophantine problems" by Ralph Jones The installation ceremonies of this chapter were held on "Dimensional analysis" by Francis Huber March 29, 1951, at Frazier Hall, Howard University. The instal- "The cross ratio" by F. S. McFeely. lation ceremony was conducted by the Director General of the Officers elected for 1951-1952 are: Director, Professor W. Fraternity, Professor C. C. MacDuffee. K. Smith; Vice-Director, James E. Hole; Secretary, R~~~~~~~ Dr. Mordecai Wyatt Johnson, President of the University, ac- Scheerer; Treasurer, C. Jerome Sechrist. cepted the Charter and Seal of the Fraternity for the University and Dr. Elbert F. Cox, for the Mathematics Department. Dr. Walter T. Daniels of the School of Engineering and Arch- Alpha of Kansas, University of Kansas itecture spoke on "Fundamental Research." Initiation ceremonies were held following the installation The Kansas Alpha chapter held its annual spring banquet in banquet. April, 1951. The speaker, Professor E. S. Robinson, gave an ad- The chapter held three meetings following its installation. dress on "Why some philosophers are interested in mathemat- The first and second meetings were organizational. ics." At the May meeting Dr. Butcher of the Mathematics Depart- The winter initiation and tea Was held in January, and, in ad- ment spoke on "What is a Tensor?" diti~n,there were several business meetings during the year. Officers for the chapter are: Director, Dr. George H. Butcher; The officers elected for the year 1951-1952 are: Director, President, Lt. Robert N. Smith, USAF; Secretary, Mrs. Andretta Robert Fisher; Vice-Director, David Murcray; Secretary, james Yeldell; Treasurer, Young Lee. Larkin; Treasurer, Kathleen O'Donnell; Corresponding Secretary, Wealthy Babcock. 24 0 PI MU EPSILON JOURNAL April

Beta of Illinois, Northwestern University The program of the Northwestern University chapter for the year 1950-1951 was as follows: "Theorems of the universen by Dr. W. S. Krogdahl of the Department of Astronomy "Interesting facts about numbersn by Dr. R. P. Boas "Philosophy and mathematics" by Dr. Douglas Morgan of the MEDALS, PRIZES AND SCHOLARSHIPS Department of Philosophy "Science and religionn by Dr. Edson Peck of the Department of Physics EDITOR'S NOTE. Each chapter undoubtedly will be interested in "Analogue computer" by Dr. Elliott Buell, as part of a tour learning what other chapters are doing along the line of prize competitions. So the editor makes the request that chapters of- through the Technological Institute. fering prizes, scholarships, or other awards, write up their plans The officers elected for 1951-1952 are: President, Richard for such contests and submit them for publication in this journal. Goldberg; Vice-president, Franklin Peterson; Treasurer, David Whitehouse; Secretary, Grant Steffen. In the Treasurer's Report of the New York Delta chapter there appeared this item: "Prizes for Mathematics Club, $35.00." The chapter should be congratulated on giving so generously to a ERRATA most worthy cause. The following errata in Volume 1, Number 5, The Michigan Alpha chapter annually makes the L. C. Plant have been called to the attention of the editors. awards. These are given to the students who have in the past Page 180: In reference (1) the author's name is year contributed the most to mathematics through scholarship, R. G. Archibald. interest in mathematics and help to the mathematics department. Page 191: In the second line from the bottom of Last year's awards were presented to Charles Parker and R. the page, change "Sco" to "Socn. Douglas Behr by Dr. J. H. Bell.

The New York Alpha chapter this year introduced, as a part "Professor," said the student in search of knowl- of its activities, a "Problem of the Month" feature. At each of the edge, "will you try to explain to me the theory of lim- monthly meetings, the Faculty Adviser, Professor Paul Gilbert, its?" proposes one or two problems to be solved by the members. The person who hands in the best solution of the problem (or prob- "Well, young man, assume that you have called lems) is awarded a Pi Mu Epsilon key, or cash award. In the on a pretty young woman. You are seated at one end Problem Section of this issue of the journal there appear the of the divan and she is seated at the other end. You problems proposed for solution at the February meeting. The move halfway toward her. Then you move half of the prize went to a senior in this year's graduating class, Leo A. remaining distance toward her. And again you reduce Magnanti. the distance separating you from her by fifty per cent. Continue this for some time. Theoretically, you will never reach the girl. On the other hand, you will soon get close enough to her for practical purposes." - DAVIDSON SCRIPTS 'N PRANKS 1952 OPEN LETTER 24 3

12:OO Lunch, followed by a 30 minute business meeting for mem- bers and delegates.

An open letter to the members and chapters of the Pi Mu Epsilon 2:00 p.m. Sessions of the Mathematical Fraternity: Association, including the first of three one hour Earle Ray- A national meeting of the Pi Mu Epsilon Fraternity will be mond Hedrick Lectures by held at Michigan State College, East Lansing, Michigan, on Labor Professor Tibor Rado of Ohio Day, September 1, 1952, just before the meetings of the Mathe- State University on "Deriva- matical Association of America (September 1-2), and those of the tives and Jacobians." American Mathematical Society (September 2-6). The Institute of Mathematical Statistics and the Econometric Society are also 7:30 p.m. If enough student speakers are scheduling parallel meetings in East Lansing at that time. It is available, an additional pro- hoped that as many chapters as possible may send delegates. gram Monday evening may be scheduled. The postcard ballot concerning this meeting was returned by a majority of the chapters, and all but three votes favored the Each member of Pi Mu Epsilon who thinks he (or she) may meeting. Only two chapters were willing in January to promise to possibly attend the meeting (and each chapter who may possibly supply a student speaker, but others indicated that they might send delegates) is urged to send a postcard addressed to the Sec- possibly supply one. They are urged to inform the Secretary retary General (207 P. M., Michigan State College, East Lansing, General by May 15. Michigan) before June 1 giving his (or her) name, summer ad- dress, and chapter. These cards do not imply a definite commit- Although the program is not definite at this time, the follow- ment to attend, but serve as a mailing list. The Secretary will ing tentative outline is being considered. then mail final announcements of the program, (including the Mathematical Association program) and room reservation cards Sunday, August 31, 8:00 p.m. Informal discussion meeting to those people and chapters before the end of June. about programs for local chap- ter meetings, to serve as a ba- This is a real opportunity for students interested in mathe- sis for the round table discus- matics to become acquainted with each other and with their future sion Monday morning. colleagues in the profession. The cooperation of each of you is requested to make this meeting a success. Meeting of the National Coun- cil. February 28, 1952

Monday, Sept. 1, 9:OO-1l:OOa.m. Four or five 20 minute talks, including at least two by stu- J. Sutherland Frame dents and at least one by a Secretary-Treasurer General member of the National Coun- cil.

Roundtable discussion onpro- grams and activities of local chapters. 1952 NEWS AND NOTICES 24 5

the points of contact of the grains of sand and in the nar- rower channels. One of the projects of the Research In- stitute of the University of Oklahoma is to study the ge- ometry of the changing oil-water interface. It is known NEWS AND NOTICES that under certain idealized conditions, the cross sec- tion of the oil-water interface is a curve traced by the focus of one branch of an hyperbola as the hyperbola The Oklahoma Alpha Chapter of Pi Mu Epsilon is sponsoring rolls on a straight line. Two professors of the mathe- a mathematical news letter addressed to high school students in- matics department are devoting time to a study of the terested in mathematics. It seems to be a most worthy undertak- idealized geometry of this problem through the use of ing. Professor R. V. Andree, a contributor to and a loyal support- differential equations. It is hoped this study will aid in er of our own publication, the PI MU EPSILON JOURNAL, is act- a more complete understanding of the physics of residu- ing as adviser to the OKLAHOMA UNIVERSITY MATHEMATICS al oil recovery. LETTER. He has sent us a copy of Volume 1, Number 1, pub- lished December 1951, at Norman, Oklahoma. The following par- Copies of the Pi Mu Epsilon constitution and by-laws are agraph is an excerpt from Professor Andree's communication to available to the chapters as long as the present supply lasts at a the editor: charge of one dollar a dozen prepaid, or one dollar a dozen plus postage if payment is not made in advance. HIGH SCHOOL MATHEMATICS LETTER A large supply of official key order blanks is available upon request. A mathematical letter will be mailed twice a se- mester to interested high school teachers by the Univer- A letter from Secretary-Treasurer Harry M. Gehman of the sity of Oklahoma Chapter of Pi Mu Epsilon. Each letter Mathematical Association of America called our attention to the will contain mathematical news, a short article and a availability of extra copies of the booklet entitled "Professional problem section. High school students are invited to Opportunities in Mathematics," and he requested that we either submit solutions to the problems. A list of correct solv- print a review (for which he enclosed a copy) or at least give ers will be carried in the next letter. High school teach- some space to the booklet in the Pi Mu Epsilon Journal. Though ers wishing to receive these letters should send their we are glad to meet the latter request, we are limited in re- n name and school address to Professor Richard Andree, sponding to the former as we have not yet set up a "Review sec- Dept. of Mathematics, University of Oklahoma, Norman, tion in the Journal. Okla." We have, however, read the booklet carefully and recommend it strongly to our readers. Indeed, it is a timely article similar He states that each "Lettern will contain a short article. Be- in intent to "Why Study Mathematics? ," a condensation to a pam- lieving that our readers will be interested in the short article on phlet published by the Canadian Mathematical Congress and pre- "oil recovery," which appeared in Number 1, and with permission pared by Professor Norman Miller (Queen's College) and others, of the 0. U. Math. LETTER, we reprint the article here. which we published in Number 3 of this journal. While the Cana- dian pamphlet was addressed to the interests of Canadian high school students, in the main, yet it was well received outside of 0. U. MATHEMATICAL RESEARCH IN OIL RECOVERY Canada, not only by high school students but by those in college and university as well. The booklet published by the American Many of you realize that when residual oil is recov- Mathematical Monthly should be helpful to any young student who ered from underground .sand, water or gas is pumped is considering mathematics as a career. into the sand to "flood outn the oil. Only about half of the oil is recovered in this manner; the rest remains around 246 PI MU EPSILON JOURNAL April

It was prepared by a committee of the Association, composed of W. H. Brinkman, Z. I. Mosesson, S. A. Schelkunoff, S. S. Wilks, and Mina Rees, Chairman. Reprints of this article (booklet) may be obtained from Professor H. M. Gehman, Mathematical Associ- ation of America, University of Buffalo, Buffalo 14, N. Y.; the cost is 259 for single copies and 109 each for orders of ten or more.

We are in receipt of a commmication from Professor Akit- sugu Kawaguchi, Faculty of Science, Hokkaido University, Sapporo, DIRECTORY Japan, asking us to settle an exchange relation between the publi- of cation, TENSOR, of the Tensor Society and our publication, Pi Mu Epsilon Journal. TENSOR is now in its New Series, Volume 1, PI MU EPSILON FRATERNITY, INC. Number 1, having been published in 1950, only a few months after our Journal made its first appearance. It is the official organ of the Tensor Society for the publication of original researches of members of the Society. General Officers (1951-1954)

Notice to Chapter Secretaries from the Business Manager Director General: Professor C. C. MacDuffee, 202 North Hall, University of Wisconsin, Madison 6, Wisconsin Will you please co-operate with me in the following three Vice-Director General: Professor W. M. Whyburn, Department ways: of Mathematics, University of North Carolina, Chapel Hill, 1. When the list of new initiates is sent to the Secretary North Carolina General, send a copy of the names and addresses of initiates with the date of initiation to Pi Mu Epsilon Journal, 15 Smith College, Secretary-Treasurer General: Professor J. S. Frame, 207 Phys- Syracuse University, Syracuse 10, N. Y. This will enable us to ics-Mathematics Bldg., Michigan State College, East Lans- distribute the Journal immediately after it is printed, thus avoid- ing, Michigan ing a delay such as has been caused in some cases in the past, Councilors General: due to our lack of complete information. Professor S. S. Cairns, Department of Mathematics, Univer- 2. Send, also, the name and address of the Permanent Sec- sity of Illinois, Urbana, Illinois retary, or the Corresponding Secretary, or some other officer of your chapter, with whom we may communicate not only during the Professor Tomlinson Fort, Department of Mathematics, Uni- summer but also during the following academic year. versity of Georgia, Athens, Georgia 3. Advise each initiate that when he gives us a change of ad- Professor Sophia McDonald, Department of Mathematics, dress he should state also the name of the chapter into which he University of California, Berkeley, California was initiated. Professor Ruth W. Stokes, Department of Mathematics, Syra- Howard C. Bennett cuse University, Syracuse 10, New York Mr. Howard C. Bennett (ex officio), Department of Mathe- matics, Syracuse University, Syracuse 10, New York PI MU EPSILON JOURNAL April 1952 DIRECTORY 249

Corresponding Secretaries1 (7) Illinois Alpha, 1924, University of Illinois, Urbana, Illinois; 1951-1952' Prof. Echo Pepper, 1005 S. Sixth, Champaign, Illinois (42) Illinois Beta, 1944, Northwestern University, Evanston, Ill.; (5) Alabama Alpha, 1922, University of Alabama, University, Ala.; Mr. Richard R. Goldberg, Department of Mathematics Dr. H. S. Thurston, Box 1453, University, Alabama (6) Iowa Alpha, 1923, Iowa State College, Ames, Iowa; (40) Arizona Alpha, 1941, University of Arizona, Tucson, Arizona; Mr. Donald L. Klippenstein, Department of Mathematics Dr. R. F. Graesser, Department of Mathematics (16) Kansas Alpha, 1928, University of Kansas, Lawrence, Kansas; (22) Arkansas Alpha, 1931, University of Arkansas, Fayetteville, Prof. Wealthy Babcock, 209 Strong Hall, Department of Arkansas; Mathematics Faculty Adviser: Professor S. L. Hull, Department of Mathematics (31) Kansas State College, Manhattan, Kansas; Prof. W. T. Stratton, Department of Mathematics (12) California Alpha, 1925, University of California, Los Angeles 24, California; (49) Kansas Gamma, 1950, University of Wichita, Wichita, Kansas; Faculty Adviser: Prof. W. T. Puckett, Department of Professor C. B. Read, Department of Mathematics Mathematics (13) Kentucky Alpha, 1927, University of Kentucky, Lexington, Ky.; (19) California Beta, 1930, University of California, Berkeley 4, Dr. H. H. Downing, Department of Mathematics California; (38) Louisiana Alpha, 1939, Louisiana State University, Baton Mrs. Sophia L. MacDonald, Room 453, Wheeler Hall Rouge 3, Louisiana; (33) Colorado Alpha, 1936, University of Colorado, Boulder, Colo.; Professor H. T. Karnes, Department of Mathematics Professor Jack R. Britton, Department of Mathematics (39) Michigan Alpha, 1940, Michigan State College, East Lansing, (50) Colorado Beta, 1950, University of Denver, Denver, Colorado; Michigan; Miss Katherine C. Garland, Department of Mathematics Dr. L. M. Kelly, Department of Mathematics (41) Delaware Alpha, 1941, University of Delaware, Newark, Del- (4) Missouri Alpha, 1922, University of Missouri, Columbia, Mis- aware; souri; Professor Russell Remage, Jr., Department of Mathe- Professor Mary Cummings, 213 Engineering Building matics (11) Missouri Beta, 1925, Washington University, St. Louis 5, Mis- (52) District of Columbia Alpha, 1951, Howard University, Wash- souri; ington 1, D. C.; Professor Jessica Young Stephens, Dept. of Mathematics Dr. G. H. Butcher, Jr., 2641 Myrtle Avenue, N. E. (43) Missouri Gamma, 1945, St. Louis University, St. Louis, Mo.; (51) Florida Alpha, 1951, University of Miami, Coral Gables 46, Professor Francis Regan, Department of Mathematics Florida; (9) Montana Alpha, 1925, Montana State University, Missoula, Mrs. Georgia Del Franco, Department of Mathematics Montana; (29) Georgia Alpha, 1934, University of Georgia, Athens, Georgia; Permanent Secretary: Prof. A. S. Merrill, 533 E. Beckwith Professor W. S. Beckwith, 731 Cobb Street (15) Nebraska Alpha, 1928, University of Nebraska, Lincoln, Ne- braska; -when known, the name and address of the Permanent Secretary, Faculty Adviser: Dr. Edwin Halfar, Dept. of Mathematics or Corresponding Secretary, is given; otherwise, that of the Fac- ulty Adviser or Department Chairman. (45) New Hampshire Alpha, 1948, University of New Hampshire, u umber appearing before chapter designation indicates chrono- Durham, N. H.; logical order in which charter was granted. Miss Elinor Burleigh, Alpha Chi Omega 250 PI MU EPSILON JOURNAL April DIRECTORY 251

(1) New York Alpha, 1914, Syracuse University, Syracuse 10, (36) Oregon Beta, 1938, Oregon State College, Corvallis, Oregon; New York; Professor G. A. Williams, Department of Mathematics Professor Nancy Cole, Department of Mathematics (3) Pennsylvania Alpha, 1921, University of Pennsylvania, Phila- (10) New York Beta, 1925, Hunter College, 695 Park Ave., New delphia 4, Pennsylvania; York 21, N. Y.; Dr. R. D. Shafer, Department of Mathematics Prof. Jewel Bushy and Miss Leila Singh, Department of Mathematics (8) Pennsylvania Beta, 1925, Bucknell University, Lewisburg, Pennsylvania; (26) New York Gamma, 1933, Brooklyn College, Bedford Ave. and Professor W. K. Smith, Department of Mathematics Ave. H., Brooklyn 10, N. Y.; Prof. Moses Richardson, Department of Mathematics (17) Pennsylvania Gamma, 1929, Lehigh University, Bethlehem, Pennsylvania; (28) New York Delta, 1933, N. Y. University, 100 Washington Sq. Professor R. R. Stoll, Department of Mathematics East, New York 3, N. Y.; Prof. Van Heijenoort, Department of Mathematics (20) Pennsylvania Delta, 1930, Pennsylvania State College, State College, Pennsylvania; (30) New York Epsilon, 1935, St. Lawrence University, Canton, Mr. Walter J. Harrington, Department of Mathematics New York; Dr. Ruth Peters, Department of Mathematics (44) Pennsylvania Epsilon, 1947, Carnegie Inst. of Tech., Pitts- burgh 12, Pennsylvania; (53) New York Eta, 1951, University of Buffalo, Buffalo 14, N. Y.; Professor David Moskovitz, Department of Mathematics Mr. Howard W. Baeumler, Department of Mathematics (47) Virginia Alpha, 1948, University of Richmond, Richmond, Va.; (24) North Carolina Alpha, 1932, Duke University, Durham, N.C.; Professor E. S. Grable, Box 45, University of Richmond Prof. F. G. Dressel, 309 Frances Street, Durham, N. C. (25) Washington Beta, 1932, University of Washington, Seattle 5, (46) North Carolina Beta, 1948, Univ. of N. C., Chapel Hill, N. C.; Washington; Dr. W. M. Whyburn, Department of Mathematics Dr. Roy Leipnik, Department of Mathematics (2) Ohio Alpha, 1919, Ohio State University, Columbus, Ohio; (27) Wisconsin Alpha, 1933, Marquette University, Milwaukee 3, Mr. Pat Sterbenz, Brand Road, Worthington, Ohio Wisconsin; (13) Ohio Beta, 1927, Ohio Wesleyan University, Delaware, Ohio; Dr. H. P. Pettit, Department of Mathematics Prof. Rufus Crane, 269 West William St.,Delaware, Ohio (37) Wisconsin Beta, 1939, University of Wisconsin, Madison 6, (32) Ohio Gamma, 1936, University of Toledo, Toledo, Ohio; Wisconsin; Dr. Wayne Dancer, Department of Mathematics Professor R. E. Fullerton, Dept. of Mathematics (48) Ohio Delta, 1949, Miami University, Oxford, Ohio; Dr. H. S. Pollard, Upham Hall, Oxford, Ohio (18) Oklahoma Alpha, University of Oklahoma, Norman, Oklahoma; Professor Dora McFarland, Department of Mathematics (35) Oklahoma Beta, Oklahoma A. and M. College, Stillwater, Oklahoma; Professor James H. Zant, Department of Mathematics (21) Oregon Alpha, 1931, University of Oregon, Eugene, Oregon; Dr. K. S. Ghent, Department of Mathematics 1952 INITIATES, ACADEMIC YEAR 1950-1951 253

NEW YORK DELTA, New York University (January 5, 1951) INITIATES, ACADEMIC YEAR 1950-1951 Harold L. Berkowitz Jerrold Goldman Theodore C. Salzman (Continued from Vol. 1, No. 5) Richard Chrenko Anita Goldrich Alfred Schulman Robert F. Davis Walter Koppelman Aaron Shinbein Elinor Duncan Isaac Levi Lawrence C. Turk Stanley Weitzner COLORADO ALPHA, University of Colorado Marilyn Flom Eleanore Lux Joseph Zapadinsky (spring, 1951) Susan Fuchs Frank Marasa - Miles A. Galin Willard Miranker William G. Zoellner Jack Angevine William L. Cox Marion Lang Ira Batt Jean B. Cropley Edward Leonard Eugene Barrows Thomas G. Donaldson Samuel Maley (May 12, 1951) Carrol D. Beaman Rueben H. Gablehouse Sherwin Miller Kenneth D. Brunelli Robert Gunning Harold Nylander Jerome Sidman J. Richard Collins Robert Harland Frederick P. Storke Richard Cordingly William J. Kellar Larry R. Travis NEW YORK EPSILON, St. Lawrence University (December, 1950) COLORADO BETA, University of Denver (March, 1951) Robert Lambert Mary Plumer Mrs. Harold Lyon Alison Provoost Oliver Lee Kingsley Bonnie Krogh Charles Robertson Jean Koch. Patrick Panfile William Scott (March 19, 1951) William Richter Robert George Cromie Jean Marie Kane

MISSOURI BETA. Washindon University NEW YORK ETA, University of Buffalo (April 2, 1951) Sefik Altay Cornelius Eldert Gilbert Walter Meier William Earnest Ball August William Geise, Jr. Gustav Mesmer Charter Members Sam J. LaMancusa Leonard R. Schaer Loren Allen Benson Richard Allen Glenn Paul Edward Muckerman Howard W. Baeumler Charles J. Lindsay Paul J. Schillo Edith R. Schneckenburger John Elwood Bertram Charles William Hargrave Max Martin Nathan Vernon N. Behrns Bernadine M. Lippert (old member) Joseph Edward Blanke, Jr. Richard Lee Hedden Edwin Raynold Rabin Albert G. Fadell Charles E. Maley Arthur Mark Bolsterli Edward John Heising Eugene Richard Rodemich Lorraine W. Farber June M. McArtney Phyllis C. Schwartz Norman Meyer Esther Seiden (old member) George Mark Brelias Robert Allen Hilker Charles Edward Russell Jean B. Feidner Harriet F. Montague Harlan R. Stevens Charles Dean Briner Barbara Bartels Hixon Frederick William Seubert Harrv M. Gehman David D. Strebe Thomas Maynard Burford, Jr. James Noble Holsen Donald Milton Stern (old member) Mabel D. ~ont~omary Frederic C. Warner William Frank Burggrabe, Jr. Richard Charles Huning Morris Stern Lillian Gough Lloyd J. Montzingo Everett T. Welmers Alfred Alden Cook George Montgomery Johnson Tadao Takayesu Leonard J. Grzankowski Ruth B. Noller Hadassah M. Nomof (old member) Leon Cooper Herbert Edmond Johnson Louis Joseph Tichacek Roger G. Hill Ina W. Welmers John Wallace Diesel Donald Michael Koch Cay George Weinel Joseph E. Kist V. Ellsworth Pound Erick Reeber Norma J. Wilson Robert Carrel Drews William Koenig . Thomas W. Wood Robert C. Kroeger Lester Herman Krone, Jr. NORTH CAROLINA ALPHA, Duke University NEW HAMPSHIRE ALPHA, University of New Hampshire (October 31, 1950) (November 16, 1950) Blair Bowers Alden B. Gorham Joyce McAfee Robert M. Price Rene H. Biron John Dutton Robert Leavitt William Chapman John W. Haskens Emily West Elinor J. Burleigll Frank Gagliuso Robert Louttit Thomas Edelbute ConradCaron Gilbert Gallant Robert L. Pease Donald Childs William Goldthwaite Francis Penney (May 8, 1951) Edward A. Demers John Jacobsmeyer Harry Plumb Neil R. Andon W. D. Climenson Lyle B. Connor Eugene Bernstein Thomas Cole G. H. Coppala 254 PI MU EPSILON JOURNAL April 1952 INITIATES, ACADEMIC YEAR 1950-1951 255

Edrnond Dewan Donn McGeihan Sidney Trundle William Marvin Blinker Barney Lawrence King Wayne Edward Richardson H. J. Dukes Douglas Montgomery R. M. Tucker Yu Why Chen Anthony Edward Labarre, Jr. Donald Ray Rutan Albert Erwin R. L. Padgett R. V. Warden Walter Dale Compton Mark Aldridge Melton Howard Wesley Prier C. C. Johnston Douglas Schafroth Martha Woolery Thell Lu Hilburn George Elva Neal Alva Taylor Stair, Jr. Eleanor Lake Bill Wright Robert Draper Hill George Richard Pickett Lyla Katherine Tisdale John Edward Hoffman Frank Raunikar Leon Tisdale Chia Ying Hsueh Charles William Reich Ted S. Webb NORTH CAROLINA BETA, University of North Carolina Norman Gene Rhode (Date of initiation not given)

H. Leroy Adams John Jones Jack Padgett OKLAHOMA BETA, Oklahoma Agricultural and Mechanical College L. P. Burton David Kerley Tullio J. Pignani Maurice Clayton Peter Kotsch Kloeppel Samuel Jackson Scott David L. Johnson Galyn D. Pownell David L. Weeks Richard Robney Croxton Mary Nunn Morrow Frank Wilhelm Stallard Arthur W. Liles Richard F. Stewart Zenon S. Zannetos Meyer Dwoss Thomas L. Nabors Jeron Thomas Terrell OREGON ALPHA, University of Oregon OHIO ALPHA, Ohio State University (May 4, 1951) (May, 1951) Kermit G. Clemens Gerald F. Owsley Donald R. Stark Walter Albrecht John Kettelle Sol Matt Carolyn Diane Ford William J. Parker Jochanan H. Stensch Beryl Cloffelter R. Conway LeCraw Norman Pomerantz Bruce A. Koppe Donald E. Robison Karl R. Stromberg Basil Curnutte Alfred Baker Lehman Richard Sapp David D. Lanning Sam C. Saunders John R. Toomey Henry Diehl Curt A. Levis Richard Shame1 Calvin T. Long Robert J. Silva LeRoy J. Warren Douglas Drake Walter Loewenstein Pater Stephan Cyrus R. McAUister James H. Skelton Thomas W. Williams Karl Eichhorn W. C. Lowry Robert Thompson Frank G. Snyder John Garing Barbara Manning Claude Walston Otto Cleve Juelich Richard Zemlin OREGON BETA, Oregon State College (May 17, 1951) OHIO GAMMA, University of Toledo Kenneth W. Andreson John M. Gerhauser Gerald S. Racanelli Donald Ewing Jacquelyn McLain Richard Pio Phillip Anselone William W. Gribble Ronald R. Runge James F. Machen Phillip Miller Steven Szabo Christian G. Backman Dale M. Holme John W. Schultz Robert I. Beecroft Dale E. Kirk Robert F. Sharp John V. Blankenbaker Gail Elaine Kromer Robert G. Snyder OHIO DELTA, Miami University Robert W. Brown Walter M. Lovett Warren C. Steele (November 2, 1950) Burton C. Carlson Charles P. Luehr Robert S. Steffanson Arthur N. L. Chiu William Maillard Matti S. Siukola Allan Branch J. Stanley Fouch Edmund Lis William R. Cromwell Don G. McKinstry Sirkka Liisa Siukola Kenneth Brinkman James Harris David Perry Monte V. Davis Marshall W. McMurran Ralph A. Sundberg Marilyn Copeland Samuel Harris Phyllis Rinderknecht Glen E. Dillon LeVern E. Marsters John Sur Margaret Dennison Robert Hefner, Jr. Robert Rowntree Sterling Ditsworth Donald R. Miedaner George C. Turrell Francis Edmunds Edwin Hunger Richard Schmunk David LeRoy Espey Henry T. Norton Richard L. Whitely Donald Emerick Eleanor Keffer Marylois Scrafford Nancy Ann Gentle John D. Worthman Kenneth Last PENNSYLVANIA ALPHA, University of Pennsylvania (March 15, 1951) Jack R. Webb Fred W. Aron, Jr. Lee Frantz Thomas Mott Anita Bredt Lawrence Frishkopf Bernice Newman Fu-Ho Chan H. Newton Garber Alexander LeF. Pugh III OKLAHOMA ALPHA, University of Oklahoma Robert B. Dickson Walter Hess Leser George Rosen (May 3, 1951) Sheldon J. Einhorne David J. Lobe1 Girard K. Rosenblum David Eugene Field Pinchas Mendelson Judith Richman Bill Atldnson Jean Marie Baldwin Carl Edward Bleil Bruce S. Fisher John M. Miller Harvey S. Safren Kenneth Lee Austin Richard Park Bland Tom Kelman Boehme PI MU EPSILON JOURNAL April INITIATES, ACADEMIC YEAR 1950-1951

Elizabeth Schwarz Matthew London Herbert J. Vassian Bruce L. Lercher Gertrude Nissenbaum Jean H. Snover Thomas Scott James S. McCaughan Kurt F. Wisebrun Eugene T. Leverentz William H. Peirce Willia Soper Charles C. Sweley Donald R. McIlvain Milton Wollman Ely S. Levinsky Duane A. Rohde James E. Thompson Werner B. Teutsch Clifford L. Maier Lee A. Rube1 Roger N. Van Norton Orville J. Marlowe Peter D. Welch

PENNSYLVANIA BETA, Bucknell University (December 4, 1950) (May 24, 1951)

Eugene L. Baker Rose Hostetler Richmond S. Murdough John F. Bartholomew Eugene Hough Roger B. Russell, Jr. Dorothea Bohling Francis C. Huber William L. Mickolls Robert P. Bell Millard Johnson Mary Tsingou Elinor Childs Paul M. Hurst, Jr. George Polinko Colleen Byrne Margaret H. Jolly Peter H. Wackman Fred E. Cloud Kirk Kazarian Eugene B. Price Leslie W. Gunter William J. Ochalek David B. Wittry Robert A. DeGroot Martin W. Kwasnoi Samuel R. Pulford Margaret L. Garrett Rocco L. Mascioli , Rosemary Scheerer William G. Hendrixson Edwin R. Mignell Jerome C. Sechrist James E. Hole John G. Morrison Donald J. Skovholt Joan G. Hoshauer Daniel Woodside INITIATES, ACADEMIC YEAR 1951-1952

VIRGINIA ALPHA, University of Richmond CALIFORNIA ALPHA, University of California, Los Angeles (November 1, 1950) (January, 1952)

Sterling Clark Marian Jeffries William C. Martin Fred Baskin De~iSonA. Curtiss Barrett O'Neill Grace Collins William F. Herget Willard E. Meador Jordan J. Bloomfield Richard L. Dunn Eugenia Houg Kiser Charles E. Buie Martin Emil Fuller U Richard E. Tuck Enrique Cansado Kaspar P. Grimm Maria Weber (May 2, 1951) Noriko Chiwaki Eugene Levin David F. Weinberg

James L. Judson CALIFORNIA BETA, University of California, Berkeley (December 18, 1951) WASHINGTON BETA, University of Washington Robert Lee Besag Nasrollah Javahertarash Richard Montague Richard Byrne James McKay Gerald S. Rogers Robert S. Freeman Jan Kalicki Leon Orloff Ted R. Jenkins Arthur Moskin Robert J. Wisner James B. Herreshoff John Killeen Donn B. Parker Douglas Newton Arnold M. Matscke

WISCONSIN ALPHA, Marquette University DELAWARE ALPHA, University of Delaware (April 30, 1951) (Winter, 1951-52)

John E. Callan Willard E. Lawrence Dave A. Rux Vincent Bacchetta Elam K. Hertzler Morris Glatt Mrs. Dale S. Godbarsen John W. Lindner Walter P. Rynkiewicz Fred Q. Barnett Verna Frances Lair John B. Lyon L. Beverly Hauk Matthew A. McCormick Jerome J. Soboleski John H. Barrelt Donald B. Clement Esther Simon Leonard F. Herkowski John E. Quinlan Dorothy J. Stodola Girard Golden William V. Smith Bencion Kampos Fred Ray Terrance E. Utz Carl G. Koch Dave R. Rhyner Felix J. Veliath IOWA ALPHA, Iowa State College (Fall, 1951) WISCONSIN BETA, University of Wisconsin (October 12, 1950) Sigurds Arajs Edwin Gatz, Jr. Stanley Petrick Marts D. Blue Lawrence Heyerdahl James Talboy, Jr. Roy C. Alverson Robert C. Carson Trevor Evans Mary Ethel Buxton Arthur Karwath Robert Tweedy Edward H. Batho John 0. Danielson Walter A. Harman, Jr. Elaine Kobylanski Byrd Gary Meisters Martin B. Wilk Chris C. Braunschweiger Stephen A. Dickinson William Hemmerle Bruce E. Deal Kalliope Papantonis Joan Wilson Jean-Francois Canu Ralph J. Eckert Melvin Henriksen Howard Peterson QUALITY LITHOPRINTING

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