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Pi Mu Epsilon Journal Tee Official Publicationof the Honorary Mathematical Fraternity

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Pi Mu Epsilon Journal Tee Official Publicationof the Honorary Mathematical Fraternity 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.
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