-0ltl'du : ,"fp(23)(24)ly the form of equation (1),: first condition of (24) if and(2s)rnow that in this case theseI;econd condition of (24) andrvalue (25) is thereforeH"(u), (26)rterest in the detailed pro-:ver, the problem is only anpolynomials, so we will notntrng out that formula (25)of the harmonic oscillator.only these discrete values,he corresponding classicalncrete application of thesems in a diatomic molecule.tally. the observed energies901), one of the most eminent, was particularly distinguished'ork. As a student, he courtedto study the classic masters ofPOWER SER-ltrS sot-Ul'loNs AND SPITCIAL FTJNC-TIONS 221mathematics; and though he nearly failed his examinations, he became a first-ratecreative mathematician himself while still in his early twenties. In 1870 he wasappointed to a professorship at the Sorbonne, where he trained a wholegeneration of well known French mathematicians, including Picard, Borel, andPoincar6.The unusual character of his mind is suggested by the following remark ofPoincard: "Talk with M. Hermite. He never evokes a concrete image, yet yousoon perceive that the most abstract entities are to him like living creatures." Hedisliked geometry, but was strongly attracted to number theory and analysis, andhis favorite subject was elliptic functions, where these two fields touch in manyremarkable ways. The reader may be aware that Abel had proved many yearsbefore that the general polynomial equation of the fifth degree cannot be solvedby functions involving only rational operations and root extractions. One ofHermite's most surprising achievements (in 1858) was to show that this equationcan be solved by elliptic functions. His 1873 proof of the transcendence of e wasanother high point of his career.Several of his purely mathematical discoveries had unexpected applicationsmany years later to mathematical physics. For example, the Hermitian forms andmatrices he invented in connection with certain problems of number theoryturned out to be crucial for Heisenberg's 1925 formulation of quantum mechan-ics, and we have seen that Hermite polynomials and Hermite functions are usefulin solving Schrtidinger's wave equation. The reason is not clear, but it seems tobe true that mathematicians do some of their most valuable practical work whenthinking about problems that appear to have nothing whatever to do with physicalreality.APPENDIX C. GAUSSCarl Friedrich Gauss (Illl-1855) was the greatest of all mathematiciansand perhaps the most richly gifted genius of whom there is any record.This gigantic figure, towering at the beginning of the nineteenth century,separates the modern era in mathematics from all that went before. Hisvisionary insight and originality, the extraordinary range and depth of hisachievements, his repeated demonstrations of almost superhuman powerand tenacity-all these qualities combined in a single individual presentan enigma as baffiing to us as it was ttt his contemporaries.Gauss was born in the city of Brunswick in northern Germany. Hisexceptional skill with numbers was clear at a very early age, and in laterlife he joked that he knew how to count before he could talk. It is saidthat Goethe wrote and directed little plays for a puppet theater when hewas six, and that Mozart composed his first childish minuets when he wasfive, but Gauss corrected an error in his father's payroll accounts at theage of three.ra His father was a gardener and bricklayer without eithello See W. Sartorius von Waltershausen, "Gauss zum Geddchtniss." These personalrecollections appeared in 1856, and a translation by Helen W. Gauss (the mathematician'sgreat-granddaughter) was privately printed in Colorado Springs in 1966.I':;isll222 DIFFERENTTAL EeuAnoNSthe means or the inclination to help develop the talents of his son.Fortunately, however, Gauss's remarkable abilities in mental computationattracted the interest of several influential men in the community, andeventually brought him to the attention of the Duke of Brunswick. TheDuke was impressed with the boy and undertook to support his furthereducation, first at the Caroline College in Brunswick (1792-1795) andlater at the University of Gottingen (1795-1798).At the Caroline College, Gauss completed his mastery of theclassical languages and explored the works of Newton, Euler, andLagrange. Early in this period-perhaps at the age of fourteen orfifteen-he discovered the prime number theorem, which was finallyproved in 1896 after great efforts by many mathematicians (see our noteson Chebyshev and Riemann). He also invented the method of leastsquares for minimizing the errors inherent in observational data, andconceived the Gaussian (or normal) law of distribution in the theory ofprobability.At the university, Gauss was attracted by philology but repelled bythe mathematics courses, and for a time the direction of his future wasuncertain. However, at the age of eighteen he made a wonderfulgeometric discovery that caused him to decide in favor of mathematicsand gave him great pleasure to the end of his life. The ancient Greekshad known ruler-and-compass constructions for regular polygons of 3, 4,5, and 15 sides, and for all others obtainable from these by bisectingangles. But this was all, and there the matter rested for 2000 years, untilGauss solved the problem completely. He proved that a regular polygonwith n sides is constructible if and only if n is the product of a power of 2and distinct prime numbers of the form po : 22' + 1. In particular, whenk:0,1,2,3, we see that each of the corresponding numbers po :3,5,17,257 is prime, so regular polygons with these numbers of sides areconstructible. r5During these years Gauss was almost overwhelmed by the torrent ofideas which flooded his mind. He began the brief notes of his scientificdiary in an effort to record his discoveries, since there were far too manyto work out in detail at that time The first entry, dated March 30. 1796,states the constructibility of the regular polygon with 17 sides, but evenearlier than this he was penetrating deeply into several unexploredcontinents in the theory of numbers. ln 1795 he discovered the law ofquadratic reciprocity, and as he later wrote, "For a whole year thistheorem tormented me and absorbed my greatest efforts. until at last Irs Details of some of these constructions are given in H. Tietze, Famous Problems ofMathematics, chap. IX,