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By R. C. ARCHIBALD, Brown University.

Presidential Address1 delivered 2 before the Mathematical Association of America;
September 6, 1923.

"Mathematics and Music, the most sharply contrasted fields of intellectual activity which one can discover, and yet bound together, supporting one another as if they would demonstrate the hidden bond which draws together all activities of our mind, and which also in the revelations of artistic genius leads us to surmise unconscious expressions of a mysteriously active intelligence." In such wise wrote one supremely competent to represent both musicians and mathematicians, the author of that monumental work, On the Sensations of Tone as a physiological basis for the Theory of Music.

"Bound together?" Yes! in regularity of vibrations, in relations of tones to one another in melodies and harmonies, in tone-color, in rhythm, in the many varieties of musical form, in Fourier's series arising in discussion of vibrating strings and development of arbitrary functions, and in modern discussions of acoustics.

This suggests that the famous affirmation of Leibniz, "Music is a hidden exercise in arithmetic, of a mind unconscious of dealing with numbers," must


1 To the address as delivered a number of footnotes, mainly with a few references to the vast literature of the subject, have been added. A fundamental work in this connection is H. L. F. Helmholtz, On the Sensations of Tone as a Physiological Basis for the Theory of Music, and the best edition is the second English edition translated with many additions from the fourth (last) German edition by A. J. Ellis, London, 1885; the third edition was reprinted from the second in 1895, and the fourth from the third in 1912. While some mathematical discussion occurs in this work, the standard treatise on the mathematical theory is Rayleigh, Theory of Sound, 2 vols., second ed., London, 1894. Another work of high order is H. v. Helmholtz, Vorlesungen über die mathematischen Akustik, 1898, vol. 3 of Vorlesungen über theoretische Physik, Leipsic. H. Lamb, The Dynamical Theory of Sound, London, 1910, was intended as a stepping stone to the writings of Helmholtz and Raleigh. Between 1898 and 1915, 38 papers by various authors appeared at Leipsic in 8 Hefte of Beiträge zur Akustik und Musikwissenschaft herausgegeben von C. Stumpf. For the most part, they are reprints of articles in Zeitschrift für Psychologie, Zeitschrift für Psychologie und Physiologie der Sinnesorgane, and 6. Kongress der Gesellschaft für experimentelle Psychologie. Another very valuable general work, discussing the writings of mathematicians on musical matters, is F. J. Fétis, Biographie Universelle des Musiciens et Bibliographie générale de la Musique, 8 vols., second ed., Paris, 1 (1873), 2 (1867), 3-4 (1869), 5–8 (1870). R. Eitner's Biographisch-bibliographisches Quellen-Lexikon der Musiker der christlichen Zeitrechnung bis zur Mitte des 19. Jahrhunderts, 10 vols., Leipsic, 1900-1904, is also sometimes useful. 2 At a joint session of the Mathematical Association of America and of the American Mathematical Society, Vassar College, Poughkeepsie, N. Y.

H. v. Helmholtz, Vorträge und Reden, Braunschweig, vol. 1, 1884, p. 82. See also Helmholtz, Popular Lectures on Scientific Subjects, London, 1873, p. 62.

4"Musica est exercitium arithmeticæ occultum nescientis se numerare animi," which occurs in a letter dated April 17, 1712, and addressed to Goldbach. It is letter 154 in Leibniz, Epistolæ ad diversos, vol. 1, Leipsic, 1734, p. 241. The quotation is in the section dealing with the question "Vnde oritur ex musica voluptas?" This is preceded and followed by two other sections on

be far from true if taken literally. But, in a very general conception of art and science, its verity, may well be granted; for, in creating as in listening to music, there is no realization possible except by immediate and spontaneous appreciation of a multitude of relations of sound.

Other modes of expression and points of view were suggested by that great enthusiast to whom America owed much, him who called himself 1 "the Mathematical Adam" because of the many mathematical terms he invented; for example, mathematic-to denote the science itself in the same way as we speak of logic, rhetoric or music, while the ordinary form is reserved for the applications of the science. He referred to the cultures of mathematics and music "not merely as having arithmetic for their common parent but as similar in their habits and affections." 2 "May not Music be described," he wrote, "as the Mathematic of Sense, Mathematic as the Music of reason? 3 the soul of each the same! Thus the musician feels Mathematic, the mathematician thinks Music,Music the dream, Mathematic the working life,-each to receive its consummation from the other when the human intelligence, elevated to the perfect type, shall shine forth glorified in some future Mozart-Dirichlet, or Beethoven-Gauss --a union already not indistinctly foreshadowed in the genius and labors of a Helmholtz"!4


But such intimacies in these cultures are not discoveries and imaginings of a later day. For two thousand years music was regarded as a mathematical science. Even in more recent times the mathematical dictionaries of Ozanam,5 Savérien, and Hutton, contain long articles on music and considerable space is devoted to the subject in Montucla's revised history,—which brings us to "Quibus musicis proportionibus homines delectantur?" and "Quando rationes surdæ in musica commode locum inueniunt?"

T. W. Preyer corrupted Leibniz's sentence into "Arithmetica est exercitium musicum occultum nescientis se sonos comparare animi" (compare M. Lecat, Pensées sur la Science, la Guerre, et sur des Sujets très variés, Brussels, 1919, p. 438). Preyer's thought in this connection will be apparent on turning to his monograph, "Ueber den Ursprung des Zahlbegriffes aus dem Tonsinn und über das Wesen der Primzahlen," pages 1-36 of Beiträge zur Psychologie und Physiologie der Sinnesorgane, Hermann von Helmholtz als Festgruss zu seinem siebzigsten Geburtstag. Gesammelt und herausgegeben von A. König, Hamburg and Leipsic, 1891.

1 J. Sylvester, in a footnote to "Note on a Proposed Addition to the Vocabulary of Ordinary Arithmetic," Nature, vol. 37, p. 152, 1887.

2 J. Sylvester, British Assoc. for the Adv. of Science, Report, 1869, page 7 of Notices and Abstracts.

3 Compare "Die Mathematik ist die Musik des Verstandes, die Musik die Mathematik des Gefühls" as employed by Josef Petzval, Jahresbericht der deutschen Mathematiker-Vereinigung, vol. 12, 1903, p. 327.

4 This passage occurs in a footnote in the midst of Sylvester's memoir "Algebraical researches containing a disquisition on Newton's rule for the discovery of imaginary roots . . .," Philosophical Transactions for 1864, vol. 154, 1865, p. 613.

J. Ozanam, Dictionaire Mathematique, Amsterdam, or Paris, 1691.

• A. Savérien, Dictionnaire Universel de Mathematique et de Physique, Paris, 1753, vol. 2.

7 C. Hutton, A Philosophical and Mathematical Dictionary, London, 1795, vol. 2; new edition, 1815.

8 J. F. Montucla and J. de La Lande, Histoire des Mathématiques, Paris, vols. 1 and 4, 1799, and 1802. Compare D. E. Smith, "The threatened loss of the second edition of Montucla's History of Mathematics" in this MONTHLY, 1921, 207–208.

the threshold of the nineteenth century. It is, therefore, not surprising that many mathematicians wrote on musical matters. I shall presently consider these at some length. But certain other facts may first be reviewed.

The manner in which music, as an art, has played a part in the lives of some mathematicians is recorded in widely scattered sources. A few instances are as


Maupertuis was a player on the flageolet and German guitar and won applause in the concert room for performance on the former. At different times William Herschel served as violinist, hautboyist, organist, conductor, and composer (one of his symphonies was published) before he gave himself up wholly to astronomy.2 Jacobi had a thorough appreciation of music.3 Grassmann was a piano player and composer, some of his three-part arrangements of Pomeranian folk-songs having been published; he was also a good singer and conducted a men's chorus for many years.4 János Bolyai's gifts as a violinist were exceptional and he is known to have been victorious in 13 consecutive duels where, in accordance with his stipulation, he had been allowed to play a violin solo after every two duels. As a flute player De Morgan excelled. The late G. B. Mathews knew music as thoroughly as most professional musicians; his copies of Gauss and Bach were placed together on the same shelf." It was with good music that Poincaré best liked to occupy his periods of leisure. The famous concerts of chamber music held at the home of Emile Lemoine during half a century exerted a great influence on the musical life of Paris. And in

1 Basler Jahrbuch, 1910, p. 46 in "Maupertuis" (pp. 29-53) by F. Burckhardt; see also "Maupertuis' Lebensende" by the same author in Basler Jahrbuch, 1886, pp. 153-159. These very interesting articles contain new material concerning the closing days of Maupertuis at the home of his friend Johann Bernoulli the second. Compare D. E. Smith, "Maupertuis and Frederick the Great" in this MONTHLY, 1921, 430-432. The first memoir presented to the French Academy by Maupertuis was "Sur la forme des instruments de musique," Mémoires de l'Academie Royale des Sciences, 1724, Paris, 1726, pp. 215-226 + 1 plate; Histoire, pp. 90-92. This memoir is not contained in Oeuvres de Mr. Maupertuis published at Dresden in 1752 and at Lyons in 175€.

2 Concerning the musical activities of Herschel, see especially The Scientific Papers of Sir William Herschel, vol. 1, London, 1912, pp. xiv–xxii; F. J. Fétis, Biographie, etc., vol. 4, loc. cit., and A. Noyes, The Torch-Bearers, Watchers of the Sky, New York, 1922, p. 231 f. Noyes perpetuates the old error about Herschel "deserting from the army." Herschel was born in 1738 and died in 1822.

S. Hensel, Die Familie Mendelssohn, 1729-1847, second ed., Berlin, 1880, vol. 2, pp. 364–365; English edition, London, 1881, vol. 2, p. 324.

4 Compare F. Engel, Grassmann's Leben in Hermann Grassmann's Gesammelte Mathematische und Physikalische Werke, vol. 3, part 2, Leipsic, 1911, pp. 250–253, 371–372. Grassmann was born in 1809 and died in 1877.

F. Schmidt, "Lebensgeschichte des ungarischen Mathematikers Johann Bolyai," Abhandlungen zur Geschichte der Mathematik, Heft 8, 1898, p. 141. Bolyai was born in 1802 and died in 1860; sketches by G. B. Halsted appeared in this MONTHLY, 1896, 1–5, and 1898, 35–38. Sophia E. De Morgan, Memoir of Augustus De Morgan, London, 1882, p. 16. See also A. M. Stirling, William De Morgan and his Wife, London, 1922, p. 61. De Morgan was born in 1806 and died in 1871.

* Proceedings of the Royal Society, London, vol. 101 A, the London Mathematical Society, 2 series, vol. 21, 1923, p. l. in 1922.

1922, p. xiv; also in Proceedings of Mathews was born in 1861 and died

L. Augé de Lassus, La Trompetle. Un demi-siècle de Musique de Chambre, Paris, 1911. See also D. E. Smith, "Emile Michel Hyacinthe Lemoine," in this MONTHLY, 1896, 29-33. Lemoine was born in 1840 and died in 1912.

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