¹ Herschel, J. F. W., the article ‘Sound’ in the Encyclopaedia Metropolitana, London, 1830, iv, § 236.Google Scholar

² Sauveur, J., ‘Rapport des sons des cordes d'instruments de musique aux flèches des cordes, et sur une nouvelle détermination des sons fixes’, Paris, Histoire de l'Académie Royal des Sciences, Mémoires, 1713 (Paris, 1716), 324–48Google Scholar. On Sauveur, see DSB, xii, 127Google Scholar; for more on beats see Tyndall, J., Sound, London, 1867, 263–75Google Scholar. Not surprisingly, earlier reports of the phenomenon of beats are to be found. De Morgan drew attention to one such by Holder, William (1616–1698)Google Scholar, whom he named as the immediate predecessor of Sauveur, ‘because he distinctly opposes the sympathy of consonant vibrations, and its effects, to the clashing of dissonant vibrations’. De Morgan, who never balked at an interesting digression, quoted, chiefly for its quaintness and charm, a passage from Holder, 's Natural Grounds and Principles of Harmony, published in 1694Google Scholar. See De Morgan, A., ‘On the beats of imperfect consonances’, Cambridge Philosophical Society Transactions (1858), 10, 129–45.Google Scholar

³ Smith, Robert, Harmonics, or the Philosophy of Musical Sounds, Cambridge, 1749, 2nd edn, London, 1759Google Scholar. ‘It is worthy of note that at this period the book bears the name of the place where it is printed, not of the place where the publisher sells it. Both these editions are printed for Cambridge publishers (the Merrills).’ Note by A. De Morgan.

⁴ MrsHerschel, John, Memoir and Correspondence of Caroline Herschel, London, 1878, 35.Google Scholar

⁵ Robison, J., the article ‘Temperament’ in Encyclopaedia Britannica, Suppl, to 3rd edn, 1801, 652Google Scholar. Robison's praise is somewhat depreciated in value when we learn (from De Morgan) that he too had not fully understood his author's meaning. See note 30, below.

⁶ Peacock, George (ed.), Miscellaneous Works of Thomas Young, London, 1855, i, 136.Google Scholar

⁷ De Morgan, , op. cit. (2), 135.Google Scholar

⁸ Smith, , op. cit. (3), p. ix.Google Scholar

⁹ As quoted by Kassler, J. C., The Science of Music in Britain 1714–1830: A Catalogue of Writings, Lectures and Inventions, New York and London, 1979, ii, 952.Google Scholar

¹⁰ De Morgan, , op. cit. (2), 131n.Google Scholar

¹¹ Quoted by Chenette, L. F., Music Theory in the British Isles during the Enlightenment, a Dissertation presented for the degree of Doctor of Philosophy to the Ohio State University, 1967Google Scholar. Order number: AAC 6802965. University Microfilms International, 1967, 157.

¹² De Morgan, , op. cit. (2), 133.Google Scholar

¹³ De Morgan, , op. cit. (2), 129.Google Scholar

¹⁴ De Morgan, , op. cit. (2), 137–8.Google Scholar

¹⁵ De Morgan, , op. cit. (2), 140n.Google Scholar

¹⁶ Stephen, Leslie, ‘Augustus De Morgan’, DNB, xiv, 331–4.Google Scholar

¹⁷ De Morgan, , A Budget of ParadoxesGoogle Scholar. Reprinted, with the author's additions, from the Athenaeum, London, 1872.Google Scholar

¹⁸ Copy in the author's possession.

¹⁹ In the Herschel Collection at the Harry Ransom Humanities Research Center of the University of Texas at Austin. The original letter to Whewell is in the Whewell archive at Trinity College, Cambridge, Add. MS A.202, 135. I am grateful to the Master and Fellows of Trinity College, as owner, for their permission to publish.

²⁰ Horace, , Ars poetica, 287Google Scholar. To celebrate domestic facts. De Morgan was himself a former Fellow of Trinity, so perhaps this tag is more suitably rendered: A comic vein! the poet may decide: His College annals then the theme provide.

²¹ Truth is the daughter of Time. Gellius, xii, 11, 7Google Scholar. The whole passage reads:

Therefore he said that one should have on his lips

these verses of Sophocles, the wisest of poets:

See to it lest you try aught to conceal;

Time sees and hears all, and will all reveal.

Another one of the old poets, whose name has escaped my memory at present, called Truth the daughter of Time [The Attic Nights of Aulus Gellius, xii, 11, 6–7].Google Scholar

²² Smith, was Master of Trinity College, Cambridge, 1742–1768.Google Scholar

²³ ‘I have been informed, on the best authority, that Dr. Smith restricted the organist of Trinity College to such keys and modulations as were best suited to the system by which the organ was tuned; and that the organ, as well as the instruments that were made for Dr. Smith, has long been tuned according to the more common method.’ Young, Thomas in Peacock, , op. cit. (6) i, 137.Google Scholar

²⁴ Schmidt, Bernard (1630?–1708)Google Scholar, known as Schmidt, Father, built the organ of Trinity College Chapel, which was completed in 1694.Google Scholar

²⁵ To do brown, slang: to be completely fooled.

²⁶ Mehercle! A Roman oath: So help me, Hercules!

²⁷ To come in pat, slang: to utter or otherwise provide something peculiarly apt.

²⁸ Please the pigs, slang: DV (i.e. Deo volente), God willing.

²⁹ To break the neck, slang: to accomplish the first essential of a task.

³⁰ This formula is illustrated by several examples in von Helmholtz, Hermann's On the Sensations of Tone, 2nd English edn (tr. Ellis, A. J.), London, 1885, 184Google Scholar. What De Morgan styled ‘Smith's beats’ are now simply called ‘beats’, and his ‘Tartini beats’ are now called combination or difference tones, which were erroneously considered by De Morgan, as also by Young and Lagrange, as the blending of beats that are sufficiently rapid to become a tone in itself. Romieu, Jean-Baptiste (1723–1766)Google Scholar was the first to suggest this hypothesis in 1751. A historical account of these tones is given by Jones, A. T., ‘The discovery of difference tones’, The American Physics Teacher (1935), 3, 49CrossRefGoogle Scholar. Failure to distinguish clearly between the ‘Tartini beats’ and the Smith beats is the fault that De Morgan imputes to Young, Chladni, Robison and others. The modern explanation of ‘Tartini beats’ was first advanced by Helmholtz, , Poggendorff's Annalen (1856), 99, 497Google Scholar. The effect is due to an aural response that contains elements introduced by the non-linearity of the ear. As beauty lies in the eye of the beholder, so consonance lies in the ear of the listener.