John Couch Adams: From Senior Wrangler to the Quest for an Unknown Planet

Abstract

As Eugène Bouvard’s attempts stalled to successfully revise his uncle’s Tables of Uranus, a young British mathematician, John Couch Adams, became interested in the problem of Uranus. As an undergraduate at St. John’s College, Cambridge, Adams achieved enormous academic success, including becoming Senior Wrangler and First Smith’s Prizeman. Though he was kept extremely busy with his college work, and could not afford any significant distractions, in June 1841, he became intrigued by the problem of Uranus’s wayward motions, and interested in finding out to what extent the hypothesis of an exterior planet could furnish a solution. The Cambridge mathematics curriculum helped him to develop the world-class expertise he needed to tackle this high-level problem in analysis.

Adams’s success in the Mathematical Tripos exam led to his becoming a Fellow of St. John’s in 1844. His duties as a Fellow included lecturing and tutoring, but mainly during the extended vacations, he did take up the Uranus problem, and in October 1845 he attempted to present his famous Hyp I solution to George Biddell Airy personally during ill-fated visits to the Royal Observatory, Greenwich. He failed to meet Airy, and only left a paper summarising his results (including a position for the planet). Contrary to statements in prior historical writings, Airy did not lack interest and—despite being fiendishly busy and distracted at the time—followed up with a query to Adams about the radius vector, which was his own particular hobbyhorse. Despite beginning a new term, and being once more swamped with lectures and tutoring. New research reveals Adams did attempt to respond to Airy. However, the letter was never sent, and the trail went cold. There followed a hiatus until June 1846 when, as described in a later chapter, Urbain Jean Joseph Le Verrier’s independent calculation was published, setting off a race to find the mysterious planet….

Appendix 4.1 A Brief History of Mathematics at Cambridge

Though the Tripos did not encourage “original synthesis, analysis or a developing fountain of knowledge,” and even the present-day Cambridge University Mathematics Department (Johnstone, 2016) admits on its web page that “it may be doubted that a system in which the best students spent three years training to solve problems against the clock represented the ideal way to teach mathematics,” for all its flaws, “the Mathematical Tripos offered the best mathematical foundations for the advancement of physical science available in Britain.” (Smith and Wise 1989:57). According to Bristed (1852:95–96), the top wranglers had to be “remarkably clever,” show “considerable industry,” and “read mathematics professionally”. During the time Adams was there, this system had already succeeded in becoming “the nursery for the great flowering of British physics in the 19^th century.” (Johnstone, 2016). To take only a few examples, the two Senior Wranglers ahead of Adams were George Gabriel Stokes (1841) and Arthur Cayley (1842), while the Second Wrangler in 1845, the year of Stephen Parkinson’s (1823–1889) success, was William Thomson (1824–1907), later Lord Kelvin, all of whom went on to great distinction in mathematics or mathematical science. However, it must be emphasized that Cambridge was not, at the time, about research; good research was admired, but the 19th century Cambridge system concentrated on undergraduate teaching, and research “was not viewed as a professional duty and the university was not expected to provide support for it.” (Johnson, 2016).

During the time Adams was at Cambridge, the quality of mathematics instruction was first-rate. But this had not always been the case. Throughout the 18th century, it had a been a rather lacklustre affair. This was at least in part owing to the paralysing effects of the near-idolatry in which Isaac Newton was held. The spectacular success of the British hero had had “the fortunate effect of establishing the prestige of mathematics in Britain and Cambridge and the unfortunate effect of blinding British mathematicians to the progress in mathematics elsewhere.” (Johnstone, 2016). While British mathematics lagged, mathematics on the Continent (as described in Chap. 1) had made rapid progress, owing in good part to the adoption by mathematicians such as Euler, Clairaut, d’Alembert, Lagrange and Laplace of the Leibnizian notation dy/dx for differentials instead of the Newtonian notation of dots for fluxions. The methods of course are at bottom equivalent but the Leibnizian form of notation is much more suggestive and convenient to use.

The standard mathematics texts used at Cambridge in the 18th century and into the nineteenth consisted of a series of sturdy volumes produced between 1795 and 1799 by the Rev. James Wood (1820–1901), President and Master of St. John’s until his death in 1839, and Samuel Vince (1749–1821) , Plumian professor of Astronomy and Experimental Philosophy at Cambridge from 1796 until his death in 1821, and known collectively under the title The principles of mathematics and natural philosophy. Needless to say, in all these volumes Wood and Vince used the Newtonian notations wherever later writers would have used differentials.

Not until the late 1810s did the inertia of established practice and anti-French sentiments owing to the French Revolution and the Napoleonic wars subside enough to allow the winds of change to blow with ever-increasing fury. A first significant, though premature, effort was made by Robert Woodhouse (1773–1827), a Fellow of Gonville and Caius College (later Lucasian professor of mathematics and Plumian professor at Cambridge Observatory, a position in which he preceded George Biddell Airy and held until his death in 1827). In his Principles of Analytical Calculation in 1803, he advocated for the differential notation. (Woodhouse, 1803).

Unfortunately, the book had the misfortune of appearing at almost the exact moment when Cambridge was probably close to its intellectual nadir. As Adam Sedgwick’s biographer Colin Speakman writes (1982:47), “It was conservative, even by its own standards, complacent and singularly remote from any genuine habits of research or intellectual invention.” When serious change began to come, after 1810, it was largely a result of student enterprise rather than faculty initiative. Charles Babbage (1791–1871), an undergraduate at Trinity, even before coming to Cambridge had developed an intense passion for the Continental (i.e., analytic) methods through self-study of the French mathematician Sylvestre François Lacroix’s (1765–1843) textbook, Traité élémentaire de calcul differential et de calcul integral. (Lacroix, 1802). Babbage shared his enthusiasm with John Hudson (1773–1843), his private tutor (and Senior Wrangler for 1797) but Hudson was not encouraging: in his view, questions in Leibnizian form would never appear on the Tripos, so there was no point in learning about it. Realising at once that he would never gain ground for his cause in the college of Newton, Babbage transferred to Peterhouse, where he continued a determined effort, even promoting the differentials in broadsides on the Cambridge town walls. More helpfully, he began a translation of Lacroix’s textbook into English (Lacroix, 1816).

Babbage soon found important allies in John Herschel of St. John’s, son of William Herschel, and George Peacock (1791–1858) of Trinity. A mutual friend, Edward Thomas Ffrench Bromhead (1789–1855) (later Sir Edward Bromhead, 2nd Baronet of Thurlby Hall) “…invited a small group to meet in his rooms to discuss the formation of a society whose aim would be to encourage analytical methods in Cambridge. It was then that Babbage, Herschel and Peacock first met each other. Feeling at the meeting was favourable to the formation of a society and the formal inaugural meeting of the Analytical Society took place very shortly afterwards.” (Wilkes, 1990).

Though Bromhead played no further role, Babbage, Herschel and Peacock devoted themselves with furious energy to “the propagation of D’s; and consigned to perdition all who supported the heresy of dots.” (Snyder, 2011:32). They strove to produce a complete translation, with annotations, of the Lacroix text that Babbage had begun. At the Mathematical Tripos of 1813, Herschel was Senior Wrangler, and Peacock Second Wrangler, which increased the Analytical Society’s influence and prestige, since it was from high wranglers that the moderators who wrote the examination questions on the Tripos were then and afterwards selected. Indeed, in 1817, a year after the Babbage-Herschel-Peacock translation of Lacroix appeared, Peacock, as one of the moderators of the Tripos, included some questions using the Continental mathematics in the examinations. His decision to do so was not entirely without controversy, mainly because the problems were given in general form rather than applied to the solution of actual physical problems. William Whewell of Trinity, Second Wrangler for 1816, wrote to Herschel on 6 March 1817:

You have I suppose seen Peacock’s examination papers. They have made a considerable outcry here and I have not much hope that he will be moderator again. I do not think he took precisely the right way to introduce the true faith. He has stripped his analysis of its applications and turned it naked among them. Of course all the prudery of the University is up and shocked at the indecency of the spectacle. The cry is “not enough philosophy.” Now the way to prevent such a clamor would have been to have given good, intelligible, but difficult physical problems, things which people would see that they could not do their own way and which would excite their curiosity sufficiently to make them thank you for your way of doing them. (RS:HS18:158)

Nominated moderator of the Tripos in 1818–19, and further influenced by a visit to Cambridge by the French physicist, mathematician and astronomer Jean-Baptiste Biot (1774–1862), who also argued for the importance of applying the calculus to the solution of practical problems, Peacock used the Leibnizian notation for the 1819 Tripos examinations, and included a number of problems of applied mathematics. The following year, he published a Collection of Examples of the Application of the Differential and Integral Calculus. (Peacock, 1820).) The triumph of Leibniz was now nearly complete, as from 1820, all of the questions on the Tripos were given in the form of differentials, the Newtonian dots having completely disappeared without anyone shedding a tear. Babbage’s close friend Richard Jones (1790–1855) told Whewell, “I hear the old mathematics have died and faded away with scarcely an audible groan before the bright flood of analytical love.” (Snyder, 2011:35).

Though as late as 1830, Babbage (1830:1), comparing the British scene with that on the Continent, declared, “We are fast dropping behind. In mathematics we have long since drawn the reins, and given over a hopeless race” by the end of the decade, when Adams came up, the reform of Cambridge mathematics was largely complete. The geriatric texts of Wood and Vince had been (mostly) retired, and newer texts had taken their place. There was George Biddell Airy on trigonometry; Lacroix (as annotated and improved by Babbage, Peacock, and Herschel) on differential and integral calculus. William Whewell had published An Elementary Treatise on Mechanics, A Treatise on Dynamics, and The Mechanical Euclid, on mechanics and hydrostatics.

In addition, and particularly relevant here, there were Airy’s Mathematical Tracts on the Lunar and planetary Theorie, the Figure of the Earth, Precession and Nutation, the Calculus of Variations and the Undulatory Theory of Optics. In addition, the Cambridge Mathematical Journal had been founded in 1837 by the Scots Duncan Gregory (1813–1844) and Archibald Smith (1813–1872) with the Englishman Samuel S. Greatheed (1813–1887), which gave Cambridge mathematicians a journal of their own to compare with the prestigious Journal für die reine und angewandte Mathematic (Crelle’s Journal) in Germany and the Journal de Mathématiques Pures et Appliquées (Liouville’s Journal) in France. England was finally catching up.