by William W C Scott © 3rd May 2024

One evening, in December 1959, I went to a sparsely attended lecture given by Professor A.C. Aitken in the Mathematics Institute in Chambers St. Edinburgh. He spoke about James Stirling [1836-1916] [Senior Wrangler in the Cambridge Tripos of 1860 ] "the highest intellectual achievement attainable in Britain" at that time. This James Stirling, son of a church minister, was born in Aberdeen, graduated MA there in 1855. At Trinity College, Cambridge he could not become a fellow because he was not Church of England. Thus he became a High Court Justice (1886-1900), Lord Justice of Appeal (1900-1906), yet continued to study mathematics. He was knighted, elected FRS 1902 and Vice President of the Royal Society 1909-1910.

Another James Stirling [1692-1770] was even more extraordinary. Third son of a long line of Stirlings who owned property about 20 miles from Stirling at an estate called Garden. Not much was known about this James Stirling until he entered Balliol College, Oxford, where he was a Snell exhibitioner 1710 and a Warner exhibitioner in 1716, when he had to leave Balliol because he refused to swear the Oath of Hanover, a new king, George 1, having been crowned in 1715. He lost his scholarships and could not continue or graduate. This Stirling was a mathematician who corresponded with Colin MacLaurin, Professor of Maths in Edinburgh and Leonard Euler, the prodigious Swiss mathematician poached by Catherine the Great for her court in Russia. Oddly, after publishing an important book on mathematics, he became mine manager at Leadhills where he was given a fine house and became wealthy. MacLaurin, born in Glendaruel, had been appointed Professor of Mathematics at Aberdeen at the age of 19 and corresponded with Isaac Newton who, though retired, wielded huge influence over all Science here and on the continent because he was President of the Royal Society, besides being the author of Principia Mathematica, (vol 1 1686, vol 2, 1687), the greatest advance in Science in the world for two centuries. Stirling also corresponded with Newton.Stirling was admired by Aitken for his original work. Aitken was my professor of Mathematics and everyone thought him a genius, stories going round the college the night before we went for our first lecture. His ability to calculate (before any kind of computer) was legendary. In his class I heard him do this every day. The answer was available instantly; the calculation effortless and accurate every time. Aitken was our Ramanujan, whose gift for numbers and every kind of calculation was extraordinary. And yet, one day Aitken began his lecture in Latin, as some of us realised. One seized his satchel and drew out a book which he began to read: 'By God', the student said: 'he is reciting Virgil's Aeneid from memory!' Aitken had taught Latin in New Zealand before taking his violin through the trenches of Gallipoli and the Somme. It now reposes in Otago university as a memorial of that time. Aitken told us of Stirling's contributions to numerical analysis. He told us about going to a dinner at which he would be asked to recite the value of π to a thousand decimal places. To prove it, he wrote it to 500 places on the blackboard, his violinists' fingers moving like lightning. When finished, he faced us and then repeated it slowly so that we could follow it. Of course it was correct. To begin with, Aitken's lectures were interrupted. A student from India would sit in a front seat reading the Guardian, the newspaper spread across his desk, while Aitken talked. After a few days, Aitken stopped the lecture to explain to him that failing to pay attention would have consequences for his success. The Indian replied that 'when he heard something new, he would listen carefully.' Aitken decided, I believe, to ignore him but to say such interesting things in his lectures that even the Indian would pay attention. Probably, that course was the highlight of Aitken's skill, as he dealt with the 'interloper'. For there were occasions when the Indian's jaw would drop in surprise, the newspaper forgotten, falling to the floor. After a month or so, unhappy with the genius in Edinburgh, he left for Cambridge to continue his education there.Every day, Aitken would challenge us to produce a question or a problem and instantly he would break it down and deal with it. When the first Comet aircraft were crashing, he left us for a short time, to join the team of luminaries collected to solve it. The answer was metal fatigue due to vibration. Yet there was another early Scottish mathematician whom Aitken spoke about that night. It may have been John Napier (born 1550 at Merchiston Tower, died 1617) with his logarithms. These were discoveries of immense importance for calculations, still used every day until about 1960, when slide rules and hand-held calculators made their first appearance. Every student had his set of logarithmic tables which turned a difficult multiplication into a much easier addition. They were a godsend. Napier was a landowner as well as a mathematician, physicist and astronomer. Napier University incorporated part of the original building in which Napier had lived. The doyen of the mathematical-physicists of that time was William Thomson who had attended Glasgow University from the age of ten, learnt all his six uncles, who taught there, set before him, and then left for Peterhouse College, Cambridge, in 1841 to prepare for the Tripos exams of 1845. J.J. Thomson, nobel laureate, et al, told the story of how Thomson sent his servant to discover 'who was second in the Tripos'. Back came the report: 'You were!' It had not occurred to Thomson that he could fail to be senior wrangler, as the best mathematician was known. Technically, Thomson was Irish, because born in Belfast in 1824, but educated in Scotland when his father became Professor of Mathematics at Glasgow. Having won the Smith's Prize (considered harder than the Tripos) in 1846), Thomson was appointed Professor of Natural Philosophy at Glasgow. He was then 22.The Tripos then consisted of 16 papers over 8 days, lasting 44.5 hours with 211 questions. Seven years younger than Thomson was James Clerk Maxwell who attended Edinburgh Academy when Gloag, the great maths teacher there in that period, would be inspiring his students. In the year beneath Maxwell was Peter Guthrie Tait who would reach Cambridge before Maxwell, enter Peterhouse in 1848 and be senior wrangler in 1852, aged 20. He too, would be Smith's Prizeman like Thomson and Maxwell but would never make FRS which the other two did. (Maxwell was Second Wrangler in 1854)

In his lifetime, Thomson published 640 papers. When Head of section A in the British Association for the Advancement of Science, Thomson once read out all six papers he had recently written. Doubtless they were received with smiles: what an imposition it was! It must have taken over an hour. Tait became Professor of Nat. Phil. at Edinburgh (Maxwell being deemed a poor teacher who could not control or inspire his classes and was liable to engage in flights of imagination which, though wondrous, were beyond the minds of most students to follow.) I do not recall anyone reading papers in section A of the British Association for the Advancement of Science when I attended the conferences in the 70s and 80s. However, once, a small man stood up unannounced and began to tell the assembly of his experiences taking the secret of the atomic bomb to Copenhagen, Britain and then America, where Einstein wrote his letter to Roosevelt, which precipitated the building of Los Alamos out in the desert north of Santa Fé; and the frantic search for the means of producing the bomb in three years under the inspired leadership of J.R. Oppenheimer. The name of the man who spoke? Otto Frisch FRS, whose aunt, Lise Meitner, was the creator of the physical chemistry with the main idea of the bomb; but who had written the paper and, idealistically put at its head the name of Kurt Hahn because, despite contributing little, he deserved the position because he was at least a man. In time, Hahn was therefore awarded the Nobel for the bomb, a travesty of justice. Frisch was spellbinding. He had been in charge of Geiger counters at Los Alamos. He told us what it was like to work there and even about sailing Einstein's yacht near Princeton where the great man lived at 12 Mercer St and worked every day in The Institute for Advanced Study. Such excitements were inspiring; made one return to one's students to set fires of ambition among them. Note: Tait occupied 17 Drummond Place, Edinburgh where Thomson would stay at times. They wrote a textbook together on Natural Philosophy which would have educated Britons and continentals for years. A century later, I lodged in 18 Drummond Place next door, for a year, unaware of the tall ghost who had lived there when not playing golf. Tait's dominating presence had been forgotten. Only his name continued: I studied each day at the Tait Institute of Mathematical-Physics (named after him), under Professor Kemmer. Most mornings I would see Compton McKenzie totter around the square recovering from an evening of talk with literati over John Barleycorn. He had bought two adjacent houses and filled them with books. Twenty yards away, on my left, every evening, my tutor Peter Higgs, would be sitting at the window of his study on the first floor, writing the papers that first mentioned the heavy particle, or boson, he believed existed to make the new quantum mechanics fit the facts. For this he was awarded a Nobel Prize (shared) in 2013.

Thomson was by far the most colourful character of the trivium, known as T. Tait was designated T' and Maxwell dp/dt which some wag completed the equation with JCM, Maxwell's initials. John Tyndall, a materialist/atheist also from Belfast, was called T''', though viewed as not quite equal. Thomson was knighted in 1866 at Windsor Castle by Queen Victoria and raised to the peerage (Baron Kelvin of Largs) in 1892. In 1870, his first wife died and he found himself four years later at Madeira, laying Atlantic cables (from which he would make a great deal of money). There he met the two sisters, daughters of Charles Blandy (owner of The Marquis of Blandy, a fine wine) and became enamoured of one of them: Anna Francis. When he reached home, smitten, he decided to sail his schooner, 'Lalla Rookh', from the Clyde to Madeira, a journey completed in 6¾ days sailing. The lady accepted him, he married her, and bore her off to the Clyde where he built a huge red sandstone house with four conical towers a hundred feet or so above the ferry terminal at Wemyss Bay. This is called Netherhall and still stands. Thomson continued to lecture from 1846 until 1899 when he retired. He died in 1907. His yacht, Lalla Rookh, was his pride and joy. His grandson and biographer, Sylvanus P. Thomson, shows the schooner with 6 headsails, two mainsails and a crew of 17 in a photo on p 617 vol 2. It is 126 tons and about 200 ft long, a ship of great beauty, built for speed. Thomson employed a captain to look after it while he escaped below to work on physics and mathematics leaving others to crew the ship. He would often sail from Glasgow to Arran at week-ends or vacations and stay in a house he had there. The crew were often relatives, shanghied for the purpose. Once with Maxwell, he visited Jemima Wedderburn, the artist, at her home with her husband, Professor Blackburn of Glasgow (also mathematics), at Kinlochailort. On another occasion he planned to take his friends T.H. Huxley, Helmholtz, Maxwell and Tait for a sail around the western isles of Scotland. Clearly, Thomson was a tall powerful person who would be a fine navigator and seaman. Huxley was the famous professor who defended Darwin against Wilberforce in 1859. Thomson approved of the use of Del in the Maxwell Equations and of the speed of light appearing in them but not in the failure to distinguish the theory of light from magnetism in fields of force. Maxwell had taken them to be equivalent, for the equations suggested it. Not until his 1905 paper did Einstein follow Maxwell. Therein is the equation e=mc² which is one of the greatest statements in the history of science. But before it, came the transformations with c, the speed of light, which is the most surprising feature of the Maxwell Equations. The first time the speed of light appears in equations. It is easily seen in the Einstein-Lorentz transformations which make Relativity Theory possible. These are very simple. My best students could understood them and prove them. Einstein's argument in the 1905 paper is now considered circular. He even rewrote it, aided by a paper of Poincaré, the great mathematician of France of that time. Even so, the insight is correct, as a century of further advances have shown. Thus Einstein was right, could 'see' it but could not quite prove it. He produced two short, straight forward proofs of an elementary nature; one first published in 1946, the other in 1950. Both depend upon a few conservation laws.

The Maxwell Equations (reduced to four; once twenty) are now quoted using the Del operator which Maxwell got from Sir William Hamilton, Professor of Logic and Metaphysics at Edinburgh (not to be confused with Sir William Rowan Hamilton, author of Quaternions). The vector cross product and the dot product provide simple expressions. Maxwell also used the Laplacian, Del squared. Maxwell, seven years younger than Thomson, was a man who at his father's estate Glenlair near Parton, Galloway, began his life as an experimenter when still a child. He was deeply curious of everything, incessantly demanding: 'what was the go of it?' He made a laboratory for himself in the large house and then moved to Heriot Row to stay with relatives while attending Edinburgh Academy. Like Thomson, Maxwell published papers before leaving school for Edinburgh University which he soon left to enter Peterhouse College, Cambridge before transferring to Trinity where he became a fellow. Maxwell was quickly recognised as a genius (despite episodes of chaotic thinking) and elected to the Apostles, a club which Whitehead (4th Wrangler, 1883), Bertrand Russell (7th Wrangler, 1893), the six foot six John Maynard Keynes (12th Wrangler, 1905), and F.P. Ramsey (Senior Wrangler, 1923) (brother of the Archbishop of Canterbury), would eventually join. Membership by invitation based on genius. Maxwell was a man of faith, a small man of five feet four, a very quiet, kindly and gentle Christian who, desperately concerned to understand goodness, wished above all to be as good a person as he could. He was hugely obliging to others' needs but rarely put himself forward. Yet when giving inaugural lectures at Aberdeen, King's College, London and Cambridge, where he rebuilt the Cavendish, he was original and in his element. At Trinity, he was inclined to work with furious concentration until 2 am when he would run upstairs and along corridors for half an hour, waking sleeping students. Though Maxwell was a man of faith it did not impinge upon his science. When Thomson finished the Tripos in 1845, he set off for the continent to meet Liouville and Regnault, from whom he learnt the importance of experimental work and especially the efforts to produce exact numerical values. Faith had no part to play in science. Indeed, the success or failure of it would depend upon the most rigorous rendition on all occasions. The ambition in lab work was the greatest accuracy at all times; speculation, still less faith, had nothing to do with science. His juniors would be told all about this.T.F. Torrance, a theologian who added a doctorate in science to 'take on the scientists and meet them on equal terms', believed Maxwell's faith was the source of his scientific achievement. Not so. The very concentration on numbers (and not God, singular or plural) shows this: The Theoretical had to be confirmed by the Experimental. Maxwell was an ace at both. The Treatise on Electromagnetism by Maxwell is full of experimental data and conclusions as well as the best mathematical-physics of the time. There is nothing in it of theology or even faith. Maxwell began with Faraday whose lectures at the Royal Institution he often attended. There he learned about fields of force. Then, after preliminary dances with the ether as the ever present material which contained everything else, Maxwell began to assemble all the equations he could think of for the various aspects of reality. Finally he came to the problem of Electromagnetism. The scientist, Maxwell, said 'uses objective measurements which are independent of his or her feelings: eg a thermometer with its precise markings rather than just feelings of hot or cold. …'All material things are capable of exact measurement, that each leads to laws of nature, and that the objective criteria of number, measure, and weight, are the elements of all knowledge of the material world.'[1] On the face of it, Electricity is one aspect of reality and Magnetism another. When Maxwell wrote down his equations for each, the symbols, the mathematics were nearly identical. Later, he told Faraday that he had found the velocity of transverse vibrations to be 193,088 miles per second which was remarkably close to the then accepted velocity of light of 193,118 miles per second. Writing to Thompson, Maxwell said 'I have reason to believe that the magnetic and luminous media are identical. That visible light was simply a particular manifestation of electromagnetic phenomena. They were both part of an electromagnetic field travelling with the same finite speed.''Whether you are moving toward or away from a beam of light, the light will always approach you at the same speed.'[2]

This is easily seen from one of the Einstein-Lorentz Equations: m=m'/√1 -v²/c²

If v is greater than c, the Right hand side of the equation is m'/√negative number

ie there is no real solution. Therefore, v is never greater than c.

What a mathematician is nowadays, is very different from what Aitken was in 1959. Calculation is not valued: computers deal with it in many ways. The current mathematician is one who either extends the discoveries of others or, better, derives a completely novel system of ideas which, inevitably, has surprising connexions to the others in mathematics and the current view of reality. Einstein was a mathematical-physicist of this type. Apart from the Riemann Hypothesis, all the problems set by David Hilbert in 1900 have been solved, joined by six others more recently.

**© William W. C. Scott**

Philippa Fawcett [1868-1948] won the Tripos in 1890, scoring 13% more in marks than the best man. She was not declared Senior Wrangler because women had not been admitted to full membership of Cambridge University until 1848 and not allowed to sit the Tripos until 1881. Philippa won the Marion Kennedy Scholarship which enabled her to study fluid dynamics. For ten years she was a lecturer in mathematics at Newnham. She then went to Johannesburg where she taught teachers. She obtained an ad eundum degree from Trinity College, Dublin but had to go there to receive it. She was thought the best maths teacher of her generation.

Ruth Hendry was Senior Wrangler in 1992.

The youngest Senior Wrangler was Arran Fernandez, Fitzwilliam College, aged 18 years and 0 months in 2013.

Jacob Bronowski, of Jesus College, star of the Sunday T.V. brainstrust was Senior Wrangler in 1930 and Hermann Bondi (whom I knew) of Trinity, was Senior Wrangler in 1940.

[1] James Clerk Maxwell, by Bruce Ritchie, 2024

[2] Steven Hawking, 2007, 'The Essential Einstein' p1x