While he was still an undergraduate, Muirhead became a member of the Royal Philosophical Society of Glasgow in 1879. He published the paper On a Substitute for Euclid's Third Postulate in the Proceedings of the Royal Philosophical Society of Glasgow in 1880. In the same year he was appointed the Society's delegate to the meeting of the British Association for the Advancement of Science held in Swansea in August-September. He reported back to the Royal Philosophical Society of Glasgow and his report was published in the Society's Proceedings.

Muirhead graduated with a B.Sc. from the University of Glasgow in 1879 and with an M.A. in 1881, with the highest honours in mathematics and natural philosophy (physics). He was one of five students who graduated with an M.A. with Honours in Mathematics and Natural Philosophy in that year, two with First Class Honours and three with Second Class Honours. Muirhead and Alexander Russell were the two First Class students. The professor of mathematics was Hugh Blackburn when Muirhead began his studies but, from 1879, it was William Jack. The professor of Natural Philosophy was William Thomson with Andrew Gray as his assistant. Muirhead was awarded a Ferguson Scholarship, which was open to all graduates from the four Scottish universities (there were only four at this time). He was also awarded a George A Clark Scholarship which funded him for four years at the University of Cambridge.

On 12 June 1881 Muirhead was admitted as a pensioner at St Catharine's College, University of Cambridge. Being a pensioner means that he paid for his own tuition. He matriculated at the start of the Michaelmas term of 1881 (at the beginning of October). At this time Arthur Cayley was the Sadleirian professor of Pure Mathematics and John William Strutt was the Cavendish Professor of Physics. In 1884 J J Thomson became the Cavendish Professor of Physics. William Henry Young was studying the mathematical tripos in the same years as Muirhead, having also matriculated in 1881. Muirhead graduated M.A. in 1884 as 19th Wrangler and was 1st class in Part III of the mathematical tripos in 1885. He then spent the year 1885-86 at the University of Göttingen where he attended the lectures of Hermann Schwarz.

It was in 1886 when I first met Edward Carpenter, who was giving some lectures in Glasgow for the Glasgow Branch of the Socialist League, of which I was a member. I had read with delight England's Ideal and was not disappointed when I met its author. He was my guest (at my lodgings, 22 Arlington Street) on that occasion, and the friendship he and I formed then was never interrupted. From year to year I paid visits of a few days or a week to Millthorpe, where at first Albert Fearnehough and his wife and daughter, later the Adams', and then George Merrill were Edward's house-mates. There I met and formed friendships with a number of the Sheffield Socialists, as well as others from a distance, such as Philip Dalmas, that remarkable American composer and singer, who set some of Whitman's chants to music. There also I met Dr Cecil Reddie and a project was mooted to start a new school on more or less Socialist lines with Reddie as head master and with the co-operation of Carpenter and myself and possibly others.

It was not till after a year or two's incubation that this project actually materialised. At one stage it was contemplated that a site in Yorkshire within easy reach of Millthorpe should be chosen, and that the new school should be run by four partners, Dr Reddie, Carpenter, William Cassels, and myself, Carpenter giving part time and the three others their whole time to the work of the school. But finally a mansion house in Derbyshire was fixed on, and its original name "Abbot's Clownholme" shortened to "Abbotsholme".

There the new School started in 1889. The lines on which it started were no doubt due to the very definite ideas of Dr Reddie on school education, and were somewhat different from what Carpenter had in mind, and partly on this account and partly because the site chosen was not quite near to Millthorpe, Edward Carpenter did not enter the partnership but remained as adviser and occasional visitor, a strong supporter of the School. The original staff included the three partners with Dr Reddie as head master, Mr J H Badley (who afterwards founded Bedales School), Herbert Pearson (who managed the farm), Miss Aitchison, and Mrs Walters, and a workshop instructor furnished by the London "Guild of Handicraft". C R Ashbee and Lowes Dickinson who were both much interested in the project, were appointed "visitors".

Towards the end of the first year, however, differences arose between Dr Reddie and Mr Cassels, which seemed to make it impossible for them to continue co-operation. I took the view that the blame for this rested on Dr Reddie, and after some consultation with Edward Carpenter, Cassels and I decided to break up the partnership. Thereupon it was arranged that Cassels and I should leave at the end of the term and allow Dr Reddie to carry on the School on his own lines.

A more complete account of the early history of Abbotsholme School might be interesting, but I think I have recorded all that is essential to explain the part taken by Edward Carpenter in connection with it.

About the same time I was indebted to him for the opportunity of making the acquaintance of that fiery genius Olive Schreiner, and of forming a friendship with her which was lasting. When the war broke out she was much disappointed that Edward did not come out more strongly as a pacifist, and I am afraid that because of that the long-standing friendship between them was latterly somewhat clouded.

When Edward moved from Millthorpe in Derbyshire to the later "Millthorpe" at Guildford, I think it was because of the breaking of the strongest link that bound him to the North, by the death of George Hukin, whose sweet nature inspired more than one of the poems in Towards Democracy.

About the late 'eighties I had the privilege of making acquainted with each other Edward Carpenter and James M Brown, the latter one of the most well-loved and influential of the Glasgow Socialists of that time, though on account of his retiring disposition not known to the outer world. Edward was delighted with James M. Brown, and when the latter's poor health required that he should find quarters in the country, found lodgings for him near Millthorpe, where he stayed during his last years. After Brown's death in 1893, Carpenter had published a slim booklet of poems by Brown, with prefatory notes by himself and by Bruce Glasier.

One of the later poems in Towards Democracy is recognisable as a poetic version of an episode of James M Brown's life. Another aspect of my friend was revealed to me when he and I climbed the easily accessible peak of Goatfell in Arran one summer. Not content with that exploit, though at that time well on in years, Carpenter led me along the top of the narrow ridge stretching south-east from the summit, including some rather dangerous bits, the risk of taking which he seemed to enjoy.

I hope these somewhat disjointed notes may be of some little interest to readers who cherish the memory of my late friend.

A socialist meeting had been announced for 3 p.m. in Trafalgar Square, the authorities, probably thinking Socialism a much greater terror than it really was, had vetoed the meeting and drawn a ring of police, two deep, all round the interior part of the Square.

The three leading members of the SDF - Hyndman, Burns and Cunninghame Graham - agreed to march up arm-in-arm and force their way if possible into the charmed circle. Somehow Hyndman was lost in the crowd on the way to the battle, but Graham and Burns pushed their way through, challenged the forces of "Law and Order", came to blows, and were duly mauled by the police, arrested, and locked up.

I was in the Square at the time. The crowd was a most good-humoured, easy going, smiling crowd; but presently it was transformed. A regiment of mounted police came cantering up. The order had gone forth that we were to be kept moving. To keep a crowd moving is I believe a technical term for the process of riding roughshod in all directions, scattering, frightening and batoning the people.

I saw my friend Robert Muirhead seized by the collar by a mounted man and dragged along, apparently towards a police station, while a bobby on foot aided in the arrest. I jumped to the rescue and slanged the two constables, for which I got a whack on the cheek-bone from a baton, but Muirhead was released.

The case came into Court afterwards, and Burns and Graham were sentenced to six weeks' imprisonment, each for "unlawful assembly". I was asked to give evidence in favour of the defendants, and gladly consented - though I had not much to say, except to testify to the peaceable character of the crowd and the high-handed action of the police. In cross-examination I was asked whether I had not seen any rioting; and when I replied in a very pointed way "Not on the part of the people!" a large smile went round the Court, and I was not plied with any more questions.

The University of Glasgow, like the other universities of Scotland, is passing through a period of growth and change. The Universities' Commission of 1889 (the outcome of the movement for reform that became active about half a dozen years earlier), was occupied up till 1897 in issuing new Ordinances for the regulation of university affairs, and scarcely more than a year ago the Court exercised its power in making a new Ordinance involving further important changes. It is, I think, fairly obvious that the period of change has not yet ceased, and I think no one who has the welfare of the University at heart, and who is well acquainted with its present condition would desire the cessation of reform until considerable improvements have been effected. The time is, then, opportune for the consideration of the present state and future prospects and possibilities of the University by a Society like ours which aims at being a focus for the intelligence and wisdom of the inhabitants of this city, and which can furnish, and from time to time has furnished an arena for serious and yet free and unprejudiced discussion of matters of practical and theoretic interest. It is with the hope of demonstrating the urgency of the problem to the members of this Society and others whom my words will reach, that I have undertaken tonight to address to you "Some Considerations on Scottish Universities, with Special Reference to Needed Reforms."

I conclude this section by mentioning a man who was a fine mathematician but less successful in a worldly sense than those referred to above. Robert Franklin Muirhead (1861-1941) was a graduate of Glasgow and Cambridge Universities and spent some time at the University of Göttingen. He took a great interest in the Society and was twice President, in 1899 and again in 1909. He was elected an Honorary Member in 1912. He held lectureships in Glasgow and Birmingham for brief periods and tutorships in London, Edinburgh and Glasgow, but never held any permanent position worthy of his talents. It is possible that this may have been because of his outspoken views on home rule and socialism. In his latter years he was head of a coaching establishment, the Glasgow Tutorial College. I first heard of Muirhead forty years ago from my supervisor, the late Professor G H Hardy, who had a high opinion of Muirhead's abilities, and some of his work on convexity which is of interest to statisticians has recently come into greater prominence. Men like Muirhead, or his younger colleague John Dougall (1867-1960), who was President in 1925, would easily have obtained university posts in the golden 25 years following the last war, but they lived in times, rather like those we have moved into recently, when appropriate positions were not available to everyone of ability.

Robert Franklin Muirhead was born at Shawlands, Glasgow, in January 1860. From 1868 to 1872 he was educated under tutors at his father's home, Meikle Cloak, near Lochwinnoch, and at the village school. He was at Hamilton Academy in 1872-73, and at Paisley Grammar School in 1873-75 (dux, 1875). From 1876 to 1881 he studied at Glasgow University. He graduated as M.A., B.Sc., with highest honours in Mathematics and Natural Philosophy, and gained the Ferguson Scholarship.

With the George A Clark Scholarship from Glasgow he went up to Cambridge, to St Catharine's College. He was nineteenth wrangler in 1884, was classed Division I, Part III, in 1885, and was awarded a Smith's Prize in 1886 for an essay on Newton's Laws of Motion.

During his four years tenure of the George A Clark Scholarship he conducted supplementary classes at Glasgow University both for pass and for honours men. The present writer well remembers his lectures at this period on Theory of Equations, and on Electrostatics, and can vouch for the high quality of his teaching of these subjects. The friendship thus formed continued unbroken throughout Muirhead's life. A truer and more helpful friend no one could possibly have.

About 1885 he went to Germany and spent a year at Göttingen University where he attended the lectures of Schwartz. The next few years were spent in teaching work of various kinds and at various places - Mason College, Birmingham, Keith and Edinburgh. Characteristically the first school to which he was attracted was one founded by Dr Reddy at Abbotsholme, the special feature of which was the absence of all corporal punishment. Muirhead had a rooted aversion to compulsion of any kind - an "anarchist" was what he used to call himself.

He married in 1893 and soon afterwards settled down in Glasgow as a coach in Mathematics, Physics and Engineering. His classes were very successful, and those in Mathematics were recognised under the University extra-mural scheme as qualifying a student to sit his degree examinations. About forty years ago he founded the Glasgow Tutorial College, which he continued to supervise almost up to the time of his death.

Muirhead published many papers on mathematical subjects, most of them in our own Proceedings and Mathematical Notes, and in the Mathematical Gazette. The topics were mainly elementary - for example, he gave a large number of proofs of the Binomial Theorem for a positive integral index. Among longer papers may be mentioned "On the number and nature of the solutions of the Apollonian contact problem" in Vol. 14 of our Proceedings; and "On a method of studying displacement" in Vol. 15. Akin to the latter paper was one on the Foundations of Geometry read at the International Congress at Cambridge in 1912.

He also wrote a number of papers on Inequalities, notably one on "Inequalities relating to some algebraic means" (Vol. 19). Prof G H Hardy, in a lecture on the Theorem of the Arithmetic and Geometric Means, has directed attention to some very important but much neglected work by Dr R F Muirhead on inequalities of a more general type.

Muirhead took a deep interest in the Edinburgh Mathematical Society. He was elected a member in its second session, in February 1884. He became President in 1899, and again in 1909; and was elected an Honorary Member in 1912.

He was a keen politician, with a strong bias towards the left wing, especially in his youth. One of his most intimate friends was Edward Carpenter, author of "Civilization: Its Cause and Cure." When the Russian "nihilist," Prince Kropotkin, held a meeting in Glasgow, about 1895, Muirhead was his chairman. He had an idea that it might have been better for his professional career had his political sympathies been more orthodox, or less outspoken. In later years he was a prominent supporter of the movement for Scottish Home Rule.

His social sympathies were by no means merely theoretical. His helping hand had a wide reach, though you had to know him well to find it out.

Dr Muirhead's wife, who was an accomplished musician, died last year. They leave a family of two sons and two daughters.

The first example centres around Bob Muirhead, a Glasgow mathematician and member of the Scottish Socialist League, who formed a long-term friendship with both Carpenter and Olive Schreiner. Between 1889 and 1890, the "wonderfully handsome and athletic" Muirhead stayed with Carpenter at Millthorpe, enjoying what Carpenter termed a "romance of affection" with James Brown, a tailor, poet and fellow member of the Scottish Socialist League, as well as sleeping with Carpenter, before three years later marrying and settling down in Glasgow.