Michael Barany (@mjb.mathstodon.xyz.ap.brid.gy)

The Young mathematician couple William Henry Young **courtesy:** MacTutor History of Mathematics Archive | Senate Hall of the University of Calcutta in 1910. Wikimedia Commons ---|--- The universities of Bombay, Calcutta and Madras (now respectively Mumbai, Kolkata and Chennai) were established in 1857, starting in January with the University of Calcutta. These functioned more as examining bodies in their initial years, rather than institutions of higher learning. The academic activity in Calcutta began to flourish after the appointment of Asutosh Mookerjee as Vice-Chancellor in 1906. The visit of King-Emperor George V in 1911–1912 provided the occasion to create the Hardinge Professorship Chair for higher mathematics, and William Henry Young became the first Hardinge Professor shortly thereafter. Here, _Bhāvanā_ explores the context of Young’s brief stints in Calcutta by republishing obituary articles for W.H. Young by G.H. Hardy and for the other half of Young’s famous mathematician couple, G.C. Young, by M.L. Cartwright. They are prefaced by a contextual preamble by _Bhāvanā_ corresponding editor Michael Barany. ### Contextualizing William Henry Young in Calcutta By **Michael J. Barany**1 For Raghavan Narasimhan, the brief appointment of William Henry Young to a part-time professorship in Calcutta from 1913–1916 represented a refreshing exception to the rule of British mediocrity in the leadership of colonial-era university mathematics. Narasimhan’s account of “The Coming of Age of Mathematics in India,’’2 the occasion for this reflection, departs briefly from the narrative of mathematics in India to contextualize Will Young’s mathematical eminence with reference to his uneven European career and his remarkable marriage to the accomplished mathematician Grace Chisholm Young. Historians have devoted considerable attention to the Youngs’ mathematical partnership and to their decision to attempt to maximize their academic success within the biases and prejudices of their time by publishing the considerable results of their joint work under Will’s name alone, on the expectation that he would best be able to make a prestigious career on the basis of taking full credit for their shared efforts. Though Grace never joined Will in Calcutta, she is very much a part of the story of Will’s appointment and its significance. Beyond that brief digression, Narasimhan says virtually nothing about what Will Young’s visit meant, mentioning it again only as fleeting evidence of Calcutta’s early primacy for university mathematics in India. This reflection indicates some of the historical scholarship on the Youngs and suggests some of the implications this scholarship may have for appreciating the 1913–1916 Calcutta professorship. It emerges that this short interval was pivotal, mathematically and otherwise, for the Youngs’ marriage and careers. It was, in particular, the beginning of Will’s concerted interest in international comparison and cooperation and also a watershed for Grace’s recognition as a first-rate mathematician independent of Will. The circumstances and effects of the professorship, while of fleeting importance to the coming of age of mathematics in India, thus had rather more notable consequences in Europe. As a starting point for contextualising Will’s Calcutta professorship, _Bhāvanā_ is here republishing the 1942 obituary of Will Young written by G.H. Hardy and the 1944 obituary of Grace Chisholm Young by Mary Cartwright. Calcutta figures in exactly one sentence of Hardy’s obituary, as the initial example of the kind of “ `occasional’, though honourable’’ posts as a professor that characterized his belated establishment as a recognized and productive research mathematician capable of being taken seriously in the field. One must look even more closely between the lines for Calcutta’s place in Cartwright’s obituary: again, the reference is a single sentence referring to the period “About 1914’’ when Grace resumed publishing mathematics independently under her own name and produced “her most important work.’’ Both the Youngs were accomplished students of mathematics at Cambridge University. Will’s success as a student positioned him to tutor in the subject at Girton College, Cambridge’s first women’s college, where he met Grace. While Will was reputedly a charismatic and well-regarded tutor, Grace established the earlier interest and capacity for mathematical research and was the primary driver of both of the couple’s early research activity. Following their marriage, Grace maintained their household and reared their children on the European continent, initially in Göttingen and eventually in Switzerland, while Will worked in a variety of academic positions, mostly in Britain. The circumstances and effects of Will Young’s professorship had rather more notable consequences in Europe Historians have access to quite a lot of information about the Youngs’ partnership because of the large share of time they spent apart, including, of course, during long stretches in the period of 1913–1916. In 1972, Ivor Grattan-Guinness published a detailed account of their lives, marriage, and mathematics based primarily on their voluminous written correspondence. The Youngs’ letters have since been the foundation of further notable examinations of their distinctive partnership, including analyses by their descendant Sylvia Wiegand (1996), Claire Jones (2009), Elisabeth Mühlhausen (2020), and Patricia Auspos (2023). Grattan-Guinness, Wiegand, and Auspos each give further details of Will’s experience of Calcutta and the professorship’s ramifications for the Youngs’ marriage and partnership. Will, it would seem, shared Narasimhan’s dismay at the prevailing mediocrity of the British presence in colonial institutions (Grattan Guinness 1972, p. 149). His contempt for his compatriots was exceeded by his racism toward the rest of the people he encountered in Calcutta, a major motivation for his insistence that his wife and children not join him there (Auspos 2023, p. 150). His disdain for the state of university education and his own living experience in Calcutta were both likely motives in his successful appeal to spend some of the time committed to his professorship, already a mere few months per year, travelling through Europe and preparing a report on mathematical training to guide reforms in India. War in Europe necessitated changes to this project, which eventually involved travel to the United States and Japan, and the report was never completed. After the war, presumably supported at least somewhat by the international perspectives and credibility established from his Calcutta years and his comparative project, Will became involved with the short-lived first International Mathematical Union. He served as president from 1929 to 1932, as the union’s proponents struggled unsuccessfully to overcome the challenges of its troubled origins (see Curbera 2021). The enforced separation over longer distances of Will’s Calcutta professorship and associated travels appears to have had a transformative effect on the couple’s mathematical partnership. As Cartwright’s obituary indicated, in this period Grace finally established herself with notable (indeed, prize-winning) results published under her own name. Auspos considers how the distance between the couple, combined with the strains and pressures of war, created the conditions for the couple’s son Frank to enlist in the British military over Will’s strenuous objections. Frank became a pilot and was killed in action in 1917, a grave tragedy for Grace and Will’s family. Grace’s striking emergence from Will’s professional shadow during the Calcutta years underscores her independent capabilities as a mathematical researcher. The couple together made the conscious decision to work as a partnership and deliberately to court a public misperception of Will’s talents and productivity. This decision was a reasoned response to a profession that systematically rendered women as institutional exceptions to the extent it accommodated their participation at all (see Barany 2022). Recent scholarship, especially that of Jones and Auspos, has elaborated the complexity, tension, and ambivalence accompanying the decision and its changing manifestations across their work and marriage, evident in their letters and papers. Auspos vividly documents how Will was a demanding partner, prone to insecurity and jealousy, who repeatedly created difficult conditions for Grace to pursue her ambitions even as he benefited enormously from her persistence. Figalli remarked “there is one problem that I really hope to solve soon, and this is me and my wife living in the same city” Though the decision for subsumed authorship and its intended effects have been well recognized, recent scholarship has also stressed further how Grace’s contributions to the partnership’s mathematics have been under-credited even by later parties aware of the subterfuge. Jones, in particular, has challenged many of the assumptions that coloured earlier interpretations of the Youngs’ mathematical relationship. Grace, not Will, should be understood as the mentor in their mentor-mentee research partnership. There is considerable evidence of the extent to which Will relied on Grace for ideas, exposition, and quality control, often misleadingly summarized as simply editing or writing up (cf. Edwards and Gillies 2024). Even with Grace’s considerable interventions, Will had a reputation as an inconsistent and sometimes disorderly expositor. Jones observes evidence that Will’s mathematical exposition deteriorated markedly in circumstances where Grace had diminished involvement. Mühlhausen gives a detailed analysis of an exchange predating the Calcutta professorship where Grace took responsibility behind the scenes for defending a paper they had published under Will’s name in 1903. The editors of the volume on scientific couples in which Wiegand’s analysis appeared take the Youngs’ story as an example of the repeated pattern in the history of science of women’s contributions being devalued and men receiving credit for their work (Pycior, Slack, and Abir-Am 1996). They also note the significance of the Youngs’ deliberate sense of joint ownership of their work and of the countervailing perception that Grace’s brilliance (where recognized) diminished the estimation of Will’s merits. In a 1993 article that quite strongly influenced the 1996 edited volume, Margaret Rossiter proposed to call the pattern of women’s devaluation and men’s appropriation the Matilda Effect, marking its distinction from what sociologists of science identified as the Matthew Effect (from a famous Bible verse) where compounded attention produces distorted recognition. Rossiter flagged the misallocation of credit in scientific marriages as a special case of the Matilda Effect, of particular disservice to wives who in fairer circumstances would clearly outshine their husbands. Building from this critical intervention, historians interested in scientific and mathematical marriages have considered the multiplicity of contexts and dynamics involved in such relationships (Dunning and Stenhouse 2023). Among other functions, marriages could offer talented and ambitious women ways of subverting gendered roles and expectations in mathematics, clearing the way to some (often delayed and distorted) recognition (cf. Stenhouse 2021). The particular constellation of difficulties facing couples aiming to sustain academic careers together, often called the “two-body problem,’’ remains a matter of human concern as well as scholarly investigation (e.g. Wolf-Wendel, Twombly, and Rice 2003). A particularly poignant moment in the 2018 Fields Medal presentations in Rio de Janeiro came at the end of a video introducing medallist Alessio Figalli. After describing what he hoped would be several decades of open problems for his mathematical work, Figalli remarked “there is one problem that I really hope to solve soon, and this is me and my wife [mathematician Mikaela Iacobelli] living in the same city’’ (Old Bridge Media 2018). Iacobelli was appointed to a Titular Professorship at Figalli’s institution, ETH Zürich, the next year. In American slang, a `beard’ is someone who takes on a role in public that is meant to conceal another’s actions or relationships. Will Young’s very full beard dominates his portrait that appeared in _Bhāvanā_ ‘s initial republication of Narasimhan’s article. Charitably, one might think of Will as having acted as a beard for Grace’s relationship with elite mathematical research, enabling her to carry on somewhat clandestinely with aspects of the life of a research mathematician that would otherwise have been foreclosed to her on the basis of gender. Will’s appointment in Calcutta, won on the basis of a mathematical reputation that owed so much to Grace’s uncredited work, opened up a temporary distance between the couple that let the beard slip. The Calcutta professorship represented a first tentative culmination of their plan that joining their mathematics under Will’s name could result in a remunerative professional standing that would benefit both. Grace’s brief flourishing in this period as a named author in her own right confirmed for many her independent stature as a mathematical researcher, facilitating the retrospective re-evaluation of the mathematics she did under Will’s name (in addition to the above-cited analyses, see also Rothman 1996). Jones observes that in some respects the Youngs’ arrangement was a precursor to the famous authorial partnership of G.H. Hardy and J.E. Littlewood, who as established male mathematicians in Oxford and Cambridge found their reputations mutually enhanced by systematically sharing authorial credit in ways that often significantly distorted their respective contributions (2009, pp. 107–108). Their capacity to manipulate by private conspiracy the conventional relationship between publication and authorial credit took advantage of new roles for publication in the professional ecosystem of mathematics on an international stage. The denouement of the Youngs’ lives and careers came at the cusp of an efflorescence of pseudonymity that pushed such changing norms around authorship in mathematics to glorious extremes (Barany 2020). While “W. H. Young’’ was not exactly a pseudonym for Grace, nor exactly a collective pseudonym for the partnership of Grace and Will, it represented an investment in authorial bylines and a projection of the relationship between bylines and careers that was at the time in a period of dynamic reconfiguration, engaging the same phenomena that drove the era’s pseudonyms. (A false beard figures iconically in the origin story for that era’s most famous product, the collective pseudonym Nicolas Bourbaki.) In another sense, Will Young was perhaps a beard for the hidebound colonial university apparatus represented in Calcutta. His superficial and comparatively highly remunerated role as guarantor of metropolitan British mathematical credibility was in obvious tension with his precarious professional and rather misleading mathematical standing at home. It is telling that, for Narasimhan, Young telegraphed international prestige and excellence without leaving any ostensible trace of this in his mathematical activity in India. Such telegraphy surely mattered: as the Youngs’ authorial contrivances proved, perception counted for a lot in the era’s mathematics. In bylines and professorships, perception was frequently shaped by partnerships that, though complex and consequential beneath the surface, hinged on fictions that were ever paper deep. ### Acknowledgments I would like to thank C.S. Aravinda, Jan Vrhovski, and Elisabeth Mühlhausen for valuable suggestions. ### References * [1] Patricia Auspos, “A `Two Person Career’: Grace Chisholm Young and William Henry Young’’, in Auspos, _Breaking Conventions: Five couples in search of marriage-career balance at the turn of the nineteenth century._ Open Book, 2023. pp. 93–173. * [2] Michael J. Barany, “Impersonation and personification in mid-twentieth century mathematics.’’ _History of Science_ 58(4), 2020, 417–436. * [3] Michael J. Barany, “Mathematical Institutions and the `In’ of the Association for Women in Mathematics,’’ in Beery, Greenwald, and Kessel (eds.), _Fifty Years of Women in Mathematics_. Springer, 2022. pp. 325–341. * [4] M.L. Cartwright, “Grace Chisholm Young,’’ _Journal of the London Mathematical Society_ 19, 1944, 185–192. * [5] Guillermo P. Curbera, “William Henry Young, an Unconventional President of the International Mathematical Union,’’ in Mazliak and Tazzioli (eds.), _Mathematical Communities in Reconstruction After the Great War 1918–1928_. Springer, 2021. pp. 1–29. * [6] David E. Dunning and Brigitte Stenhouse, “Bringing the history of mathematics home: Entangled practices of domesticity, gender, and mathematical work.’’ _Endeavour_ 47, 2023, 1–10. * [7] Ros Edwards and Val Gillies, _Thanks for Typing_ (podcast). The Sociological Review, 2024 (series 1). https://thesociologicalreview.org/podcasts/thanks-for-typing/ * [8] Claire G. Jones, _Femininity, Mathematics and Science, c. 1880–1914_. Palgrave, 2009. esp. ch. 4. * [9] Ivor Grattan-Guinness, “A mathematical union: William Henry and Grace Chisholm Young.’’ _Annals of Science_ 29(2), 1972, 105–186. * [10] G.H. Hardy, “William Henry Young,’’ _Journal of the London Mathematical Society_ 17, 1942, 218–237. * [11] Elisabeth Mühlhausen, “Grace Chisholm Young, William Henry Young, Their Results on the Theory of Sets of Points at the Beginning of the Twentieth Century, and a Controversy with Max Dehn,’’ in Kaufholz-Soldat and N.M.R. Oswald (eds.), _Against All Odds: Women’s_ Ways to Mathematical Research Since 1800. Springer, 2020. pp. 121–132. * [12] Raghavan Narasimhan, “The Coming of Age of Mathematics in India,’’ in _Miscellanea Mathematica_. Springer, 1991. pp. 235–258. * [13] Old Bridge Media, “Fields Medal Video: Alessio Figalli.’’ Simons Foundation, 2018. https://www.simonsfoundation.org/2018/08/01/fields-medal-video-alessio-figalli/ * [14] Helena M. Pycior, Nancy G. Slack, and Pnina G. Abir-Am, “Introduction,’’ in Pycior, Slack, and Abir-Am (eds.) _Creative Couples in Science_. Rutgers, 1996. pp. 3–35. * [15] Margaret W. Rossiter, “The ~~Matthew~~ Matilda Effect in Science.’’ _Social Studies of Science_ 23, 1993, 325–341. * [16] Patricia Rothman, “Grace Chisholm Young and the division of laurels.’’ _Notes and Records of the Royal Society of London_ 50(1), 1996, 89–100. * [17] Brigitte Stenhouse, “Mister Mary Somerville: Husband and Secretary.’’ _Mathematical Intelligencer_ 43, 2021, 7–18. * [18] Sylvia Wiegand, “Grace Chisholm Young and William Henry Young: A Partnership of Itinerant British Mathematicians,’’ in Pycior, Slack, adn Abir-Am (eds.) _Creative Couples in Science_. Rutgers, 1996. pp. 126–140. * [19] Lisa Wolf-Wendel, Susan Twombly, and Suzanne Rice, _The Two-Body Problem: Dual-Career-Couple Hiring Practices in Higher Education._ Johns Hopkins, 2003. * * * ### William Henry Young3 by **G.H. Hardy4** W.H. Young (1863–1942) **courtesy:** MacTutor History of Mathematics Archive William Henry Young, who died at Lausanne on 7 July, 1942, at the age of 78, was one of the most profound and original of the English mathematicians of the last fifty years. He was born in London on 20 October, 1863. His ancestors were Ipswich people, but had been bankers in the city for some generations. His early education was at the City of London School; the headmaster, Edwin A. Abbott, had been a schoolfellow of Young’s father. Abbott was the author of the entertaining mathematical fantasy _Flatland_ , and, though he left the actual teaching in mathematics to others, was enough of a mathematician to recognize Young’s exceptional talents. Young seems indeed to have been understood much better at school than at home, and he always spoke of Abbott with gratitude and admiration. He came up to Cambridge, as a Scholar of Peterhouse, in 1881. He came with a reputation to sustain and, if we are to judge him as an undergraduate and by the standards of the time, he hardly lived up to it. He was expected to be Senior Wrangler, but was fourth, Sheppard, Workman and Bragg being above him; and he did not send in an essay for a Smith’s Prize (though the new regulations, which should have suited him exactly, had just come into force). It is easy now to see reasons for Young’s comparative failure. The whole system of mathematical education in Cambridge was deplorable. The college teaching was negligible, the professors were inaccessible, and an undergraduate’s only chance of learning some mathematics was from a private coach. Young, like nearly all the best mathematicians of his time, coached with Routh, from whom he could learn a lot. But he had many other interests, and no doubt he wasted much of his time. He was a good, though unsystematic, chess player, and an enthusiastic swimmer and rower; and his greatest disappointment as an undergraduate seems to have been his failure to get a place in the college boat. He had always immense physical as well as mental energy, and remained an ardent oarsman all his life. Whatever disappointments he may have had, Young seems to have been happy as an undergraduate: he said afterwards that “he never _began_ to live until he went to Cambridge’’, and he formed one friendship which was important throughout his career. This was with George and Foss Westcott, the sons of Professor Westcott, afterwards Bishop of Durham. Both of the Westcotts became bishops themselves later, and their influence led Young to turn his attention to theology. His family were Baptists, but he was baptized into the Church of England, and became for a time superintendent of a Sunday school. These preoccupations help to explain his neglect of the Smith’s Prize competition; he was occupied at the time in winning a College theological prize. Young was elected a Fellow of Peterhouse in 1886. There was no dissertation or examination; it was a matter of course because of his place in the Tripos. He remained a Fellow until 1892, but was never given any permanent position either by the College or the University. It was not until he was an old man that his College, in 1939, elected him an Honorary Fellow. The next thirteen years of Young’s life were spent, almost exclusively, in teaching and examining. It was common enough then for Cambridge mathematicians to earn quite large sums by private coaching, and Young set himself resolutely to do so. Here the Westcott connection was a help, bringing him a good many pupils. He also went twice to Charterhouse as a temporary assistant master. It is difficult for anyone who knew Young only later to imagine him as a schoolmaster, but he seems to have enjoyed the experience. He did much examining, at Eton (where he awarded the “Tomline Prize’’ to P.H. Cowell) and at other big schools; but primarily, through all these years, he was a coach. His position became a little more official in 1888, when Girton made him a lecturer in mathematics. “Lecturing’’ at Girton meant, in effect, more coaching, and after this coaching absorbed practically all his time and energy. He was working from early morning till late at night, sometimes taking two classes simultaneously in adjacent rooms, and often going without lunch. There is general agreement that Young was an excellent coach, and here again I find something surprising in the testimony to his merits. It is easy enough to imagine him a great inspiration to any first-rate pupil, but he did not have the chance of teaching more than a very few. He had the monopoly of Girton, but Girton wranglers were rare; the second whom he taught was his future wife. I should hardly have expected him to have had the patience necessary for success with less gifted pupils, but apparently it was just there that he excelled. His preferences in mathematics also seem very surprising to anyone familiar with his later work. He had read widely, and could teach anything in reason, but astronomy was his pet subject. “Astronomy’’ was the mathematical astronomy required for the Tripos of those days, and a man who could make that stimulating must have been a teacher indeed. This interest lasted, and his first suggestion for “independent work’’, in the early days of his marriage, was one of a text-book of astronomy to be written in collaboration with his wife. And all this time we hear not one word of research. Young was the most original of the younger Cambridge mathematicians; twenty years later he was the most prolific. Yet no one suggested to him that he might have it in him to be a great mathematician; that the years between twenty-five and forty should be the best of a mathematician’s life; that he should set to work and see what he could do. The Cambridge of those days would seem a strange place to a research student transplanted into it from to-day. I still find it difficult to visualize Young’s own attitude during these early years of unproductivity. The productivity, when it did come, was so astonishing; it seems at first as if it must have been the sequel to years of preparation, by a man who had succeeded at last in finding his subject and himself. One would have supposed that anyone so original, however he might be occupied, must surely have found something significant to say, but actually the idea of research seems hardly to have occurred to Young. Mr. Cowell says that Young once told him that he “deliberately accepted ten years of drudgery’’, that he “fancied his knowledge of the Stock Exchange’’, and that he thought that he could “win his leisure’’ by thirty-five; but “leisure’’ meant freedom, comfort, reading, and travel, not a life of mathematical research. The truth seems to be that Young had really no time to think of much but his teaching; that the atmosphere of Cambridge was mathematically stifling; that no one was particularly anxious to look out for or encourage originality; and that he was too much absorbed in his routine, in his pupils and their performance, to dream of higher ambitions. However that may be, the dreams were to come and the “drudgery’’ to end, and the end came quickly after Young’s marriage. In 1896 he married Grace Chisholm, the second of his wrangler pupils. Mrs. Young’s father was H.W. Chisholm, for many years Warden of the Standards;5 and her brother, Hugh Chisholm, was editor of the _Encyclopaedia Britannica_. The family carries on this tradition of distinction, and two of Young’s six children are well known to us as mathematicians. The eldest son, Frank, was killed as an airman in France in 1917. The great break in Young’s life came, quite suddenly, in 1897; and here perhaps I had better quote Mrs. Young’s own words. “At the end of our first year together he proposed, and I eagerly agreed, to throw up lucre, go abroad, and devote ourselves to research’’: it seems to imply a revolution in Young’s whole attitude to mathematics. But Mrs. Young had studied in Göttingen before her marriage, and knew what the air of a centre of research was like, so that possibly the revolution was a little less abrupt than it appears. At any rate, the Youngs left Cambridge for Göttingen in September. “Of course all our relations were horrified, but we succeeded in living without help, and indeed got the reputation of being well off’’: Young’s “banker’s instincts’’ had served him well. Young’s permanent home was abroad for the rest of his life, in Göttingen until 1908 and then in Switzerland, first in Geneva and afterwards in Lausanne; but the continuity of home life was much broken by his many activities. In 1901 he came back to Cambridge, and had rooms in Peterhouse during term time for some years, returning home for vacations. During 1902–05 he was Chief Examiner to the Central Welsh Board, and seems to have thrown himself into the work with all his usual enthusiasm. His reputation was now rising, and he became a Fellow of the Royal Society in 1907, but it was not until 1913 that he obtained any definitely academic position. He was still not properly appreciated, and I can remember that, when he was a candidate for the Sadleirian chair in 1910, no one in Cambridge seemed to take his candidature very seriously. The next few years were his years of greatest activity. He wrote a great deal—there are forty papers of his in Vols. 6–18 of our _Proceedings_ only; and in 1913 he at last became a professor. His first posts were of an “occasional’’, though honourable, kind. He was the first Hardinge Professor of Mathematics in Calcutta; this involved residence in India for the three winter months of the next three years. He also became Professor of the Philosophy and History of Mathematics in Liverpool, and lectured there during the summer. This was a special post created for him, and he held it until 1919, when he was appointed Professor of Pure Mathematics at Aberystwyth. Here he was as energetic as ever, but his residence abroad was sometimes a source of trouble, and disagreements about this, and about appointments to the staff, led to his resignation in 1923. By this time Young had almost ceased his activity as an original mathematician. It had slackened for a time about 1915, but the death of his son in 1917 caused him great distress and drove him back to mathematics “as a drug’’. It was in this year that the Society awarded him the De Morgan Medal. He did good work after, but none quite equal to his best, and after 1923 he wrote little. He was President during 1922–24, and his last papers (including his Presidential address) were printed in the _Proceedings_ in 1925. It was a little later, in 1928, that the Royal Society gave him the Sylvester Medal in recognition of a life of invincible mathematical activity. Young was well over sixty now and regarded his career as a constructive mathematician as finished, but there were other openings for his activities. He had always been keenly interested in the international organization of mathematics; and this brings me to the one controversy where I found myself in active opposition to him. We had no quarrel and, so far as I know, lost no respect for one another: our differences concerned means rather than ends but, within their limits, they were irreconcilable. They had first appeared in 1922–24, when he was President and I was Secretary, and the Society refused to send delegates to the mathematical congress at Toronto. In 1929 Young became President of the “International Union of Mathematicians’’, one of the unions formed, under the aegis of the “International Research Council”, in 1919–20. It seemed an honourable position, and there is no doubt that Young thought that he could use it to do real service to the cause of international cooperation, and worked whole-heartedly to that end. I am afraid (though my judgment is no doubt biassed) that it brought him little but worry and disappointment. The truth seems to me to be that Young, though the objects of his activity were irreproachable, was carrying them on under an impossible handicap. The Union had been so much prejudiced by its previous history that all his efforts were foredoomed to failure. It had been founded too soon, before the passions of the war had had any time to cool; it was shackled by the statutes of the I.R.C.; and these statutes had been largely inspired by men anxious to direct them towards a boycott of ex-enemy nations. These feelings (with which Young had never sympathized) gradually weakened, and the opposition was always strong; so that in 1926 the Council declared itself ready to open its ranks. But then, as might have been anticipated, the “enemy nations’’ showed no desire to accept favours from such a quarter. It was in these discouraging circumstances that Young succeeded to the Presidency of the Union. He did his best, but the case was hopeless; the majority of mathematicians had made up their minds to scrap all this machinery and to start again. The Zürich Congress of 1932 broke away from the Union, and that was effectively its end. I think that Young himself ended by feeling that he was well quit of thankless job. He turned back to his other old interests: law (he had been a member of Lincoln’s Inn from early days), finance, and above all languages. These included Jugoslav and Polish, two of the most intricate of Central Europe. The end of Young’s life was rather tragic, since he was cut off from his family completely by the war. Mrs. Young had left him, as they meant for a few days only, and the collapse of France prevented her return; and his children were settled in London, South Africa, and Paris. He had always been the centre of a family, and found himself imprisoned in Switzerland and practically alone. Little had been heard from him, but it is known that he died quite suddenly, that he “just went out’’, and that the University of Geneva, of which he was an honorary doctor, did him every honour. It is not particularly difficult to estimate Young’s rank as a mathematician. There is no mystery about him; his work, three books and over 200 papers, is entirely characteristic; one may get a little lost in it at times, but both appreciation and criticism are straightforward tasks. Two features of it stand out on almost every page, intense energy and a profusion of original ideas. Indeed it is obvious to any reader that Young has a superabundance of ideas, far too many for any one man to work out exhaustively. One feels that he should have been a professor at Göttingen or Princeton, surrounded by research pupils eager to explore every bypath to the end. It may be that he had hardly the temperament or the patience to lead a school in this way, but he never had a chance to try. Two features of his work stand out on almost every page, intense energy and a profusion of original ideas. His style is better in his books than in his papers, which are sometimes rather rambling and diffuse—faults natural in the writing of a man with many ideas, anxious to press on in a field which is developing rapidly and where there are many rivals. He makes astonishingly few mistakes, and the critical passage will almost always be found to be accurate and clear; but his repetitions are sometimes rather trying to a reader anxious to dig out the kernel of what he has to say. A theorem will be proved, in varying degrees of generality, in half a dozen different papers, with continual cross-references, and promises of further developments not always fulfilled. It is not surprising that a good many of Young’s theorems should have been missed and rediscovered. At his best, however, he can be as sharp and concise as any reader could desire; and he (or he and his wife together) could write an excellent historical and critical résumé, with just the right spice of originality.6 There is one particular compliment which I find it easy to pay to Young. His work stands up stoutly to critical examination. There are men who seem to me admirable mathematicians so long as they write about geometry or physics—it is easy to be impressed by what one does not understand very well. Young’s best work seems to me to be his work on the subjects which I myself know best, on the theory of Fourier and other orthogonal series, on the differential calculus, and on certain parts of the theory of integration. I will say something first about the last of these subjects, not because his work here (except that on the “Stieltjes integral’’ in [27]) seems to me his very best, but because it is the most widely known, and because it was the occasion of a disappointment which, coming as it did right at the beginning of his active career, might easily have broken the spirit of a weaker man. The theory of functions of a real variable has been written afresh during the last forty or fifty years. In particular, the foundations of the integral calculus have been entirely remodelled; and it is acknowledged by everyone that, among those who have reconstructed them, Lebesgue stands first. The “Lebesgue integral’’ opens the blocked passages and smooths the jagged edges which disfigured the older theories, and gives the integral calculus the aesthetic outlines of the best “classical’’ mathematics. In particular it brings integration and differentiation into harmony with one another. It is Lebesgue’s theorems about integrals and derivatives, the core of any modern treatment of the subject, which are his greatest achievement. It is easy to be impressed by what one does not understand very well Young, working independently, arrived at a definition of the integral different in form from, but essentially equivalent to, Lebesgue’s. He had not made Lebesgue’s applications: the great theorems about integrals and derivatives are Lebesgue’s and his alone. But naturally Young’s integral, being equivalent to Lebesgue’s, “has them in it”. If Lebesgue had never lived, but the mathematical world had been presented with Young’s definition, it would have found Lebesgue’s theorems before long, In the definition itself Young was anticipated by about two years, and it must have been a heavy blow to a man who was just beginning to find himself as a mathematician; but he recognized the anticipation magnanimously, and set himself whole-heartedly to work at the further development of the theory. The phrase “the Lebesgue integral” is Young’s. It may seem a paradox, but it is possible that Young’s work on integration, fine as it was, actually impeded his recognition. These subjects were not popular, even in France, with conservatively minded mathematicians. In England they were regarded almost as a morbid growth in mathematics, and it was convenient for men out of sympathy with Young’s interests, and perhaps a little jealous of his growing reputation, to dismiss him as “the man who was anticipated by Lebesgue’’. It is easy enough now to recognize the absurdity of such a view: if Young had never given his definition of an integral, his reputation would not be very materially affected. Most of this work is set out in [23], [24], and [25]; it is in [25] that the “Young integral’’ is actually defined. All these papers were written in ignorance of Lebesgue’s work, and recast when Young discovered it. This spoils their continuity a little, and it is perhaps a pity that he did not leave them as they stood and add the acknowledgments necessary in appendices; the genesis and progress of his own ideas would then have stood out more clearly. It is plain that Young, in his first essays at a theory of integration, was rather hypnotized by the familiar Darboux sums. He wished, at first, to _preserve_ their properties, and came only gradually to see that the essential thing is to get away from them. There is another important generalization of the Riemann integral, the integral first defined by Stieltjes in 1894. The Stieltjes integral covers sums as well as ordinary integrals, and has come rapidly into vogue since about 1909, primarily because of its outstanding importance in the theory of “linear functionals’’. In Stieltjes integration we integrate one function f with respect to another function g: the classical case is that in which f is continuous and g monotone. It was inevitable, after Lebesgue’s work, that mathematicians should try to combine the two generalizations, and define the integral of any Lebesgue-integrable f with respect to any monotone g. In particular, Lebesgue had tried, but his results were not altogether satisfactory. Young solved the problem with complete success: he showed that his method of monotone sequence could be applied to this more general problem with little more than verbal changes. The best tribute to Young’s work that I can quote is that of Lebesgue himself.7 Referring to his own attempt, he says “En réalité, je n’avais que très imparfaitement compris ce rôle [that of monotone sequences], sans quoi je n’aurais pas écrit … qu’il serait très difficile d’étendre la notion d’intégrale de Stieltjes par un procédé différent de celui que j’employais. Peu de temps après que j’eus commis cette imprudence, Mr.W.H. Young montrait que mon procédé était loin d’être indispensable, et que l’intégrale de Stieltjes se définit exactement comme l’intégrale ordinaire par le procédédes suites monotones …. Ce travail de M. Young est le premier de ceux qui ont finalement bien fait _comprendre_ ce que c’est qu’une intégrale de Stieltjes …”. [In reality, I had only very imperfectly understood this role [that of monotone sequences], otherwise I would not have written that it would be very difficult to extend the notion of the Stieltjes integral by a process different from the one I was using. Shortly after I had committed this imprudence, Mr. W.H. Young showed that my process was far from being indispensable, and that the Stieltjes integral is defined exactly like the ordinary integral by the process of monotone sequences. This work by Mr. Young is the first of those which have finally made it clear what a Stieltjes integral is.] The phrase “the Lebesgue integral” is Young’s Young’s work on integration, which reaches its peak in the paper on the Stieltjes integral, was preceded and accompanied by a whole flood of papers on the theory of sets of points and its application to the general theory of functions. A considerable part of the contents of these papers is incorporated in the Youngs’ book [21], published in 1906 and, unfortunately, never revised and reprinted. It is curious that Young should never have written a really successful book. He wrote three, alone or in collaboration with his wife; the _Sets of points_ , this tract (both “classics’’ which somehow hung fire), and a third book which must have seemed to have every chance of popularity, but which was, to the publisher at any rate, a complete disappointment, This was the _First book of geometry_ ([20]), written by the Youngs jointly, and published by Dent in 1905. The book is a genuine “book for children’’ of a very interesting and original kind. The central idea is that children should be encouraged to think of geometrical objects in three dimensions, to think of a plane, for example, as a boundary of a solid, and of a line as an intersection of two planes, or as a fold in one. I am no authority on such a matter, but I should have thought that the idea was sound, and that a book based on it should be more concrete and more stimulating, for most learners, than those of the more conventional and abstract pattern. The authors, however, were asking too much of English teachers. It appeared that they could not, or would not, fold paper, and the book fell absolutely flat in England. It was much more successful abroad, has been translated into German, Italian, Magyar, and Swedish, and used with success in German schools. I must end by saying something about the series of papers ([28]–[32]) which dominated Young’s last period of activity, though here I am not a very well qualified critic.8 The central problem is that of the area of surfaces, a problem of notorious difficulty still not completely solved. The most interesting and characteristic paper is the short paper [29]. “No one’’, Young says, “has hitherto succeeded in giving a definition of the area of a surface’’ \\[\begin{equation} \notag x=x(u, v), y=y(u, v), z = z(u, v), \end{equation}\\] “which allows us to state a sufficient condition of a general character that the surface should possess an area …”, given by the familiar formula \\[\begin{equation} \notag S = \iint \sqrt{(J_1^2 + J_2^2 + J_3^2)} \, du \, dv, \end{equation}\\] where the J are the three Jacobians of x, y and z. Lebesgue had attacked the problem in his thesis, defining the area as the lower bound of that of polyhedra which tend to the surface, without necessarily being inscribed in it; but the results were not altogether satisfactory, and Young suggests a new procedure. He takes a network in the plane (u, v); this defines a network of curved quadrilaterals on the surface. He then defines _the area of a skew curve_ (and this is his most characteristic contribution to the problem). Inscribe polygons in the curve, and imagine that (to use the language of statics) their sides represent forces. The forces are equivalent to a couple, and the area of the curve is the limit, if it exists, of the magnitude of the couple. For the surface, sum the areas of the network of curved quadrilaterals, and again proceed to the limit; if this limit exists, it is the area of the surface. The definition differs fundamentally from Lebesgue’s in two respects; it uses only auxiliary figures _inscribed_ in the surface, and it is based on limits and not on lower bounds. The test of the definition lies in its results, and these are in some ways very satisfactory. Thus Young’s main theorem (stated only in its simplest and most striking form) is that _if all of_ x_u, x_v, y_u, … _are of integrable square, then the area exists and is given by the aforementioned familiar formula_. The theorem demands a detailed preliminary study of the formula \\[\begin{equation} \notag A=\iint J \ du dv \end{equation}\\] for the plane area defined by x=x(u, v), y=y(u, v) when (u, v) varies over a rectangle. The first results concerning the auxiliary formula for the plane area are set out in [28]: thus this plane area formula is true whenever x_u, … are of integrable square. Young returned to them and generalized them in the later papers. Young’s definition has drawbacks. It is insufficiently intrinsic, depends too much on the particular parametric representation, and does not lend itself to proofs of invariance. His work was developed a little later by Burkill. Burkill’s _definition_ is much the same, but his theory of (non-additive) functions of intervals enables him to define the J quite differently and so to reach more satisfactory results. Since 1926 Lebesgue’s definition has come again into favour, as a result of Tonelli’s work. Tonelli solves the problem definitely for surfaces z=z(x, y),9 but not in the general case. This has attracted many writers, but no final solution has been found. This was Young’s last work, and very remarkable work for a man of sixty, though it may not have quite the quality of the best work of his best years. He knew that it was his last, and said so rather dramatically in his Presidential Address ([33]) to us a little later. “What I have been able to do, I have done— this rough magic I here abjure” There is a touch of grandiloquence in it, but it was excusable, for few of our Presidents could claim to have done more. ### References * **[1]** _The first book of geometry_ (with Grace Chisholm Young; London, Dent, 1905). * **[2]** _The theory of sets of points_ (with Grace Chisholm Young; Cambridge, University Press, 1906). * **[3]** _The fundamental theorems of the differential calculus_ (Cambridge Tracts in Mathematics, No. 11, 1910). * **[4]** “Open sets and the theory of content’’, _Proc. London Math. Soc. (2)_ , 2 (1905), 16–51. * **[5]** “On upper and lower integration’’, _Proc. London Math. Soc. (2)_ , 2 (1905), 52–66. * **[6]** “On the general theory of integration’’, Phil. Trans. Royal Soc. (A), 204 (1905), 221–252. * **[7]** “On the theorem of Riesz-Fischer’’ (with Grace Chisholm Young), Quarterly Journal, 44 (1913), 49–88. * **[8]** “On integration with respect to a function of bounded variation’’, _Proc. London Math. Soc. (2)_ , 13 (1914), 109–150. * **[9]** “On a formula for an area’’. _Proc. London Math. Soc. (2)_ , 18 (1920), 339–374. * **[10]** “On the area of surfaces’’, _Proc. Royal Soc. (A)_ , 96 (1920), 71–81. * **[11]** “On the triangulation method of defining the area of a surface’’, _Proc. London Math. Soc. (2)_ , 19 (1921), 117–252. * **[12]** “On a new set of conditions for a formula for an area’’, _Proc. London Math. Soc. (2)_ , 21 (1923), 76–94. * **[13]** “Integration over the area of a curve and transformation of the variables in a multiple integral’’, _Proc. London Math. Soc. (2)_ , 21 (1923), 161–190. * **[14]** “The progress of mathematical analysis in the twentieth century’’, _Proc. London Math. Soc. (2)_ , 24 (1926), 421–434. * * * ### Grace Chisholm Young10 By **M.L. Cartwright** G.C. Young (1868–1944) Wikimedia Commons Grace Chisholm Young was a true pioneer; she was one of the very few women mathematicians of her generation to achieve an international reputation, and the repercussions of her enthusiasm and her ideas on the mathematical world extend far beyond her own individual achievements. Something of her life and work has already been told in the account of her husband, W.H. Young, published in this Journal;11 but that is by no means the whole story, and I shall try to avoid repetition. Grace Chisholm was born on 15 March, 1888, and her early education was at home; she took the Senior Cambridge examination in December, 1885. In April, 1889, she entered Girton College as Sir Francis Goldsmid Scholar; and she was a wrangler in Part I of the Mathematical Tripos in 1892. Immediately after the Tripos she and I.M. Maddison went to Oxford, and sat for the Final Honours School of Mathematics, obtaining a first and a second class respectively. I believe that they were the first women to sit for the Final Honours School of Mathematics, and that they did it to refute a suggestion from one of their coaches that it was more difficult for a woman to obtain a first at Oxford than at Cambridge. Their names do not appear in the lists in the Oxford University Calendar, probably because they took the examination by some unofficial arrangement; but two or three years later the names of women, chiefly students of Royal Holloway College, become quite usual in the Oxford Finals list, and often in the first class. Miss Chisholm then proceeded to take Part II of the Mathematical Tripos, which was a most unusual thing for a woman to do in those days. According to the “Vita’’ in her PhD dissertation she was taught chiefly by Berry, Richmond, and W.H. Young, her future husband. She also attended lectures by Smith and Webb, and later (when reading for Part II presumably) she attended lectures by Forsyth, Darwin and Cayley. After Part II, since there was no possibility of a Smith’s Prize or Fellowship for a woman, she was advised to go to Göttingen. By whom I have failed to discover, but it would be most interesting to know in the light of her subsequent career and its influence on Cambridge pure mathematics. Berry and Forsyth wrote to Klein asking him to admit her to lectures, and it seems likely that it was the latter but it is also worth observing that her PhD dissertation is dedicated to her father, Henry Williams Chisholm, “von welchem sie frühzeitig gelernt hat Deutschland und die mathematische Wissenschaft hochzuschätzen’’. [from whom she learned early on to value Germany and mathematical science] Since there was no possibility of a Smith’s Prize or Fellowship for a woman, Chisholm was advised to go to Göttingen Klein replied12 that the decision did not rest with him but with the Faculty in general, and ultimately with the Ministry in Berlin. He made no promises, but advised her to come to Göttingen early in October. Klein returned from Chicago a few days after she arrived, and she seems to have been immensely impressed by his height, his long capable-looking hands, and above all by his smile. Two American ladies had also just arrived on the same errand, independently of Miss Chisholm and of each other. Miss Winston, a graduate of the University of Wisconsin, former Fellow of Bryn Mawr, had been at Chicago for a year; she went to the same lectures as Miss Chisholm at Göttingen. Miss Maltby, of Wellesley College and the Boston Institute of Technology, was an experimental physicist and worked under Nernst on conductivity. She was an MA and BSc, and held a travelling fellowship from the Institute. Cover page of Grace’s book designed by Alice B. Woodward. Wikimedia Commons These three, following Klein’s advice, made an application to the Minister of Education to be admitted to lectures. Klein told them that it would be hopeless to ask for permission to matriculate, but he arranged for them to be admitted unofficially to lectures until the reply came. On the first day they went to the Room of the Mathematical Models before eleven, so as to avoid the crowd of students wandering about during the quarter of an hour’s grace between lectures; and at 11.15 they followed Klein into the lecture room. They seemed to have been needlessly apprehensive about the attitude of men students towards the presence of women in the auditorium, and found themselves made welcome. By the third lecture official permission for them had come, but only as exceptional cases. Miss Chisholm wrote to the Girton Mathematical Club: “There are lectures given here by University professors outside the University itself to women, and there are about thirty women who go to these lectures; naturally some of these would like to be admitted to the University and allowed to go to any lectures they please. In a German University there is none of that organization of colleges, and tutors, and coaches, and examinations which makes it easy at Cambridge to systematize and control the studies of the students, and this makes the question of the admission of women here one of much greater difficulty than the corresponding one at home. Prof. Klein’s attitude is this, he will not countenance the admission of any woman who has not already done good work, and can bring him proof of the same in the form of degrees or their equivalent, or letters from professors of standing; and, further, he will not take any steps till he has assured himself by a personal interview of the solidity of her claims. Prof. Klein’s view is moderate. There are members of the Faculty here who are more eagerly in favour of the admission of women, and others who disapprove altogether. But the chief difficulty is in Berlin. Were not Hanover reduced to the condition of a province of Prussia, a condition very much disliked by a strong party here, I should have very little doubt of the success of the cause in a few years’’. Miss Chisholm and Miss Winston went to Klein’s and Weber’s seminars. She wrote: “The latter simply gives us problems to solve, and one of the students works them out on the board at the next meeting. The problems are interesting enough to solve, but Prof. Weber always seems to choose out the dullest person to work out the problems on the board, and it is nonetheless wearisome because the function takes place at 8 o’clock in the morning, and necessitates breakfast at 7.30.’’ “Prof. Klein’s seminary is quite different; it takes place every Wednesday at 11 o’clock, and lasts about two hours, and the members make `Vortrag’s [Lectures] on their special subjects on different Wednesdays. The students who have been here some time, and some of the new students who have come from other Universities, have already got their special subjects; for the others, Prof. Klein has always suggestions as to special lines of work which they might take up, generally in connection with the lectures. Miss Winston made her Vortrag on the last Wednesday before the Christmas holidays. It would be nervous work in any case to make a Vortrag before an audience of about a dozen men, half of whom are Doctors, and one Prof. Klein; but the strain is considerably increased by having to speak German. There are about a dozen of us in our lectures; we are a motley crew: five are Americans, one a Swiss-French, one a Hungarian, and one an Italian. This leaves a very small residuum of German blood’’. The position of women must have improved quite soon. For Miss Chisholm obtained her PhD degree “magna cum laude’’ [with great distinction] in 1896 for a dissertation on the algebraic groups of spherical trigonometry, a subject evidently suggested by Klein. He was so much interested in the problem that he discussed it at length,13 and her treatment of it, twelve years later in _Elementary mathematics from an advanced standpoint_ , where he referred to her as the first woman in Prussia to pass the normal examination for the doctor’s degree. Things had moved a long way since 1874, when Sonja Kowalewski14 took the degree at Göttingen _in absentia_ , having been refused permission to attend Weierstrass’ lectures. Things had moved a long way since 1874, when Sonja Kowalewski took the degree at Göttingen _in absentia_ In 1896 Grace Chisholm married W.H. Young, and from this point the story of her life and work is closely bound up with that of her husband’s career, which has already been told.15 Until his marriage he had done practically no research, although he was by then about thirty-three; but a year later he gave up most of his coaching and examining, and they went to live abroad and do research. There is a strong impression, in spite of Mrs. Young’s statement16 that he suggested the move, that the idea of it came from her; the impression is based partly, no doubt, on her own already established record of research done abroad. If we accept the view of Hobson’s career as a pure mathematician drawn by Hardy,17 that his interest in the modern theory of functions was largely due to his intercourse with W.H. Young, and that the present position of real function theory at Cambridge is very largely due to Hobson, then it all began with the Youngs’ move from Cambridge to Göttingen. No doubt the ideas of real function theory would have found a place in the Cambridge Tripos sooner or later, but perhaps never such an important place if Mrs. Young had not pushed her way into Göttingen. Not that either of the Youngs began immediately to work on real function theory when they arrived at Göttingen. Mrs. Young at that time seemed to be more interested in algebraic, geometrical, and even astronomical topics; and W.H. Young began to write on vectors in dimensions; but about 1901 both W.H. Young and Hobson turned to the theory of functions of a real variable, and from that time it has been the main interest of the mathematical members of the Young family. I do not know who or what directed their attention that way. None of the people connected with Göttingen at that period were particularly interested in the subject, so far as I know, except perhaps Osgood, though the French mathematicians were already writing a great deal about it. After a period of comparative mathematical inactivity in early married life, when the children were quite young, Mrs. Young’s name began to appear, first in 1906 as joint author with her husband of the book _The theory of sets of points_ , then as joint author with him of several important papers from 1909 onwards. About 1914 she once more began to write mathematics independently; and the next few years saw the appearance of her most important work–-work which has given her a permanent place among those mathematicians who were developing the modern theory of real functions.18 Her special topic was the theory of differentiation and of derivates. The first of this series of papers was one in _Acta Mathematica_ , containing the theorem _Except at an enumerable set of points, the lower derivate of any function on either side is not greater than the upper derivate on the other side._ Mrs. Young’s paper was written independently of a slightly earlier Habilitationsschrift of Rosenthal19 which includes the special case of this theorem in which the function is continuous. This special case lends itself to a treatment which is more geometrical than Mrs. Young’s, and it is an analytical discussion substantially on her lines which would normally be given to-day. In 1915 Girton College awarded Mrs. Young the Gamble Prize for an essay “On infinite derivates’’ which was published in a modified form in the _Quarterly Journal_ in 1916. The introduction to this essay contains a general survey of the theory; the style is in parts exuberant, and effervesces into fantasy on the subject of the ultra-microscope and the atom—“Away with your ordinary curves, the wild atom will none of them.’’ The main theorem proved in the essay is that _The points at which the upper right-hand derivate of a continuous function is + \infty and the lower left-hand derivate is different from - \infty form a set of measure zero._ The second part of the essay contains a detailed investigation of Weierstrass’s and Cellérier’s non-differentiable functions. The complete statement of the relations between the derivates of an arbitrary function was given in Mrs. Young’s paper in Volume 15 of the _Proceedings of the London Mathematical Society_ , namely, _Except at a set of measure zero, there are three possible dispositions of the derivates of a measurable function f(x), either_ 1. _they are all equal, and there is a finite differential coefficient,_ or 2. _the upper derivates on each side are + \infty, and the lower derivates on each side are - \infty,_ or 3. _the upper derivate on one side is + \infty, the lower derivate on other side is - \infty, and the two remaining extreme derivates are finite and equal._ This statement—for a continuous f(x)—was contained in the first part of Denjoy’s exhaustive study of differentiation and integration, which appeared at about the time of the award of the Gamble Prize, and so the first published account is his. Mrs. Young’s work is, however, the more general in that she assumes only that f(x) is measurable, and these striking results are fittingly associated with the names of both of them. Many years later Mrs. Young wrote another substantial paper on the foundations of the differential calculus (_Fundamenta Mathematicae_ , 14), but it is marred by a mistake (in Theorem 5). Her energy and enthusiasm must have been quite extraordinary. She was a good tennis player in her younger days, and her interests, recorded in _Who’s Who_ , include music, domestic occupations, vine-culture, literature and languages, history (especially the sixteenth century), philosophy, chess, and formerly tennis, croquet and billiards. The domestic occupations involved in bringing up six children (two of whom, L.C. Young and R.C. Young, are mathematicians) would have been sufficient for most women, not to mention her collaboration with her husband in so much of his work, and her independent work on derivates. But she studied medicine at Göttingen and Geneva; she wrote articles in _Nature_ , and poems; and her paper “On the solution of a pair of simultaneous Diophantine equations connected with the nuptial number of Plato’’ shows a more than superficial knowledge of the history of Greek mathematics. Mrs. Young’s educational ideas are worth some consideration. For the fact that cheaper education for the six children, and education more in harmony with her ideas, could be obtained abroad than in England was probably an important factor in determining the Youngs to settle in Geneva permanently. She wrote three educational books for children, _The first book of geometry_ with her husband, and _Bimbo_ and _Bimbo and the frogs_ by herself. The two latter are really lessons on the elementary biology of plants and animals, including cell structure seen under a microscope, with a story about a family with marked resemblances to the Young family as jam to make the pill go down. They apparently had some success, and were probably a product of the days when she was studying medicine. The geometry was not successful in England, but went better in translation in Germany. All the books were in some ways in advance of their time, but unequal and only suited for children taught individually, with the inclination and sufficient intelligence to study things scientifically at an earlier age than most. For instance, I think that it is generally recognized now that the teaching of geometry should begin with very elementary solid geometry and the use of models as in _The first book of geometry_ , but an average child in a class could not follow the book to the later proofs, which are also based on models and paper folding, until he was of an age to learn the grown-up proofs; and the English examination system was quite sufficient to kill the book. The end of her life was saddened by the tragic separation from her husband by the collapse of France. She had intended to leave him for a few days only, but she was never able to rejoin him. He died in July, 1942, in Switzerland, and she on 29 March, 1944, at Croydon. The Fellows of Girton College had just recommended her for election to an honorary fellowship, and it is a matter of great regret to them that she died before the Governors were able to elect her.\blacksquare ### Footnotes 1. Michael J. Barany teaches and researches the history of modern science and mathematics at the University of Edinburgh, Scotland. He is a corresponding editor of _Bhāvanā_. ↩ 2. “The Coming of Age of Mathematics in India’’ by Raghavan Narasimhan, published in _Bhāvanā_ Vol 1 Issue 1 Jan 2017, pp. 36–50. ↩ 3. This article (G.H. Hardy, William Henry Young, J. Lond. Math. Soc. (1) 17 (1942), no. 4, 218-237. https://doi.org/10.1112/jlms/s1-17.4.218) is republished here [with contextual edits] with the permission from _London Mathematical Society._ ↩ 4. I have received much help in writing this notice from Mrs. Grace Chisholm Young and Dr. J.C. Burkill. ↩ 5. The post afterwards occupied, though with less responsibilities, by MacMahon. ↩ 6. For example, [7]. ↩ 7. _Lecons sur l’intégration_ , ed. 2, p. 263. ↩ 8. What I say about them is based on notes given to me by Dr. Burkill. ↩ 9. See Saks, _Théorie de l’intégrale_ , ed. 2, ch. 5. ↩ 10. This article (M.L. Cartwright, Grace Chisholm Young, J. Lond. Math. Soc. (1) 19 (1944), no. 3, 185-192.) https://doi.org/10.1112/jlms/19.75_Part_3.185 is republished here [with contextual edits] with the permission from _London Mathematical Society._. ↩ 11. JLMS, Volume s1-17, Issue 4, October 1942, 218–237. ↩ 12. Much of what follows about Göttingen is taken from a letter from Miss Chisholm to the _Girton Review_ published in March, 1894. I follow her terminology as regards American colleges and universities. ↩ 13. See _Elementary mathematics from an advanced standpoint, arithmetic, algebra, analysis_ , translated by Hedrick and Noble (London, 1932), 177–180. ↩ 14. E.T. Bell, _Men of mathematics_ (London, 1937), 474–475. ↩ 15. _Journal London Math. Soc._ , 17 (1942), 218-237. ↩ 16. Loc. cit., 221. ↩ 17. _Journal London Math. Soc._ , 9 (1934), 225–237 (227 and 236). ↩ 18. I am indebted to Dr. Burkill for much help in the discussion of Mrs. Young’s mathematical work. ↩ 19. A. Rosenthal, “Uber die Singularitäten dor reellen ebenen Kurven’’, _Habilitations-schrift_ (München, 1912). ↩ https://bhavana.org.in/the-young-mathematician-couple/