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  • Writer's pictureDale DeBakcsy

Spherical Triangles and Domineering Males: The Saga of Grace Chisholm Young

When Grace Chisholm, at age 28, married the mathematician William Henry Young, she had every prospect of a brilliant career before her. The previous year she had been the first woman to earn a doctorate in Germany thanks to her influential work on the categorisation of spherical triangles, and a few years prior to that she had scored first place in the gruelling Oxford mathematical exam and earned first-class placement in the infamous Cambridge Tripos exam. A student under the wing of Felix Klein, one of history’s greatest mathematicians, she could look forward to a lifetime of first-rank discoveries.


You may think you know this story after having heard it so many times over the course of the other books in this series: a brilliant woman on the verge of masterful greatness marries unwisely and suddenly abandons all intellectual activity on the say-so of a condescending, self-centred husband. It was the story of Isabel Morgan, of Harriet Brooks, of Mileva Maric Einstein, and of Mary Somerville before her first husband did the world a massive favour by dying young. And while there are elements of that tale in Chisholm’s story, the web of expectations and neglect she found herself in upon marrying was a truly complicated tangle, giving her alternately complete creative freedom and complete subservience to the demands of her husband’s mathematical visions and personal demons.


For as complicated as her adulthood was, the first stages of Chisholm’s life were cheerfully stuffed with simple good fortune. Her father was Warden of the Standards and had been the head of Britain’s Department of Weights and Measures, a man of influence and wealth who also believed fully in the talents of his daughter and in providing opportunities for their fostering. Her mother was a gifted pianist who regularly played duets with Chisholm’s father, an amateur violinist. Chisholm’s most treasured early memories were of exploring the British Mint with her father and working alongside him in his shop at home.



Happily ensconced in a creative, intellectual family, Chisholm did have an unhappy tendency towards terrifying nightmares and headaches which had one overwhelmingly positive outcome: the family doctor recommended that, so as not to overtax her, she should only be educated in the things that she actually wanted to learn. Freed at the stroke of a doctor’s pen from the smothering expectations that characterised most wealthy young ladies’ early years, Chisholm set a course for herself that concentrated on her two great loves: mathematics and music.


Eventually, as the nightmares subsided, a governess was employed to fill out her education, and in spite of not having the benefit of a formal grammar school education, she managed to easily pass the Cambridge Senior Examination at age 17. She at first wanted to go on to study medicine but her mother sternly vetoed that plan (later Chisholm would characteristically study it anyway on the side, passing all the requisites except the internship), and so Chisholm returned to her long-standing companion, mathematics, thanks to a scholarship to Girton College, where she enrolled in 1889.


Girton, like Cambridge, relied heavily on a tutor system into which occasional formal classes were enmeshed, and Chisholm’s tutor was a young man of fearsome reputation by the name of William Young. Gruff, sarcastic and demanding, he regularly reduced students to tears in the name of preparing them for the overwhelming Tripos exam. Their relationship at the time did not go beyond that of tutor and student, and Chisholm excelled in her program of study, gaining a first-class placing in her 1893 exam.


It was a triumph, but it was also all England could offer her. There was no road to graduate studies available for women at the time, and so Chisholm faced either settling down to respectable charity work, as was widely expected of someone of her status, or finding an institution on the continent at which to study. Being Chisholm, she elected for the latter and set her sights on Göttingen, where the legendary Sofia Kovalevskaya had earned a degree in absentia in 1874. As it turned out, then, England’s oppressive graduate system worked to Chisholm’s advantage as it forced her out of England, where mathematics was caught in one of its recurring bouts of stagnation, and into Germany, where the makings of several mathematical revolutions were in the works.


One of the great names of Göttingen at the time was Felix Klein, whose work on the links between group theory and geometry was paving the way towards a new, modern, and more rigorously founded geometry. Importantly, he had no objection to taking on women students as long as they had already done work of value, and his influence was to be central in removing the bureaucratic obstacles to Chisholm’s earning of a doctorate degree. Her work with him culminated in her dissertation, ‘Algebraisch-gruppentheoretische Untersuchungen zur sphärischen Trigonometrie’.


Klein was so taken with her work that he included it in his classic text, Elementary Mathematics from an Advanced Standpoint. Chisholm investigated spherical triangles – objects formed by connecting any three non-diametrically opposed points lying on a sphere. Each of these has three sides, labelled say a, b and c, and three angles, labelled say α, β and γ. Taking sines and cosines of these six quantities gives twelve values that characterise that triangle, three of which are free to be whatever they want, and nine of which are dependent on the values chosen for those other three. Of all the combinations of sine and cosine values, the set of all those that describe an actual, possible spherical triangle is called M3. Klein’s question was how do we characterise this group M3? What algebraic equations are satisfied by the points in it? How can we represent any surface passing through it as a series of equations? How ‘big’ is it?



These questions struck at the heart of modern mathematic’s attempt to marry group theory to geometry, and Chisholm’s contribution was to think not in terms of sines and cosines of a, b, c, α, β, and γ, but rather in terms of the cotangents of a/2, b/2, and so on. This representation allows M3 to be represented in a six- instead of twelve-dimensional space, and thereby led to the answer that M3 is an order eight space completely representable as the intersection of three surfaces of degree two in six-dimensional space.


Chisholm’s approach to spherical triangles represented a neat and original shift in the framing of the problem that provided basic answers to an emerging set of questions. Chisholm sent her work to her former tutor, who soon proposed marriage. She refused him at first, stating that she was afraid of what marriage might do to her nascent mathematical career, but Young, heedless – as usual – of anybody’s concerns but his own, kept pressing his case and in 1896 they married.


And now the story gets complicated. Young had tremendous difficulty his whole life in securing a permanent post that would allow his family to settle, resulting in a yo-yo existence in which Young would find temporary work, leave home for an undefined amount of time, and then return again for another undefined amount of time before repeating the cycle. A brilliant man, he lacked focus and the slightest awareness that other people’s problems might possibly be more important than his own, and when he was home he demanded every drop of Chisholm’s time to organise his life for him and put his thoughts in order. He was so domineering in his usurpation of her attention that, whenever he left to assume a new post, Chisholm would habitually lie in bed for days, paralysed by headaches as her body sought to reset itself from the juggernaut of his presence.


When he was gone, however, she could carve out some time for herself in between the raising of six children, each of whom was to obtain some degree of fame as a mathematician or scientist (well, except the one who died in the First World War, but five out of six is still pretty good you’ll admit). Chisholm still had to manage the demands for her mathematical insight that came from Young by post, including a self-serving decree that, for a time at least, their joint work should be published under his name only, as he had a greater need of the laurels than she. But letters can be folded and stuffed in a drawer in a manner that humans typically cannot, and so during those absences she managed to learn six languages, undertake medical studies, and, most importantly, work on her own mathematical problems, including her part in the Denjoy-Young-Saks Theorem.


DYS is a tremendously useful theorem which investigates that central question to all of calculus – at what points can the slope (or steepness) of a given graph be calculated? This is important, because by knowing the slope of a graph at a point, you know what the rate of change is there for the quantity depicted by the graph – How fast are these icecaps melting? How quickly is this asteroid speeding up? How precipitously is the bee population declining? These are all questions about rates and thus about slopes. The DYS theorem relies on objects called Dini derivatives which are great fun but as I have probably already exhausted everybody’s patience with that spherical triangle bit earlier, I’ll just say that they describe the extremes of what is happening to the slopes of a curve as you approach a point from the left or right. In 1915, Denjoy proved that, for any continuous function, there are only three possible ways for these derivatives to behave, only one of which allows the slope to be found. So, if you have eliminated the two options that don’t allow the slope to be found, then it must be the case that the slope is calculable. In 1917, Young extended this result to a broader class of functions, so-called ‘measurable’ functions, which can be very weird and theoretically tricky to deal with, but which, looked at through the Dini derivatives, can be analysed much more simply. And that is pretty great.


All told, between her solo papers and her joint papers with Young, she published some 220 articles over four decades, in addition to the First Book of Geometry (1905) which aimed, through foldable cut-out figures, to reacquaint children with the physically graspable truths of solid geometry, and a book called Bimbo (the nickname of her baby at the time – bimbo being a simple diminutive of ‘bambino’ which didn’t take on its modern connotations until the 1920s), which described cell division via a children’s short story.



By 1940, Chisholm had spent forty years in continental Europe at the whim of Young’s shifting academic fortunes, but the rise of Hitler was a clear indication that it was time to return to England. Chisholm ventured back in 1940, but Young vacillated, afraid that he would be judged for his many years of close work with German universities. As it happened, he waited too long, and the window of opportunity for return closed. Young died in occupied Europe in 1942, senile and unbalanced in his isolation. Chisholm died two years later, at the age of 76, just before Girton College was to unveil its plan to award her an honorary fellowship in recognition of her life’s work.


Grace Chisholm Young’s life is equal parts inspiration and cautionary tale. Her solo achievements speak for themselves, but there is a great lurking What If to be tackled. What If Grace Chisholm had resisted William Young? Certainly, his career would have suffered – by his daughter’s accounts he was utterly dependent on Chisholm to sift, focus and organise his ideas – with the result that mathematics would lose many of his published works and all their jointly published ones. But in compensation we would have been treated to forty years of a uniquely multifaceted mind entering whatever fields might catch her fancy and I cannot help but think that, in the summing, that would have worked towards a net positive. As it is, we have several remarkable individual results, hundreds of joint works, two thoroughly individual books, and a small gaggle of offspring each of whom (well, five out of six of whom) advanced scientific knowledge in their own right, and while that might not represent all that ought to have been, it is still a staggering testament to a mind capable of bringing mathematical order from the chaos of reality.


FURTHER READING: The main biographical source for Chisholm Young is Grattan-Guinness’s eighty page ‘A Mathematical Union: William Henry and Grace Chisholm Young’ in Annals of Science 29 (1972). But those are hardly lying around anyone’s local bookshop, so for shorter accounts that have quite different focuses you can pick up either the omni-useful Women in Mathematics: A Biobibliographic Sourcebook (1987) by Grinstein & Campbell, or Teri Perl’s unique Math Equals: Biographies of Women Mathematicians (1978). Klein’s book is available in an easy-to-find Dover reprint and is thoroughly wonderful.


If you'd like to read more about women mathematicians like this one, check out my History of Women in Mathematics, launching in October 2023 from Pen and Sword Books!

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