In 1884, Sophie Bryant’s paper, *On the ideal geometrical form of natural cell structure*, was published by the London Mathematical Society (LMS). It was ambitious, logical and descriptive: it looked at the phenomenon of the honeycomb.

Her insight was that the complex and beautiful honeycomb shape was a product of the natural activity of bees. All that was needed was for each bee to excavate its own cell at approximately the same rate as the others, and to use the excavated material to build up the walls of its cell. Bryant’s conclusion, that elongated rhombic semi-dodecahedra are the natural form of honeycomb cells, had been observed by Kepler.

A rhombic semi-dodecahedron can be made by putting square-based pyramids on the faces of a cube

In the eighteenth century, it was believed that the honeycomb was the most efficient cell shape possible, but this is now known not to be the case. In 1964, the Hungarian mathematician Fejes Tóth observed in his paper,

*What the bees know and what they do not know*, that there are in fact more efficient cell shapes which have yet to be determined.

Kepler conjectured in 1611 that no packing of balls of the same radius in three dimensions has density greater than the face-centred cubic packing—the cannonball packing—with a density of about 74%. Bryant’s paper assumed this conjecture to be true, as it had appeared obvious for centuries and many had attempted proofs.

Cannonball packing. Image: Wikimedia Commons user Greg L, CC BY-SA 3.0

The conjecture was eventually proved by Hales et al in 1998. Their computer-assisted proof was so huge that it took 12 referees to check

it. After five years, the referees said that they were 99% sure that the proof was correct. Unusually, Annals of Mathematics published the paper in 2005 without complete certification from the referees. It was finally accepted as proven in 2014, and then only with the aid of massive amounts of computer time.

Bryant’s approach to the subject was not unusual at that time. Abstract proofs, so essential to us now, were not as common as general discussion of mathematical phenomena. She wrote “The form of a natural structure is a logical result of its mode of genesis, and that form is ideal of which the mode of genesis is perfectly regular”. She states that there are only three possible arrangements without explaining why these are the only ones.

Bryant’s paper is notable since it is the first published paper written by a woman member of the LMS. However, she was not the first woman to be elected to membership, being preceded by two remarkable women, Charlotte Angas Scott and Christine Ladd Franklin.

## Charlotte Angas Scott

Though Bryant was the first woman member of the LMS to publish a paper, she was not the first woman member. That honour goes to Charlotte Angas Scott (1858—1931), an algebraic geometer, who became a member in 1881.

Scott had been aided in her mathematical education by an enlightened father. This resulted in her obtaining a scholarship to Girton College, Cambridge. However, women in Cambridge were not granted degrees until 1948 and she had to be content with the accolades of her peers.

She was appointed a lecturer at Girton and received an external BSc degree from London University, and later a doctorate. Scott moved to the newly opened Bryn Mawr College for women in the USA where she was appointed head of mathematics, and remained for forty years.

## Being first

Being the ‘first woman’ was not unusual for Bryant. She was the first woman to receive a DSc degree in England, studying what was then mental and moral philosophy, but today would be referred to as psychology and ethics. She was also one of the first three women to be appointed to a Royal Commission—the Bryce Commission on Secondary Education in 1894–95—and she was one of the first three women to be appointed to the senate of London University.

While on the senate she advocated setting up a day training college for teachers, which eventually became the Institute of Education. Later in 1904, when Trinity College, Dublin opened its degrees to women, Bryant was one of the first to be awarded an honorary doctorate. In Cambridge, she was also instrumental in setting up the Cambridge Training College for Women which eventually became Hughes Hall, the first postgraduate college for women in Cambridge.

She was also, it seems, one of the first women to own a bicycle.

## Beginnings and early widowhood

Bryant was born in Ireland, and was fortunate to learn mathematics as well as other academic subjects with her five siblings in a very natural way from their father, the Rev WA Willock DD. A keen educationalist, he had been a fellow and tutor at Trinity College, Dublin and had gained high honours in mathematics and mental sciences.

When Bryant was about thirteen, her family moved to England and her family education continued until she attended Bedford College, where she was awarded the Arnott scholarship for science in 1866. She sat the Cambridge local examination for girls in 1867 and was the only one to be placed in the first class of the senior division.

In 1869, Bryant married the surgeon Dr William Hicks Bryant, only to be widowed the following year when he died of cirrhosis at the early age of 30.

## Christine Ladd Franklin

The second woman member of the LMS, who also joined in 1881, was Christine Ladd Franklin (1847—1930), an American mathematical logician.

Though Johns Hopkins University was not open to women, UCL’s JJ Sylvester, then professor of mathematics, urged not only that she be admitted, but arranged for her to do graduate work under his supervision and to be granted a fellowship. As Johns Hopkins did not award degrees to women, she left without a PhD for her dissertation on symbolic logic.

She was finally awarded a PhD by Johns Hopkins forty-four years after she submitted her dissertation, when she was seventy-eight years old.

## Schoolteacher and doctorate

After a short interval, Sophie Bryant returned to her studies. While she had been sitting her examinations, she was introduced to Frances Buss, the headmistress and founder of North London Collegiate School (NLCS), an excellent school then and still highly regarded today. It had been founded in 1850, the year of Bryant’s birth.

Bryant arranged to meet Buss who, in 1875, invited her to teach mathematics at NLCS and encouraged her to take a training course as well. Three years later, London University opened its degrees to women. As Bryant had not had a conventional education, she had to learn Latin and biology to matriculate before she could sit for her degree. In 1881, she earned a BSc degree, gaining a first class in mental and moral science and second in mathematics.

In 1884, she received a science doctorate. The NLCS, where she had continued to teach, presented her with scarlet doctoral robes. Bryant was influential in improving the education system and introduced a scheme of enlightened and serious study.

In 1885, Buss died and Bryant became the headmistress of NLCS until her retirement.

## Psychology

Bryant receiving her

doctorate. Image: North London Collegiate School

Meanwhile, she continued to publish ambitious papers. In her 1884 paper in

*Mind, The double effect of mental stimuli; a contrast of types*, Bryant attempted to analyse the difference between reflex actions, which are performed without conscious thought, and consciously controlled actions.

She was grappling with a contemporary problem: the understanding of consciousness. Unfortunately, her arguments are too diffuse to shed much light on the problem.

In 1885, she published a paper in the Journal of the Anthropological Institute, *Experiments in testing the characters of school children*. This study, undertaken at the suggestion of Francis Galton, produced an early account of the use of open-ended psychometric tests to deduce personality types. Bryant claimed that her results agreed with the observations of teachers familiar with the children but did not provide any supporting evidence. Despite incomplete analysis, this was a pioneering study.

## Later life

Bryant was interested in Irish politics, and wrote books on Irish history and ancient Irish law. She was an ardent Protestant Irish nationalist and was active in the Home Rule movement, which pressed for Irish self-government within the United Kingdom. She wrote on women’s suffrage in 1879 but later advocated postponement until women were better educated in politics.

She enjoyed mountain climbing and she summited the Matterhorn twice. Her death in 1922 was both tragic and unexpected. Only four years after retirement, she was on a mountain hike near Chamonix, in France, when she went missing. Her body was found thirteen days later with several injuries.

## Epilogue

Although Bryant’s direct contribution to mathematical scholarship was not substantial, her influence as a teacher and educationalist was immense. The rising number of women mathematicians today is a lasting tribute to her work.