Grace Chisholm Young
Shelley St. John
History of Mathematics
March 27, 2007
Grace Emily Chisholm Young helped to pave the way for women in all fields of
education. She was talented, smart, brave, a mother, wife, and a good role model. After
she received official permission from the German government to pursue her studies, she
became the first woman in Germany to receive a doctorate in any field. She took all the
necessary exams and fulfilled the necessary requirements for the degree. She made
contributions to both pure and applied mathematics and is known for the Denjoy-SaksYoung theorem. She left behind many talented children and grandchildren who also
became incredible mathematicians and scientists.
Grace was born on March 15, 1868 near London in the small town of Haslemere,
Surrey. She is the youngest of four children and her father, Henry William Chisholm was
fifty-nine years old when she was born. Her father was a senior civil servant and was in
charge of the Weights and Measures Department.1 He was a very intelligent man and due
to his early retirement, he was able to help with the education of Grace along with her
mother, Anna Louisa Bell. She was forty-four years old when Grace was born and her
other three children were Hugh, Marion, and Helen. Hugh went to Oxford and went on to
achieve fame as the editor of both the eleventh and twelfth editions of the Encyclopedia
Britannica.2 He was also highly influential as the city editor of The Times, a British
Periodical. All of the children were taught as young children by both of their parents.
Grace was also taught by her governess. When she was only 17 years old, she passed the
Cambridge Senior Examination. This was in 1885 when it was extremely uncommon for
Rothman, Patricia. “Grace Chisholm Young and the Division of Laurels”
Notes and Records of the Royal Society of London. 50 (1996): 89-90.
2
O’Connor, J.J. and E.F. Robertson. Grace Chisholm Young.
February 2005. School of Mathematics and Statistics, U of St. Andrews, Scotland.
25 Mar. 2007.
<HTTP:// www.groups.dcs.st-and.ac.uk/~history/biographies.chisholm_young.html>.
1
2
women to attend a university. Girton College opened in 1869 and was associated with
Cambridge University. It was the first institution in England dedicated to educating
women at the university level. Grace had previously been encouraged to help the poor
people and “make herself useful” but she had a yearning for education and she had a
brilliant mind. Therefore in 1889, she entered Girton College. She was awarded the Sir
Frances Goldschmid scholarship which her father matched in value. At this time the
women could listen to lectures, receive tutoring and take the exams, but they were graded
separately from the men. During her first year at
Girton, Grace attended a lecture given by Arthur
Caley, a well-known mathematician who played
a central role in founding the Modern British
School of Pure Mathematics. He is also the
author of more than 900 published articles
addressing almost every aspect of modern mathematics.3 Grace received her degree in
June, 1892, after taking the Cambridge Tripos Part I examination. The 112 male
graduates were listed in order in the usual categories of Wrangler, Senior Optimes, and
Junior Optimes. The women were on a separate list and Grace graduated between 23 and
24 of the male graduates, which was equivalent to a first class degree. Grace and a friend,
Isabel Maddison received a challenge to take the Oxford examination. Grace scored
higher than all the students that attended Oxford that year as well as she became the first
man or woman to obtain a First in any subject at both Oxford and Cambridge.4
3
Gillespie, Charles C., Ed. Dictionary of Scientific Biography.
New York: Charles Scribner’s Sons. 1976. p. 327
4
Rothman, Patricia. “Grace Chisholm Young and the Division of Laurels”
Notes and Records of the Royal Society of London. 50 (1996): 91.
3
It was impossible at the time for Grace to obtain a Fellowship at Cambridge and
she was unsuccessful with her application for a scholarship at Cornell University so her
options in England were quickly depleted. She decided to attend Gottingen University in
Germany. At this time Gottingen was one of the well-known mathematical institutions in
the world. There was a man there by the name of Felix Klein. He was a well-known
mathematician who is remembered for the Klein bottle, a four-dimensional figure with no
inside surface.5 Klein didn’t let just any woman into the University. He required a
personal interview and solid proof that the woman had mathematical talent before
admitting her into the program. The University was doing a test to see if women could
succeed in mathematics. If they did succeed, this would prove that women could be
admitted to any program because mathematics was thought to be the most challenging.
Up to this point letting women into a post graduate studies program was unheard of.
Many of the male students and professors disapproved the idea of female students. Klein
wanted to be sure that the women he admitted would succeed so as not to give him a bad
name. At first Grace could only attend informally but shortly after her arrival she
received special formal permission to be admitted. There were two American women,
May Winston and Isabel Maltby, who also entered the program.6 Grace was preparing for
her dissertation on the application of Klein’s new theory of groups to spherical
trigonometry. Grace had already learned German with her father so she had no problem
understanding the language there. She completed her dissertation on ‘The Algebraic
5
Grinstein, Louise S. and Paul J. Campbell, Eds. Women of Mathematics.
New York: Greenwood Press. 1987.
6
Rothman, Patricia. “Grace Chisholm Young and the Division of Laurels”
Notes and Records of the Royal Society of London. 50 (1996): 91.
4
Groups of Spherical Trigonometry’ and she graduated with her Ph.D. magna cum laude
at age twenty-seven. 7
While Grace studied at Gottingen
she met an Englishman by the name of
William Henry Young. He was the
oldest child of Henry Young and
Hephzibah Jeal and was born on October
20, 1863. From 1886 to 1892, William
earned money as a private tutor for the
undergraduate students. During this time is when he first met Grace and their friendship
began. This was only the beginning of their professional and romantic relationship they
would have for the rest of their lives. William is known for his discovery of the integral
as well as his work on Fourier series and cluster sets. His most significant contribution
was the development of a calculus approach that has been adopted by nearly all authors
of advanced calculus textbooks since 1910.8
William and Grace decided to get married in 1896, one year after she graduated.
They enjoyed married life together in Cambridge and they were soon joined by a baby
boy. They named their first baby Francis Chisholm Young. Meanwhile she was following
up some work she had started on the solution of an equation whose roots determine the
orbit of a comet. The results of this research were later published and her reputation as a
creative thinker was established.9 They traveled in Italy and did some serious
7
Case, Bettye Anne and Anne M. Leggett, Eds. Complexities.
New Jersey: Princeton UP. 2005.
8
Young, Robyn V. and Zoran Minderovic, Eds. Notable Mathematicians.
Detroit: Gale Publishing. 1998.
9
Rothman, Patricia. “Grace Chisholm Young and the Division of Laurels”
Notes and Records of the Royal Society of London. 50 (1996): 92.
5
mathematical research together under Klein’s guidance and this is where they discovered
the somewhat new field of set theory. Following the birth of their first son Francis, they
had five other children. Their first daughter, Rosalind (Cecily) Tanner was born in 1900.
Only one year later came their second daughter, Janet Michael. A third daughter was born
in 1903 by the name of Helen Marion Canu. The last two children were both boys.
Lawrence Chisholm Young was born in 1904 and the last child, Patrick Chisholm was
born in 1908.10 Grace now had her hands full with six children born in only ten years.
She had five of these children in only seven years. After the birth of their last child they
moved to Switzerland, first to Geneva and then to Lausanne, where they would raise their
children.
Before William and Grace were married, Grace was by far a better researcher of
the two. After they had their children, William began his career in mathematical research.
They filled the typical roles of a mother and father. Grace stayed at home with the
children and William was busy traveling to and from their home in Switzerland to earn a
living. He was a lecturer, a coach and an examiner. Eventually in 1913, William was
appointed the first Hardinge Professor of Mathematics at the University of Calcutta. He
was also a Professor of Philosophy and History of Mathematics at Liverpool University.
He was later a Professor of Pure Mathematics at the University of Wales at
Aberystwyth.11 While Grace raised her children she also studied medicine, music, and
languages. She taught each of her children a different instrument and they would have
informal concerts in their home. She made sure that her children were well educated.
10
Grinstein, Louise S. and Paul J. Campbell, Eds. Women of Mathematics.
New York: Greenwood Press. 1987.
11
Rothman, Patricia. “Grace Chisholm Young and the Division of Laurels”
Notes and Records of the Royal Society of London. 50 (1996): 90.
6
Grace also completed all the requirements for a medical degree except for the internship
so her medical practice was limited to just her family. She knew six languages and she
taught them to her children.
While the children were young, Grace would help William with the writing of his
mathematical papers. She would fill in the proofs and correct the mistakes. She was the
scribe of the two and William did the research. He was very creative and had many ideas
but sometimes had a hard time putting it on paper. Grace was excellent with helping him
put the ideas he had on paper and coming up with conclusions. There are letters that
William wrote to Grace where he admits that it would only be fair to put both of their
names as co-authors of the papers but he knew that if he were to do this the papers would
receive no credit. He told her that he would be the official author to the outside world but
he knew that she was in large part the brains behind the paper. Not only did Grace exceed
the role of a mother, but she was an integral part to her husband with his publications of
his papers.12
Together William and Grace wrote 214 mathematical papers. Of course the papers
were mostly in William’s name. However, 13 of the publications were jointly authored
and 18 have Grace as the sole author.13 The papers were in French, Italian, German, and
English. They also wrote four mathematical books. The papers that were in Grace’s name
alone began after 1914 when William was doing a lot of traveling and her children were
getting to an age where Grace had more time to write. The range of her papers was wide
and varied. She wrote on Pythagorean and Platonic mathematics, including an intriguing
12
Grinstein, Louise S. and Paul J. Campbell, Eds. Women of Mathematics.
New York: Greenwood Press. 1987.
13
Riddle, Larry. Grace Chisholm Young.
4 May 2006. Agnes Scott College. 25 Mar 2007.
<HTTP://www.agnesscott.edu/lriddle/women/young.htm
7
paper on the nuptial numbers of Plato which proves the uniqueness of the solution of the
two simultaneous Diophantine equations. These are as follows,
x2 + y2 = z2
and
x3 + y3 + z3 = t3
In 1915, Grace won the Gamble Prize for Mathematics at Girton College, Cambridge for
her essay ‘On infinite derivates’. This essay discussed the foundations of calculus. In this
paper she discussed continuous functions, differentiation, infinite derivatives, and
derivatives of a real function. She was interested in this subject because of the invention
of the ultramicroscope. She wrote other papers on this topic as well. The most famous
result with which Grace Young is identified is the Denjoy-Saks-Young Theorem that
explains the relations between the derivates of an arbitrary function.
Theorem: Except at a set of measure zero, there are three possible dispositions of the
derivates fo a function f(x), either
(i)
they are all equal and the function is differentiable or,
(ii)
the upper derivates on each side are +  and the lower derivates on each
side are
(iii)
-  , or
the upper derivate on one side is +  , the lower derivate on the other side
is -  and the two remaining extreme derivates are finite and equal.14
Grace and William also found some time to write books for children. They
published a book on paper folding for children. The title of this book is A First Book of
Geometry. This book was published in 1905 in their joint names in the English edition
but only in Grace’s name in the German edition.15 It was very possible that Grace was the
sole author of this book. Grace wrote two children’s books that are nicknamed the
14
Grinstein, Louise S. and Paul J. Campbell, Eds. Women of Mathematics.
New York: Greenwood Press. 1987 p.251
15
Rothman, Patricia. “Grace Chisholm Young and the Division of Laurels”
Notes and Records of the Royal Society of London. 50 (1996): 96.
8
‘Bimbo’ books, which was the pet name for their oldest son Frank. These books cover the
elementary biology of plants and animals, including cell structures seen under a
microscope. The first book is titled Bimbo and was published in 1906, and the second is
Bimbo and the Frogs and was published in 1907. These books were inspired by her
medical studies and her interest in education.16 Bimbo and the Frogs was one of the first
books written for children on the topic of reproduction. The book was tasteful yet
scientific and did not talk down to children.
The most popular and well-known book that
Grace and William wrote together is called The Theory of
Sets of Points. This book was published in 1906. Set
theory was not popular with most mathematicians at that
time. Grace and William used the first book to prepare a
second edition with corrected mistakes and side notes that
were mostly written in Grace’s handwriting. Unfortunately, they were not able to print
the second edition. During the First World War, William and Grace’s oldest son Frank,
who was an RAF pilot, was killed in action. Frank had received an engineering degree at
the young age of 18 before his death one year later.17 Both William and Grace were
deeply saddened. After the death, there wasn’t near as much research from either of the
two. However, in 1929, Grace completed a vast historical novel set in the sixteenth
century called The Crown of England. This book was never published.
O’Connor, J.J. and E.F. Robertson. Grace Chisholm Young.
February 2005. School of Mathematics and Statistics, U of St. Andrews, Scotland.
25 Mar. 2007.
<HTTP:// www.groups.dcs.st-and.ac.uk/~history/biographies.chisholm_young.html>.
16
17
Riddle, Larry. Grace Chisholm Young.
4 May 2006. Agnes Scott College. 25 Mar 2007.
<HTTP://www.agnesscott.edu/lriddle/women/young.htm.>
9
William received the De Morgan Medal of the London Mathematical Society in
1917 and was its President during 1922-24. In 1928 the Royal Society awarded him the
Sylvester medal in recognition of “a life of invincible mathematical activity.” Grace,
however, tried to stay away from any kind of recognition. She says in a letter to her sister,
“I don’t want to be mistaken for the modern ambitious female, ambitious for herself and
her own glorification.”18
With the outbreak of the Second World War, Grace traveled from her home in
Switzerland to England with two of her grandchildren (children of her daughter Janet),
that had been staying with them. After Germany occupied France, Grace was not able to
return to Switzerland. Because of this she would never see her beloved husband again.
William died on July 7, 1942 in Lausanne, Switzerland. Two years later, on March 29,
1944, Grace suffered a heart attack and died in England at the home of her daughter Janet
in Park Road, Croydon.19 She was 76 years old. The Fellows at Girton College
Cambridge had recommended her for an honorary degree but Grace died before it was
given to her.
In 1965, Sussex University Professor David Larman told the following story
regarding Grace and William’s intra-marital rivalry. In his narrative, he explained that
another fellow and William were out swimming one day, and William appeared to be
struggling. The fellow went over to help him and William’s response was, “Are you one
of those people who think my wife is a better mathematician than I am?”20 This may have
Rothman, Patricia. “Grace Chisholm Young and the Division of Laurels”
Notes and Records of the Royal Society of London. 50 (1996): 97.
19
O’Connor, J.J. and E.F. Robertson. Grace Chisholm Young.
February 2005. School of Mathematics and Statistics, U of St. Andrews, Scotland.
25 Mar. 2007.
<HTTP:// www.groups.dcs.st-and.ac.uk/~history/biographies.chisholm_young.html>.
20
Rothman, Patricia. “Grace Chisholm Young and the Division of Laurels”
Notes and Records of the Royal Society of London. 50 (1996): 97.
15
10
been a joke but it also had underlying truth. Many people who truly knew William and
Grace, including their children report that Grace was truly the brilliant mind behind their
works.
The Young’s created a family full of mathematicians and scientists. Cecily
Tanner, the oldest daughter received a doctorate in mathematics at Girton but was still
not able to receive a formal degree from Cambridge because in 1929 such degrees were
not yet being granted to women. Despite this, she went on to become a tenured member
of the University of London and also won the Gamble Prize in 1939. Janet fulfilled
Grace’s dream of becoming a physician. She was the first female member of the Royal
College of Surgeons. Helen Marion completed her undergraduate studies in mathematics
at the University of Lausanne. She did additional graduate work in other subjects at Bryn
Mawr. Laurence, who was also a mathematician, taught at the University of Cape Town,
South Africa, and later at the University of Wisconsin-Madison. He was also a chess
grandmaster. Their youngest child, Patrick, did his academic training in science and
earned his Ph.D in chemistry at Oxford University. However he pursued a career in
public finance and diplomacy. William and Grace have fifteen grandchildren. Of the
grandchildren, Sylvia Wiegand, who is the daughter of Laurence Young, is a
mathematician at the University of Nebraska and a past president of the Association for
Women in Mathematics.21
21
Riddle, Larry. Grace Chisholm Young.
4 May 2006. Agnes Scott College. 25 Mar 2007.
<HTTP://www.agnesscott.edu/lriddle/women/young.htm.>
11
Through her mathematical genius and pioneering spirit, Grace Chisholm Young
was able to inspire generations of women to pursue higher education. Her expansive
work in mathematics shaped the science in ways that influences mathematicians today.
All of this she did while raising a family. It is through her family that she leaves her
legacy, her pioneering spirit, and her drive for greatness in mathematics.
12
Bibliography
Case, Bettye Anne and Anne M. Leggett, Eds. Complexities.
New Jersey: Princeton UP. 2005.
Gillespie, Charles C., Ed. Dictionary of Scientific Biography.
New York: Charles Scribner’s Sons. 1976.
Grinstein, Louise S. and Paul J. Campbell, Eds. Women of Mathematics.
New York: Greenwood Press. 1987.
O’Connor, J.J. and E.F. Robertson. Grace Chisholm Young.
February 2005. School of Mathematics and Statistics, U of St. Andrews, Scotland.
25 Mar. 2007.
<HTTP:// www.groups.dcs.st-and.ac.uk/~history/biographies.chisholm_young.html>.
Riddle, Larry. Grace Chisholm Young.
4 May 2006. Agnes Scott College. 25 Mar 2007.
<HTTP://www.agnesscott.edu/lriddle/women/young.htm.>
Rothman, Patricia. “Grace Chisholm Young and the Division of Laurels”
Notes and Records of the Royal Society of London. 50 (1996): 89-100.
Young, Robyn V. and Zoran Minderovic, Eds. Notable Mathematicians.
Detroit: Gale Publishing. 1998.
Young, W.H. and Grace Chisholm Young. The Theory of Sets of Points.
New York: Chelsea Publishing. 1972.
13