You are here

Mathematical Treasure: A Letter of James Joseph Sylvester to Leopold Kronecker

Sid Kolpas, Delaware County Community College

James Joseph Sylvester (1814-1897)
(Source: MacTutor History of Mathematics Archive)

James Joseph Sylvester, a short biography

James Joseph Sylvester was born James Joseph on September 3, 1814 in London; he died March 15, 1897. James assumed the last name Sylvester when his older brother chose that name upon immigrating to the United States; at the time, immigration required a first name, middle name, and last name. When he was 14, Sylvester became a student of Augustus De Morgan (1806-1871) at the University of London. He withdrew from the University after being accused of stabbing a student. 

Sylvester formally began his study of mathematics a few years later, in 1831, at St. John’s College, Cambridge. At St. John’s he ranked second in Cambridge's rigorous mathematical examination, the Tripos. Despite his stellar performance, he was not awarded a degree because, as a Jew, he could not see his way clear to the mandatory swearing to the Thirty-Nine Articles of the Church of England. Moreover, because of his refusal on religious grounds, he was unable to compete for a Fellowship at Cambridge. In 1838 he became Professor of Natural Philosophy at University College London. In 1841, he was awarded a BA and an MA by Trinity College, Dublin. That same year he moved to the United States where he briefly served as a professor at the University of Virginia, returning to England in November 1843.

On his return to England he pursued law, studying at the Inns of Court in London at the same time as fellow mathematician Arthur Cayley (1821-1895). The two became friends and together they made major contributions to linear algebra, particularly in matrix theory. It is interesting to note that Sylvester’s private students, taken on to supplement his income, included Florence Nightingale (1820-1910), one of the founders of modern nursing. In 1855, he was appointed Professor of Mathematics at the Royal Military Academy, Woolwich; he left in 1870 because he had reached the required retirement age of 55.

In 1872, Sylvester finally received his B.A. and M.A. from Cambridge. In 1876 he returned to the United States to become the first Professor of Mathematics at the newly formed Johns Hopkins University in Baltimore, Maryland; the university wanted him to start a graduate program in mathematics. While at Johns Hopkins he founded the American Journal of Mathematics. The website of the Johns Hopkins University Mathematics Department has an image of the letter featured in this article; see

In 1880, the Royal Society of London presented the Copley Medal, its highest award for scientific achievement, to Sylvester. In 1883, Sylvester returned to England to assume the Savilian Professorship of Mathematics at Oxford University, which he held until 1894. As Savilian Professor of Geometry beginning in 1884, Sylvester was required to deliver a public lecture, and he initially chose the topic “Descartes as Geometer.” He was anxious about delivering the lecture and thus solicited suggestions from his mathematical colleagues; in the letter featured in this article Sylvester asked Leopold Kronecker (1823-1891) for help with the public lecture.

Sylvester invented a great number of mathematical terms such as “matrix” and “discriminant”. His collected scientific work fills four volumes; The Collected Mathematical Papers of James Joseph Sylvester are available on Google Books.

For more information:

J. J. O'Connor and E. F. Robertson, "James Joseph Sylvester," MacTutor History of Mathematics Archive, 2005

Karen Hunger Parshall, James Joseph Sylvester: Jewish Mathematician in a Victorian World, Johns Hopkins University Press, 2006

Letter from Sylvester to Kronecker (1884)

The three-page letter reproduced here is in the collection of the author. You may use these images in your classroom; all other uses require the permission of the author.

Transcription of Letter from Sylvester to Kronecker

Keswick Cumberland 4th Septr 1884

Dear Professor Kronecker – 1

Having chosen “Descartes as a Geometer” as the subject of a Public Lecture which I am compelled by the Statutes to deliver at Oxford next term2 if you could suggest any moderately sized book or books or article in some mathematical notice or review which would be likely to be useful in enabling me to say anything worth hearing on the subject (with which I have no special acquaintance) you would be rendering me a great service. I have Baillet, Millet and Liard3 (the last I think of no value except to contradict).

Anything in the way of a connected history of Analytical Geometry from the time of Descartes down to the present day would I think be likely to be very useful. Does anything of the kind exist in German or any other language that you could bring to my knowledge? Until this business is accomplished I am obliged to suspend all mathematical work.

I have 3 articles to appear in the Comptes Rendus sent in some time ago on the Solution of the Linear Equation in Matrices or Quaternions.4

Yours very truly

J. J. Sylvester

Please present my kind regards to Madame Kronecker and your daughter.5

1. Leopold Kronecker (1823-1891) was a German mathematician who worked on number theory and algebra. Like Sylvester, he was Jewish.

2. In his assumption of the Savilian Chair of Mathematics at Oxford, Sylvester was required to give a public inaugural lecture. He seemed very nervous about giving the lecture, and began to solicit help from his friends. Because the philosophical works of René Descartes were then popular at Oxford, he chose “Descartes as a Geometer” as his topic. See Karen Hunger Parshall's James Joseph Sylvester: Life and Work in Letters (Oxford University Press, 1998), pp. 292-293, for details. The position of Savilian Professor of Geometry was established in 1619 by Sir Henry Savile, a mathematician and classicist. Henry Briggs (of common logarithm fame) was the first Savilian Professor.

3. Adrien Baillet wrote a biography of Descartes, published in Paris as La Vie de Monsieur Des-Cartes in 1691 and in London as The Life of Monsieur Des Cartes in 1693: see "Adrien Baillet" in Catholic Encyclopedia and WorldCat. J. Millet wrote a biography of Descartes, published in Paris as Histoire De Descartes Avant 1637 in 1867. Louis Liard wrote a biography of Descartes, published in Paris as La méthode et la mathématique universelle de Descartes in 1882.

4. The publication Comptes rendus de l'Académie des Sciences (Proceedings of the French Academy of Sciences) was founded in 1666. Currently it is split into seven sections, published on behalf of the Academy by Elsevier: Mathématique, Mécanique, Physique, Geoscience, Palevol, Chimie, and Biologies. In 1884, Sylvester published the following articles in Comptes rendus:

  • Sur la resolution generale de l’equation lineaire en matrices d’un ordre quelconque.
  • Sur les deux methods, celle de Hamilton et celle de l’auteur, pour resoudre l’equation lineaire en quaternions.
  • Sur l’achevement de la nouvelle method pour resoudre l’equation lineaire la plus generale en quaternions.
  • Sur l’equation lineaire trinome en matrices d’un orde quelconque.

Sylvester made major contributions to linear algebra, and coined the term "matrices." Quaternions were first introduced by William Rowan Hamilton in 1843.

5. Kronecker's wife and daughter were, respectively, Fanny Kronecker and Elisabet Kronecker.

Sid Kolpas, Delaware County Community College, "Mathematical Treasure: A Letter of James Joseph Sylvester to Leopold Kronecker," Convergence (October 2015)