# From Servant to Queen: A Journey through Victorian Mathematics

The title of this book is an echo of a very similar title used by E.T. Bell for his book Mathematics: Queen and Servant of Science (MAA, 1997). Bell explains the unreasonable effectiveness of mathematics in the physical sciences. The meaning of this title is that mathematics, has the historic role of being a tool, a servant, for other more applied sciences, but at some point, what is usually called pure mathematics, evolved into a separate discipline, where mathematics is studied for its own right, a science that is the queen of intellect and a source of beauty. And yet, as we have experienced on several occasions already, what was originally assumed to be pure game of abstraction, devoid of any practical use, turned out later to have very practical applications.

In this book, Heard describes how this transition happened in England during the Victorian period (approximately 1830-1900). He sketches how mathematics, traditionally in its servant role for the other sciences, like physics, astronomy, economics, and even the invention of practical mechanical machinery, gradually appeared in its queen role of pure mathematics.

The book starts with sketching the situation in England in the 18th and first half of the 19th century. Since calculus was introduced by Newton and Leibniz, mathematics started to take off in a different direction. A mathematician used to be someone who applied mathematics to practical use, without being a "professional mathematician" in the modern sense of the word. It could be anyone occasionally using some computational technique, and the plural in mathematics may have referred to all these applications. But gradually mathematical knowledge became the ruler of all other sciences, just like queen Victoria was ruling the British Empire.

Britain was still under the spell of Newton, and the controversy with Leibniz had started an aversion for the continental approach to mathematics. There were two main scientific centres in Britain. Oxford which was considered to be the university of choice if you wanted to specialize in the classics. Theology and classics had been the dominant studies for many centuries. And the alternative was Cambridge with its system of of tripos that generated the wrangles, a prestigious honours degree in mathematics, that opened many doors to public positions. So this was the place to be if you were interested in mathematics. Perhaps because of Newton's spirit still dwelling in the premises there, it had more prestige in the eye of some beholders.

Pure mathematics and the mathematical profession was definitely much more accepted on the continent. There was much more exchange of ideas and results were published in professional mathematical journals. England had a tradition of publishing popular science magazines with puzzle sections, but no proper mathematical journals. The more practical notation in the Leibniz approach to calculus, which was more popular on the continent, may have given an advantage. So British mathematics lingered behind, and a lack of communication made that they had difficulties understanding the more advanced continental mathematics. In Britain, only from around 1830, a similar movement of pure mathematicians started to emerge. It became gradually accepted that the square root of a negative number could be studied for its mathematical properties, without the necessity of it representing some physical quantity. Although it was still generally belief that even pure mathematics was developed to the benefit and the advantage of science and technology.

Starting the London Mathematical Society has been a strong driving force in this evolution. Founded in 1964 at the University College London, it officially started a year later with August De Morgan as its first president. At first it was just a local community, but from the beginning it was keeping up a high standard for its members, as well as for the publications in its Proceedings and for the winners of the De Morgan medal that they awarded. The number or members was relatively low, although slowly growing, but its high standard eventually attracted many foreign members. In fact, here the British took a leading role, because the LMS became an example for other societies abroad (SMF, DMV, AMS....).

In the remaining chapters, Heard explains what it actually meant to be a "professional mathematician" for Victorians. In the chapter with the title "The pure mathematician as hero", he introduces the biographies of several British mathematicians who produced some "pure mathematics" and that usually were somehow linked to the LMS. James Whitbread Lee Glaisher who worked on number theory and who was editor of the Messenger of Mathematics (now Quarterly Journal of Mathematics); Henry J.S. Smith (known for example for the Smith normal form of a matrix, and the first to introduce the Cantor set); Percy MacMahon (combinatorics); and others. The British did not have the tradition of seminars as that was usual on the continent. Perhaps for this reason, several of the missionaries of pure mathematics were not the "heroes", the charismatic leaders, that had a school of followers like their continental counterparts had.

Then there is a chapter about the mathematics that was required to solve the problem of light and electromagnetism. Unlike Newton's corpuscular approach to optics, light (and later other electromagnetic quantities) seemed to be propagating like waves. But waves had to propagate in some medium that was termed aether. So there was much ado about the mathematics of the aether. This is a somewhat strange subject to be discussed extensively in the context of this book, but it definitely was a hot topic in those days, and it illustrates that in applied mathematics, Britain was not at all a backwater area. It shows that "pure mathematics" can also be produced by engineers, and non-professional mathematicians. Here we meet big names like Faraday, Stokes, Maxwell, Heaviside, W. Thomson (lord Kelvin), but also mathematicians got involved like Airy, and Clifford. The latter is known from the Clifford algebra, but he is also the one who linked gravity to non-Euclidean geometry which later inspired Einstein for his relativity theory.

The last two chapters are more of a social nature. With G.H. Hardy's A mathematician's apology in mind, (Hardy is the standard example of a thoroughbred pure mathematician abhorring any practical application) there is a discussion about mathematics as a profession, and what it meant to be a professional mathematician. It was quite different from a scientist. A typical British scientist was supposed to be a gentleman, free and independent, whose conclusions were not questionable. If you had a profession, then that was supposed to be at public service (which was not exactly what pure mathematics was pursuing), and you had an organisation that defended your rights (which was not the role of the LMS). So the idea came up that also a pure mathematician can be creative and explore unknown domains, which was also to the eventual benefit of society. This transformed a pure mathematician into an artist striving for beauty. Heard gives an account of the geometer George Salmon who in Nature praises the work of Cayley as an artist which was quite opposite the ideas about aesthetics of Walter Pater (an Oxford essayist).

The book includes many short (in line) and long (displayed) quotes and a few illustrations. There is also a long list of extra literature at the end of the book and many references at the end of each chapter to refer to the source of the quotes in the text. However if you are not an historian you can safely ignore these and keep on reading, because the story is told in a smooth and entertaining way. It is really interesting to read how long the Leibniz-Newton dispute had serious consequences, how ideas changed in about seventy years, and the important role that was played by the LMS in this process. It also illustrates that Britain was (and it still is) an island and that British were in many ways different from the rest of the world. They are probably less so today than they were in the Victorian period. Globalisation has made national identities more fuzzy worldwide, but it will take many more generations before this will be erased completely.

**Submitted by Adhemar Bultheel |

**1 / Jul / 2019