**Nicolas Vilant**was Regius Professor of Mathematics at St Andrews University in Scotland during 1765-1807. Plagued by ill-health, he was unable to teach for much of this time, and employed a series of assistants. Well-versed in the British analytical tradition, he was, like his contemporaries, largely unaware of developments in the rest of Europe. However, he was a mathematician of some skill, and his textbook

*The Elements of Mathematical Analysis, abridged*,

*for the use of students*is of considerable interest. Though unable to complete for publication a more comprehensive work, many manuscripts survive in St Andrews University Library. The only known likeness of Nicolas Vilant is a small drawing by his colleague Professor John Cook, who often avoided the difficulty of faces by drawing rear views - perhaps appropriately in Vilant's case.

The Vilant family, originally from France, had long associations with St Andrews University and with the surrounding neighbourhood, where several members held academic and church positions. Nicolas Vilant's parents were William Vilant, the minister at Ferryport-on-Tay (now Tayport) and his second wife Jean Wilson. The date we have given above as his date of birth, is actually the date on which he was baptised. He first matriculated at St Andrews in 1752, studying mathematics under David (II) Gregory.** **He graduated M.A. in 1756 and worked for a time as a mathematical teacher at Watts' Academy in London. He returned to St Andrews to succeed David Gregory in 1765, the same year that John Playfair graduated M.A. He held the mathematics chair until his death, but for much of the time he was unable to teach owing to severe rheumatic pains, probably rheumatoid arthritis, that set in when he was in his mid-thirties. As a private arrangement, he employed a series of assistants to teach in his stead; but he gave occasional lectures, supervised the work of his assistants and took an interest in the best students. Among his assistants were the former St Andrews students James Glenie, John West, James Brown, Thomas Duncan and Thomas Chalmers; and other students of this time include James Ivory, John Leslie and Adam Anderson. Though Vilant's colleagues complained of his inability to perform his duties, the historian David Masson [Masson, 1911; 45] later drew attention to "a tradition of unusual mathematical excellence and ardour" dating from the time that "the nominal incumbent of the mathematical chair, Professor Vilant, finding himself disqualified by ill-health, had committed the duties ... to well-chosen assistants." Indeed, it is reasonable to claim that a considerable portion of the mathematical and physical activity in late eighteenth-century and early nineteenth-century Britain emanated, at least indirectly, from St Andrews.

Vilant's *The Elements of Mathematical Analysis, abridged. For the Use of Students* is perhaps the first book in English to use the phrase "Mathematical Analysis" in its title. A large part was first printed in 1777 and then used as a textbook from 1783. The Preface begins:-

But most extant copies of the book were published in 1798 with additional Notes and a separately-paginated 28-pageIn this small Treatise on Mathematical Analysis, the most general Propositions only are given; and in Notes at the bottom of the pages, are pointed out the principal authors to be consulted by students. The whole up to page129, was printed in1777; but owing to the Author's bad health, etc it was not until1783, that the same with the addition of the quarter sheet from page129to page133was used here as a Text Book.

*Synopsis of Book V. of Euclid's*

*Elements*(printed in St Andrews in 1797). Though much briefer than other contemporary texts, it nevertheless covers a great deal of ground; but it often lacks clarity and is written in an already old-fashioned style. Those parts for which some originality may be claimed are: (

*a*) a method for finding the cube root of binomials of form

*R*± √

*S*, where

*S*may be positive or negative, and (

*b*) a method for finding rational and whole-number solutions of indeterminate problems involving linear, quadratic and cubic equations.

Vilant states in his preface that he "proposes hereafter to publish a complete System of the Elements of Mathematical Analysis demonstrated", but this would have to wait until "his health will allow him to arrange properly what he hath prepared on this subject." His health did not allow this, but thirteen volumes of manuscripts on this theme, and some other subjects, are preserved in St Andrews University Library.

Vilant was an able synthesiser and summariser; perhaps the *last* analyst steeped in the British tradition that followed Isaac Newton, but largely ignorant of important developments elsewhere in Europe.

Around 1770 he married Elizabeth Brand; two of their sons later matriculated at the University of St Andrews.

The picture of Vilant is from *Cook Drawings* [c.1797]. Rear view of Nicolas Vilant, drawn by Professor John Cook. Cook-25-3, St Andrews University Library Special Collections.

**Article by:** Alex D D Craik, University of St Andrews.