**Sketch of the History of Mathematics in Scotland to the end of the 18th Century: Part II**

By Professor G A Gibson.

(Read 7th August 1926. Received 4th January 1927.)

The centre of interest now shifts from St Andrews and Edinburgh to Glasgow. The troubles that afflicted Scotland during the 17th Century bore heavily on Glasgow University and more particularly on the position of Mathematics in the University; but in 1691 a distinct Professorship of Mathematics was founded, and from that date the old system of Regents disappeared from Glasgow so far as Mathematics was concerned. The first occupant of the Chair was George Sinclair, who is now chiefly remembered by the controversy in which James Gregory held up Sinclair's *Treatise Ars nova et magna* to ridicule. It is not fair however to take Gregory's pamphlet as a final estimate of Sinclair's contributions to science; Sinclair laid himself open to attack, but he rendered great service to the mining industry of Scotland and deserves the gratitude of posterity in spite of his many eccentricities. His contributions to mathematics however are of no importance, but during his tenure of the Chair the number of students grew rapidly and the new professorship made a good start.

George Sinclair was succeeded in 1699 by Dr Robert Sinclair of whose work or attainments [in 1704 Sinclair was intrusted by the Faculty with the teaching of Hebrew] I can say nothing. In 1711 Robert Simson was elected, and we must give more attention to him as he left an enduring mark on the development of mathematics in Scotland.

Robert Simson was born at Kirtonhill, West Kilbride, Ayrshire, in 1687. When he entered Glasgow University Trail in his life of Simson states that "at this time, from temporary circumstances, it happened that no Mathematical Lectures were given in the College; but young Simson's inquisitive mind, from some fortunate incident having been directed to Geometry, he soon perceived the study of that science to be congenial to his taste and capacity. This taste, however, from an apprehension that it might obstruct his application to subjects more connected with the study of theology, was anxiously discouraged by his father, though it would seem with little effect. Having procured a copy of Euclid's *Elements,* with the aid only of a few preliminary explanations from some more advanced students, he entered on the study of that oldest and best introduction to Mathematics. In a short time he read and understood the first six with the eleventh and twelfth books; and being delighted with the simplicity of language and accuracy of reasoning in Euclid, notwithstanding the discouragements he met with, he persevered in his Mathematical pursuits; and by his progress in the more difficult branches he laid the foundation of his future eminence." As Trail was a pupil and, later, a close personal friend of Simson, I think we may accept the above statement as a true account of Simson's mathematical education. In any case Simson had acquired a reputation, for in 1710 he was offered the Chair of Mathematics. He asked however to be allowed to spend at least one year in London for the purposes of study; the Faculty granted his request, kept the position open, and in 1711 definitely appointed him Professor of Mathematics. He retired in 1761 and died in 1768. In 1746 he received the degree of M.D. from the University of St Andrews.

Simson's University course extended over two sessions of seven months each, and seems to have preserved the same general plan throughout his professorship. The fullest description of it that I have met with is by Professor Robison who was a student under him. Robison states *(Encycl. Brit.,* 3rd Ed., vol. 17, pp. 504-509): he "made use of Theodosius as an introduction to spherical trigonometry. In the higher geometry he prelected from his own Conics, and he gave a small specimen of the linear problems of the ancients by explaining the properties sometimes of the conchoid, sometimes of the cissoid, with their application to the solution of such problems. In the more advanced class he was accustomed to give Napier's mode of conceiving logarithms, i.e. quantities as generated by motion and Mr Cotes's view of them as the sums of ratiunculae; and to demonstrate Newton's lemmas concerning the limits of ratios and then to give the elements of the fluxionary calculus; and to finish off his course with a select set of propositions in Optics, gnomonics and central forces. His method of teaching was simple and perspicuous, his elocution clear, and his manner easy and impressive. He had the respect, and still more the affection, of his scholars." Both Trail and Robison note his readiness to encourage any students who showed special ability and the trouble he would take in guiding them to authorities. Robison adds that he treated the trigonometrical functions as ratios before Euler had made the method popular.

Simson is known almost solely as an exponent of the Greek geometry, but I think it is worth noticing that when he was still a young man he showed a full command of one branch of analysis. In a letter to Dr Jurin, of date 1st February 1723, he sent proofs of the addition formula for the tangent and deduced the expression (for *a* > 1)

tan

^{-1}(1/a) = 2^{n}tan^{-1}(1/(2^{n}a)) - ∑ 2^{k-1}tan^{-1}{1/(2^{k-1}a[2^{2k}a^{2}+ 3])} (where the sum is overkfrom 1 ton)

from this he deduced several series for π, among them the series known as Machin's Series. Simson did not claim any originality for his work, but Jurin's letter in reply shows that he had given much more general results than were then known in Machin's circle. But analysis of this kind did not really interest Simson, and it is for his devotion to the ancient geometry that he is specially memorable.

Though not the first to be published his best-known work is the translation of Euclid (Books i-vi, xi, xii) from the edition of Commandine [Commandino]. This was published in 1756, separate editions in Latin and in English being issued; a second edition in English, along with a translation of Euclid's *Data,* appeared in 1762. It is perhaps within the mark to say that every English edition of Euclid till near the close of last century (with the exception of Williamson's) was not merely influenced by Simson's work but was in all essentials based on Simson's text and not on the Greek text. For good or for ill it was Simson's conception of Euclid that prevailed in this country till our own days. If we are to form a fair judgment we should consider not merely what has been done since Simson's time for the study of elementary geometry but, quite as much, what was the state of the geometrical textbook before the issue of Simson's *Euclid.* Judging from my own reading I am of opinion that the welcome given to Simson's text was to a large extent due to the very unsatisfactory character of the texts then current in this country - in England as well as in Scotland. It is a very striking fact not merely that Euclid became predominant but that one particular edition ousted all rivals for so long. Simson had no really good original to work upon, and he was in no real sense of the word a competent textual critic; but he had a clear conception of the general trend of Euclid's development, and he stuck to that with almost fanatical tenacity. He was simply steeped in the ancient geometry and one should be very sure of one's ground before questioning any deliberate judgment of Simson's on the facts of any Greek textbook.

The only other textbook published by Simson was his *Conic Sections. Sectionum Conicarum Libri Quinque* (Edinburgh, 1735); a second, improved and enlarged, edition appeared in 1750. The book was designed to stem the tide that had begun to set in in favour of Analytical or Algebraic Geometry (on the lines of de I'Hôpital's well-known work); though it was based on Apollonius the cone was not used in defining the conic. For the parabola the focus and directrix property, for the ellipse and the hyperbola the constancy of the sum and the difference of the focal radii to a point are used to define the curves. For many years English translations of the first three books were in use in many of the Scotch schools but it is now seldom met with. Of course all the proofs are strictly Euclidean and very little is taken for granted; no important property is shoved into a Corollary. But the book is quite a good exposition and worthy of its author; it is the first textbook which contains the theorems of Desargues and Pascal.

To the study of Greek geometry Simson may almost be said to have dedicated his life, and he found ample scope for his ingenuity in his effort to recover some of the more important of the treatises of Euclid and Apollonius that had been lost but whose contents had been to a certain extent described by Pappus. The text of Pappus' *Collection* was itself in a very unsatisfactory state so that the opportunities for conjecture were endless. Though I have at various intervals spent a considerable time in the study of these lost treatises I cannot profess to have mastered the subject, or indeed to have reached any decided conclusion on the most debateable points, so that I must content myself with little more than a list of titles.

In 1749 he published his Restitution of the *Loci Plani* of Apollonius. In this case he had predecessors in Fermat and Schooten, but he added considerably to what they had given. Trail was neither unprejudiced nor very critical, but he had a competent knowledge of Greek geometry and was thoroughly familiar with Simson's work, and I quote his estimate of the edition:- "Such is the elegance of method and the ingenious contrivance of demonstration in this work that he has truly exhibited a copy, or at least very nearly a copy, of the work of Apollonius, that little regret need be had for the loss of the original." We need not indorse this eulogium in its entirety but it is not altogether wide of the mark.

Simson's *Opera Reliqua,* published in 1776 from his MSS under the editorship of Professor Clow (Professor of Logic in Glasgow University), contains besides two short tracts on *Logarithms* and *Limits* the restoration of Apollonius's *Determinate Section* and Euclid's *Porisms.* In recent times the geometry developed in the *Determinate Section* has often been represented as in many respects an equivalent of the modern theory of Involution, though it is not at all from that standpoint that Simson considered it. There are however many propositions that can be readily adapted to the geometry of involution.

The mystery of Euclid's *Porisms* however seems to have fascinated Simson, and from the very early days of his professorship he seemed to brood over it. Till he produced his article in the *Philosophical Transactions* in 1723 there had been no elucidation of the mystery that had baffled every inquirer, and even then there was only an approach to a solution, not the solution itself. Simson worked at the subject to the day of his death and was very unwilling to publish the MS which he had completed, though he left it in such a state that it could be sent to the press. Whether he has succeeded in solving the mystery of the porisms completely is still a moot point. Chasles, the next in importance of those who have thoroughly investigated the subject, agrees in the main with Simson's conceptions, but develops certain views of his own. Heiberg is less enthusiastic, though I do not attach quite the same weight to Heiberg's views in this connection as in other fields in which he has rendered such great service to Greek geometry. I cannot here enter at all into the matter; the literature of the subject is considerable, and I may refer to Heath's *Greek Mathematics,* Vol. I, pp. 431-438, for a short description of the more important conceptions of a porism and for references to the literature.

Simson may justly be described as a "great geometer"; but while each man must be allowed to follow his bent I agree with one of his distinguished pupils that it is a matter of regret that he devoted himself so exclusively to the restoration of the lost books of the ancients and took such a slight interest in the development of the new analysis. He had a competent knowledge of fluxions but he never really set himself to master algebraical analysis, and he held views that were completely antiquated on the nature and possibilities of algebra and of algebraic geometry. It is much to be regretted that he did not apply his profound knowledge of the Aristotelean logic to the somewhat crude reasonings of the founders of the modern algebra; had he devoted to this branch of mathematics a tithe of the labour he expended on the restoration of mutilated texts he would probably have had a more beneficial influence on the development of mathematics in Scotland.

JOC/EFR April 2007

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