*A Study of the Life and Writings of Colin*Maclaurin

*.*(Mathematical Gazette. October 1915.)

(2) The *"The Geometria Organica " of Colin Maclaurin: A Historical and Critical Survey.* (Proc. R.S.E. Vol. 36. Part I (No. 5). 1916.)

*Elements,*was fascinated by the subject, and in a few days mastered the first six books. He got into touch with Simson and, under his guidance, set himself to the serious study of mathematics. Though Maclaurin, fortunately for the progress of mathematics, diverged into lines of research that were alien to Simson's bent he always maintained a warm respect, for his old professor - in spite of the fact that he in later years, not quite justly I think, gave vent to the expression that Simson was "lazy."

In 1717 he was chosen, after a ten days' competitive examination, Professor of Mathematics, in Marischal College, Aberdeen. Shortly after his appointment he contributed two papers to the *Philosophical Transactions* the substance of which was incorporated in the *Geometria Organica* which appeared in 1720, Newton's Imprimatur being dated Nov. 12, 1719. The somewhat slack conditions that were apt to show themselves in the Scottish Universities of that period are exemplified in the fact that in 1722 Maclaurin acted as tutor to a son of Lord Polwarth during a visit to the Continent; the visit lasted till the end of 1724, when Maclaurin returned to Aberdeen to resume the professorial duties for which he had apparently made no provision during his absence. Not unnaturally the College authorities felt aggrieved at his conduct, but possibly the fact that Maclaurin had while in France been awarded a prize by the Académie Royale des Sciences of Paris for his thesis on the *Percussion of Bodies* (1724) helped to effect a reconciliation. It was a happy circumstance however that he was freed from the difficulties of the position by his appointment in 1725, on the recommendation of Newton, to the Chair of Mathematics in Edinburgh University.

All the accounts that have been handed down of Maclaurin's work in Edinburgh show him to have been a man of lofty ideals and generous outlook, combined with untiring energy and business capacity. The programme of his classes for a session is published in the *Scots Magazine* for August 1741, and it seems proper to reproduce it as an indication of the state of mathematics under his regime. "He gives every year three Colleges; and sometimes a fourth, upon such of the abstruse parts of the Science as are not explained in the former three. In the first he begins with demonstrating the grounds of vulgar and decimal arithmetic; then proceeds to Euclid; and after explaining the first six books, with the plane trigonometry and use of the tables of logarithms, sines, etc., he insists on surveying, fortification and other practical parts and concludes this college with the elements of algebra. He gives geographical lectures, once in a fortnight, to this class of students.

In the second college he repeats the algebra again from its principles and advances further in it; then proceeds to the theory and mensuration of solids, the spherical trigonometry, the doctrine of the sphere, dialling and other practical parts. After this he gives the doctrine of the conic sections, with the theory of gunnery, and concludes this college with the elements of astronomy and optics.

He begins the third college with perspective; then treats more fully of the astronomy and optics. Afterwards he prelects on Sir Isaac Newton's *Principia* and explains the direct and inverse method of fluxions. At a separate hour he begins a college of experimental philosophy, about the middle of December, which continues thrice every week till the beginning of April; and at proper hours of the night describes the constellations and shows the planets by telescopes of various kinds."

Even though the academical year was only from the 1st of November to the end of May the strain implied by this course must have been severe; in one letter to Stirling he speaks of "teaching six hours daily," and in another "I have so much drudgery in teaching that I am commonly so fatigued at night I can do little business." Added to the heavy official duties were the calls which his popularity in the social circles of the city made on his leisure.

He was a man of many interests. The establishment of an astronomical observatory, the proposals for more accurate surveys of the northern coast of Scotland, the subject of Arctic Expeditions, memoirs on the proper gauging of vessels for the use of the Excise, calculations for the basis of a Pension Fund for the widows of ministers and of professors, all claimed and received his attention. His devotion to the general interests of the city was specially manifested during the "forty-five," but his efforts to stimulate the civic authorities to put Edinburgh in a state of defence were unsuccessful, and he had to leave the city. He found a refuge with Dr Herring, Archbishop of York, but his health, never robust, was so much impaired by the hardships he encountered in his flight that he returned to Edinburgh to die. He passed away on the 14th of June 1746 and was buried in the Greyfriars' Churchyard.

In estimating Maclaurin's place in the history of mathematics we should, I think, interpret the word "history" in a wide sense. From one point of view additions to mathematical knowledge by memoirs that embody the results of original research must take the first place; on the other hand research alone is not sufficient to make mathematics a living subject. The stimulus that comes from a capable and enthusiastic teacher is of the highest value in creating a suitable atmosphere as well as in providing the supply of competent workers. The original contributions made by Maclaurin during the twenty years of his Edinburgh professorship are not numerous though they are valuable, but his work as a teacher and administrator had a far-reaching influence on the position of mathematics as an essential element in general culture, and the fact that he took such pains with his classes in preparing suitable courses seems to me to indicate that he was alive to the importance of this aspect of his work.

Maclaurin's *Account of *Newton*'s Philosophical Discoveries,* his *Physical and Literary Essays,* and his work on mathematical physics - some of it of great value - I can do nothing more than mention. From the standpoint of pure mathematics the important contributions are his *Geometria Organica* of 1720, his *Treatise of Fluxions,* published in 1742, and the *Treatise on Algebra,* with the valuable Appendix on the *General Properties of Geometrical Lines* which was issued in 1748 after his death.

The *Geometria Organica* has been subjected to a most careful and exhaustive analysis by Mr Tweedie, and the main result of Mr Tweedie's investigations has been to prove that Maclaurin's treatise has been strangely neglected and that in this work he has anticipated many of the discoveries of a much later date. It is quite impossible in a short note to summarise the contents of the treatise, but it may be pointed out that many of the well-known properties of Circular Cubics are due to Maclaurin, that the whole theory of Pedals, and more particularly of the Pedals of the Conic Section is given in the *Geometria Organica,* and that he discovered "a whole host of new curves never before discussed and which have since have been named and investigated with but scant acknowledgment of their true inventor." Regarding the work as a whole Mr Tweedie thus characterises it. After noting that Maclaurin's use of the Cartesian geometry is, as compared with modern developments, somewhat cumbersome he emphasizes the "consummate skill" with which Maclaurin applies the methods of the ancient geometry. Of the two Parts into which the treatise is divided the first treats the cases in which the loci along which the vertices of constant angles are made to move are straight lines. In the second Part the curves so found in the first Part are added to the loci to obtain curves of higher order. It contains in particular the theory of pedals and the epicycloidal generation of curves by rolling one curve on a congruent curve. The last section contains some general theorems in curves forming the foundation of the theory of Higher Plane Curves. It also contains what is erroneously termed Cramer's Paradox; in fact Cramer quotes Maclaurin as his authority.

The *Treatise of Fluxions* had its origin in the desire of Maclaurin to defend the Newtonian doctrine of fluxions against Berkeley's attack in his essay *The Analyst.* As he proceeded with the work however he was induced, for various reasons, to expand it into a treatise that would contain an account of all the more important applications of fluxions so that it grew into a bulky volume of over 760 pages. As an exposition of the validity of the theory of limits, and of the fallacies in Berkeley's statement, the discussion is thoroughly sound. After an interesting introduction in which he reviews the methods of exhaustion of the Greek geometers and the method of indivisibles of Cavellerius [Cavalieri] - an exposition marked by accuracy and breadth of view - he proceeds in Book I to explain and develop the general theory, making use of the conception of a velocity and keeping algebraic symbolism and calculation as far as possible in the background. The range covered is very wide; many of the theorems, for example, respecting areas can be easily interpreted as theorems in integration: his test for the convergence of a series (pp. 289 et seq.) is a case in point. The disadvantages however of the plan on which the treatise is written make themselves felt when in Book II he comes to deal with the *Computations in the Method of Fluxions.* The proofs of various theorems have either to be repeated from Book I or to be merely sketched, with reference to Book I for complete demonstration. It is interesting to note that he puts the method of infinitesimals on a sound basis, and in fact develops in a rigorous way the theory of differentials; I have no doubt at all that Cauchy's definition of the differential was fully and consciously given by Robins and Maclaurin. Maclaurin also points out (p. 578) that there is no necessity for the introduction of the notion of the generation of quantities by motion, and one has the feeling that much would have been gained by a frank adoption of the Leibnizian and Bernoullian notation for differentials and integrals. The *Treatise,* in spite of the handicaps imposed, partly by the limited scope which it was at first designed to serve, and partly by too rigid adherence to fluxional notations, is a great storehouse of theorems and applications; the investigations in attractions, the Euler-Maclaurin Summation Theorem and the special form of Taylor's theorem that goes by Maclaurin's name are frequently quoted, but these give only a slight indication of the wealth of results and of the quite exceptional rigour (for the day) of the demonstrations of fundamental theorems.

The *Treatise on Algebra* was published after his death and gives clear proof of his aptitude as a teacher. It passed through many editions and, while a good commentary on Newton's *Arithmetica Universalis,* it can hold its own as an excellent introduction to the subject - as that subject was understood by the best mathematicians, of his time.

It would be out of place to pass from Maclaurin without referring to another instance of his many-sided activity. In 1731 a Society had been formed in Edinburgh for the promotion of medical knowledge by collecting and publishing Essays on medical subjects, and had proved its value by the volumes it produced in a few years. In 1739, Maclaurin was instrumental in broadening the scope of the Society by the inclusion of Philosophy and Literature; the Society was remodelled, a set of laws and regulations was drawn up, the number of members increased and the title of "The Philosophical Society of Edinburgh" adopted, Maclaurin being one of the Secretaries. He took an active part in the work of the new Society and contributed papers that were incorporated, according to Murdoch's statement, in the *Treatise of Fluxions* and in his *Account of *Newton*'s Philosophy.* After Maclaurin's death, the Society continued in existence though the spark of life was nearly extinguished when, largely owing to the action of Principal Robertson, it was again remodelled and in 1783 took the form of "The Royal Society of Edinburgh." The important part that The Royal Society has played in the development of scientific knowledge in Scotland is too well known to be insisted upon, but it is pleasant to remember that the foundation of the Society is so, directly associated with the activities of Colin Maclaurin. It may perhaps be added that Stirling's name occurs in the List of Original Members of the remodelled Society of 1739.