Turnbull lectures on Colin Maclaurin

To celebrate the bi-centenary of the death of Colin Maclaurin, Herbert Westren Turnbull, Regius Professor of Mathematics in the University of St Andrews, gave a lecture in Aberdeen on 4 February 1947. Below is the first part of his lecture on Maclaurin's life.

For Turnbull's description and assessment of Maclaurin's mathematics see: Turnbull on Maclaurin, Part 2

For Maclaurin's publications see: Maclaurin's publications


Mathematician and Philosopher

COLIN MACLAURIN [his name is Cailean MacLabhruinn, that is, Colin the son of Laurence, in the Gaelic] was born at Kilmodan, Argyll, in the month of February, 1698. His grandfather, Daniel Maclaurin, who held ancestral estates on the island of Tyree, a western. outpost of the county, had moved to Inverary where he helped to restore the town after the civil wars. His son, John, who inherited his literary ability, became minister of the parish of Kilmodan in Glen Daruel. He was a faithful pastor, public spirited and able at his work, and was employed by the provincial synod in completing a version of the Psalms in Gaelic which came into regular use. He married a gentlewoman of the family of Cameron, and, of their three sons, John, the eldest, became a minister in the city of Glasgow, Daniel died young, after having signs of extraordinary genius, and Colin, who is the subject of this lecture, and whose bi-centenary we have just commemorated, became the most distinguished disciple of Isaac Newton. Colin was six weeks old when his father died, and the family came under the care of an uncle, Daniel Maclaurin, minister of Kilfinan, Argyll. When Colin was nine years old his mother died, in 1707, and the care of the children thereafter devolved entirely on their uncle.

At the age of eleven Colin was sent to the University of Glasgow where his brother John was already a student. It was in his first year, when he was in the rooms of a friend, that a copy of Euclid came into his hands, which attracted and held his mathematical interest. In this he resembled Pascal, the great French geometer, for both lads in their teens not only rapidly mastered Euclid but proceeded at once to make essential and novel discoveries in geometry. Many properties which Maclaurin treated in his first book, the Geometria Organica, were invented at college. He graduated Master of Arts in his fifteenth year when he also wrote and publicly defended a thesis On the power of gravity, a method of examination still obtaining in continental universities. After spending a year reading divinity he left Glasgow to live with his uncle in their Highland home, continuing his researches in mathematics and philosophy, reading the classic authors, and growing in awareness of the natural beauty around. Like many a student to-day he enjoyed climbing the hills, as he actively sought out the scientific secrets of their stones and plants, and gave himself to the joy of wandering in high places. Unfinished scraps in his notebooks reveal the sensitivity of his nature, as he would sometimes break out into a hymn or poetic rhapsody on the beauties of the scene, and the perfections of its Author. Such was the background of the startlingly mature work upon the organic description of plane curves which he completed in 1719, at the age of twenty-one.

We may pause for a moment to consider what a boon this interval of unhampered leisure would bring to the lad upon the threshold of manhood: the simple life in the manse of Kilfinan upon the open easterly shore of Loch Fyne, but a few miles over the hill from Glendaruel, the home of his childhood: the opportunity for his thoughts upon geometry to ripen, the fruit of the teaching that he received from Robert Simson, his Professor at college. It should be said of this Professor that he exerted an influence in the teaching of geometry that has reached to the beginning of the twentieth century. Simson had been inspired by Halley, the astronomer, to study the geometry of the Greeks at first hand, a task which he readily undertook; and so fully did he enter into the spirit of their work that he reached an explanation of what Euclid and Pappus meant by certain obscure allusions to their method of porisms. For the books in which Euclid had developed his method had long been lost.

But the greatest of the formative influences upon the life of young Maclaurin was that of Isaac Newton. It was Halley who had persuaded Newton to write the Principia: and though the masterpiece had been before the world for thirty years, it was still a closed book to all but a handful of the acutest scholars. David Gregory was one, and perhaps the first to study it with care and understanding: David, the nephew of James Gregory the astronomer and mathematician, the greatest alumnus of your ancient University.

Roger Cotes was the second, a young English scholar of whom Newton once wrote Had Cotes lived we might have known something. He died in the year 1716 at the age of thirty-four and at the height of his powers: a brilliant mathematician and astronomer who was already acclaimed as a worthy successor to Newton at Trinity College, Cambridge. We shall see later how one single golden thread of his discoveries drew Maclaurin to fresh geometrical adventures. Maclaurin was the third and greatest of these disciples. How far he had developed in a grasp of the Newtonian philosophy, during the years of quiet preparation for his life work, we do not know: but we may suppose that through the influence at college of Robert Simson he had become acquainted with both the geometry and the natural philosophy of Newton, thereby receiving a grounding in these fields of mathematics that paved the way for his peculiar gifts in later years.

At the age of nineteen Maclaurin was appointed to the Chair of Mathematics in Marischal College, Aberdeen, obtaining it after a ten days' trial against a very able competitor, Walter Bowman. According to an old report both men:

were capable to teach Mathematics anywhere. In Euclid Bowman was much readier and distincter, but in the last trial M'Laurine plainly appeared better acquainted with the speculative and higher parts of the Mathematics.

In the vacation of 1719 he visited London where he left his Geometria Organica to be printed and where he met Sir Isaac Newton. During this visit he was admitted Fellow of the Royal Society at the remarkably early age of twenty-one. Two years later, and during a second visit to London, he met Martin Folkes, who became his intimate friend. Folkes (1690-1754), who eventually succeeded Newton as a President of the Royal Society, was an antiquary as well as a man of science - a choice youth of penetrating genius and master of the beauties of the best Roman and Greek writers. He is described as upright, modest and affable, was eight years senior to Maclaurin and outlived him eight years. The two friends had much in common, and we are indebted to Folkes for preserving to posterity some of the mature writings of Maclaurin which were left unpublished when he died.

In 1722, Lord Polwarth, the plenipotentiary of our King at the Congress of Cambrai, engaged Maclaurin as a tutor for his son. He accepted the post, which involved travelling abroad during the long vacation: but with a fine disregard for his duties in Aberdeen, he actually extended his absence to three years. After a short stay in Paris, which would have brought him into contact with leading men of science, the tutor and pupil settled in Lorraine. It was there that Maclaurin wrote a thesis On the Percussion of Bodies which gained for him one of the three prizes of the Academy of Sciences at Paris for the year 1724, other recipients being Leonard Euler and Daniel Bernoulli. The substance of this work was eventually incorporated by Maclaurin in his two-volume treatise on Fluxions. In Lorraine, we are told:

besides the advantage of a good academy, they had that of the conversation of one of the most polite courts of Europe. Here Mr Maclaurin gained the esteem of the most distinguished persons of both sexes, and at the same time quickly improved that easy genteel behaviour which was natural to him, both from the temper of his mind, and from the advantages of a graceful person.

The tour proceeded with much intellectual profit to Maclaurin; but one wonders what was happening to his unshepherded classes in Marischal College. Had he forgotten all about them; did he turn a deaf ear to all calls to return; was there something in him, akin to the impenetrable aloofness of Newton, which shut him off from his fellows and his duties at times of mental creativity? But a shock came which brought the travels to an end when his pupil was suddenly taken ill with a fever and died at Montpelier in southern France.

Maclaurin returned to Aberdeen, and in April 1725 appeared before the council, to whom he expressed his regret at his long absence without leave. He was "reponed" for a time, but in the following January his Chair was declared vacant; whereupon he sent in his resignation. He had in fact during the previous November removed to the University of Edinburgh to deputise for James Gregory who was an old man and no longer capable of teaching. This Professor Gregory was the younger brother of David, the friend of Newton, and was the nephew of his more famous namesake James. Long ago Newton had recommended David to the Chair of Mathematics at Edinburgh: now he did the same for Maclaurin. He showed his interest in the appointment by writing both a letter of encouragement to Maclaurin:

not only because you are my friend, but principally because of your abilities, you being acquainted as well with the new improvements of mathematics as with the former state of those sciences;

and also a letter to the Lord Provost of Edinburgh, in which he proposed Maclaurin as the eventual successor of Gregory and was:

ready (if you please to give me leave) to contribute twenty pounds per annum towards a provision for him, till Mr Gregory's place becomes void; if I live so long, and I will pay it to his order in London.

Two years later Newton died. The friendship which had grown between the ageing natural philosopher and his young disciple bore fruit in after years when Maclaurin wrote his work on Fluxions and also his account of Newton's philosophical discoveries.

For twenty years Maclaurin occupied the Chair at Edinburgh with distinction. His classes were well attended, and from November till June for four or five hours a day he was occupied with teaching. The subjects ranged from Euclid and elementary algebra to conics, fluxions, probability and Newton's Principia. Maclaurin attained a position of wide influence, made many friends, and used his opportunities for public and scientific service. Many things which we are apt to take for granted owe their origin to his imagination and initiative. He laid sound actuarial foundations for the insurance society that has ever since helped the widows and children of Scottish ministers and professors.

In 1739 a perusal of the volumes published by the Medical Society of Edinburgh eight years previously prompted Maclaurin to extend its usefulness to include physics and antiquities. It was this enlarged society that eventually became the Royal Society of Edinburgh. Though he did not live to see this, one of the first to contribute to the newly formed society was his son John (1734-1796), Lord Dreghorn, a Scottish judge and advocate, a man of considerable literary ability and satirical wit, who published in the first volume of the Transactions an ingenious proof that the Greeks never took Troy. As an ancient record tells us:

Mr Maclaurin had lived a bachelor to the year 1733; but being formed for society as well as for contemplation, and desirous of mixing more delicate and interesting delights with those of philosophy, he married Anne, daughter of Mr Walter Stewart, Solicitor General to His late Majesty for Scotland; by whom he had seven children, of which, two sons John and Colin, and three daughters, have survived him.

Colin Maclaurin had great skill also in experimental physics: he proposed an astronomical observatory for Scotland and offered, as a contribution towards the cost, the fees which he received from his lectures upon practical physics. He also helped a friend by decidedly improving the existing maps of Orkney and Shetland. He was a firm believer in the existence of a north polar passage - he objected to the name north-west passage - and expressed his readiness to undertake the voyage should an opportunity occur.

In 1745, when Prince Charles Stuart raised a Highland army to dispute the Hanoverian succession and marched on Edinburgh, Maclaurin took the leading part in rallying the townsfolk and organising the defences of the city against the rebel troops. Night and day he spent in planning the hastily raised fortifications and in the oversight thereof, a task of devotion which damaged his health. When the city was taken he escaped and withdrew to England, travelling to York where he became the guest of Thomas Herring, the Archbishop who had been active on the Hanoverian side. Here, wrote Maclaurin in a letter, I live as happily as a man can do who is ignorant of the state of his family, and sees the ruin of his country.

He returned the next year to Edinburgh where he resumed his duties, hoping to complete the considerable volume that he had undertaken on the Newtonian philosophy. But the rigours of the journey, including a fall from his horse, further broke his health; and in 1746 he died of a dropsy at the early age of forty-eight.

Only a few hours before his death he was engaged in dictating to an amanuensis a concluding passage which summed up his philosophy; the argument in favour of a future life contained in the last sentences of this unfinished chapter is now well known: it proceeded from the lips of a dying man.

As man is undoubtedly the chief being upon this globe, and this globe may be no less considerable, in the most valuable respects, than any other of the solar system, and this system, for aught we know, not inferior to any in the universal system; so, if we should suppose man to perish, without ever arriving at a more complete knowledge of nature, than the very imperfect one he attains in his present state; by analogy or parity of reason, we might conclude, that the like desires would be frustrated in the inhabitants of all the other planets and systems; and that the beautiful scheme of nature would never be unfolded, but in an exceedingly imperfect manner, to any of them.

This, therefore, naturally leads us to consider our present state as only the dawn or beginning of our existence, and as a state of preparation or probation for further advancement, which appears to have been the opinion of the most judicious philosophers of old. And whoever attentively considers the constitution of human nature, particularly the desires and passions of men, which appear greatly superior to their present objects, will easily be persuaded that man was designed for higher views than of this life. These the Author of nature may have in reserve to be opened up for us, at proper periods of tine, and after due preparation.

Surely it is in His power to grant us a far greater improvement of the faculties we already possess, or even to endow us with new faculties, of which, at this time, we have no idea, for penetrating farther into the scheme of nature, and approaching nearer to Himself, the first and supreme cause. We know not how far it is proper or necessary that we should not be let into knowledge at once, but should advance gradually, that, by comparing new objects, or new discoveries, with what was known to us before, our improvements might be more complete and regular; or how far it may be necessary or advantageous, that intelligent beings should pass through a kind of infancy of knowledge.

For new knowledge does not consist so much in our having access to a new object, as in comparing it with others already known, observing its relations to them, or discerning what it has in common with them, and wherein their disparity consists. Thus our knowledge is vastly greater than the sum of what all its objects separately could afford; and when a new object comes within our reach, the addition to our knowledge is the greater, the more we already know; so that it increases not as the objects increase, but in a much higher proportion ...

For Turnbull's description and assessment of Maclaurin's mathematics see: Turnbull on Maclaurin, Part 2

For Maclaurin's publications see: Maclaurin's publications

JOC/EFR April 2007

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