STEWART (the Rev. Dr. Matthew)

, late professor of mathematics in the university of Edinburgh, was the son of the reverend Mr. Dugald Stewart, minister of Rothsay in the Isle of Bute, and was born at that place in the year 1717. After having finished his course at the grammar school, being intended by his father for the church, he was sent to the university of Glasgow, and was entered there as a student in 1734. His academical studies were prosecuted with diligence and success; and he was so happy as to be particularly distinguished by the friendship of Dr. Hutcheson, and Dr. Simson the celebrated geometrician, under whom he made great progress in that science.

Mr. Stewart's views made it necessary for him to attend the lectures in the university of Edinburgh in 1741; and that his mathematical studies might suffer no interruption, he was introduced by Dr. Simson to Mr. Maclaurin, who was then teaching with so much success, both the geometry and the philosophy of Newton, and under whom Mr. Stewart made that proficiency which was to be expected from the abilities of such a pupil, directed by those of so great a master. But the modern analysis, even when thus powerfully recommended, was not able to withdraw his attention from the relish of the ancient geometry, which he had imbibed under Dr. Simson. He still kept up a regular correspondence with this gentleman, giving him an account of his progress, and of his discoveries in geometry, which were now both numerous and important, and receiving in return many curious communications with respect to the Loci Plani, and the Porisms of Euclid. Mr. Stewart pursued this latter subject in a different, and new direction. In doing so, he was led to the discovery of those curious and interesting propositions, which were published, under the title of General Theorems, in 1746. They were given without the demonstrations; but they did not fail to place their discoverer at once among the geometricians of the first rank. They are, for the most part, Porisms, though Mr. Stewart, careful not to anticipate the discoveries of his friend, gave them only the name of Theorems. They are among the most beautiful, as well as most general propositions, known in the whole compass of geometry, and are perhaps only equalled by the remarkable locus to the circle in the second book of Apollonius, or by the celebrated theorem of Mr. Cotes.

Such is the history of the invention of these propositions; and the occasion of the publication of them was as follows. Mr. Stewart, while engaged in them, had entered into the church, and become minister of Roseneath. It was in that retired and romantic situation, that he discovered the greater part of those theorems. In the summer of 1746, the mathematical chair in the university of Edinburgh became vacant, by the death of Mr. Maclaurin. The General Theorems had not yet appeared; Mr. Stewart was known only to his friends; and the eyes of the public were naturally turned on Mr. Stirling, who then resided at Leadhills, and who was well known in the mathematical world. He however declined appearing as a candidate for the vacant chair; and several others were named, among whom was Mr. Stewart. Upon this occasion he printed the General Theorems, which gave their author a decided superiority above all the other candidates. He was accordingly elected professor of mathematics in the university of Edinburgh, in September 1747.

The duties of this office gave a turn somewhat different to his mathematical pursuits, and led him to think of the most simple and elegant means of explaining those difficult propositions, which were hitherto only accessible to men deeply versed in the modern analysis. In doing this, he was pursuing the object which, | of all others, he most ardently wished to attain, viz, the application of geometry to such problems as the algebraic calculus alone had been thought able to resolve. His solution of Kepler's problem was the first specimen of this kind which he gave to the world; and it was perhaps impossible to have produced one more to the credit of the method he followed, or of the abilities with which he applied it. Among the excellent solutions hitherto given of this famous problem, there were none of them at once direct in its method, and simple in its principles. Mr. Stewart was so happy as to attain both these objects. He founds his solution on a general property of curves, which, though very simple, had perhaps never been observed; and by a most ingenious application of that property, he shows how the approximation may be continued to any degree of accuracy, in a series of results which converge with great rapidity.

This solution appeared in the second volume of the Essays of the Philosophical Society of Edinburgh, for the year 1756. In the first volume of the same collection, there are some other propositions of Mr. Stewart's, which are an extension of a curious theorem in the 4th book of Pappus. They have a relation to the subject of Porisms, and one of them forms the 91st of Dr. Simson's Restoration.

It has been already mentioned, that Mr. Stewart had formed the plan of introducing into the higher parts of mixed mathematics, the strict and simple form of ancient demonstration. The prosecution of this plan produced the Tracts Physical and Mathematical, which were published in 1761. In the first of these, Mr. Stewart lays down the doctrine of centripetal forces, in a series of propositions, demonstrated (if we admit the quadrature of curves) with the utmost rigour, and requiring no previous knowledge of the mathematics, except the elements of plane Geometry, and of Conic Sections. The good order of these propositions, added to the clearness and simplicity of the demonstrations, renders this Tract perhaps the best elementary treatise of Physical Astronomy that is any where to be found.

In the three remaining Tracts, our author had it in view to determine, by the same rigorous method, the effect of those forces which disturb the motions of a secondary planet. From this he proposed to deduce, not only a theory of the moon, but a determination of the sun's distance from the earth. The former, it is well known, is the most difficult subject to which mathematics have been applied, and the resolution required and merited all the clearness and simplicity which our author possessed in so eminent a degree. It must be regretted therefore, that the decline of Dr. Stewart's health, which began soon after the publication of the Tracts, did not permit him to pursue this investigation.

The other object of the Tracts was, to determine the distance of the sun, from his effect in disturbing the motions of the moon; and his enquiries into the lunar irregularities had furnished him with the means of accomplishing it.

The theory of the composition and resolution of forces enables us to determine what part of the solar force is employed in disturbing the motions of the moon; and therefore, could we measure the instanta- neous effect of that force, or the number of feet by which it accelerates or retards the moon's motion in a second, we should be able to determine how many feet the whole force of the fun would make a body, at the distance of the moon, or of the earth, descend in a second of time, and consequently how much the earth is, in every instant, turned out of its rectilineal course. Thus the curvature of the earth's orbit, or, which is the same thing, the radius of that orbit, that is, the distance of the sun from the earth, would be determined. But the fact is, that the instantaneous effects of the sun's disturbing force are too minute to be measured; and that it is only the effect of that force, continued for an entire revolution, or some considerable portion of a revolution, which astronomers are able to observe.

There is yet a greater difficulty which embarrasses the solution of this problem. For as it is only by the difference of the forces exerted by the sun on the earth and on the moon, that the motions of the latter are disturbed, the farther off the sun is supposed, the less will be the force by which he disturbs the moon's motions; yet that force will not diminish beyond a fixed limit, and a certain disturbance would obtain, even if the distance of the sun were infinite. Now the sun is actually placed at so great a distance, that all the disturbances, which he produces on the lunar motions, are very near to this limit, and therefore a small mistake in estimating their quantity, or in reasoning about them, may give the distance of the sun infinite, or even impossible. But all this did not deter Dr. Stewart from undertaking the solution of the problem, with no other assistance than that which geometry could afford. Indeed the idea of such a problem had first occurred to Mr. Machin, who, in his book on the laws of the moon's motion, has just mentioned it, and given the result of a rude calculation (the method of which he does not explain), which assigns 8″ for the parallax of the sun. He made use of the motion of the nodes; but Dr. Stewart considered the motion of the apogee, or of the longer axis of the moon's orbit, as the irregularity best adapted to his purpose. It is well known that the orbit of the moon is not immoveable; but that, in consequence of the disturbing force of the sun, the longer axis of that orbit has an angular motion, by which it goes back about 3 degrees in every lunation, and completes an entire revolution in 9 years nearly. This motion, though very remarkable and easily determined, has the same fault, in respect of the present problem, that was ascribed to the other irregularities of the moon: for a very small part of it only depends on the parallax of the sun; and of this Dr. Stewart seems not to have been perfectly aware.

The propositions however which defined the relation between the sun's distance and the mean motion of the apogee, were published among the Tracts, in 1761. The transit of Venus happened in that same year: the astronomers returned, who had viewed that curious phenomenon, from the most distant stations; and no very satisfactory result was obtained from a comparison of their observations. Dr. Stewart then resolved to apply the principles he had already laid down; and, in 1763, he published his essay on the Sun's Distance, where the computation being actually made, the parallax of the sun was found to be no more than 6″ 9, | and consequently his distance almost 29875 semidiameters of the earth, or nearly 119 millions of miles.

A determination of the sun's distance, that so far exceeded all former estimations of it, was received with surprise, and the reasoning on which it was founded was likely to undergo a severe examination. But, even among astronomers, it was not every one who could judge in a matter of such difficult discussion. Accordingly, it was not till about 5 years after the publication of the sun's distance, that there appeared a pamphlet, under the title of Four Propositions, intended to point out certain errors in Dr. Stewart's investigation, which had given a result much greater than the truth. From his desire of simplifying, and of employing only the geometrical method of reasoning, he was reduced to the necessity of rejecting quantities, which were considerable enough to have a great effect on the last result. An error was thus introduced, which, had it not been for certain compensations, would have become immediately obvious, by giving the sun's distance near three times as great as that which has been mentioned.

The author of the pamphlet, referred to above, was the first who remarked the dangerous nature of these simplifications, and who attempted to estimate the error to which they had given rise. This author remarked what produced the compensation above mentioned, viz, the immense variation of the sun's distance, which corresponds to a very small variation of the motion of the moon's apogee. And it is but justice to acknowledge that, besides being just in the points already mentioned, they are very ingenious, and written with much modesty and good temper. The author, who at first concealed his name, but has now consented to its being made public, was Mr. Dawson, a surgeon at Sudbury in Yorkshire, and one of the most ingenious mathematicians and philosophers this country now possesses.

A second attack was soon after this made on the Sun's Distance, by Mr. Landen; but by no means with the same good temper which has been remarked in the former. He fancied to himself errors in Dr. Stewart's investigation, which have no existence; he exaggerated those that were real, and seemed to triumph in the discovery of them with unbecoming exultation. If there are any subjects on which men may be expected to reason dispassionately, they are certainly the properties of number and extension; and whatever pretexts moralists or divines may have for abusing one another, mathematicians can lay claim to no such indulgence. The asperity of Mr. Landen's animadversions ought not therefore to pass uncensured, though it be united with sound reasoning and accurate discussion. The error into which Dr. Stewart had fallen, though first taken notice of by Mr. Dawson, whose pamphlet was sent by me to Mr. Landen as soon as it was printed (for I had the care of the edition of it) yet this gentleman extended his remarks upon it to greater exactness. But Mr. Landen, in the zeal of correction, brings many other charges against Dr. Stewart, the greater part of which seem to have no good foundation. Such are his objections to the second part of the investigation, where Dr. Stewart finds the relation between the disturbing force of the sun, and the motion of the apses of the lunar orbit. For this part, instead of being liable to objection, is deserving of the greatest praise, since it resolves, by geometry alone, a problem which had eluded the efforts of some of the ablest mathematicians, even when they availed themselves of the utmost resources of the integral calculus. Sir Isaac Newton, though he assumed the disturbing force very near the truth, computed the motion of the apses from thence only at one half of what it really amounts to; so that, had he been required, like Dr. Stewart, to invert the problem, he would have committed an error, not merely of a few thousandth parts, as the latter is alleged to have done, but would have brought out a result double of the truth. (Princip. Math. lib. 3, prop. 3.) Machin and Callendrini, when commenting on this part of the Principia, found a like inconsistency between their theory and observation. Three other celebrated mathematicians, Clairaut, D'Alembert, and Euler, severally experienced the same difficulties, and were led into an error of the same magnitude. It is true, that, on resuming their computations, they found that they had not carried their approximations to a sufficient length, which when they had at last accomplished, their results agreed exactly with observation. Mr. Walmsley and Dr. Stewart were, I think, the first mathematicians who, employing in the solution of this difficult problem, the one the algebraic calculus, and the other the geometrical method, were led immediately to the truth; a circumstance so much for the honour of both, that it ought not to be forgotten. It was the business of an impartial critic, while he examined our author's reasonings, to have remarked and to have weighed these considerations.

The Sun's Distance was the last work which Dr. Stewart published; and though he lived to see the animadversions made on it, that have been taken notice of above, he declined entering into any controversy. His disposition was far from polemical; and he knew the value of that quiet, which a literary man should rarely suffer his antagonists to interrupt. He used to say, that the decision of the point in question was now before the public; that if his investigation was right, it would never be overturned, and that if it was wrong, it ought not to be defended.

A few months before he published the Essay just mentioned, he gave to the world another work, entitled, Propositiones More Veterum Demonstratæ. It consists of a series of geometrical theorems, mostly new; investigated, first by an analysis, and afterwards synthetically demonstrated by the inversion of the same analysis. This method made an important part in the analysis of the ancient geometricians; but few examples of it have been preserved in their writings, and those in the Propositiones Geometricæ are therefore the more valuable.

Doctor Stewart's constant use of the geometrical analysis had put him in possession of many valuable propositions, which did not enter into the plan of any of the works that have been enumerated. Of these, not a few have found a place in the writings of Dr. Simson, where they will for ever remain, to mark the friendship of these two mathematicians, and to evince the esteem which Dr. Simson entertained for the abilities of his pupil. Many of these are in the work upon the Porisms, and others in the Conic Sections, viz, marked with the letter x; also a theorem in the edition of Euclid's Data. |

Soon after the publication of the Sun's Distance, Dr. Stewart's health began to decline, and the duties of his office became burdensome to him. In the year 1772, he retired to the country, where he afterwards spent the greater part of his life, and never resumed his labours in the university. He was however so fortunate as to have a son to whom, though very young, he could commit the care of them with the greatest confidence. Mr. Dugald Stewart, having begun to give lectures for his father from the period above mentioned, was elected joint professor with him in 1775, and gave an early specimen of those abilities, which have not been confined to a single science.

After mathematical studies (on account of the bad state of health into which Dr. Stewart was falling) had ceased to be his business, they continued to be his amusement. The analogy between the circle and hyperbola had been an early object of his admiration. The extensive views which that analogy is continually opening; the alternate appearance and disappearance of resemblance in the midst of so much dissimilitude, make it an object that astonishes the experienced, as well as the young geometrician. To the consideration of this analogy therefore the mind of Dr. Stewart very naturally returned, when disengaged from other speculations. His usual success still attended his investigations; and he has left among his papers some curious approximations to the areas, both of the circle and hyperbola. For some years toward the end of his life, his health scarcely allowed him to prosecute study even as an amusement. He died the 23d of January 1785, at 68 years of age.

The habits of study, in a man of original genius, are objects of curiosity, and deserve to be remembered. Concerning those of Dr. Stewart, his writings have made it unnecessary to remark, that from his youth he had been accustomed to the most intense and continued application. In consequence of this application, added to the natural vigour of his mind, he retained the memory of his discoveries in a manner that will hardly be believed. He seldom wrote down any of his investigations, till it became necessary to do so for the purpose of publication. When he discovered any proposition, he would set down the enunciation with great accuracy, and on the same piece of paper would construct very neatly the figure to which it referred. To these he trusted for recalling to his mind, at any future period, the demonstration, or the analysis, however complicated it might be. Experience had taught him that he might place this confidence in himself without any danger of disappointment; and for this singular power, he was probably more indebted to the activity of his invention, than to the mere tenaciousness of his memory.

Though Dr. Stewart was extremely studious, he read but few books, and thus verified the observation of D'Alembert, that, of all the men of letters, mathematicians read least of the writings of one another. Our author's own investigations occupied him sufficiently; and indeed the world would have had reason to regret the misapplication of his talents, had he employed, in the mere acquisition of knowledge, that time which he could dedicate to works of invention.

It was Dr. Stewart's custom to spend the summer at a delightful retreat in Ayrshire, where, after the academical labours of the winter were ended, he found the leisure necessary for the prosecution of his researches. In his way thither, he often made a visit to Dr. Simson of Glasgow, with whom he had lived from his youth in the most cordial and uninterrupted friendship. It was pleasing to observe, in these two excellent mathematicians, the most perfect esteem and affection for each other, and the most entire absence of jealousy, though no two men ever trode more nearly in the same path. The similitude of their pursuits served only to endear them to each other, as it will ever do with men superior to envy. Their sentiments and views of the science they cultivated, were nearly the same; they were both profound geometricians; they equally admired the ancient mathematicians, and were equally versed in their methods of investigation; and they were both apprehensive that the beauty of their favourite science would be forgotten, for the less elegant methods of algebraic computation. This innovation they endeavoured to oppose; the one, by reviving those books of the ancient geometry which were lost; the other, by extending that geometry to the most difficult enquiries of the moderns. Dr. Stewart, in particular, had remarked the intricacies, in which many of the greatest of the modern mathematicians had involved themselves in the application of the calculus, which a little attention to the ancient geometry would certainly have enabled them to avoid. He had observed too the elegant synthetical demonstrations that, on many occasions, may be given of the most difficult propositions, investigated by the inverse method of fluxions. These circumstances had perhaps made a stronger impression than they ought, on a mind already filled with admiration of the ancient geometry, and produced too unsavourable an opinion of the modern analysis. But if it be confessed that Dr. Stewart rated, in any respect too high, the merit of the former of these sciences, this may well be excused in the man whom it had conducted to the discovery of the General Theorems, to the solution of Kepler's Problem, and to an accurate determination of the Sun's disturbing force. His great modesty made him ascribe to the method he used, that success which he owed to his own abilities.

The foregoing account of Dr. Stewart and his writings, is chiefly extracted from the learned history of them, by Mr. Playfair, in the 1st volume of the Edinburgh Philosophical Transactions, pa. 57, &c.