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and will, probably, be long regarded as a specimen of the most arduous inquiry which has been attempted by mere geometry; at the same time, the mistake into which the geometrical method has betrayed this great mathematician, will serve as a proof that it is not equal to such difficult researches; and that in those cases, especially, where approximation is to be used, it is necessary to sacrifice the rigour of the ancient demonstration for the accuracy of the modern analysis.
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 Geometrica More Veterum Demonstratæ. This title, I have been told, was given it by Dr Simson, who rejoiced in the publication of a work so well calculated to promote the study of the ancient geometry. It consists of a se
ries of geometrical theorems, for the most part new; investigated, first, by an analysis, and afterwards synthetically demonstrated by the inversion of the same analysis. In the former, the proposition to be investigated is supposed true; from thence consequences are deduced, and the reasoning is carried on till some consequence is drawn that is already known to be true. A necessary connection is thus traced between the proposition that was supposed true, and another that is certainly known to be so; and, thus, an ingenious method is laid down for making the knowledge of any truth subservient to the discovery of its demonstration. This method made an important part in the analysis of the ancient Geometers; but few examples of it have been preserved in their writings, and those in the Propositiones Geometrica are, on that account, the more valuable.
Dr 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. In the preface to his Conic Sections, in which he acknowledges, that all the theorems, distinguished by the letter
x, were communications from Dr Stewart, he calls him, "egregiæ indolis et peritiæ virum ;" and in that to his Porisms, after pointing out many propositions that had been suggested by Pappus, and a few that had been adopted from Fermat, he adds, "Alia quædam adjecta sunt quorum præcipua mihi proposuit, et aliquorum constructionem dedit eximius Geometra Matthæus Stewart, a quo materia hæc jam egregie est exculta, postea, ut spero, mul
There is also a theorem of Dr Stewart's published in Dr Simson's edition of Euclid's Data, which I take notice of, chiefly as it affords me an opportunity of paying a tribute to the memory of a man, whose high rank did not prevent him from cultivating a science, which it enabled him to patronize. In the note, where Dr Simson acknowledges that communication, he mentions another theorem, also published among the Data; "These propositions (says he) were communicated to me by two excellent Geometers, the first by the Earl Stanhope, the second by Dr Matthew Stewart."
To this Nobleman, for whose abilities and worth Dr Stewart entertained the highest respect, he made a visit in the course of a tour through England, soon after the publication of the Essay on the Sun's Distance, and received from him very singular marks of attention. At a later period, when he lamented the loss of Dr Simson, he had the con
solation to see a lasting monument raised to the fame of his friend, by the munificence of Lord Stanhope, who, by the publication of Dr Simson's posthumous works, has obliged the world with a restoration of the most curious fragment of the Greek geometry.
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 now 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 geometer. 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 January 23, 1785, at the age of 68.
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 rarely 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 put 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