In the three remaining tracts, our au thor had it in view to determine 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 solution required and merited all the clearness and simplicity which our au thor possessed in so eminent a degree. It must be regretted, therefore, that the decline of Mr. Stewart's health, which be gan soon after the publication of the tracts, did not permit him to pursue this i nvestiga tion. 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 inquiries into the lunar irregularities had furnished him with the means of accomplishing it, as he supposed: and in 1763, he published his "Essay on the Sun's distance," where the computa tion being actually made, the parallax of the sun wasfound to be no more than 6" 9m, and consequently his distance almost 29,875 semi-diameters of the earth, or nearly 119 millions of miles. A determi nation of the sun's distance, that so far exceeded all former estimation of it, was received with surprise, and the reasoning on which it was founded was likely to tin. dergo a severe examination. But, even among astronomers, it was not every one who could judge in a matter of such diffi cult discussion. Accordingly, it was not till above five years after the publication of Dr. Stewart's work, that there appeared a pamphlet, tinder the title of " Four Propositions," intended to point out cer tain errors in Dr. Stewart's investigation, which had Riven a result much greater than the truth. From his desire of simpli fying, and of employing only the geome trical method of reasoning, he was redu ced to the necessity of rejecting quantities, which were considerable enough to have a great effect on the last result. An er ror was thus introduced, which, had it not been for certain compensations, would have become immediately obilons, by giving the sun's distance near three times as great as thatwhich has been mentioned.
The " Sun's Distance" was the last work which Dr. Stewart published ; and though he lived to see the animadversions made on it, he declined entering into any contro versy. His disposition was far from pole mical : and he knew the value of that qui et, which a literary man should rarely suf fer 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 ne ver 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, " Proposi tiones More Veterum Demonstrate." It
consists of a series of geometrical theo rems, mostly new ; investigated, first by an analysis, and afterwards synthetically demonstrated by an inversion of the same analysis. This method made an impor tant part in the analysis of the ancient geo metricians; but few examples of it have been preserved in their writings, and those in the " Propositiones Geometrica" are therefore the more valuable.
Doctor Stewart's constant use of the geometrical analysis had put him in pos session 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 friend ship of these two mathematicians, and to evince the esteem which Dr. Simson en tertained for the abilities of his pupil. Many of these are in the work upon the l'orisms, and othersin the Conic Sections, 742:. marked with the letter .1•: also a theo rem 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 becutne burdensome to him. In the year 1772, he retired to the country, where he after wards spent the greater part of his life, and never resumed his labours in the uni versity. Ile 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 men tioned, 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 amuse ment. 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 disappear ance of resemblance in the midst of so much dissimilitude ; make it an object that astonishes the experienced, as well as the young geometr:cian. To the considera tion of this analogy, therefore, the mind of Dr. Stewart very naturally returned, when disengaged from other speculations. His usual success still attended his investiga tions ; and he left among his papers some curious approximations to the areas, both of the circle and hyperbola. For some years,towards the end of his life, his health scarcely allowed him to prosecute study even as an amusement. He died the twenty-third of January, 1785, at sixty eight years of age. See vol. i. Edinburgh Transactions.