destined to serve as the basis of this investigation. We are not, however, to imagine, that Dr Stewart intended to proceed in the same direct manner that Clairault, and some other geometers, had done. It is not probable that he believed this to be within the power of pure geometry. His design undoubtedly was, to pursue that method of approximation which Sir Isaac Newton had begun, and which Callendrini, Machin, and Walmsley, had greatly improved; and, by using the methods of geometry, he hoped to reduce the problem to its ultimate simplicity. Such an undertaking was worthy of a great geometer, and of a philosopher, who considered that one of the chief obstructions to the advancement of knowledge, is the difficulty of simplifying that knowledge, which has already been acquired. We must regret, 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. The approach of the transit of Venus, which was to happen in 1761, had turned the attention of mathematicians to the solution of this curious problem. But when it was considered, of how delicate a nature the observations were from which that solution was to be deduced, and to how many accidents they were exposed, it was natural, that some attempt should be made to ascertain the dimensions of our system, by means less subject to disappointment. Such accordingly was the design of Dr Stewart; and his inquiries 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 instantaneous 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 feet the whole force of the sun would many make a body, at the distance of the moon, or of the earth, descend in a second, 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 dimi. nish 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 three degrees in every lunation, and completes an entire revolution in nine 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, as will afterwards appear, 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 his distance, of consequence, almost 29875 semidiameters of the earth. * A determination of the sun's distance, that so far exceeded all former estimations of it, was re About 118,541,428 English miles. ceived with surprise, and the reasoning on which it was founded was likely to be subjected to 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 five 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. A dispute in geometry was matter of wonder to many, and perhaps of satisfaction to some, who envied that science the certainty of its conclusions. On account of such, it must be observed, that there are problems so extremely difficult, that, in the solution of them, it is possible only to approximate to the truth; and that, as in arithmetic, we neglect those small fractions, which, though of inconsiderable amount, would exceedingly embarrass our computations; so, in geometry, it is sometimes necessary to reject those small quantities, which would add little to the accuracy, and much to the difficulty of the investigation. In both cases, however, the same thing may happen; though each quantity thrown out may be inconsiderable in itself, yet the amount of them altogether, and their effect on the last result, may be greater than is apprehended. This was just what had happened in the present case. The problem to be resolved is, in its |