According to what Mr. Simpson had intimated at the conclusion of his Doc trine of Fluxions, the greatest part of this arduous undertaking was drawn up in the year 1750. About that time M. Clai. raut, a very eminent mathematician of the French Academy, had started an objec tion against Newton's general law of gm. vitation. This was a motive to induce Mr. Simpson (among some others) to endea vour to discover whether the motion of the Moon's apogee, on which that objec tion had its whole weight and foundation, could not be truly accounted for, without supposing a change in the received law of gravitation, from the inverse ratio of the squares of the distances. The success an swered his hopes, and induced him to look further into other parts of the theory of the Moon's motion than lie at first in tended ; but before he had completed his design, M. Clairaut arrived in England, and made Mr. Simpson a visit; from whom he learned, that he had a little before print ed a piece on that subject, a copy of which Mr. Simpson afterwards received as a present, and found in it the same things demonstrated, to which he himself had directed his enquiry, besides several others.
The facility of the method Mr. Simp. son fell upon, and the extensiveness of it, will in some measure appear from this, that it not only determines the motion of the apogee in the same manner, and with the same ease as the other equations, but utterly excludes all those dangerous ki nds of terms that had embarrassed the great est mathematicians, and would after a great number of revolutions, entirely change the figure of the Moon's orbit.
From whence this important consequence is derived, that the Moon's mean motion and the greatest quantities of the several equations, will remain unchanged, un less disturbed by the intervention of some foreign or accidental cause. These tracts are inscribed to the. Earl of Mac clesfield, President of the Royal So ciety.
Mr. Simpson's extreme application in this difficult pursuit greatly injured his health. Exercise and a proper regimen were prescribed to him, but to little pur pose; for his spirits sunk gradually, till he became incapable of performing his duty, or even of reading the letters of his friends. The effects of this decay of na. ture were greatly increased by vexation of mind, owing to the haughty and insult ing behaviour of his superior, the first professor of mathematics. This person, greatly his inferior in mathematical ac complishments, did what he could to make his situation uneasy, and even to depreciate him io the public opinion ; but it was a vain endeavour, and only served to injure himself. At length his physicians advised his native air for his recovery, and he set out in February, 1761; but was so fatigued by his journey, that, upon his arrival at Bosworth, he be took himself to his chamber, and grew continually worse till the day of his death, which happened on the 14tIr of May, in the 51st year of his age.