XXii Mr. Pickering’s Eulogy on 
After noticing these historical facts in regard to the question, Dr. 
Bowditch remarks, that these late opinions of eminent mathema- 
ticians do not remove the objections which had been raised ; that is, 
that the neglect of the tangential force and other peculiarities of the 
method might possibly affect the result; and he then proceeds to 
the examination of the question, whether this fundamental theorem 
expresses, in an approximative form, the mean motion of the apsides, 
supposing, with the author, the eccentricity to be very small, or the 
orbit nearly circular. He observes, at once, in his communication, 
which was written twenty years ago, that there is now no difficulty 
in settling this point by means of the analytical expression of the 
motion of the apsides given by La Place; and, upon applying that 
method, he found that Dr. Stewart’s formula was far from being so 
correct as had been supposed by the able writers above referred to, 
but, on the contrary, that it was essentially defective. The first and 
most important term of the series is double its true value; and the 
whole formula will not give an accurate numerical result, except 
when the primary planet and satellite have a certain proportion to 
each other; and this, by a remarkable coincidence, happens to be 
the case nearly with the earth and Moon, the example taken by 
Dr. Stewart; but the same formula would not answer, if the 
Moon’s distance from the earth were much greater or much less 
than it now is; and it would require but a very small decrease of 
the Moon’s mean distance from the earth, to render the Sun’s dis- 
tance infinite when computed according to Doctor Stewart’s direc- 
tions; so that this method would have failed, if it had been applied 
to other cases, as, for instance, Jupiter’s satellites; and he then 
demonstrates, that such would be the result. 
Dr. Bowditch concludes, that the defect in the most important 
theorem of Dr. Stewart makes his method wholly fail. He attri- 
butes the mistake in this case entirely to the use of the geomet- 
