436 APPENDIX. 



has been made since the age of Newton and Leibnitz, if it 

 have not a rival in the calculus of variations, the honour 

 of which also is shared by him with Lagrange. 



12. It must be observed that when in 1771 (Berlin 

 Mem.) Lambert extended the theorem to elliptic arcs, he 

 was ignorant of Euler having anticipated him as to para 

 bolic arcs. But Lagrange truly states (Mec, Anal. ii. 28., 

 ed. 1855), what shows that all of them had been antici 

 pated by Newton. For in the IY. and V. Lemmas of the 

 Third Book he had very distinctly given the whole 

 materials of the proposition as far as parabolic arcs are 

 concerned. 



Lagrange notes the uses of the theorem, and observes 

 upon the remarkable circumstance of the time not depend 

 ing at all on the form of the ellipse, provided the trans 

 verse axis remains the same. This must have frequently 

 recurred to his recollection, when engaged in those great 

 investigations which show the connection that the trans 

 verse axis remaining unchanged, has with the permanency 

 of the system. 



13. He further remarks upon another consequence of 

 the conjugate axis, or the form of the orbit, not affecting 

 the time; namely, that the conjugate wholly disap 

 pearing, and the orbit becoming rectilinear, the theorem 

 applies to the time of falling to the centre, on the centri 

 fugal force or that of projection ceasing to act. (Berlin 

 Mem. 1778.) But Newton s Vlth Lemma, to which he 

 does not refer, in some degree anticipated this also. 



14. The great difficulty of the problem of several cen 

 tres, has been stated. Euler was clearly of this opinion, 

 and he was the first that undertook the solution. After 

 speaking of the general problem (Berlin Mem. 1760, 

 p. 228.) as alike important and difficult, he confines him 

 self to the case of two bodies in fixed positions, acting 

 upon a third, which moves in the plane of those disturbing 

 bodies ; in a word, to the motion of a body drawn towards 

 two fixed centres. He says that, whoever undertakes the 



