TO SEVERAL CENTRES. 201 



the vanishing of the expression for the conjugate axis in some 

 fundamental formulae connected with the ellipse, for example, 

 the subtangent, gives rise to other curious properties of the 

 curve similar to the one noted in this theorem, which is 

 itself related to that peculiarity. (See a porism arising from 

 this circumstance in Tract IV.) The same theorem had oc- 

 curred to Lagrange himself, in examining the problem of 

 deflecting forces to two centres ; it is indeed derivable imme- 

 diately from the case of that problem when one force vanishes 

 and the centre connected with it is in an arc of the ellipse ; 

 for then the radius vector belonging to that centre becomes 

 the chord. But Euler, long before either of them, in 1744, 

 had given the theorem for parabolic arcs, which they only 

 extended to elliptic arcs, and had published it in the Berlin 

 Mem. 1760. Yet when Lambert claimed it as his own in 

 1771, and Lagrange gave him the honour of it in 1780, Euler, 

 though he lived three years after, never thought of reminding 

 them of his prior claims. It was thus, too, with the first of 

 analysts, respecting the extension of the Differential Calculus 

 to that of Partial Differences (Tract III.), by far the greatest 

 step in mathematical science which 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 ig- 

 norant of Euler having anticipated him as to parabolic arcs. 

 But Lagrange truly states (Mec. Anal. ii. 28, ed. 1855), what 

 shows that all of them had been anticipated by Newton. For 

 in the IV. 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 depending at all 

 on the form of the ellipse, providing the transverse axis 

 remains the same. This must have frequently recurred to 

 his recollection, when engaged in those great investigations 



