664' . PHILOSOPHICAL TRANSACTIONS. [aNNO 1702. 



produced by the same rotation, and the rectification of curves, as also the cen- 

 tres of gravity of all these, depend all on the quadrature of curves; these will 

 be easily computed when they depend on any of the above curves. 



After I had put these theorems into form, and had showed them to the great 

 Newton, as the supreme judge of these matters; he was pleased to show me his 

 own manuscripts, by which it appeared that he had long been in possession of 

 a method, by which, vvhen any trinomial equation is given, expressing the 

 nature of a curve, he could either square the same, or reduce it to some sim- 

 pler curve. It is much to be wished that he would condescend to publish not 

 only what relates to these matters, but many others of his excellent inventions ; 

 which would be a great benefit to tlie republic of letters. 



I doubt not but those learned men, whose writings in the Leipsic Journals, 

 and elsewhere, have greatly contributed to the improvement of the mathema- 

 tical sciences, may have methods similar to these ; and therefore I assume no- 

 thing to myself, but that I have discovered these theorems, not knowing whe- 

 ther they may be already extant or not ; and that I have reduced them to so 

 easy a form, that all the calculation required, is performed as it were by 

 intuition. 



Before finishing this paper, it may not be amiss if, on this occasion, I add a 

 few words in answer to the animadversions of the learned M. Leibnitz, on a 

 certain series I published* for finding the root of an infinite equation. That 

 celebrated man is of opinion, that that series is not sufficiently general, as not 

 extending to the cases in which the quantities z and y are multiplied into each 

 other; and therefore he substitutes another series for mine, which he asserts is 

 infinitely more general. Now I suspect he has been led into this small mistake, 

 by having taken the quantities a, h, c, d, &c, for given quantities ; whereas they 

 may be used at pleasure for either given or indeterminate quantities. But for 

 one example to show that my series includes all cases whatever, take the equa- 

 tion nyz — z^ =: \f. Now in my theorem make a = ny, Z' = 0, c = — ], 

 g- = O, /j = O, / = 1 ; or, which is better, make g = yy, h =■ 0, / = O ; in 



either case it will bez=--|--. + -^ + — ^ &c. 



Account of several unusual Parhelia, or Mock Suns, and several Circular ArcheSy 

 seen in the Air. By E. Halley. N° 278, p. 1127. 



Aprils, this present year, 1702, about 10 o'clock in the morning, the air 

 being clear, I observed the sun shine faintly, or as we call it, waterish, and 

 perceived several arches of circles round hiin, as represented in fig. 11, pi. l6 ; 



* See p. 275 of tliis volume. 



