247 - 19. 39 s - 13:19 vo tang. V3, radius I. And the same will readily appear by transforming the numerators of the series to 5 – 3, 7–4, dently equal to one half of the following one ; namely, 1.206457, &c. the true sum of the series, which Mr. Lorgna makes more than six times as great as it should be. Mr. Clarke seems to have been strangely prepossessed in favour of his author, and his method, for he has filled many pages of close quarto letter-press, in commenting on several of the series, whose sums we have given above; and that, without discovering any one of the errors here specified. It may be proper to observe, that the method by which we have transformed the numerators and denominators above, is subjećt to some certain rules, and reasons; which to give here would be taking up too much room. Mr. C. has indeed greatly extolled the method made use of by Mr. Lorgna, in his 9th section; but, for the even powers of the reciprocals of the natural numbers, it is certainly much inferior to that made use of by Mr. John Bernoulli, in the fourth volume of his Works, and by Mr. L. Euler, in the 1st volume of his Introduction to the Analysis of Infinites; and as to the odd powers, it is far from being the thing that is wanted. The method is taken from Mr. Cotes’s account of the Newtonian differential method, published at the end of his Harmonia Menforarum. On the whole, Mr. Lorgna's is a very unequal performance, some things in it being very curious and accurate, and others as much the contrary. To his Author's Differtation, Mr. C. has added a long Appendix, intituled, the Summation of Series, exhibiting general formulae, for the summing of both a finite and infinite number of terms, of all possible numeral or literal series whatsoever, &c. Surely this is saying a great deal, and more than Mr. C. or the greatest mathematician living, could be certain of the truth of. He |