ftant signs. And from the four last examples it is obvious, that it is not necessary for the factors in the denominators of the terms of the series to be immediately consequent to each other, neither need they be in the fame arithmetical progression ; but all the authors we know of, that have confidered this subject, have thought one or other of these conditions absolutely requi. site, for the summation of the series to be exhibited in an algebraic expression.' Certainly then, if there be any thing particular and extraordinary in this, it will appear by considering these four examples : I &c. but this manis 6.12 12.20 I festly of the series &c. which 2.3 3.4 4.5 series by Mr. Bernoulli's method is equal to the difference of 1 1 the two series it &c, and + + &c. 3 4 that is to unity, and consequently is the sum of the pro I I 3.8 9.16 I + + + 12 1.2 + + 2 1 12 + posed series. The next example is + + 8.18 10.21 12.24 &c. = the fixth part of the series + + + 14.27 57 6.8 1 &c, or of &c. &c. di. 7.9 4 5 6 7 3 vided by 2. consequently X 12 4 5 80 example + of + 15 1.4 2.5 &c. where the factors in each denominator having 3 for their common difference, we have by the Bernoullian method of I 프 of the difference of the feries 1 ++ + 3 15 3 I &c: and I Х it + 5 4 5 45 3 is the sum of the proposed series. The remaining ex270 ample is the alternate series 3.6 6.8 &c. X: + &c. with 2 for the 2.4 3.5 + I I + I I I I + 2.3 I I I I I 4.6 I I 2 2 : common difference of the factors, therefore it is = 12 I I I I I &c. = 24 If now we compare this with the trouble of making each particular series that may occur agree with the general formula, and finding the sums thereby, we shall be judges of the advantages or disadvantages of our Author's method. For as to the criterion of the possibility of fummation, mentioned in the article above quoted, Mr. Landen has shewn that there is nothing in it. Let us try then one of the series with three factors, as example the 3d, section 3d, where the series is + 1.2.3 2.3.4 + &c. which is = + 3.4.5 1.3 2.3 2.4 3.5 fo &c. = + -&c. 4.5 4.5 4.6 1.3 2.4 3.5 + &c. = 4 the sum of the series. Let us try then some of series whose sums cannot be had in finite terms, the first example is + &c, evi 4.5 7.8 dently = I + &c. = by the bi4 5 7 8 å nomial theorem to the fluent of when x - X3 i+*+* =l, which is given in all books of fluxions. His next example is the most common and well known series for the Arc whose tangent is unity and radius ✓ 3. And the following one evidently from what is done above = the hyperb. Log. of 2. In the next example the series is evidently = the fix times + + &c. = 6X: 9.10 15.16 21.22 9 15 16 &c. = 6X: 1 &c.—6x: 3 4 ***--**-****** XA. + 6. A, zoo 6 x2 3 6 x3 + A. fi. I+x27x+ + I I I + + I I I &c. = I I X: X: + 3.6 ties thus expressed involve no ambiguous or infinite expressions ; and the want of this contrivance has led Mr. Lorgna here into an error, he giving, as Mr. Landen observes, a falle fum to this series ; and Mr. Landen himself has not given us this manner of ordering the fuxions..... Mr. Lorgna’s next example is evi I dently X:I + X 2 X:1 + 3 3 3 3 L. 2. 18* It may be thought perhaps that more complicated series will fhew the excellency of Mr. Lorgna's method better than these simpler ones, it may therefore be necessary to give an example or two of these. Taking then &c. 4.7.12 6.10.16 8.13.20 3 47 X: + 2.3.7 3.4.10 4.5.13 I 3 + + + &c. = 3.6.7 4.9.10 5.12.13 8 4.9 I &c. Šx: to 5.12 3.7 4.10 5.13 2.3 3.4 &c. = 13.15 9 X: &c. = X:) 16 7 9 16 16 3 9 31 9 *+ ** A. 4 6 16 x 3 64 16° I+*+ *? 31 L. rad. 1. agreeing with 32 Mr. Landen.” Mr. Lorgna having here made the fame miltake as before, by not dividing the numerator and denominator of the fluxional fraction each by I Let us then try one with four factors in the denominator, as for example, the series at Art. 53, p. 67, which is so complicated, that Mr. Lorgna himself, as Mr. Landen observes, has made more blunders than one in finding the sum. But this is evidently = 6 times the series = 6 x : 1.2.8.9 2.3.14.15 3.4.20.21 I + + 6X: + 1.2.8 2.3.14 3.1.20 1.2.9 2.3.15 + &c. but the first of these series = 3X: -3.4.21 1.2.4 + + &c. is evidently = 24 times that in the lar: 2.37 3.4.10 example X: I example here given plus 3 X: And the latter - 2x: 1.2.4 1 I + 2.3.4 4.5.6 1 4.5 2.4 2 last example above + Ź + &c. - 8 X: + + 1.2.3 2.3.5 3.4.7 1 &c. = 8X: + + + &c. - 8X: + 6.7.8 2.3 6.7 I &c. =8-8X:I to &c. -8%: - = 4.6 3 4 4 6 - 8. L. 2. consequently 24 X the sum of the series in the 3 6 + 8 L. 2. is the sum of the pre 8 sent series. The method here made use of is far more perspicu. ous than Mr. Lorgna's, and would be more fimple too, if we had a table of Auxions with their fluents in series to refer to. An example of this is Art. 45, p. 47, where the series 1.2.5 + &c. = four times the series whose denominators 2.3.7 3.4.9 are 2.4.5, 4.6.7, 6.8.9, &c. where, all the three factors in each having conitant differences, it easily and evidently resolves I 2 4 into 1 + &c. Х 3 3 4 5 3 2 I + Xi1 3 2 + 4 &c. + X:I &c. X:I-3 3 3 5 7 3 where the first infinite series is that for the hyperb. Log. of 3 2, and the second the well known one for A, the arc whose tangent and radius are each of them unity, conseq. 3 17 17 is the sum of the series ; and not which is 3 9 19 probably an error of the press at p. 48. Let this be compared with Mr. Lorgna's fummation and Mr. Clarke's comment at p. 60 and 61 of his Treatise. There are several more mistakes in this Treatise, which are neither corrected by Mr. Clarke nor Mr. Landen, but may cause no small perplexity to the Reader. It is no pleasure to us to be finding fault, where we would much rather wish to commend; at the same time we should ill perform our duty as Readers for the Public, not to point out for their service the most material errors we have found, and thew how to correct them. And in doing this it will appear, how the method of Mr. Bernoulli, 3 aided 2 I I I I 7.8 4.2 I I I + &c. = 3 2 aided by some similar artifices, may be applied to the most complex series that are summable by Mr. Lorgna's method. It is. to be observed, that the algebraic factors mentioned in the titlepage, are not the things that create difficulties in these inquiries, but are meant to obviate them; and our Author has made use of no algebraic factors, but such as ferve for elucidating the law observed in numeral ones, so that it is in these alone that all the difficulty is contained; and the method of Mr. Bernoulli is equally applicable to the series when they are thus expressed algebraically. The first instance we shall give is example the 4th, Sect. V. 3 which is evidently = twice the series + + 3.4.2 5.7.4 7.10.8 &c. which is the difference of the two series it + 3.2 5.4 I + &c. and it + + &c. or putting c= 7.4 10.8 the square root of 2, and a = the cube root of 2, the former series becomes c X: + cx hyperb. 30 cts and the latter -ax + + 7 a? a? taxt fi. of XL. 4 X A the arc *3 6 - 2 ax + x2 whose tangentis and radius when xis = 1, is 40 + 2* 2 the sum of this latter series, which taken from that of the former, and the remainder doubled, will be the true sum of the proposed series, which is wide enough from that found by Mr. Lorgna, who has committed two errors in his operation at p. 71, which are overlooked by both the gentlemen that have done him j ✓ 2 the honour to comment on his performance; for first 2- y has for its fluent L. V2 + y or only half the value that 2 - 3 ü he gives it, and secondly the fluent of is not + hyperb. log. of a -- u, but that same quantity made negative. And he commits two similar mistakes at Art. 91, p. 99; and Mr. Clarke has wrote a comment on that same article, containing more than three pages of close letter-press in quarto, without 552 discovering either of them ; for first the fraction there given should log. of |