ftant figns. And from the four laft examples it is obvious, that it is not neceffary for the factors in the denominators of the terms of the feries to be immediately confequent to each other, neither need they be in the fame arithmetical progreffion; but all the authors we know of, that have confidered this fubject, have thought one or other of these conditions abfolutely requifite, for the fummation of the feries to be exhibited in an algebraic expreffion. Certainly then, if there be any thing particular and extraordinary in this, it will appear by confidering thefe four examples:

I

The first is + + + &c. but this mani6.12 9.16 12.20

1

feftly =

12

3.8

I

I

I

of the series + + + &c. which 1.2 2.3 3.4 4.5

feries by Mr. Bernoulli's method is equal to the difference of

1 &c. where the factors in each denominator having 3 for

3.6

their common difference, we have by the Bernoullian method

of the difference of the feries 1 +

I of 15

+

+ + 3

3

1 &c. and

5

II

270'

is the fum of the propofed feries. The remaining ex

ample is the alternate feries

&c. = X:

+

&c. with 2 for the

1.3

214 3-5

4.6

com

I

common difference of the factors, therefore it is = X

I

12

3 4

If now we compare this with the trouble of making each particular feries that may occur agree with the general formula, and finding the fums thereby, we shall be judges of the advantages or disadvantages of our Author's method. For as to the criterion of the poffibility of fummation, mentioned in the article above quoted, Mr. Landen has fhewn that there is nothing in it. Let us try then one of the feries with three factors, as ex

ample the 3d, fection 3d, where the series is

I

I

+
1.2.3 2.3.4

I

+

1.3 2.3 2.3 2.4 3.5

I

I

+

+

+

&c. =

4.5

4.5 4.6

1.3

+
2.4 3.5

-&c.

4.6

finite terms, the firft example is + + &c. evi

1, which is given in all books of fluxions. His next example is the most common and well known feries for the Arc

ties thus expreffed involve no ambiguous or infinite expreffions; and the want of this contrivance has led Mr. Lorgna here into an error, he giving, as Mr. Landen obferves, a falfe fum to this feries; and Mr. Landen himself has not given us this manner of ordering the fluxions. . . . . Mr. Lorgna's next example is evi

It may be thought perhaps that more complicated feries will fhew the excellency of Mr. Lorgna's method better than these fimpler ones, it may therefore be neceffary to give an example or

Mr. Landen. Mr. Lorgna having here made the fame miftake as before, by not dividing the numerator and denominator of the fluxional fraction each by I X. Let us then try one with four factors in the denominator, as for example, the feries at Art. 53, p. 67, which is fo complicated, that Mr. Lorgna himself, as Mr. Landen obferves, has made more blunders than one in finding the fum. But this is evidently - 6 times the

I

3.4.21

I

2.3.7

+
1.2.9 2.3.15

I + &c. but the firft of thefe feries = 3 X:

1.2.4

+

+&c. is evidently 24 times that in the laft

example

4.6

2

3 4

&c. 8 X: + + + &c.

I

4.5.6 8X: +

1

2.4

4

6-8. L. 2. confequently 24 X the fum of the feries in the

༢

last example above + 3 −6 + 8 L. 2. is the fum of the pre

fent feries. The method here made use of is far more perfpicuous than Mr. Lorgna's, and would be more fimple too, if we had a table of fluxions with their fluents in feries to refer to. An

example of this is Art. 45, p. 47, where the series

I

+

I

2.3.7 3.4.9

1.2.5 &c. four times the feries whofe denominators

are 2.4.5, 4.6.7, 6.8.9, &c. where, all the three factors in each having conftant differences, it eafily and evidently refolves.

I

2

3

3

4

where the first infinite feries is that for the hyperb. Log. of

2, and the fecond the well known one for A, the arc whofe tangent and radius are each of them unity, confeq. 4. L. 2 +

probably an error of the prefs at p. 48. Let this be compared with Mr. Lorgna's fummation and Mr. Clarke's comment at p. 60 and 61 of his Treatife.

There are feveral more mistakes in this Treatife, which are neither corrected by Mr. Clarke nor Mr. Landen, but may caufe 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 fame time we should ill perform our duty as Readers for the Public, not to point out for their fervice the most material errors we have found, and fhew how to correct them. And in doing this it will appear, how the method of Mr. Bernoulli,

3

aided

aided by fome fimilar artifices, may be applied to the most complex feries that are fummable 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 fuch as ferve for elucidating the law obferved in numeral ones, fo that it is in these alone that all the difficulty is contained; and the method of Mr. Bernoulli is equally applicable to the feries when they are thus expreffed algebraically.

The first inftance we fhall give is example the 4th, Sect. V.

&c. which is the difference of the two feries 1 +

I

+ 3.2 5.4

or putting c≈

+ &c. and I 7.8

I

I

+

+

&c.

4.2

the fquare root of 2,

and a

с

feries becomes c X:

; and the latter a X

a2 + ax + x

·2ax + x2

a √3

+

+

363

2

+
a 404

+

2

2

the fum of this latter feries, which taken from that of the former, and the remainder doubled, will be the true fum of the propofed feries, 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 the honour to comment on his performance; for first

he gives it, and fecondly the fluent of is not hyperb.

a-u

log. of a-u, but that fame quantity made negative. And he commits two fimilar mistakes at Art. 91, p. 99; and Mr. Clarke has wrote a comment on that fame article, containing more than three pages of clofe letter-prefs in quarto, without

difcovering either of them; for first the fraction