again and again have committed the fame error: which, probably to this time, would have remained uncorrected by them, if it had not been pointed out to them. The confequences of which, as far as Mr. Lorgna's conclufions are affected by it, are noticed by me, with the fame ftrict regard to veracity as is obferved in the writing down every other article in the pamphlet.'

So far Mr. Landen; but it may not perhaps be, improper for us to ftop here, and endeavour to reprefent this last matter as it appears to us.

That excellent mathematician, L. Euler, in his Inftitutiones Calculi differentialis, has a whole chapter on the fubject of such fractions as this here fpecified by Mr. Landen, and fhews fo clearly and fully that the numerators and denominators have not generally the ratio of equality when they vanifh together, as in the cafe here of y=1, that one would think that neither Mr. Lorgna nor his Commentator could poffibly be ignorant of it. We rather think that the errors arofe from their omitting to put down fuch quantities fraction wife as they ought to have done, and not ordering their fluential expreffions in the manner exemplified in our Review above quoted, p. 332. And we are farther confirmed in this opinion by obferving, that at p. 111 of the Summation of converging Series, where an expreffion of this kind occurs, namely, 1-y

tionwife, they immediately say =

put down properly frac1+y+y2.y3, which is its

true value, and they certainly then would not have put the finite factors downo, when y. But as we obferved in our Review, errors of this kind are far from being the only material ones in M. Lorgna's treatife.

Mr. Clarke feems at first not to have been aware of the extent of Mr. Simpson's theorems; indeed, as that Author has not given any plenty of examples to illuftrate them, they do not appear at first fight to comprehend that univerfal variety which they are applicable. And if Mr. C. had been inclines to make use of this plea, had he not turned the tables upon h very able antagonist Mr. Landen, he might have bespoke the indulgent favour of his unprejudiced readers; but fo far from this is his prefent mode of proceeding, that, beginning his bod with the motto, Damnant quod non intelligunt, which muft cou with a peculiarly bad grace from a man that, having publishe very erroneous book, was obliged to print another to comme and defend it, he pertinacioufly adheres to whatever he had before, as much as poffible, whether right and wrong.

That fuch of our Readers as have a taste for these subje&st be able to form a judgment of this controverfy, we will en

vour, as impartially and briefly as we can, to represent the matter as it appears to us.

M. De Moivre, in the first chapter of the fixth book of his Mifcellanea Analytica, has fhewn how, from fome of the moft fimple and well known feries, we may afcend to very compli cated ones, whofe fums fhall thus be given; viz. by taking such fimple feries, and equating it to its known fum, then multiplying each fide of this equation, by the fluxion and some power of the unknown quantity conftituting the feries, he finds the fluents on each fide, whence he obtains another infinite feries and its fum; and again multiplying each fide of this laft-found equation by the fluxion and fome power of the unknown quantity, and taking the fluents, he obtains another infinite feries and its fum; and in this manner it is evident he might have proceeded ad libitum, obtaining feries and their fums fuch, that if the first or original feries bad only one term or factor in its denominator, the second would have two, the third three, the fourth four, &c. The method being effentially the fame as that fince made ufe of by Mr. Lorgna, who, however, has patience to carry the matter in this form to a much greater length. This perhaps may be more apparent from M De Moivre's fummary of his conclufions, at p. 120 of his Mifcellanea Analytica.

If there be taken, fays he, the feries beginning with unity, and having its terms in geometrical progreffion, this multiplied by the fluxion of the common ratio, and the fluent taken, will give the most common logarithmic feries; and if this and its fum be multiplied by the fame fluxion as before, and the fluents on each fide of the equation taken, a new feries will arife, whofe fum will be found either by means of quantities wholly rational, or by thofe that are partly fo and partly logarithmic. If both the fides of the equation made by this laft-found feries and its fum, be multiplied ftill by the fame fluxion, a third feries and its fum will arife by taking the fluents as before, &c. and if any one of the feries fo produced be multiplied not only by the fluxion above-defcribed, but alfo by fome given power of the unknown quantity or ratio itself, another feries will arise, with a fum either wholly rational, or partly fo and partly logarithmic.

If any of the feries generated as above, or fome of them multiplied by the fame or different numbers, be joined at pleasure either by addition or fubtraction, other feries will arife, whofe terms fhall be fuch, that the law by which their numerators increafe may vary at pleafure; bút till thefe feries may be dif folved into more fimple component parts by help of the wellknown method of differences.

The numerators may alfo increafe for another reafon; for having arrived at fome feries whofe fum is exhibited in finite

U 3

terms,

termis, let it and its fum be multiplied by any power you please of the unknown quantity, then the fluxions of each being taken, there will arife, on one fide the equation, a new feries, the numerators of whofe terms fhall be in arithmetical progreffion, and on the other fide will arife the fum of the feries.

If this new ferics be again multiplied by the famé power of the unknown quantity, or by any other power thereof, and the Aluxions taken, a third feries will arife, the numerators of whofe terms fhall be fo conftituted, "that their fecond differences will be equal; and the fum thereof will be given from the fluxion of that of the preceding one, and thus we may proceed at pleasure."

And now, let any one pleafe to compare this with the account given in our Review above-quoted, p. 328, and 329, and then judge whether the foundation of Mr. Lorgna's method be not here given in the words of M. De Moivre. Only Mr. Lorgna, inftead of beginning with M. De Moivre's geometrical progreffion, multiplies that progreffion by the fluxion of the unknown quantity or ratio, and by fome power of the faid quantity itself, and finding the fluents as directed before by M. De Moivre, he makes that his initial feries. It is true, he makes the exponent of the affumed power of the unknown quantity a broken number; but this answers no end, but to keep the numerators of his feries, as they arife, clear of any common factor or multiplier; it neither renders the conclufions more general, nor the operations more fimple and clear," However, having done this once, he continues to do it again and again, in the manner directed by M. De Moivre, whofe words, though wrote about 50 years before the publication of Mr. Lorgna's book, give a remarkably good account of the method therein purfued. Now, as Mr. Clarke and Mr. Landen have contrary views, the one to fet off the method as much as poffible, and the other to deprefs it, a fair statement of the cafe was hardly to be expected from either of them. We therefore thought that an endeavour at this might prove acceptable to fuch of our Readers as are lovers of this kind of learning, and fhall therefore, beg leave to proceed a little further with our account of the

matter.

M. De Moivre, not having treated the above-defcribed method in a way fufficiently general, is obliged to have another. chapter purposely to fhew the manner of regrefs from a proposed feries to its fum: fo that the honour certainly belongs to M. Lorgna, of having rendered the method fo univerfal as to facilitate, as much as poffible thereby, the regrefs from the feries to its fum, extending to all fuch as agree with his general forms.

The property of the differences, and other relations of the factors compofing the terms of a feries, and fhewing the method of its continuation, was introduced with great fuccefs by M. De

Moivre,

Moivre, but chiefly as a foundation for folving the most difficult problems about what he named recurring feries; but as there was a confiderable difference between these recurring feries and others, it seemed doubtful whether the ways of proceeding, which fucceeded fo well with thefe, could be as fuccefsfully applied to others. But Mr. Stirling, having made the trial, found it to fucceed beyond expectation: for he found that this invention of M. De Moivre fupplied the moft general, and at the fame time the moft fimple principles, not only for recurring fe-. ries but for any others, wherein the relation of the factors of the terms varied according to fome regular law. For the relation of the terms being affignable, though variable, the fummations, interpolations and other difficult problems of this kind, are hence brought to a certain species of equations, which, befides the root that is to be extracted, involve other unknown quantities, that cannot be exterminated. But, notwithstanding this, the refolution of thefe equations is effected, fometimes with the greatest facility, but at others not without M. De Moivre's artifices for affigning the terms in recurring feries. And this is the bufinefs of the greatest part of Mr. Stirling's tract on this fubject.

Mr. Simpfon, taking up the matter as M. De Moivre and Mr. Stirling had left it, makes the most of what they had dif covered concerning the differences, and other properties of the factors of the terms of the feries: and that to fo good purpose,. that his fourth propofition, Mr. Landen obferves *, is a general formula, comprehending all the feries in Mr. Lorgna's first eight fections; and this 4th propofition is only a particular cafe of his 5th. The 6th and 7th propofitions and their corollaries contain very general and ufeful forms, for feries, where the number of factors in the numerators and denominators of the terms, varies according to fome certain law, as well as their respective values. The demonftrations of the propofitions are very true and fatisfactory; but the unfkilled and inexperienced Reader is not to fuppofe that they were found out in the manner in which they are there delivered. The firft and the other fundamental ones especially, are the confequences of repeated trials and deductions, which conftituted the real analyfis, of which the propofitions themselves are properly the fummary. Whereas, Mr. Lorgna's procefs, as far as it goes, is a complete analysis, delivered, to appearance, in the manner it was found out. At leaft, this is our opinion, and our reafon for making the remark is, that young readers may not be frighted from thefe ftudies by fuch complex operations, or by thofe ftill more compounded

This, however, is to be understood in a qualified fenfe; it does not properly comprehend them without fome additional artifices.

U 4

ones

ones deduced therefrom by Mr. Landen and Mr. Clarke... We would not be understood here to depreciate general forms, which, in compounded cafes, are eligible and ufeful; what we mean to infinuate is, that there may be bounds beyond which they may ceafe to be fo. Thus, whoever attentively confiders Mr. Simpfon's propofitions above-mentioned will fee, that if the feries contained in one or more of them, be added, fubducted, multiplied, or otherwise compounded with others, we shall obtain other forms ftill more general and more complex, and fo on að infinitum. There must therefore be fome limit or other, beyond which it may not be eligible to compound these things. For if fuch forms, as that given by Mr. Landen at the end of his Obfervations, be enough to affright an old foldier, what shall we fay to thofe that are ten times, nay a hundred or a thousand times more complex. In real practice, we seldom meet with any very compound ones, agreeing at the fame time with a preaffigned general form; thofe that occur commonly require fome previous reduction or transformation to fit them to the formula. Of this Mr. Simpfon (vide p. 99 of his Mathematical Differtations) was well aware, and exemplifies accordingly how needful it is (for avoiding trouble) firft to reduce every feries to the moft commodious form, before we fet about to determine its vaJue. And we will venture to add, that when it is fo reduced, it often appears at fight, how without the ufe of complex forms, it may be farther reduced to very fimple ones, whofe fums may be immediately apparent. His concluding example at p. 98, of his Differtations, might furnish an instance; but that at p. 85 fupplies one still more remarkable; where from the fum of the + &c. being given S, he pro

feries z+ +

5

pofes to find that of

% 9

+

9

13

y

+

312

6y3

+

10 y + 1.5.13 5.9.17 9.13.21 13.17.25 &c. But in order to adapt this to his forms, he directs to puty, and then fhews that the feries itfelf will thus be

4

4.8

+

+

+ &c.

8.12z+ 12.1628 changed to X: 32 1.5.13 5.9.17. 9.13.21 which now agrees with his forms. But it is also now very plain, when it is fo transformed, how its fum may readily be found without them. For the relation of the factors in the numerators, fo as to have conftant differences to the fame respective ones in order in the denominators, is eafily difcovered, perhaps as eafily and foon as we could, by comparing the factors of the terms, be fure that the feries was adapted to any general formula. Thus then the laft feries evidently becomes (omitting

for