in a writer whose own works are not brought into the comparison.' I do not say (p. 2.) that Mr. Simpson's method is the same as M. Lorgna's ; but I contend, that it is more perspicuous and elegant than his, and not less (if not more) general, • Mr. Simpson, in solving the problem, does not consider how the proposed series may arise by taking fluents and Auxions, as Mr. I.orgna has done ; but he resolves the series which is to be summed into as many other series (each having a simple denominator, and being summable by a known theorem) as there are factors in each denominator. And thus easily, in four or five pages, obtains a general solution to the problem ; about which Mr. Lorgna has written a treatise, without exhibiting any thing equally comprehensive. Mr. Clarke talks of my putting Mr. Simpson's theorems to the rack, in order to extort a secret from them which they do not contain: but ic is evident that, without any torture, those theorems impart such a secret as Mr. Clarke may blush to find divulged ! 'The formulæ (p. 9.) which Mr. Lorgna deduces are none of them properly adapted for finding the complete criterion of algebraic fummation, which is vainly pretended to be pointed out! Nor are they better adapted for finding, by induction, a general theorem like Ms. Simpson's : for after all Mr. Lorgna has done, though he has made a book of his labours, he has not discovered any such theorem ! nor does it appear that he had any notion of a general theorem being attainable by induction.' Will it be believed (p. 17.) shat if he had well understood the problem, he would have omitted some principal parts of its solution? And can it be supposed that if Mr. Clarke had understood it better, he would not in his notes have supplied the defects ? Truly (p. 14.) may it be said, we feldom find a writer fo little skilled in the subject he treats of, as our Author and Commentator appear to have been with respect to the clats of series whole lume mation they have attempted!'.: * Of the numerous errors (p. 17.) which I might have remarked in his creatise, I have (in my Observations) only adverted to three; which are all of one kind and luch as I consider as errors of judgment (that might, if not corrected, milead the Reader), not aş triling inaccuracies or negligences, as the Supplement-writer calls them. For, whatever he aod his Aythor may now know; it is not to be doubted (after so many proofs) that as they had in the case where yi, repeatedly considered the Log, of a's the Log. of 1, and therefore = 0; if fuch expression had again and again occurred; they would U 2011 again -y8 I again and again have committed the same error: which, probably to this time, would have remained uncorrected by them, if it had not been pointed out to them. The consequences of which, as far as Mr. Lorgna's conclusions are affected by it, are noticed by me, with the same strict regard to veracity as is observed in ihe writing down every other article in the pamphlet.' So far Mr. Landen; but it may not perhaps be improper for us to stop here, and endeavour to represent this last matter as it appears to us. That excellent mathematician, L. Euler, in his Institutiones Calculi differentialis, has a whole chapter on the subje&t of such fractions as this here specified by Mr. Landen, and thews lo clearly and fully that the numerators and denominators have not generally the ratio of equality when they vanish together, as in the case here of y=1, that one would think that neither Mr. Lorgna nor his Commentator could posibly be ignorant of it. We rather think that the errors arose from their omitting to put down such quantities fraction wise as they ought to have done, and not ordering their fuential expressions in the manner exemplified in our Review above quoted, p. 332. And we are farther confirmed in this opinion by observing, that at p. m of the Summation of converging Series, where an expression of this kind occurs, namely, put down properly frac تراو . او - 1 ity+yoyoy, which is its ity tionwise, they immediately say = true value, and they certainly then would not have put the finice factors down = 0, when y =1. But as we observed in our Review, errors of this kind are far from being the only ma. terial ones in M. Lorgna's treatise. Mr. Clarke seems 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 illustrate them, they do na appear at first sight to comprehend that universal variety which they are applicable. And if Mr. C. bad been inclina to make use of this plea, bad he not turned the tables upon to very able antagonist Mr. Landen, he might have bespoke sh indulgent favour of his unprejudiced readers; but so far from this is his present mode of proceeding, that, beginning his bod with the motto, Damnant quod non intelligunt, which must cor with a peculiarly bad grace from a man that, baving publith. very erroneous book, was obliged to print another to corre and defend it, he pertinaciously adheres to whatever he had before, as much as poffible, whether right and wrong. That such of our Readers as have a taste for these subje&s be able to form a judgment of this controversy, we will cru vour, as impartially and briefly as we can, to represent the mat., ter as it appears to us. M. De Moivre, in the first chapter of the Sixth book of his Miscellanea Analytica, has fewn how, from some of the most fimple and well known series, we may ascend to very complicated ones, whose fums (hall thus be given ; viz. by taking such fimple series, and equating it to its known fum, then multiplying each side of this equation, by the Auxion and some power of the unknown quantity conftituting the series, he finds the Auents on each side, whence he obtains another infinite series and its sum; and again multiplying each fide of this last-found equation by the Auxion and some power of the unknown quantity, and taking the fluents, he obtains another infinite Series and its sum; and in this manner it is evident he might have proceeded ad libitum, obtaining series and their sums such, that if the first or original feries had only one term or factor in its denominator, the second would have two, the third three, the fourth four, &c. The method being essentially the same as that since made use 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 rum. mary of his conclusions, at p. 120 of his Miscellanea Analytica. If there be taken, says be, the series beginning with unity, and having its terms in geometrical progression, this multiplied by the Auxion of the common ratio, and the Auent taken, will give the most common logarithmic feries; and if this and its fum be multiplied by the same Auxion as before, and the fluents on each fide of the equation taken, a new series will arise, whole fum will be found either by means of quantities wholly sational, or by those that are partly so and partly logarithmic. If both the sides of the equation made by this latt-found' series and its sum, be' multiplied ftill by the same fluxion, a third sę. ries and its fum will arise by taking the fluents as before, &c, and if any one of the series fo produced be multiplied not only by the fluxion above-described, but allo by some given power of the unknown quantity or ratio itself, another series will arise, with a sum either wholly rational, or partly to and partly logarithmic. If any of the series generated as above, or some of them mul. tiplied by the same or different númbers, be joined as pleasure either by addition or subtraction, other 'lerięs will arise, whose Cerms Mall be fuch, that the law by which their numerators increase may vary at pleafure; but itill these series may be dira folved into more simple componunt parts by help of the well300vn method of differences. The numerators" may also increase for another reason; for having arrived at some series whose fum is exhibited in hnite U 3 terms, termis, let it and its sum be muftiplied by any power you please of the unknown.quantity, then she fluxions of each being taken, there will arise, on one side the equation, a new series, the numerators of whose terms shall be in arithmetical progression, and on the other side will arise the sum of the series. If this new series be again multiplied by the samé power of the unknown quantity, or by any other power thereof, and the Auxions taken, a third series will arise, the numerators of whose terms shall be so constituted, that their second differences will be equal'; and the sum thereof will be given from the Auxion of that of the preceding one, and thus we may proceed at pleasure. And now, let any one please 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, instead of beginning with M. De Moivre's geometrical progression, multiplies that progression by the fuxion of the unknown quantity or ratio, and by fome power of the said quantity itself, and finding the fluents as directed before by M. De Moivre, he makes that his inisial series. It is true, he makes the exponent of the afiumed power of the unknown quantity a broken number; but this answers no end, but to keep the numerators of his series, as they arise, clear of any common factor or multiplier; it neither renders the conclusions 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, whose words, though wrote about 50 years before the publication of Mr. Lorgna's book, give a remarkably good account of the method therein pursued. Now, as Mr. Clarke and Mr. Landen have contrary views, the one to set off the method as much as possible, and the other to depress it, a fair statement of the case was hardly to be expected from either of them. We therefore thought that an endeavour at this might prove acceptable to such of our Readers as are lovers of this kind of learning, and shall therefore, beg leave to proceed a little further with our account of the matter. M. De Moivre, not having treated the above-described method in a way fufficiently general, is obliged to have another chapter purposely to hew the manner of regress from a proposed series to its fum : so that the honour certainly belongs to M. Lorgna, of having rendered the method so universal as to facilitate, as much as possible thereby, the regress from the series to its sum, extending to all such as agree with his general forms. The property of the differences, and other relations of the factors 'composing the terms of a series, and thewing the method of its continuation, was introduced with great success by M. De Moivre, Moivre, but chiefly as a foundation for solving the most difficult problems about what he named recurring series; but as there was a considerable difference between these recurring series and others, it seemed doubtful whether the ways of proceeding, which succeeded so well with these, could be as successfully applied to others. But Mr. Stirling, having made the trial, found it to succeed beyond expectation : for he found that this invention of M, De Moivre supplied the most general, and at the fame time the most simple principles, not only for recurring leries but for any others, wherein the relation of the factors of the terms varied according to some regular law. For the relation of the terms being affignable, though variable, the summations, interpolations and other difficult problems of this kind, are hence brought to a certain species of equations, which, belides the root that is to be extracted, involve other unknown quantities, that cannot be exterminated. But, notwithstanding this, the resolution of these equations is effected, sometimes with the greatest facility, but at others not without M. De Moivre's artifices for asfigning the cerms in recurring series. And this is the business of the greatest part of Mr. Stirling's tract on tbis subject. Mr. Simpson, taking up the matter as M, De Moivre and Mr. Stirling had left it, makes the most of what they had difcovered concerning the differences, and other properties of the factors of the terms of the series : and that to fo good purpose, that his fourth proposition, Mr. Landen observes *, is a general formula, comprehending all the series in Mr Lorgna's firit eight sections; and this 4th proposition is only a particular case of his 5th. The 6th and 7th propofitions and their corollaries contain very general and useful forms, for feries, where the number of factors in the numerators and denominators of the terms, varies according to some certain law, as well as their respective values. The demonstrations of the propositions are very true and satisfactory; but the unskilled and inexperienced Reader is not to suppose that they were found out in the manner in which they are there delivered. The first and the other fundamental ones especially, are the consequences of repeated trials and deductions, which constituted the real analysis, of which the propofitions themselves are properly the summary. Whereas, Mr. Lorgna's process, as far as it goes, is a complete analysis, delivered, to appearance, in the manner it was found out. At least, this is our opinion, and our reason for making the remark is, that young readers may not be frighted from there ftudies by such complex operations, or by thole ftill more compounded • This, however, is to be understood in a qualified fense; it does not properly comprehend them without some additional artifices, oncs U4 |