The author of this work, which is efteemed the best book of its kind, begins his account of this fect with the history of its founders; and carries it on with that of all the feveral parties into which they have been fince divided; tracing them not only from their original, down to the prefent times, but following them even from the old world to the new; and entering into many curious particulars hitherto little known. In this work we meet alfo with fome anecdotes regarding Luther, and the part he took in the reformation; and in particular a letter, omitted in the collection of Luther's epiftolary correfpondence, wherein he approves of the marriage of Mr. Schuldorp, a priest of the dutchy of Holftein, with his fifter's daughter. Among many other matters, of confequence to the ecclefiaftical hiftory of those times, we find also an account of a convocation held at Flensburg, and a provincial fynod at Strafburg, of which most hiftorians have omitted to give us any information.

I

K-n--k

To the Authors of the MONTHLY REVIEW.

Gentlemen,

Have often looked on the encomiums with which have occayou fionally honoured fome authors, rather as marks of your goodnature, and a laudable defire of exciting emulation by encouraging merit, than as the effects of your impartial judgment. Your goodnature in this point may, however, be carried too far, unless you make a proper diftinction between the ardour of true genius and the arrogance of pretenders. If a tyro, for want of reading, fhould happon to publish what is already known, as a new invention, it is in iome degree exculable; he might be commended for his ingenuity, and advited to read before he should write again; but-when a profefied adept Aarts up, and pretends to difcoveries that have efcaped the fagacity of his predeceitors, the merit of his pretenfions ought turely to be ftituly examined. I fhould be forry to fufpect you, of facrificing the duty you owe to the public, to any partiality for parti cular writers; but really, gentlemen, as to your account of Mr. Landen's difcovery of the Refidual Analyfis, I scarce know what to think, The fuperiority of genius which you hint is difplayed in his invention, and the great importance of the object of it, were enough to make me break out into congratulations of the prefent age, on the appearance of lach a phenomenon, as that of a mathematician, not only of the fegt ef f, but who, at one leap could over-top the heads of all his predications and cotemporaries, and even jump higher than the very apex g in thematical harming: but, alas! when I came to confider the Latter, how much realon did I find to cry out with Horace:

wid dignum tanto fer et hic promiffor biatu? Parturient montes ;,nafcetur ridiculus mus.

Perhaps,

Perhaps, gentlemen, the extravagant encomiums you lavished on that work, were only by way of ridicule, or burlesque; if fo, certain it is there are many who do not take the jeft. For their benefit, therefore, and to give you an opportunity of difplaying your impartiality, I have fent you the following (as I take it) true ftate of the cafe, with respect to Mr. Landen's pretended new difcovery, which I hope to fee in your next Review.

First, the title of Refidual Analyfis, is no more than a new term given to Sir Ifaac Newton's method of differences, and therefore is no new branch of algebraic art: fince it has been known, and treated of by many, in a much more eafy and familiar manner than by Mr. Landen; efpecially, befides the inventor, by Brook Taylor; by Cotes in his Harmonia Menfurarum; by Stirling in his book called Methodus differentialis; and occafionally by many others. Mr. Landen will probably fay, that he has folved many problems thereby, to which it had not been applied by any other before him; for he will hardly affirm that he has done any more. This is true, because it may be done, with much lefs labour, and infinitely clearer, by the method of fluxions, nay even by the common method of differences; and therefore it would be ridiculous to use any other, such as Mr. Landen's.

To come to particulars, in page 4. he fays, Notwithstanding the method of fluxions is fo greatly applauded, I am induced to think, it is not the moft natural method of folving many problems to which it is ufually applied.' Here the author fhould have given fome examples to prove his affertion; which I am certain he could not do. He then proceeds, the operations therein being chiefly performed with algebraic quantities, it is, in fact, a branch of the algebraic art, or an improvement thereof, by the help of fome peculiar principles.' What does the author mean by algebraic quantities? Are they the types, letters, ink, or paper? Such quantities were never heard of before; and as he is the inventor of them, he ought to have explained them. He allows, however, that the method of fluxious is an improvement upon the algebraic art, but difapproves the principles made ufe of; if this could have been done without any new principles, I fhould be of his opinion; but the query is, whether this can be done or not? it is true, he pretends to thew in his work, that most problems may be folved without them. This was known before; but nobody has pretended to folve thefe problems in fo eafy and clear a manner as is done by fluxions and I may add, that his pretended Refidual Analyfis renders the invefligations more tedious and obfcure than any other whatsoever, at least in the manner he applies it, as I fhall fhew prefently.

In the fame page he continues; We may, indeed, very naturally conceive a line to be generated by motion, but there are quantities of various kinds, which we cannot conceive fo generated.' Here are more quantities again, created by the author, without informing the reader what they are, or what they are made of; for hitherto mathematicians have known of no others than the continued and difcontinucd.

His allowing that a line may be described by motion, is certainly a very great conceffion, and more than he ought to have made for the fake of his pretended new invention. For if a point may move, a line or a furface may move likewife, for the fame obvious reason; and confequently, a furface and a folid may be generated by the motion of a line or a furface.-Fluxions require no other motion. In the fame page he fays, that the borrowing principles from motion, was done without any neceffity or advantage.' To fhew this, in the next page he produces the following theorem, viz.

then continues, It is by means of this theorem that we are enabled to perform all the principal operations in our faid Analyfis; and I am not a little furprized, that a theorem fo obvious, and of fuch vast use, fhould fo long efcape the notice of algebraifts!'

But Mr. Landen furely, dares not fay this theorem is of his own invention, or that it was not taken notice of before. He may, perhaps, imagine he has fo difguifed it by a new form, as to make it pafs for his own, amongst credulous and ignorant readers. But to fhew that this curious invention is no new one, Mr. M'Laurin fays, in his Algebra, page 109. art. 118. Generally, if you multiply amm by a"-"+a"-2m+a"-3" 62"-a"-4m3m+, &c. continued till the terms be in number equal to the product will be ab". It

n

is plain that n must be a whole number and a multiple of m, and m may be a whole number or a fraction; in the latter cafe, its numerator is always equal to z, and the least number divisible, as he declares in art. 119. Now when m=1, then a-ba-b x a +a”¬3b+a ̈ ̄ ̄ ̄3⁄4bb, &c. and confequently,

172

2 b+a" 31b+ &c. (")

The numera

tor of the fecond fide being divided by a"-1 and the denominator by

gives Mr. Landen's form: and when ab, either form gives

which is all the confequence he can draw from this

The author cannot fure plead ignorance, and fay he has not read M' Laurin's work. This would look ridiculous for one who cites in his works, L'Hofpital, Bernoullie, Act. Erud. Lipf. Archimed. &c. to pretend not to know the authors of his own country, and in his mother tongue!

Te

To shew the advantage his method has above that of fluxions, he gives the binomial rule, or theorem: in page 6. he affumes 1+x = 1 + ax + bx2+ cx3 &c. and then he says, the reader must confider x as a line generated by motion, and to exprefs the velocity of the point defcribing the line x, and taking, by the rules taught by thofe who have treated of the faid method, the feveral cotemporary velocities of the other defcribing points, or the fluxions of the several

4x3, &c. and then gives reasons for thefe operations.

The method of fluxions being known, all thefe fpecious reafons and confiderations are abfurd, and ferve only to impofe on the ignorant, by making them believe, that the manner of finding fluxions is very tedious and obfcure. Then he proceeds to fhew how this is done much easier by his Refidual analysis, as follows:

Affuming as above 1+x=1+ ax + bx2 + cx3 + dx4 +, &c.

m

Affuming again 1+y'%=1+as+by2+cy2+dy1+ &c.

And by Subtraction

„= a. x—y + b.x2—j2 + c.x3—y3 + &c.

If, now, we divide by the refidual x-y (properly called difference)

we shall get,

+c.xx+xytyy + &c.

which equation must hold true, lety be what it will; from whence,

by taking v equal to x, we find as before x1+x) ñ1=a+2bx+ 3x+44 + &c.

Ngw all these various operations are performed at once by taking the fluxion of the fuppofed equation, and dividing by x. So that the reader may judge whether the author's boaft, of rendering the operations of this theorem more clear and concife than by means of fluxions, has any foundation or not.

In the next page, the author gives a theorem, as tedious and perplexed, as it is ufelefs; being no more than the first under a different form. He proceeds to find the value of a certain line, in algebraic terms, involving the fubtangent, without any regard to the generation of the curve, and then affumes that expreffion, equal to another, ([ fuppofe he means, that he affumes another equal to that. After a long procefs, and inventing new terms and figns, he brings out at last a general equation, wherein one fide contains both an indetermined

5

index

index and an indetermined quantity. Then he fays, It is easy to prove (but I fhall not flay to do it here) that the index m will, in general, be equal to 2 in this cafe.' Methinks it is the bufinefs of an author to ftay till he has proved his affertion, for the reader's fake: however, we will imitate him, and proceed to page 13, where he determines the indetermined quantity Qin his own way. All this takes up three full pages, which by fluxions is done, by fimply proving, that the fluxion of the ordinate is to the fluxion of the abfciffa as the ordinate is to the fubtangent. He might have proved, nearly in as fhort a manner, that the ultimate or vanifhing ratio, of the differences of the ordinates to those of the abfciffas, is equal to that of the ordinate to the fubtangent: this would have faved much labour, and the framing new figns and terms; but then others did this before him, as for example, Muller, in his mathematical treatise; and therefore, he would not pafs for an original writer; which, however, no body that perufes his work will now difpute.

In the examples he gives of this method, he makes ufe of his pretended theorem at full length, I fuppofe, to fhew every step of his operations to the reader; there is at least no other occafion for it, fince the conclufion drawn from it, would have ferved full as well, and with much lefs labour. The author now proceeds to find the greatest and least ordinates of curves, with as much trouble and unintelligible arguments as in finding tangents; but in the application to examples, he partly makes use of his theorem in full length, and partly of the conclufion only, or, properly fpeaking, of the rules of fluxions, which differ only in the name; for, as I before faid, Mr. Landen's method is no other than Sir Ifaac Newton's method of differences; and it is well known, that if the differences are diminished fo as to vanish, their vanishing ratio becomes that of fluxions; which is precifely the cafe with the above given theorem; for when v and x become equal, their difference xv vanishes. The author's process to find the radius of curvatures is even difficult to read, with that attention required to underfland his meaning: I therefore pass it over, with only obferving, that he gives but one example, and the easiest that can be found: any other would, probably, have disgusted the reader, and prevented him from going any farther.

But his method, of finding general expreffions, for areas and lengths of curvelines, is fuch as would tire the most patient reader. Here he has recourfe again to new figns, and new terms, never heard of before; fuch as prime members, functions, &c. and to abridge his labour, cites Archimedes's axiom, as he calls it, which no body will grant him, fince it is a propofition that requires a very ftrict demonftration; and upon which the truth of his whole procefs depends. This would, indeed, be fuppofing what is the most difficult to prove. However, the author prudently avoids examples, excepting one, and that the most fimple, by which he faves himself from perplexed calculations, and the reader from the trouble of learning what, by a very few and eafy principles, may be deduced from fluxions.

The author finishes his difcourfe, with giving the methods of computing the laws of motion, and centripetal forces; but as his proceed