To the Authors of the MONTHLY REVIEW. GENTLEMEN, I obferve, in your laft Review, a Letter from one Mr. A.B. who feems to be much difturbed at your account of my Refidual Analyfis. To allay his vexation, he, at first, feigns a fort of a fneer at his own pretty conceit, that the encomiums you were pleased to bestow on that work, were, perhaps, only by way of ridicule, or burlefque: but, being uneafy under the apprehenfion, that no body will view what you have faid, in that ludicrous light, he prefently ceafes endeavouring to be witty, and, giving vent to his indignation, proceeds to condemn, without mercy, every article of my little effay, becaufe feveral things therein advanced are not explained fo fully as might be expected in a large treatife! Methinks I discover in Mr. A. B's pretended true state of the cafe, with refpect to the invention of the Refidual Analysis, several marks of spleen, but not one of candour. He very unfairly confiders my fhort difcourfe, not as a fpecimen only, but as the whole of what I can fay on the fubject. His cenfure is, indeed, fo ill grounded, that, perhaps, I ought not to take any notice of it; yet I cannot forbear troubling you with a few animadverfions thereupon, which you will do me the favour to infert in your next Review. Mr. A. B. fays," the title of Refidual Analyfis is no more than a new term given to Sir Ifaac Newton's method of Differences*, "which is treated of by Dr. Brook Taylor, Mr. Cotes, and Mr. Stirling."- What could induce him to affert fuch a notorious falfity when every one who is converfant in Mathematics, and has feen my fpecimen, must be fenfible, that there is no more likeness between the Refidual Analysis and the faid method of Differences, (or, indeed, any other method of computation taught by the Gentlemen I just now mentioned,) than between the method of Fluxions and that of Exhauftions? Muft I impute this ftrange and unjuftifiable affertion of his, to his ignorance, or difingenuity? or both?— May be he means fome other differential method; Dr. Barrow's, perhaps, or Mr. Leibnitz's. Pray, Gentlemen, defire him to recollect himself. Sir Ifaac Newton's, I will venture to fay, is the only differential method hitherto publifhed, that does not admit fuch exceptionable principles as were rejected by the ancient Mathematicians; to wit, the confidering quantities as compofed of infinitely fmall particles, and the rejecting certain quantities, as being infinitely lefs than others. In the Refidual Analyfis, no principles are admitted but fuch as were anciently received in Algebra and Geometry: wherefore, as by that Analysis we perform fuch things as have not before been performed, without admitting new principles; and as the me • Sir Ifaac Newton's method of Differences (which is treated of by these Gentlemen) is applicable to little more than the drawing curves through a number of given points, the approximating the areas of curves, and the fummation and interpolation of certain feries. thod thod of proceeding, in the application thereof, is, for the most part, new; I have ventured to call it a new Branch of the Algebraic Art; and, I am perfuaded, the candid and judicious will allow it to. be fuch,, whatever they may think of it, as to perfpicuity and elegance. Upon my declaring, that I think the method of Fluxions is not the most natural method of refolving many problems to which it is ufually applied ;--Mr. A. B. remarks, that I fhould have given fome examples to prove the affertion; which, he fays, I could not do Is not this an unaccountable oversight in him? or, what is it ?I have given two examples; one of which is the investigation of the Binomial Iheorem, (which you, Gentlemen, allow is more naturally investigated by the Refidual Analyfis than by Fluxions,) and the other is, the affigning the value of the fraction, or quotient √2rx-xx3 r-√rx when x is equal to r; which, I prefume, the moft ftrenuous advocate for Fluxions will not fay is so naturally determined by that method, as by the Refidual Analysis: for can it be thought natural to bring motion into confideration in refolving a problem purely algebraical?—I could have given many more examples, had I not ftudied to be brief. Nay, perhaps, fome other articles in my effay are not improper examples to prove fuch affertion For, if it be true, as I prefume it is, that the principles. borrowed from the doctrine of Motion, with a view to improve the algebraic art, are not the genuine and proper principles of that art; it will follow, that in the investigation of geometrical and phyfical propofitions by means of algebraic proceffes, the Refidual Analyfis, which is founded entirely on the anciently received principles of Algebra, is more properly applicable than the method of Fluxions, which is founded on thofe borrowed principles. Mr. A. B. exults vaftly at having difcovered, that the theorem which enables us to perform certain operations in the Refidual Analyfis, may be obtained from one which Mr. Mac Laurin has given in his Algebra.- I acknowlege it may; and that a theorem fimilar to Mr. Mac Laurin's is to be found in Mr. Simpfon's Algebra: yet still may mine be a new one.-It follows from Divifion, (as any one who is but very little acquainted with Algebra may perceive) that a-t a-b -is = a + a2 b+ar-362 (), r being an Integer; which is an old, and well-known theorem. From whence, by writing am and m, instead of a and b respectively, we have "ta" 112 '+a": 2 b" +a""3mm () being an Integer; which agrees with Mr. Simpfon's theorem. And hence, by fubfti ger; which agrees with Mr. Mac Laurin's theorem, tho' not expreffed exactly in the fame manner. Each of these theorems refults from a bare divifion; or each may be obtained from the other, by only altering the Exponents, as above. My theorem + is very different from thefe, it cannot be fo obtained; it is true, it may be deduced from either of them, by taking a few fteps, and making a certain fubftitution; but the procefs neceffary for that purpose, tho' eafy enough, is, I conceive, much lefs obvious than the method of obtaining either of the theorems above fpecified. My theorem im mediately gives the quotient of a "-b" divided by a-b, in finite terms, whem m and n are any integers; (which quotient is frequently required in the Refidual Analyfis ;) but fuch quotient cannot be immediately affigned by either of thofe other theorems, alter the Exponents how you pleafe. Mine, therefore, being effentially different from thefe, and not taken notice of in any book of Algebra that I know of, may, I prefume, be called a new one. Add to this, I had mine by me, and had inveftigated feveral theorems by means thereof, before I faw either Mr. Simpfon's or Mr. Mac Laurin's. However, had even my theorem been inferted in every book of Algebra, without applying it as I have done, the Refidual Analyfis would neverthelefs have been a new invention; for, in that Analyfis, the principal thing is not the difcovering that obvious theorem, but the contriving to bring it into use. 'I have faid in my eflay, that there are quantities of various kinds, which we cannot conceive to be generated by motion :-To this our Critic objects; calls them quantities of my own creating; and fays, *Mathematicians have hitherto known of no others than the con"tinued and difcontinued."-I grant, that quantities are either continued or discontinued; yet, (to name no more,) are not weight, denfity, and tenfion, quantities of various kinds? and can they be conceived to be generated by motion? The Letter-Writer afks what I mean by Algebraic Quantities; and fays, fuch quantities were never heard of before: and, not confidering, that, in a new method of computation, new terms, &c. may 72 *Mr. A. B. fays, "It is plain that must be a whole number;" but that is a mistake, for n may be a fraction as well as m. be unavoidable, (witness the new terms and characters in the method of Fluxions) he objects to prime member, function, &c. as terms never heard of before.-Alas! how egregioufly does he betray his ignorance! ridiculous enough, indeed! to fet up for a Critic in Algebra, and never heard of Algebraic Quantities, and the term Function! I am under no apprehenfions, that the opinion of fuch a Critic can have any weight with the public: therefore, to the fuppofed faults he has accused me of, with refpect to my method of proceeding in the investigation of propofitions, I fhall not give myself the trouble to fay any thing. Walton, near Peterborough, I am, Gentlemen, Your most humble Servant, MONTHLY CATALOGUE. For AUGUST, 1759 MISCELLANEOUS. Art. 1. A Difcourfe delivered in the Theatre at Oxford, in the Senate-houfe at Cambridge, and at Spring-Garden in London, by Thomas Sheridan, M. A. Being introductory to his Course of Lectures on Elocution and the English Language. 8vo. 15. Millar. WE E find very little in this Difcourfe either to applaud or cen fure: it being mere declamation, calculated to recommend the study of oratory,-under fo able a profeffor as it is infinuated Mr. Sheridan would prove to be, if placed in either University, and encour aged by a handfome falary. Mr. Sheridan talks much of the evils attending our neglect and difufe of elocution, and the great good that would follow the cultivation of its study and practice. He does not, however, point thefe out very clearly: and, indeed, while we admit the force of oratory, and know that its bufinefs is rather to inflame the paffions than inform the judgment; to perfuade rather than to convince; the art of eloquence, however effential to the character of a polite and learned nation, will be made alternately the political inftrument of good and evil, according as its profeffors and adepts are actuated by good or evil purposes. Art. 2. The Doctrine and Practices of the Jefuits. Containing their Grounds, and a Detection of their fecret Designs, and bloody Projects. In a Letter to the Editor, from a Member of the English College of Secular Priests at Lisbon. 8vo. 1 s. Dilly. 3 We We have no perfuafion that this is really a Letter from an English fecular at Lisbon, but rather look upon it as the pious fraud of fome zealous Proteftant Writer at home, whofe intention is to inform us of the inftitution, doctrines, progrefs, and practices of the Jefuits; and which,' he says, will ferve to throw a light upon their part in this late confpiracy; [the attempt on the life of his Portuguese Majefty] to give thofe who are not well acquainted with the religious inftitutions in the Church of Rome, a proper hint to know how to diftinguish a member of this order from all others, and to caution every ftation and degree of mankind, to avoid a Jefuit, as the poifon, the pest, and destruction of individuals, focieties, and governments.' Page 6. Art. 3. An Account of the Methods used to defcribe Lines on Dr. Halley's Chart of the terraqueous Globe; fhewing the varation of the Magnetic Needle about the Year 1756, in all the known Seas; their application and ufe in correcting the Longitude at Sea; with fome occafional Obfervations relating thereto. By William Mountaine, and James Dodfon, Fellows of the Royal Society. Mount and Page. The chart, of which this little pamplet gives fome account, was originally conftructed by the celebrated Dr. Halley; and is, perhaps, one of the mott ufeful difcoveries, that has hitherto been made in the practical part of navigation: and therefore, we were furprized to find the ingenious gentlemen, who have re-constructed this chart, complaining, That it has never returned its firft charge." This is, indeed, poor encouragement for thofe who undertake fuch laborious tasks, for the good of the public. Its utility is allowed by all; and yet it seems very few have brought it into practice. Surely, a perfon must have little regard for his reputation, his fortune, his life, who fhall undertake to conduct a fhip through the pathlefs ocean to fome diftant coast, without taking with him a chart, which may prove of the greatest service in his voyage. It is well known to all, who have made obfervations on the magnetical variation, that its mutation is remarkably irregular in diffe. rent parts of the world; and confequently, the only method of conftructing a variation chart, is by collecting a large number of obfervations made by artifts in their voyages, and transferring them to the chart. This is the method thefe gentleman purfued, and the reader may form fome idea of the pains they must have taken, from the following account of the materials they were favoured with, and the number of journals they perufed for this purpose. The honourable the Commiffioners of the Navy,' fay they, obliged us with an order of free access to all their mafters log-books and journals. The Directors of the honourable the Eaft India Company, in dulged us with the like privilege. The Committee of the honourable Hudfon's-Company, gave us a fet of observations made in the tracks of their trade, and tabulated by fome of their captains. • Dr. |