equations are reduced so as to exterminate mathematicians that ever existed; for y, if the terms of the greatest dimensions ofy otherwise they would not have published contained in the two equations be relpec. the same. It remains for me to give tively n and modi by reasoning on some account of them ; particularly those the same principles, it is since proved, contained in the algebra, as it is the book that the dimensions of the equations, which was first published, and has been whose root is y, will be nm-rs, if one principally read. of the terms in which y has the greatest In the first chapter, are delivered feveral dimensions in the respective equations be elegant (as appears to me) rules, for find ing the sum of any functions of the roots * y in the one equation, and of a given or given equations.-I may particularize one, as it has had the honour in the other. of being published in the Paris Acts, by Mr. Bezout deduced this proposition Mr. Le Grange, one of the greatest and extended it to more (b) equations, Mathematicians that ever existed, and involving (b) unknown quantities x, y, perhaps is fuperior, in fome respects, to %, &c. of which the terms of the greatest every rule yet published in Algebra : I dimensions of y, &c. are correspondent. must mention, atto, another rule, for findFrom the principles mentioned above may ing the film of the powers of the roots of be deduced whether the equations resulting a given equations, in terms of the cowill afcend to the dimensions given by efficients of the given equation ; Sir J.N. Mr. Bezout or to less. before found the sums of the subsequent (n) The propositions given by Mr. Bezout from the sum of the preceding n-1, 1-2, may perhaps with equal or greater fa. 1-3, . 3, 2, 1 powers; but my rule, cility be deduced from the principles pub- when the feries converges, that is when lished in 1762. one possible root is much greater than any In the same book 1762, a rule is given other, not only finds the sum of the from having (n) independent equations powers, but also the sum of the roots; containing only nếm unknown quall from it has been given by me the law, tities of so reducing them, that there may result (m) equations, since called which the reversion of a series s j=axtbike equations of condition. As also from one or more (n) fimple equa tikt &c. obferves, and some other rions having two or more (n+r) unknown problems, by no other method, yet dirquantities x, y, z, &c. which he di- covered. Perhaps, the rule for finding mension in each is only one, of finding of the roots of a given equarion, may an equation, whose roots are any power their integral correfpondent values in terms of as B, y, &c.; where a, b, y, &c. properly be mentioned, as by it any irdenote any whole numbers whatever ; rational quantities may be exterminated. these have been both (since they were I shall, also, add the transformation of given by me) published by others. equations, as Mr, Le Grange prefers it Several other new propofitions, or rules to any other,--and lastly, the method of mine, have been since published by given, of finding the coefficients of the foreign mathematicians, and fonie by the terms of the transformed equation, from English. particular cases, a method in these subIn the above mentioned book, 1762, jects superior to any other, and of great was published a method of finding a quan utility; all of which were published in tity, which multiplied into a given ir the year 1762. rational quantity will produce a rational The second chapter treats of the affir. product, or which will consequently ex mative and negative, and impoffible roots, terminate irrational quantities out of a and the limits of equations, &c. I fall given equation—this is performed from particularize the rule for finding whether the roots of an equation xt1=0: ano the two possible roots of a biquadratic ther method in the said book was given the other two are impossible; because equation are affirmative or negative, when of reducing a given equation to as to exterminate its irrational quantities ; Mr. Mr. Le Grange has done me the honour Bezout has fince also given a rule for ing the number of impossible roots, from an to demonstrate it; also, the rule for find. exterminating irrational quantities. Having sufficiently thewn that many of the differences of any two roots of equation of which the roots are the squares mathematical inventions of mine were held in some esteem by the principal of whether all the roots of a giveu equation a given equation, and thence deducing the present time, and some of the first art 1799.). Original Letter of the late Dr. Waring. 309 are poffible or not; and whether 2, 6, 10, than the abovementioned fum 28. The &c. or 0, 4, 8, &c. are impossible, from number of inventions contained in the the last terın of it being possible or inore ; Meditationes Analyticæ, on the Modern this rule was generally given in 1762, Analysis, Fluxions, Serieses, &c. is many and the latter obfervation inserted in the more than the number in the Algebra; and Philosophical Transactions for the year the number of properties of curve lines 1764.-A rule is given for finding the deduced is not greatly less; to these add number of impoffible roots, from deducing the number in the addenda, Philosophical an equation, whose roots are the results of Transactions, and translation of algebraic the roots of the equation, commonly called equations into propable relations, and the the equation of limits, substituted in the number will be more than four hundred. given equation, &c.; this finds all that It would be too much trouble to review the preceding method does, and equally such a number of propositions, particularwith that finds the true number of im- ly as they are for the most part enumerated possible roots contained in any equations in the prefaces to the books themselves. of 3, 4, or s dimenlions; and perhaps I might equally particularife my inmore generally finds the true number of ventions in all other branches of matheimposible roots than any rule yet given: matics: in properties of algebraical it may easily be rendered more general. curves and solids, in conic lections, &c. I shall mention the following rules, be there are more new properties, and several of cause they have been fince published by them, as appears to me, of the firit degree eminent Mathematicians.-I, A rule for of elegance, contained perhaps in them than finding impossible roots from the equation in the works of any other writer, and in whose roots are the squares of the roots many the algebra and principles from of the given cquation, &c. 2. The finding which they are deduced, were allo invented the number of affirmative, negative and by me. Since the publication of the book, impoflible roots in an equation, whose I have given in the addenda several new roots bar any assignable relation to the properties, and extended most of the proroots of the given equation, from the perties of circles of Archiinedes and numbers of them in the given equation. Pappus to conic sections, and rendered 3. In the common resolutions of cubic fome more general, and given in the Philoand biquadratic equations, by the differ- fophical Transactions a rule for the demonent roots of the given equations, are ex. Itration of several propositions contained pressed the roots of the reducing one: in them and similar ones, from principles and vice verfâ from them are discovered, of algebra. how many roots of the given one are im Many new series are derived from difpoffible, &c. There are given more rules ferent principles ; some of which are the for finding the nuinber of impossible roots most converging, but in these cafes it is in equation, containing one or more un- cominonly necessary that a near approxiknown quantities, one of which always dit. mate frould be given, which will be the covers them, when Sir Isaac Newton's rule firft term of the series.-Rules are given does, and oftentimes when it does not. for finding the convergency of series, and Two elegant theorems are given for finding for rendering them converging when they when tome possible quantities are never express the Auents of Auxions, contained farily greater than others; these may be between different values of the variable demonstrated, by proving their difference quantities ; new integrals of increments ; to be the aggregate of certain squares. sums of series, of which the terms are given. There are fome theorems, which give the From approximations to the different ratin of the contents of all the quantities roots, or to two or more of algebraical resulting from substituting the roots of equations, art adduced more near approxione equation, for the unknown quantity mations.—Something is added of the in the other respectively, and multiplying difficulties which occur in finding the their results : this is inserted, becaule feries for the fluents of fluxional equations. several elegant properties of parabolas The rule of falle is rendered more general, have been deduced from it-Some truths by finding nearer approximations, when are deduced concerning equations, of two or more approximations are given, which the roots are the limits of each other, and the errors of their results. The same In my preface to the algebra, the num- is applied to more unknown quantities ber of inventions enumerated in these two contained ; a new method of differences chapters are 9 and 19 respectively, their and correspondent values is added, with fum 28; in each of the three fucceeding some problenis thence deduced, &c. chapters, the number enumerated is inore Many new propositions are invented, MONTHLY MAC, No. XLIV. in RE .n the science of Auxions and integrals of Has he ever read or underftood the 'increments; whoever would see them writings of the English mathematicians : enumerated, may consult the prefaces of and, as the question comes from me, I the different books, from which they seem subjoin particularly of mine? If the more in number than perhaps given by answer be in the negative, as it is my any other writer whatever ; but it may be opinion, if his answer be the truth, that added that several new additions to the it will; then there is an end of all further algebra, properties of curve lines and controversy ;-but, if he asserts that he folids, fluents and fluxions, integrals and has, which is more than Condorcet did by increments, seriefes have been given since his own acknowledgment, then he may their publications in the addenda, printed know, from the enumeration of inventions leveral years paft, or in the Philosophical made in the prefaces, with some fubfeTransactions. quent ones added, that they are said te Some inventions are given in my amount to more than four hundred of one pamphlet on the translation of algebraic kind or other ; let him try to reduce quantities into probable relations, and those to as low a number as he can, with some in the paper, inserted in the Phi- the least appearance of candour and truth ; losophical Transactions on central for- and then let him compare the number, ces, attractions, &c. treated in a more with the number of inventions of any general manner than before given by any French mathematician or mathematicians, writer. Mr. Le Grange has since pub- either in the present or part times, and lished something similar on the same there will result, a comparison (if I mis. subject, and also applied the principles take not) not much to his liking; and, to centers of gravities, oscillations, &c. further, let him, compare some of the first On the whole, let the name of any inventions of the French mathematicians French writer be mentioned, who has dil. with some of the first contained in my covered a greater number of inventions in works, both as to utility, generality, pure mathematics ; ; he may see much the novelty, difficulty and elegance, but wisegreater part of them enumerated in the ly as to utility, there is little contained in respective books; about fifty or fixty are the deep parts of any science; he will find publised in the addenda, &c. which are their difficulty and novelty, from his not contained in the enumeration above- difficulty of understanding them, and his mentioned. I have no mathematical never having read any thing similar bebooks at present in my possession; but fore; their generality, by the application perhaps, I could from memory enu- of them; principles of elegance will difmerate the principal inventions, given by fer.in different persons.- I must say, that the different authors, before the publica. he will probably not find the difference tion of my books. expected. After or before this enquiry General assertions without proof af. is instituted for mine, let him perform the ford no conviction, and merit no re same for the other English mathematicians, gard; and particularly, when made by and when he has completed such en. persons ignorant of the truth or fallhood quiries, and not before, he will become a of the proposition asserted; for then they judge of the justice of his affertion ; but deserve silence and contempt. I am afraid, that he is not a sufficient I know that Mr. Lalande is a first rate adept in these studies, to institute such astronomer, and writer of astrononiy ; enquiries ; and if he was, such enquiries but I never heard that he was much con are invidious, troublesome, and of small versant in the deeper parts of mathematics utility. I am, Sir, with very great regard for which reason I take the liberty to ask and esteem, Yours affectionately, him the following questions ; E. WARING. TO CORRESPONDENTS. The writers on controversial points, and answerers of queries, are respectfully apprized, that our readers generally think topics fully discuffed long before all the letters we receive under these heads are inserted; whence we find it necessary to exclude some communications, perhaps fully as valuable as those we have admitted, merely to avoid tedious repetitions. Several defences of the principles of th: Quakers, and explanations of the word bitch, are now under this predicament. We have determined to close all difcuffion relative to the Northampton Academy, and Disienting Ordinations. Long political discussions are at no time well suited to our Miscellany, and are at present what we would peculiarly avoid. Some communications under this head lie ready for return to their respective writers. We beg leave to decline engaging in a controverty relative to Collier's patent filtering machine. W. H. is informed that a letter for hun is lying at No. 71, St. Paul's Church-yard, conClining President Bradshaw. ALPHABITICAL LIST of BANKRUPTCIES and DIVIDENDS announced between the 2016 of March, and the 20th of April, extra&ted from the London Gazettes. BANKRUPTCIES. Chaffers, R. Tooley-ftreet, merehant, May, 18. Dale, c. Aldersgate-itreet, banker, April, 27. Dale, R. and T. Martin, Little Britain, brokers, April, 20, Dod, J. Lime-ftreet, packer, April, 16. LLWOOD, T. Great Russell-freet, Bloomibury, currier. Dunderdale, D, Holbeck, clothier, April, 26. Donadiou, G. Hatton-ftreet, bıandy-nierchant, May, 8. Banton, E. Lancaster, merchant. Catton, Gray's-inn. Ellwood, J. Brisco, callicu-printer, April, 30. Bell, T. Woolsthorpe, miller, Wbibaw and Taylor, Gray's-i. Fenn, B, Cornbill, hosier, April, 20. Blacklin, J. King ton, York. linen-draper. Picard, Huil. Fairbone, G, King-ftreet, ironmonger, June, 15. Boyd, J. Conv ay-street, builder. Jennings, Great Sbire-lane. Forrest, J. Life-ftreet, woolen-draper, May, 14. Bott, Ġ. jun. Birmingham, dentiit. Sanderson, Paifgravi-ple Fearon, H. St. Mary Axe, factor, May, 18, Barron, w. Soulcoates, York, grocer, Allens, Furnival's-inn. Freer, R. Leiceiter, hofier, May, 16. Coles, w. Chelsea, baker. Mr. Owen, Teinple. Grainger, T, Bristol, cabinet-inaker, April, 30. Clark, J. fen. Great Totham, wheelwright. Mr. Forbes, Aly-p. Griffin, j. Fareham, draper, May, 14. Carr, W. Haltwhiftle, dyer, &c. Mountsey, Staple's-inn. Gates, W. Manchester, tallow-chandler', May, 7. Cotsford, F. Bihop': stortford, upholkerer. Husy and En Hulme, T. and J. Lancashire, Manchester, callenderers, Aprií, 16. giand, Od Broad-strett. Entwille, J. Manchester, fetian-manufacturer: Mr. Jobnson, Hunt, D. Birmingham, buckle-maker, April, 15. Mancbifter. Hanmer, T. jan. Brinól, grocer, april, 22. Entwife, A. Rivington, bleacher. Edge, Temple. Hogsfleth, G. and R. Phipps, Gutter-lane, ribbon-manufacEnfor, W. jun. Newcastle, Staffordfhire, ironmonger, Lowten, turers, April, 27. Temple. Harper, W. and j. Wilson, Caftle-court, merchants, May, 46 Felton, A. Salford, fithmonger. Mr. Edge, Temple. Hall, E. Rochester, taylor, June, 22. Fithwick, W. Duke's.court, St. Martin's-lane, taylor. Spear Holbrow, W. Dniley, scrivener, May, 22. Huweil, J. Ferrer-lane, carpenter, June, 15. inz, Wa.brooke. Guy, R. jun, Artillery-place, mariner. Mefrs. Crompton and Hewlett, H. Wallington, taniner, April, 18. Lis Curfitor.freet. Hunt, D. Birmingham, buckle-maker, May, 10. Holt, E. Bolton in the Moors, fuftian-inanufacturer, May, 14. Gowan, G. 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