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ORIGINAL LETTER OF THE LATE DR.

WARING TO THE REV. DR. MASKELYNE, ASTRONOMER-ROYAL. [The following interefting letter was written by the late Dr. WARING, in reply to a paffage contained in LALANDE's life of CONDORCET, which ftates, that in the year 1764, there was no first rate analyft in England. Had Lalande been as diftinguished a mathematian as he is an aftronomer, he would not have thus underrated the merits of fuch eminent British mathematicians as Lyons, Emerfon, Landem, and Waring, who all flourished at that period. Dr. Waring, who published his Mifcellanea Analytica" in 1762, and his "Meditationes Algebraica" in 1770. was the author of fome of the greatest difsoveries in algebra, algebraic curve lines, infinite feries, increments, and fluxions.]

DEAR SIR,

IT has been my misfortune to have had my writings attacked very early by feveral perfons; but it has been my good fortune to find that no errors in reafoning were detected, except two or three errata of the prefs, and numerous others in my errata pages; add to this fome general reflections which feem to betray both ignorance and fome envy, or malevolence of difpofition; to these I gave no answer, unlefs once, when compelled to it by a truggle for fubfiftence. My opinion is, that future ages will afcribe to writers their just merits without partiality, and if they do not, it is totally immaterial either to the writers then dead, or to their readers.

It is my cordial with, that no party fhould fubfit in fcience; names fhould be mentioned and not nations, and both paft, prefent, and future fraternifed.

Mr. Lalande, all aftronomers acknowledge, to be endued with the first rate abilities and knowledge in, and the writer of the beft fyftem of aftronomy; but of this you are the fupreme judge.

Every perfon that compares the works of different writers fhould have read and understood them.

I will confidently affert, that every intelligent reader of the inventions and writings of Harriot, Briggs, Napier, Wallis, Halley, Brancker, Wren, Pell, Barrow, Mercator, Newton, De Moivre, Maclaurin, Cotes, Stirling, Taylor, Simpfon, Emerfon, Landen, and others, will with contempt, reprobate the affertion (contained in your letter) of Mr. Lalande.

I must therefore afk Mr. Lalande, has he either underfood, cr even read the inventions and writings of the above-mentioned English mathematicians; and, in

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If he understood them, he has done more than Mr. Condorcet, for Condorcet, in a letter to me, acknowledges that he did not understand them.

I cannot but fay, that this at first gave me no great opinion of Condorcet's attention or abilities as a mathematician, for I never read any mathematical work, that I could not, with proper attention, understand the fubjects and the reasoning, though I have fometimes been almost obliged to decypher the language; but I prefume it probable, that Condorcet was at that time too much engaged in political to attend to deep mathematical matters. Perhaps, this may be faid to be owing to the obfcurity of the expreffions; of this, the readers must be the judges: I can only add in defence, that Mr. D'Alambert understood the first edition of the "Medit. Algeb. and Propr. Algebr. Curv." published in the year 1762; for he fpeaks highly of them in the Paris acts. Mr. Le Grange underftood them, for in fpeaking of my Algebra in the Berlin acts, the only book of mine he was probably then in poffeffion of; he recommends the confultation of it on the transformation of equations above all other books, and mentions it as a work full of excellent and interefting difcoveries in Algebra.

Some of the firft mathematicians in Germany and Italy understood it, for they exprefied their fentiments of it in letters to me; they elected me, without any folicitation on my part, into two of the moft refpectable focieties in the world Bononia and Gottingen; and feveral mathematicians of this country, though I cannot fay, that I know any one who thought it worth while to read through the whole, and, perhaps, not the half. of it.

Some of the greatest mathematicians that ever exifted, have paid fome regard to the inventions contained in my books, for they have published feveral of them.

In the year 1762 I published thefirst edition of my Algebra, and fent a copy of it to Mr. Euler at Petersburgh, containing the following refolution of equations,

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Mr. Condorcet affumes a for the chance of a perfons voting truly, b for the chance of his voting falfely, and c for the chance of his not voting at all, and from thence deduces a fimilar conclufion-More of thefe numbers may be added together, and more decifions inftituted, and their probabilities made

from it, i.e. more new equations have been refolved than by any other perfon. In the faid edition of 1762 was given a rule for finding impoffible roots of a given equation from an equation, of which the roots are the fquares of the roots of a given equation; a fimilar rule has been fince published in the Petersburgh acts; another rule for finding impoflible roots,, &c. or in any given ratio, &c. to was alfo given from finding an equation of which the roots are the fquares of the differences of the roots of the given equation, by which, from the change of figns are always difcovered when the roots of the given equation are all poffible or not; and from the laft term of the refulting equation, being either affirmative or negative, is difcovered whether the number of impoffible roots is either 2, 6, 10, &c. or o, 4, 8, &c.; this was published in the first edition of the Algebra, 1762, and in the philofophical tranfactions for the year 1764, and has been fince published by fome of the greatest mathematicians.

In the fame paper contained in the philofophical tranfactions for the year 1764, was introduced a new principle for finding whether the area of an algebraical curve can be expreft in finite algebraical terms, by affuming an algebraical equation, which neceffarily expreffes the algebraical relation between the area and the abfcifs, when they can be expreft in finite terms; and afterwards I published, in the Med. Analyt. the refolution of a more general problem on the faine principle: Mr. Condorcet fince did the fame for fome more algebraical equations on the fame principle above mentioned firft difcovered by me.

Mr. Condorcet did me the honour to fend me his book on the probabilities of juries differently inftituted; it contains many very fenfible reflections on political, as well as mathematical matters; I have not the book in my poffeffion, and I only fpeak from a faint memory; it contains principally the application of the binomial or trinomial feries

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each other. fome inftances have been given by De Moivre, &c. in the general reafoning, but very many on trials by Mr. Condorcet, but to me it appears very doubtful: 1. when the chance of any two perlons voting truly can be affumed equal; or the chance of any one perfon voting truly can be given and therefore what weight fuch calculations can have.

In my tranflation of algebraical quantities into probable relations, an elegant theorem is given, viz. a+b.a-fb-x

+-2x...a+t

ixa.a-x-2 X..

a_n-1x+na, a—x. —1—2x ×b+n. b.b-x+

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+P.a. a-x.

a-mx. b.k..

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bnmix + &c.
Let a and b denote the number of things cf
different kinds, A and B contained in an
urn or which may denote the fame thing,
the number of chances that an event happens
or rails; and every time that a is drawn
out of the urn, let the number of A's le
diminished by x, and every time that b
is drawn out of the urn, let the number
of B's be diminished by x; then will tle
number of chances of A's happening or
being drawn (m+1) times and B's being
drawn n-m-1 times in z trials will be

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Every thing that can be deduced on the former fuppofition that the probabilities a, and b, &c. remain the fame from the binomial, trinon.ial, &c. may with equal facility on the latter fuppofition be de

duced from this theorem.

In the year 1762, a rule was given for our two algebraical equations of n and m dimenfions, containing two unknown quantities x and y, of finding the dimen fions of the quantity x, when the two

equa

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x y

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Mr. BEZOUT deduced this propofition and extended it to more (b) equations, involving (b) unknown quantities x, y, z, &c. of which the terms of the greatest dimenfions of y, &c. are correfpondent. From the principles mentioned above may be deduced whether the equations refulting will afcend to the dimenfions given by

Mr. Bezout or to lefs.

The propofitions given by Mr. Bezout may perhaps with equal or greater facility be deduced from the principles publifhed in 1762.

In the fame book 1762, a rule is given from having (2) independent equations containing only n-m unknown quantities of fo reducing them, that there may refult (m) equations, fince called equations of condition.

As alfo from one or more (n) fimple equations having two or more (n+r) unknown quantities x, y, z, &c. of which the dimenfion in each is only one, of finding their integral correfpondent values in terms of a B, y, &c., where a, B, y, &c. denote any whole numbers whatever; thefe have been both (fince they were given by me) published by others.

Several other new propofitions, or rules of mine, have been fince published by foreign mathematicians, and fonie by the English.

In the above mentioned book, 1762, was published a method of finding a quantity, which multiplied into a given irrational quantity will produce a rational product, or which will confequently exterminate irrational quantities out of a given equation, this is performed from the roots of an equation "+10: another method in the faid book was given of reducing a given equation fo as to exterminate its irrational quantities; Mr. Bezout has fince alfo given a rule for exterminating irrational quantities.

Having fufficiently fhewn that many mathematical inventions of mine were held in fome efteem by the principal of the prefent time, and fome of the first

for

mathematicians that ever exifted; otherwife they would not have published the fame. It remains for me to give fome account of them; particularly those contained in the algebra, as it is the book which was first published, and has been principally read.

In the first chapter, are delivered several elegant (as appears to me) rules, for finding the fum of any functions of the roots of a given or given equations.-I may particularize one, as it has had the honour of being published in the Paris Acts, by Mr. Le Grange, one of the greatest Mathematicians that ever exifted, and perhaps is fuperior, in fome respects, to every rule yet published in Algebra: I must mention, affo, another rule, for finding the fum of the powers of the roots of a given equations, in terms of the coefficients of the given equation; Sir J. N. before found the fums of the fubfequent (n) from the fum of the preceding n-1, n—2, 23, 3, 2, 1 powers; but my rule, when the feries converges, that is when one poffible root is much greater than any other, not only finds the fum of the powers, but also the fum of the roots; from it has been given by me the law,

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which the reverfion of a series yax+bx +cx+

&c. obferves, and fome other problems, by no other method, yet difcovered. Perhaps, the rule for finding of the roots of a given equation, may an equation, whofe roots are any power properly be mentioned, as by it any irrational quantities may be exterminated. I fhall, alfo, add the transformation of equations, as Mr, Le Grange prefers it to any other, and laftly, the method given, of finding the coefficients of the terms of the transformed equation, from particular cafes, a method in these subjects fuperior to any other, and of great utility; all of which were published in

the

year 1762.

The fecond chapter treats of the affirmative and negative, and impoffible roots, and the limits of equations, &c. I fhall particularize the rule for finding whether the two poffible roots of a biquadratic the other two are impoffible; because equation are affirmative or negative, when Mr. Le Grange has done me the honour ing the number of impoffible roots, from an to demonstrate it; alfo, the rule for find. of the differences of any two roots of equation, of which the roots are the fquares whether all the roots of a given equation a given equation, and thence deducing

are

are poffible or not; and whether 2, 6, 10, &c. or o, 4, 8, &c. are impoffible, from the last term of it being poffible or more; this rule was generally given in 1762, and the latter obfervation inferted in the Philofophical Tranfactions for the year 1764. A rule is given for finding the number of impoffible roots, from deducing an equation, whofe roots are the refults of the roots of the equation, commonly called the equation of limits, fubftituted in the given equation, &c.; this finds all that the preceding method does, and equally with that finds the true number of impoffible roots contained in any equations of 3, 4, or 5 dimenfions; and perhaps more generally finds the true number of -impoffible roots than any rule yet given: it may easily be rendered more general. I fhall mention the following rules, because they have been fince published by eminent Mathematicians.-1, A rule for finding impoffible roots from the equation whofe roots are the fquares of the roots of the given equation, &c. 2. The finding the number of affirmative, negative and impoffible roots in an equation, whofe roots bear any affignable relation to the roots of the given equation, from the numbers of them in the given equation. 3. In the common refolutions of cubic and biquadratic equations, by the different roots of the given equations, are expreffed the roots of the reducing one: and vice verfâ from them are difcovered, how many roots of the given one are impoffible, &c. There are given more rules for finding the number of impoffible roots in equation, containing one or more unknown quantities,one of which always difcovers them, when Sir Ifaac Newton's rule does, and oftentimes when it does not. Two elegant theorems are given for finding when fome poffible quantities are neceffarily greater than others; these may be demonftrated, by proving their difference to be the aggregate of certain fquares. There are fome theorems, which give the ratio of the contents of all the quantities refulting from fubftituting the roots of one equation, for the unknown quantity in the other refpectively, and multiplying their refults: this is inferted, becaufe feveral elegant properties of parabolas have been deduced from it-Some truths are deduced concerning equations, of which the roots are the limits of each other. In my preface to the algebra, the number of inventions enumerated in these two chapters are 9 and 19 respectively, their fum 28; in each of the three fucceeding chapters, the number enumerated is more MONTHLY MAC, No. XLIV.

The

than the abovementioned fum 28. number of inventions contained in the Meditationes Analytica, on the Modern Analysis, Fluxions, Seriefes, &c. is many more than the number in the Algebra; and the number of properties of curve lines deduced is not greatly lefs; to these add the number in the addenda, Philofophical Tranfactions, and tranflation of algebraic equations into propable relations, and the number will be more than four hundred. It would be too much trouble to review fuch a number of propofitions, particular- ly as they are for the most part enumerated in the prefaces to the books themselves.

I might equally particularife my inventions in all other branches of mathematics: in properties of algebraical curves and folids, in conic fections, &c. there are more new properties, and several of them, as appears to me, of the first degree of elegance,contained perhaps in them than in the works of any other writer, and in many the algebra and principles from which they are deduced, were allo invented by me. Since the publication of the book, I have given in the addenda feveral new properties, and extended most of the properties of circles of Archimedes and Pappus to conic fections, and rendered fome more general, and given in the Philofophical Tranfactions a rule for the demonftration of feveral propofitions contained in them and fimilar ones, from principles of algebra.

Many new feries are derived from different principles; fome of which are the moft converging, but in these cafes it is commonly neceffary that a near approximate fhould be given, which will be the firft term of the feries.-Rules are given for finding the convergency of feries, and for rendering them converging when they exprefs the fluents of fluxions, contained between different values of the variable quantities; new integrals of increments; fums of feries, of which the terms are given. From approximations to the different roots, or to two or more of algebraical equations, are adduced more near approximations.-Something is added of the difficulties which occur in finding the feries for the fluents of fluxional equations. The rule of falle is rendered more general, by finding nearer approximations, when two or more approximations are given, and the errors of their refults. The fame is applied to more unknown quantities contained; a new method of differences and correfpondent values is added, with fome problems thence deduced, &c.

Many new propofitions are invented,
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n the fcience of fluxions and integrals of 'increments; whoever would fee them enumerated, may confult the prefaces of the different books, from which they feem more in number than perhaps given by any other writer whatever; but it may be added that feveral new additions to the algebra, properties of curve lines and folids, fluents and fluxions, integrals and increments, feriefes have been given fince their publications in the addenda, printed feveral years paft, or in the Philofophical Tranfactions.

Some inventions are given in my pamphlet on the tranflation of algebraic quantities into probable relations, and fome in the paper, inserted in the Philofophical Tranfactions on central forces, attractions, &c. treated in a more general manner than before given by any writer. Mr. Le Grange has fince pub. lifhed fomething fimilar on the fame fubject, and alfo applied the principles to centers of gravities, ofcillations, &c.

On the whole, let the name of any French writer be mentioned, who has dif covered a greater number of inventions in pure mathematics; he may fee much the greater part of them enumerated in the refpective books; about fifty or fixty are published in the addenda, &c. which are not contained in the enumeration abovementioned. I have no mathematical books at prefent in my poffeffion; but perhaps, I could from memory enumerate the principal inventions, given by the different authors, before the publication of my books.

General affertions without proof afford no conviction, and merit no regard; and particularly, when made by perfons ignorant of the truth or falfhood of the propofition afferted; for then they deferve filence and contempt.

I know that Mr. Lalande is a firft rate aftronomer, and writer of aftronomy; but I never heard that he was much converfant in the deeper parts of mathematics for which reafon I take the liberty to ask him the following questions ›

Has he ever read or understood the writings of the English mathematicians and, as the queftion comes from me, I fubjoin particularly of mine? If the antwer be in the negative, as it is my opinion, if his anfwer be the truth, that it will; then there is an end of all further controverfy ;-but, if he afferts that he has, which is more than Condorcet did by his own acknowledgment, then he may know, from the enumeration of inventions made in the prefaces, with fome fubfequent ones added, that they are faid to amount to more than four hundred of one kind or other; let him try to reduce those to as low a number as he can, with the leaft appearance of candour and truth; and then let him compare the number, with the number of inventions of any French mathematician or mathematicians, either in the prefent or past times, and there will refult a comparison (if I miftake not) not much to his liking; and, further, let him compare fome of the firft inventions of the French mathematicians, with fome of the first contained in my works, both as to utility, generality, novelty, difficulty and elegance, but wifely as to utility, there is little contained in the deep parts of any fcience; he will find their difficulty and novelty, from his difficulty of understanding them, and his never having read any thing fimilar before; their generality, by the application of them; principles of elegance will differ in different perfons.-I must say, that he will probably not find the difference expected. After or before this enquiry is inftituted for mine, let him perform the fame for the other English mathematicians, and when he has completed fuch enquiries, and not before, he will become a judge of the juftice of his affertion; but I am afraid, that he is not a fufficient adept in thefe ftudies, to inftitute fuch enquiries; and if he was, fuch enquiries are invidious, troublefome, and of small utility. I am, Sir, with very great regard and esteem, Yours affectionately,

TO CORRESPONDENTS.

E. WARING.

The writers on controverfial points, and anfwerers of queries, are refpectfully apprized, that our readers generally think topics fully difcuffed long before all the letters we receive under thefe heads are inferted; whence we find it neceffary to exclude fome communications, perhaps fully as valuable as thofe we have admitted, merely to avoid tedious repetitions. Several defences of the principles of the Quakers, and explanations of the word bitch, are now under this predicament.

**

We have determined to clofe all difcuffion relative to the Northampton Academy, and Diffenting Ordinations.

Long political difcuffions are at no time well fuited to our Mifcellany, and are at present what we would peculiarly avoid. Some communications under this head lie ready for return so their refpective writers.

We beg leave to decline engaging in a controverfy relative to Collier's patent filtering machine.

W. H. is informed that a letter for him is lying at No. 71, St. Paul's Church-yard, concoining Profident Bradshaw.

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