ing them. Those, however, who are fond of scientific demonstrations, object to this method, because it is merely tentative, and because the solution has no connection with the original equation. By trial, 2 may be found to be the root of x32x-4 but there is no connection between 2 and the form -2x-4; whereas, from Cardan's solution, 32+ 100 +32-100 27 we may re-ascend to the original form P. 293, the Baron considers the case in which the equation bx-xc has two roots. All his processes are conducted with the greatest accuracy and circumspection; and the quantity is always considered as a real positive quantity. The solution of the biquadratic +—bx2+box=3b, by the method of Lewis Ferrari, is given at p. 412. This solution is very intelligible and complete: but it is to be remarked that the principle of it is precisely the same as that of Waring. In Ferrari's method, let the biquadratic be x4qx2+rx+s; add 2nxn to each side; then x++ 2nx2 + n2 = (2n+q) x2+rx ++" but, in order that (2n+q) x2 + rx + (s+n1) may be a a complete square, 4× (2n + q) × (s +n2) must equal r2: whence results the cubic 8n3 +49n2+8sn+4sq—r2 = 6; which, solved, gives n. 4 3 In Waring's solution, x++2px3 qx2+rx+s: let (p2+2n) **+2pnx+n' be added to each side; then x++2px 3 + (p2+ 2n) x2+2pnx+n2= ( p2+2n+q) x2+ (2pn+r)x+(s+n2)=o. ' Let 4X (s+n2) (p2+2n+q) = (2pn+r): then results the Eubic equation 8n3 +49n+ (8s-4rp) +49s+4p2s—r2=0: whence n &c. The step which Dr. Waring made, in extending the solution to equations that have all their terms complete, was extremely easy; since the real and great difficulty, the invention of the principle of solution, had been surmounted by Ferrari. -The methods of Waring and Ferrari are, according to Baron Maseres, more tedious and incommodious than the method of Raphson by approximation. We have already said, however, that a distinction is to be made between a solution by a tentative method, and a solution by a strict and scientific method. Mathematicians, who are fond of speculative truth, do not much regard practical commodiousness. In the same volume with this Appendix, is bound up a small tract that has already appeared before the public *, intitled "Observations Indeed most of the contents of the present volume are to be found in the prior works of B. Maseres; in his Scriptores Logarith N 2 mici, a Observations on Mr. Raphson's method of resolving affected Equations of all degrees by Approximation."-In this tract, and in the Appendix, the Baron renews his old complaints against the nonsense and absurdity' of negative quantities; which quantities, he says, he and Mr. Frend have expunged from their works. This decisive tone and language, we think, can only be justified by a strict proof of the absurdity and inutility of negative quantities; a proof for which we seek in vain in the works of these two authors; since we do not deem the question decided because we cannot conceive an abstract negative quantity, or because a mathematician has defmed them to be nihilo minores," or because problems can be solved without their aid. We reserve what we have farther to say on this subject, for the consideration of the second part of Mr. Frend's Algebra, in the subsequent Article. ART. XIII. The Principles of Algebra: or the true Theory of Equations established by Mathematical Demonstration. Part II. By William Frend. 8vo. pp. 119. 3s. Boards. IN 1799. Robinsons N this second part, Mr. Frend classes equations according to the number of their unknown terms: "+"—ax"=k being an equation of the second, and x-ax"+bx-k of the third class. He observes that one general rule pervades all equations, namely that none in any class can have more roots than it has unknown terms.-It must be recollected that, according to Mr. Frend and Baron Maseres, negative and impossible quantities are degraded from the duties and the dignity of roots. In chapter II. equations of the first class are solved by the aid of logarithms, and approximation.-In chapter III. the author proceeds to the consideration of equations of the second class, and proves that the equation ax-xm+ may have two roots, in a manner similar to that of Baron Maseres. (Treatise on negative sign, Appendix, &c.) Mr. Frend (we know not from what motive) has changed the term co-efficient for co-part; and he demonstrates the relation between the co-part and roots of equations thus: Let equation be ax-x-k; roots, c and d; then, acc c2-d2 ad-d2. a=d=c+d, and ... ac—c2=dXc=k.— Again; let xbx-ax`k, and roots be d, e, fi mici, and Dissertation on the use of the negative sign, Memoir in Philosophical Transactions, &c. then, then, d+bd-ad-k e3+be-ae2-k ƒ3+bf—af1=k .. d'e3 +b (d—e) —a (d'—e2) In the same manner, b may be shewn de+df+ef and def. This certainly is a direct, but it is not a new demonstration. In page 220 of the Meditationes Algebraica, (Ed. 3tia) is this problem : "Datá æquatione (x"—px”~'+qx”—2 = 0) &c. unam incognitam quantitatem (x) habente, invenire ejus coefficientium (p, q, r, &c.) con stitutionem. “Sint a, ß, y, &c. radices datæ æquationis, quarum numerus sit në quibus pro suo valore (x) substitutis, resultant equationes "Reducantur hæ æquationes in unam, ita ut exterminentur omnes coeffi cientes præter unam (p), cujus constitutio quæritur ; et invenietur p=a+ß +y+&c. at eodem modo q=aß+ay + By &c.; r=aßy +&c. et ita deinceps." This is precisely what Mr. F. has done, and in the same manner. In the preface, the author speaks of having discovered the limiting equation by a simple principle, which he says admits of great extension. The principle is this: let a and b be cer tain numbers; z and y variable numbers; then, if q=b+AzBy+Cz-Dy' &c. The difference between the variable terms must either be equal to a given number or to nothing: now the difference cannot equal a given number (says the author) in my demonstrations, for both x and y may be taken less than any assignable number.' Let us observe the nature and efficacy of the principle, in its application to an individual case. Let the equation be ax-x3, and let m be the number which, substituted for x, gives ax-x3- G the greatest possible: then, if x be made =m+z, or m-y, the product will be less, and = some quantity k. Substitute for x, in equation ax-x3-k, m+z: likewise m-y; and equate the two expressions that arise from such substitutions: then there results a=3m2+3mxz−y+z2—zyty2. But, says the author, 3mxz-y+z-zyty must 0% a=3m2 and ma 3 Now 3mxz-z-zy+y2 musto, according to Mr. F.; because, by decreasing z and y, the quantity may be made less that any assignable number. We confess that, to us, this principle does not appear very evident. If ax-xk, z and y cannot each be taken at pleasure: but z being assumed, y is necessarily determined from the equation (a=3m2 xz−3+22 -zy+y2). It is granted that, if in any equation ax” — xm+n k where k is less than the quantity that results from putting m for x, x may be made m÷z, and z may be taken less than any assignable number: but it is no evident consequence that y is less than any assignable quantity. We do not say that is not less, only that it is not demonstrated to be so. What the author advances may be truth, but it does not appear to us to be science. In our opinion, if Mr. Frend's principle be thoroughly exa mined and firmly established, it will be found to be identical with the one used for the determination of the maxima and minima of quantities; and the problems for finding the greatest value of x in the equations ax-x3=k, ax”—xm+n=k &c. are problems which have been long since solved by the aid of the fluxionary and differential calculus. The objection which Mr. F. makes against the solution of these problems, that the foreign and unnecessary principle of velocity has been there introduced, is an objection properly against the fluxionary me. thod: but the maxima and minima of quantities have been determined on purely algebraical principles. To state once more our opinion, we think that the principle of Mr. F. as it now stands is by no means simple nor evident; and that, if it be more fully considered and verified, it will not differ materially from the principle on which the maxima of quantities have heretofore been established. In page 69, Mr. Frend applies his principle to discover the limits of the roots of equations; for instance, in the equation ax2+bx—x3=k, he finds the value of x that makes the quantity ax2+bx-x3 the greatest possible, namely a+√a1+36 : 3 this value, it is known, is a limit of x in the original equation, determined by solving the limiting equation 3x—2ax—b=0, according to the common processes of algebra. The author of the present treatise has undoubtedly thought for himself; and his work deserves notice for its freedom from absurd absurd notions and indirect demonstrations, and for the prac tical information relative to the solution of equations. It is not, we trust, agreeable either to our principles or our practice, "to transmit the sacred depositum of error from age to age:" we wish neither to defend inveterate prejudices, nor to palliate absurd misconceptions; yet we cannot find sufficient reason for that acrimonious and irreverent censure which has lately been poured forth on the inventors and abettors of the old system of algebra. We disregard contemptuous terms; they properly are only to be used against those who obstinately persevere in error after conviction. From a mathematician, formal proof is to be expected; he is required not to inveigh against the nonsense of negative quantities, but manifestly to expose their inutility. He is to be admonished that, instead of a flippant comparison of the doctrine of impossible quantities with. the "stupid dreams of Athanasius," he would have done better if he had shewn that it is unintelligible and useless in expediting mathematical reasoning. It is easy to affix an absurd notion to a thing, and then to ridicule it but, unless ridicule be justly applied, it is no test of truth. The misconceptions of one or two authors, concerning negative quantities, constitute no just and sufficient reason for their exclusion; and many distinguished mathematicians have persevered in using negative quantities in their calculations, after they had animadverted on the absurd notions. formed of them. The true idea of negative quantities, and the exposure of the unintelligible definition given of them by some authors, are not of late date; and cannot be attributed to that flood of light which has been poured on modern times, to dissipate the clouds that yet hang over human science. 66 "Il n'y a donc point réellement et absolument de quantité negative isoleé: 3. pris abstraitement, ne presente aucune idée."-Encyclopedie, year 1751. Qu'il me soit donc permis de rémarquer, combien fausse l'idée qu'on donne quelquefois des quantités negatives, en disant que ces quantités sont ou dessous de zero. Independamment de l'obscurité de cette idee envisagée metaphysiquement," &c.-Opuscules D'Alembert, 1761. "C'est le calcul, il faut l'avouer, qui a induit certains Geometres en erreur sur la valeur des quantités negatives. Ils ont remarqué que a ◄ 2a donnoit a za< o, ou — a <o: d'ou ils ont conclu que les quantités negatives etoient au dessous de zero. Mais ils ne seroient pas tombes dans cette erreur, s'ils avoient consideré qu'une quantité au dessous de zero est une chose absurde, et que -ao ne signifie autre chose que B—a B, Betant une quantité quelconque sousentendue, et plus grande quea. La sim plicite et la commodité des expressions Algebraiques, consiste a representer à la fois, et comme en racourci, un grand nombre d'idees; mais ce laconisme expression, si on peut parler ainsi en impose quelquefois à certains esprits, et leur donne des notions fausses.” Id. |