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d' ted testb=aXd te
a=dtetf. In the same manner, b may be shewn = de + df + ef and k=def. This certainly is a direct, but it is not a new demonstration. In page 220 of the Meditationes Algebraicæ, (Ed. 3tia) is this problem :
“ Datâ equatione (x*—37-' +9x*~*=0). &c. ungm incognitam quantitatem (x) habente, invenire ejus coefficientium (q, r, &c.) constitutionem.
" Sint a, B, Y, &c. radices date æquationis, quarum numerus sit n; quibus pro suo valore (x) substitutis, resultant equationes
ga-opynet+q7*?-&c. “ Reducantur he equationes in unam, ita ut exterminentur omnes coefficientes prater unam (), cujus constitutio quæritur ; et invenietur prate +r+&c. et eodem modo q=aß + ay +By &c.;' raaby + &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 6 be certain numbers; z and y variable numbers; then, if a = 6+ 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 z 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-*3, and let m be the number which, substituted for x, gives ax-x} = G the greatest possible: then, if * be made =m+z, or mấy, the product will be less, and = some quantity k.
Substitute for x, in equation ax=-xi-k, m+x: likewise m-y; and equate the two expressions that arise from such substitutions: then there results a=3m2 +3mxz-ý+z:-zyty?. N 3
But, says the author, 3mxzmy+z*-zytyä must = 0; .. a=3m2 and m=va
3 Now 3m x Z-y+z'—xy+y? must = 0, 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-_x=k, z and y cannot each be taken at pleasure : but z being assumed, y is necessarily determined from the equation (a=31n? xz-+za -2y+y?). It is granted that, if in any equation ax 4m+ =k where k is less than the quantity that results from putting m for 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 g 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 ta be science.
In our opinion, if Mr. Frend's principle be thoroughly examined 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 greatese value of x in the equations ax---x}=k, axk4m+=é &c. are problems which have been long since solved by the aid of the fuxionary 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 mee 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 ax?+bx-x=k; he finds the value of x that makes the quan. tity ax? +bx-x} the greatest possible, namely
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 notions and indirect demonstrations, and for the practical 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.
“ 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 au dessous de zero. Independamment de l'obscurité de cette idee envisagée metaphysiquement," &c. --Opusculez 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 donnoit & -20 < 0, ou a Coi d'ou ils ont conclu
les quantités negatives etoient au dessous de zero. Mais ils ne seroient pas tombés dans cette erreur, s'ils avoient consideré qu'une quantité au dessous de zero est, une chose absurde, et que a co ne signifie autre chose
-a <B, Betant une quantité quelconque sousentendue, et plus grande quea. La simplicité et-la commodité des expressions Algebraiques, consiste a representer à la fois, et comme en racourci, un grand nombre d'idees ; mais ie laconisme d expression, si on peut parler ainsi en impose quelquefois à certains esprits, e leùr donne des notions fausses." Id.
We have never tortured our minds to conceive the idea of abstract negative or impossible quantities, but have considered them as means used to abridge and facilitate calculation, Until a clear and decisive proof of their inutility be exhibited, we must follow the antient system of things : still talking the same jargon which Newton and Des Cartes have babbled, and adhering to the faith of that doctrine which has verified the law of gravitation in every phænomenon of the universe.
We have said thus much, because we are of opinion that Mr. Frend has been more bold and vehement in his assertions, than his proofs warrant him to be ; and because he has abused a most excellent privilege, the freedom of inquiry. The concluding sentence of his book * contains a sublime thought: but, recollecting some of the events of the author's life, we fancy that we see it contaminated with a small mixture of human infirmity.
Art XIV. The Hop Garden, a didactic Poem. By Luke Booker,
LL. D. 8vo. Pp. 118. 35. sewed. Rivingtons. 1799. IT
I seems rather extraordinary that Dr. Booker should not
have known that Mr. Christopher Smart, so memorable for his genius and his misfortunes, had written a poem on the same subject with that which is treated in the work before us ; but we conclude that he was unacquainted with that production, because he has not mentioned it; as we doubt not, from the liberality of his sentiments, that he would have been happy in giving it that applause which it so well deserves. Though, however, these two poets have written on the same subject, and both in blank verse, the plan and structure of the poems are so different, that each may be considered as an original. Smart is lively, easy, and poetical ; his imagery is bright; and his descriptions are animated and appropriate. Dr. Booker is in many parts stiff and laboured ; and he indulges perhaps too. much in his turn for moral sentiment, and his love of digression : yet it must be acknowleged that his digressions are frequently beautiful, and that his moral and religious sentiments are such as every good man must approve. Indeed, when interwoven in some pathetic tale or interesting narrative, they will not only be read with pleasure, but will leave impressions on the mind that are highly favourable to our improvement in
*: How great then must be that Being to whom the thoughts of all these orders of Beings are known at a moment's glance ; and how. insignificant in the eye of season are those nations which lay down fules for thought, and persecute for opinions.'
virtue and real wisdom. Of this sort, we conceive the follow,
• Thou who, solicitous of Nature's boon,
- With troubled step she walks,-now stops, and turns,