ferved for fome other occafion." He prefents his readers, however, in the work before us, with fome general tables of conclufions, accompanied with proper explanations and judicious remarks. The whole of this article is curious and important; but we wish that it had been made more full.

It gave us a fincere pleafuse to find a few articles, in the courfe of the work, communicated by fome perfons well known at prefent, and juffly effeemed in the literary world. Of thefe auxiliaries, the Rev. Dr. Mafkelyne, the prefent Aftronomer Royal, and Mr. Baron Maferes, deferve particular mention; and others are certainly entitled to much praise for their ingenious and feientific labours.

In fo large a field for the exertion of attention, as the fcientific part of the work before us, candour will doubtlefs be ready to make allowance for fome overfights, and trivial omiffions. As it is natural to fuppofe, that an author's.views mult be influenced, not only by his general zeal for his fubject, but alfo by his predominant attachment to fubordinate branches of it; in an extenfive work, we are to reckon upon meeting with real or apparent deficiencies, as the author's abilities may be inadequate to fome parts of his undertaking, or, as his inclinations and our own may differ. With thefe impreflions upon our minds, we profecuted our examination of the work before us; but are forry to fay, that we fometimes looked for what we could not find, and fometimes found what we could not approve.

Under the word approximation, we found a theorem for extracting the roots of numbers, which, to our furprife, is inferted again under the title extraction of roots, and a third time under the word root; and, what we as little expected, Dr. H. calls it new, and fays it was invented by himfelf. But this theorem is the very fame, in effect, as fome that were published near a century ago, by M. de Laguy, the difference being only in notation, in which, indeed, this nominal new theorem differs from itself and when Dr. H. afferts, as he does, in p. 131 of vol. i, and again, in p. 388 of vol. ii, that this theorem contains all the particular rational formula of Halley and De Lagny, it is only juftice to reply, that either of De Lagny's general rational theorems, contains Dr. Hutton's. All the kill in algebra required to make this new theorem out of the old one, is the reduction of a mixed quantity, to an improper fraction. This will prefently appear. Putting N to denote any number out of which the root, whofe index is m, is to be extracted, and a to denote a near value of that root, but fomewhat lefs than the

true

true value of it, M. de Lagny finds "VN to be very nearly

=a+2aX

N- am
m−1 · N+m+1.aTM

; which, by the reduction

m+1⋅N+m

before-mentioned, becomes

ax

m- 1.N++ 1, a

1.am
, the

very theorem in the Dictionary, vol. ii, p. 388, n there being the index which is here denoted by m.

M. de Lagny's other rational theorem is,

N-a-2aX

am N

1

N+m+1.aTM

very nearly, a be

ing taken a near value of the mth root of N, but fomewhat greater than the true value. This algebraic expreffion, like-, wife, when reduced, becomes exactly the fame as the other. The procefs, indeed, by which Dr. H. obtained his theorem, is different from M. de Lagny's; but we do not think it better. Dr. H. converts the algebraic expreffion into a proportion, from which no advantage appears to be derived; nor has it the recommendation of novelty. It had been done for the cube root, in two different ways, as may be feen in Bonnycaftle's Arithmetic, and in Burrow's Theory of Gunnery, printed at the end of his reftitution of Apollonius Pergaus, on Inclina

tions.

The account given, in the work before us, of Sir Ifaac Newton's method of approximating to the roots of equations, is, in our opinion, very incomplete. Dr. H. fays,

"Newton's method is this: as the quantity fought is fmall, its higher powers decreafe more and more, and, therefore, neglecting Newton, therefore, neglects them will not lead to any great error. all the terms, having in them the 2nd and higher powers, leaving only the ift power, and the abfolute known term from which fimple equation, he always find the value of the affumed unknown etter nearly, in Halley's method of doing the a very fimple and eafy manner, fame thing, was to neglect all the terms above the fquare or 2nd power, and then to find the root of the remaining quadratic equation; which would, indeed, be a nearer value of the alummed letter, than Newton's was, but then it is much more troublesome to perform.”

A reader of this statement would naturally conclude that the methods recommended by Sir Ifaac Newton and Dr. Halley were clearly diftinct. This, however, is far from being the cafe, as will readily appear from the following quotation. from Sir I. Newton.

* See a volume of Mathematical Tracts, in 8vo. published by Mr. Baron Maferes, in 1795, page 505, 507, and feq.

**Equationes

Equationes plurium dimenfionum nihilo fecius (alluding to an equation folved according to Dr. H.'s statement) refolvuntur, et operam fub fine, ut hic factum fuit, levabis, fi primos ejus terminos gradatin omiferis.

"Præterea notandum eft, quòd in hoc exemplo, fi dubitarem an Op veritati fatis accederat, pro 10 po, finxiffem 6 pp + 10p-10, et ejus radicis primam figuram in Quotiente fcripfiflem; et fecundam vel tertiam Quotientis figuram fic explorare convenit, ubi in quatione iftâ ultimò refultante quadratum coefficientis penultimi termini, non fit decies majus quàm factus ex ultimo termino ducto in coefficientem terinini antepenultimi.

"Imo laborem plerumque minnes, præfertim in Equationibus plurimarum dimenfionum, fi figuras omnes Quotienti addendas dicto modə (hoc eft extrahendo minorem radicum EX TRIBUS ULTIMIS TERMINIS Equationis no fimè refultantis) exquiras: ifto enim modo figuras duplo plures quâlibet vice Quotienti lucraberis." See vol. i, p. 269, of Dr. Horfley's edition of Sir I. N.'s Works.

Thefe paragraphs in the original, immediately follow the folution of the cubic equation, which Dr. H. has inferted in his Dictionary!

On reading what is inferted as a new property of the Bingmial Theorem, and an improvement on it, by Mr. Lonnycaftle, the following line of Horace occurred to us;

Indignor quandoque bonus dormitat Homerus.

We thought, that if Dr. H. had not been nearly in that state of flumber, he would not have inferted as new, what is, in effect, nothing more than a reverfion of Mercator's logarithmic feries, to find the correfponding number; which was known to Newton* above a hundred years ago, and has been so often performed, that Euler, in his Inftitutiones Calculi Integralis, vol. i, p. 111, calls it, Series notiffima.

Verum opere in longo, fas eft obrepere fomnum. This was the fame poet's allowance; and we are not difpofed to be lefs liberal.

Under the word Fluent, the examples of finding fluents, confidering the great utility of that method of computation, are, in our opinion, too few; and the table of forms of fluxions and their fluents too fcanty, as there are other forms of frequent ufe which are omitted. Of the XIXth form in the table, wa alfo think it proper to say, that there are other fluents, ope of which in particular, will, in fome cafes, be more useful than either of those which are given; and even of those two which are inferted, one will generally want a correction, of

See the Commercium Epiftolicum, pp. 86, 179, and 186.

which, however, nothing is faid. What Dr. H.'s reafons could be for omitting things which might have been fo eafily fupplied from other books, we know not. If thefe omiffions had been fupplied, the book might have been confulted with more advantage, both by the " ftudent and man of fcience," for whofe ufe it is profeffedly defigned.

When we came to the word Series, we were fomewhat furprifed to find fo little on this most extenfive and useful branch of the mathematics, All that is here faid about Series, is contained in a very few pages.

Under the word Sine, fome ufeful feries for the fines of multiple arches are omitted, which might eafily have been fupplied fupplied from Emerfon's Trigonometry, book i, fect. iv, from which those that are inferted feem to have been taken: and of the four feries for this purpose which are inferted, three, we think, fhould have been continued a term further, to fhow the alternate occurrence of the figns + and -.

The style of the work before us, in our opinion, is fuch as neither provokes cenfure, or excites applaufe. It is in general perfpicuous, but feldom elegant. Of the errata, we have obferved fome which must prove very inconvenient to young ftudents.

Upon the whole, Dr. H.'s Dictionary contains much useful and curious information; and when compared with feveral contemporary publications, it may be faid to be

Velut inter ignes

Luna minores.

Confidering, however, the abundance of excellent materials for fuch a work, which the prefent times afford, and the reputation of the author, it falls confiderably fhort of what we expected. Every friend of fcience, nor warped by private pique, like the writer of the articles Roval Society, and Tranfactions, will alfo regret with us, the unhandiome and unjuft manner in which that learned body, and their publications, are mentioned in those places. Pallages of this kind can.only be accounted for, by a recollection of the difputes which arote fome years ago in the Royal Society, wherein Dr. H. was materially concerned; but with every allowance for the refentment of fuppofed injuries, they will remain a strong and lafting blemish to the work.

ART.

ART. IX. Thoughts on Outline, Sculpture, and the Syftem that guided the Ancient Artifts, in compofing their Figures and Groupes: accompanied with free Remarks on the Practice of the M.derns, and liberal Hints cordially intended for their Advantage. To which are annexed, Twenty-four Designs of Claffical Subjects, invented on the Principles recommend d in the Effay. By George Cumberland. 40. 52 pp. With 24 Plates. Robinfons, &c. 1795.

THE

'HE prefent article would have been noticed by us fome months fince, but that we literally turned with difguft from an incoherent rhapfody, which, under the title of

Free Remarks on the Practice of the Moderns, and überal Hints cordially intended for their Advantage," contains a daring attack on a moft refpectable body (the Royal Academy) whom the author illiberally attempts to injure in the minds of the public. He has made it alfo a vehicle of felf-adulation, and of prepofterous praife, lavifhed on a few obfcure individuals, his own friends. A very clear prognoftic of the evil to come, met our eye in the title-page; which is affectedly made to differ from all the ufual forms of title-pages, being printed in full and equal lines of finall capitals, In the motto,

AINSI IO SON PITTORE," which is intended for Italian, the first word is French. But thefe inaccuracies, to eyes in a fine frenzy rolling, are objects of no moment. Deep refearch, and an elevated ftyle, are the boaft, we prefume, of this author; excellencies, which being, doubtlefs, as much efteemed by our readers as ourfelves, we thall no longer withhold from them.

Mr. C. introduces himself to the reader, by the following quotation :

"If there be a Beauty in Virtue," remarks the learned Mr. Petvin, in his Letters concerning Mind; "the mind muft have a feeling of it, whilft it has it under view, no lefs than a feeling of harmony, when prefented to the ear. It must be felt and understood together, we must be in fome measure what we behold; and a man must be tolerably good before he can have any tolerable notion of goodness." P. i.

And he proceeds,

"Thus when a ray from the univerfal mind infpired that great man. Mr. Fox, to place his happinefs in temperance, liberty, and borafty, the reflecting part of the kingdom felt the beauty of his public virtues; as during the courfe of many years we have feen them with dignity gradually unfolding.

*A mighty modeft one, by the way, for an awkward flourisher of non-defcripts, thus ranking himfelf, by implication, with Corregio.