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and the IV. Lem. Lib. II. which are included under one general enunciation, and may, with much greater brevity than is found in one fingle cafe of his, be refolved by the fame analyfis. Lem. V. alfo of the fame book is divided into four cafes, each with investigations of confiderable length; though the whole is little more than a cafe of Lem. III. and an obvious corollary from it. But there is very little occafion for the lemmata at all; as neither the refolution nor compofition of the problems is much shortened hy the use of them. How far this charge of frivolous minutenefs and difguftful redundancy may be applied to Apollonius himself, in the prefent question, we cannot determine; nor perhaps can any one elfe, for the conduct of the work, by Apollonius, may have been very materially different from Mr. Horfley's reftitution, even allowing the whole force of Pappus' account. But Mr. Horfley was under no obligation to restore the faults even of Apollonius; his genius was left to its own free operation, and he might have delivered this tract on Inclinations, to the public, with all the perfection that he conceived the fubject to be capable of, or himself of giving to it.

To this very material fault is added another, equally effential. In the reftitution of a work of the pureft geometer, we find, generally, neither the ftyle nor operations of geometry. An inelegant air, unknown to the ancients, is thrown over almost the whole work, by the introduction of the algebraic notation, which, in compofitions of this fuperior rank, ought to be as abfolutely rejected, as from polite writing the curtailed language of the compting-houfe, fo juftly defpifed by men of letters and tafte. The only excufe which can be made for it is, that it faves a little paper, for the words which the algebraic symbols reprefent are fupplied in the act of reading. But it has an ill effect upon ftudents, as it tends to vitiate their tafte, and infenfibly divert them into all the inelegance of the algebraic analyfis. This however is far from being the whole; the very operation as well as expreffion is algebraic. What are BA2.

AL2

AL-4ACxAV (pag. 59.) AB— ̧DBx

TR RS2, AL24

BA2

2

AC BDXAD (pag. 61.) гA+ZxAH-AH' (pag. 72.) and many fimilar inftances, but downright algebra? If this be to imitate the geometric analyfis of the ancients, or of any vaJuable example among the moderns, we confefs ourselves to be ignorant both of the ancients and moderns, and of the very diftinction between geometry and algebra. They are modes of expreffion and operation which might and ought to have been avoided, especially in a work which profefles to restore the pureft of geometers, and form the young mind to an habit of

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rational

rational investigation.-Neither do we think the Author to be commended for omitting fo frequently the composition of the problem, and annexing only a bare conftruction as the confequence of the analyfis. This is furely not agreeable to the manner of the ancients, who never (to the best of our remembrance) neglect the compofition, but rather feem to confider it as the very fubject which the refolution aims at.

It is a fault alfo in the demonftrations, that the folution is fometimes derived from a pofterior part of the Elements, which a much earlier propofition offers with more cafe and fimplicity. Thus, that a quadrilateral, two of whofe oppofite angles are together equal to two right angles, is infcribed; or that two equal angles ftanding upon the fame base, are in the fame circle with the bafe; are theorems admitted by our beft geometers as a part of the Elements, being only the converfe of the 21 and 22 III. Elem. and indeed are obvious corollaries from them. Of the extenfive utility of thefe theorems that moft ingenious geometer, Mr. Stewart of Edinburgh, has given abundant proof; and, by the ufe of the fame, our Author might have rendered his folution in feveral inftances, particularly in Probl. III, much fhorter, and fimpler.

To thefe confiderable faults, little is to be oppofed but the fimplicity of the conftructions, which we are perfuaded every one will admire. The ft probl. alone we would except, in the conftruction of which, two circles are applied, while one is fufficient. It is to be lamented that a work, wherein the greatest difficulty is overcome, fhould have appeared abroad, before it was digefted into its fimpleft and most elegant form, and before the Author had fufficiently formed his ftyle and habit of demonftration from the best models. Nor is it from any ill-natured cenforioufnefs that we have thus freely given our opinion of the faults in this work, but from a sense of the juftice we owe the public, a regret to find fuch confiderable blemishes in a performance which might have afforded the highest pleasure, and from the hope that this mention of them may contribute to render a future edition more perfect.

Ás fome may probably have entertained a very high opinion of the merit of this work, and may therefore apprehend our judgment to require fomething more than affertion to fupport it, we fhall fubjoin the refolution of Probl. IV. and V. in evidence of the moft difputable part of our cenfure. These two problems are feparately investigated by our Author, and have three lemmata fubfervient to them. The public will judge by the following analyfis, whether we have wantonly afferted that cafes are needlessly diftinguifhed, and lemmata needlefsly multiplied,

PROBLEM,

P.R O BLE M.

Between the fides of a given rhombus, or fquare, to infert a right line of a given magnitude, which may pafs through the oppofite angle:

Suppose it done, viz. that between the fides BC, DC of a rhombus or fquare ABCD given in pofition and magni ude,. is inferted a light line EF of a given magnitude, and which paffes through the oppofite angle D.

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Let a circle be defcribed round the triangle ECF, and join AC. Since the point A is within the circle (fig. 1.) AC will meet the circle in fome other point G, but (fig. 2.) because the angle EFC is greater than the angle FCA (16, 1.) viz. than the angle ACB, AC does not touch the circle, (32. 3.) but falling within it, muft alfo meet the circle in fome other point G. Join EG, GF. The angle EFG is equal to (the angle ECG, viz. to) the angle ACB; and the angle FEG is equal to (the angle ACD, viz. to) the angle BAC. The triangles EGF, ABC, are therefore equiangular, and ABC being given in kind, EGF is given in kind also. But the fide, EF is given in magnitude, wherefore the triangle EGF is likewife given in magnitude (52. dat.). But because the angle ACB is equal to the angle ACD, the angle GCE is equal to the angle A EG, and the angle CGE being common, the triangles CEG, EAG, are equiangular; CG is therefore to GE as GE to AG, and the rectangle CGA is equal to the fquare of GE. But GE is given in magnitude, and AC in pofition and magnitude, wherefore the point G is given. And because the point G is given, GE in magnitude, and BC in pofition, the point E is given (31. dat.). But the point A, as alfo the pofition of DC is given, wherefore EF is given in pofition.

The inquifition of the limits, and the compofition of this Problem, are equally eafy, and may be conducted in the fame general manner.

Of the Problem requiring a right line of a given magnitude to be inserted between two circles, and which fhall verge to a given point, there is ftill another cafe, of which this work

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makes

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makes no mention, viz. When the given point is in the right line joining the centres of the circles, and the distances of the faid point from the centres are proportional to the diameters of the circles.

****'s New Discoveries in ART. VI. Remarks on M. de V*** Natural History, in a late Publication entitled, Les Singularitès de la Nature, Bath printed, and fold by Robinson and I s. 6d. 1770. Roberts in London. 8vo.

T

HOSE who are aquainted with Voltaire's philofophy are no ftrangers to the tendency of the doctrines he generally advances-His avowed intention is to exclude all final causes from the fyftem of nature, and to ascribe to chance or neceffity those phenomena which indicate, to others, of founder principles, a fupreme intelligence and influence. Happily, indeed, for the interefts of truth and virtue, his pernicious tenets, however artfully disguised or confidently propofed, are fo evidently contradictory both to reafon and fact, as to bring with them their own refutation.-Seriously to confute a philofopher of his caft, would be paying him much greater respect than he deserves. A perfon, who invades a province in which he is not qualified to make any figure, and who maintains the groffeft contradictions for the fake of fingularity, or to gratify either pride or spleen, has no right to expect that he fhould be reafoned with. His vanity excites contempt, and ridicule is the only weapon with which he fhould be oppofed. We could fcarce read fome of Monf. V's late publications, in which he affumes the character of a philofopher, without laughter, were not the principles he advances fo fhocking to the human mind, and fo contradictory to found philofophy, as to excite a more ferious difpofition. It is with regret we confider, that the fine talents of this writer have been prostituted to the bafe and cruel purposes of promoting licentioufnefs both of principle and manners.-His Singularitès de la Nature has a tendency to exclude the Deity from all the operations of nature, and to invalidate the truth of revelation. It contains, however, fuch discoveries and reasonings as no man can read without mirth.-And the Author of the Remarks on this publication has admirably contributed to expose them to that contempt and ridicule, which they deserve.

We shall give our Readers two or three extracts, from whence they will be able to judge of the spirit and ftyle of this Remarker, and likewife of the fingular pofitions, which the ingenious philofopher has advanced.

Your works, fays our Author, are the only new books I can get to read in the French language.-I know not what good wind blows them hither, but I can affure you I find it

• Didionnaire Philofophique.

impoffible

impoffible to procure myself a ferap of any of the learned works from that kingdom.-Your experiments have fet all the children of our village to work. Were you here, Sir, you might have the pleasure of feeing how indefatigable they are in purfuit of your favourite infect, the fnail.-If ever I go out of my houfe, I am fure to meet some with new ground, or fome with rufty fciffars, cropping their afpiring antlers, to have the pleasure of seeing them bud forth again a fecond time. But I must inform you, that a certain natural hiftorian like yourself, who is the oracle of our village, has pretended that your discovery is not at all new, nor, fays he, is it confined to reptiles alone, for he is perfuaded that the human race is capable of the fame phenomenon. -These sentiments of my friend I communicated to feveral married ladies in our neighbourhood, all of whom seem anxious to observe the event of fuch an uncommon property in man.Several young ladies, who in the bloom of youth have thrown their pretty perfons away, for the fake of a fortune, upon gouty and decrepid batchelors, have sent to Salisbury for the best and fharpeft fciffars, intending, fhould fuch a regenerative faculty be discovered in their hufbands, to commence immediately the ftudy of androtomy. How agreeable to become young again at so easy a purchase! Or who would not linger on through feventy-four tedious revolutions of the fun, to experience the happy lot of Titan, for a rofy blooming Aurora! But however, Sir, this does not feem to be your cafe; for, if we may judge from the light and puerile ftyle of fome of your late produc tions, you are reduced once more to the state of the pap-spoon and leading ftrings. I grant your affecting the young man, will please the country farmers much better than all those pretty verfes you made about fifty years ago.-And, admirer as I am of natural hiftory, and particularly of thofe effays on that fubject, which your juvenile pen has produced, I cannot help congratulating you, that in the second state of turbulent youth, you fhould have confined your genius to fo rational a study.

Who indeed would have believed Spalanzi upon his own bare word, or who, in fact, would have believed Newton, upon his, if you and Madame de C had not been fo obliging as to verify them? But now be it known unto the world, that the experiments of Sir Ifaac Newton have been judged and verified beyond a doubt, by M. de V, the fublime Author of feveral tragedies and poems.-And thanks, be to you from this ifland in general, for the important discoveries you have made not only in optics but natural history, and the kind protection you have deigned to lend to Spalanzi and Newton.-Above all, thanks be to you for the care you have taken of the humble fnail; how flattering a condefcenfion! that he, who had learned to found the trumpet of fame, and relate the

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