with regard to the four books here introduced, we are clearly of opinion, that they cannot be made a part of an elementary course, without turning the attention of the student away from more important branches of the mathematics. We must, however, hear what Dr Horfley has to fay on this. fubject. Primo igitur, plerique eorum, qui in ufum ftudiofæ Juventutis Euclidem ediderunt, fecus ac nos fecimus, non nifi priores fex libros cum undecimo et duodecimo typis mandârunt; partim, ut opinamur, quia facile fibi perfuaferint, feptimi, octavi, et noni nullam eos jacturam facturos effe, qui vel in puerorum fcholis, vel a quocunque demum præceptore arithmeticæ elementa didicerint; partim quia omnem libri decimi utilitatem parvi penderint, præ furdorum doctrinâ, prout ab iis exponitur qui artem algebraicam tradunt-quod inerudite magis facfum fit, nefcio, an ofcitanter; tam a ratione ali num eft, juniores ad algebram amandare, priufquam geometria elementa rite calluerint, e quibus pendet etiam regularum algebraicarum five veritas omnis, five evidentia. Etenim has at artem quandam, fi placeat, abfque geometriâ quis condifcat; ut fcientiam non intelliget, nullâ geometriæ ratione habitâ, quæ et ea amplectitur, e quibus generales numerorum affectus exoriri compertum eft.' Præf. p. 2. It is plain from this, that Dr Horley confiders the books of Euclid, ufually taught in the schools, as not laying a fufficiently broad foundation for mathematical inftruction; and for that reafon would introduce the feventh, eighth, ninth, and tenth, as neceffary for demonstrating the rules of arithmetic and algebrathat the two laft are to be confidered as arts rather than fciences, which do not explain their own principles. To these positions, however, we by no means affent. With the imperfect numeral characters which the Greeks poffeffed, it would be fingular, indeed, if their methods of unfolding the properties of number were better than thofe of their. fucceffors, furnished with an' arithmetical notation, which, if any thing that men poffefs may be called perfect, is deferving of that epithet, and having befides the noble invention of algebraic language. The truth is, that the ancients wanted fo much the means of fimplifying the operations of arithmetic, that they proved, with confiderable difficulty, many truths which a better mode of expreflion has reduced to the clafs of felf-evident propofitions. It cannot be faid, with any good reason, that arithmetic and algebra do not poflefs the power of demonstrating their own principles and rules. Sufficient care in explaining the fundamental operations of thofe fciences, may not always be taken by thofe who have written of them. This, however, is not the fault of the fcience, but of the writers on it; and it is, befides, a cenfure that is by no means general. Dr Horley fays, it is abfurd to fend young men to ftudy alge bra bra before they have learnt the elements of geometry, on which depends the truth or evidence of all the algebraic rules. To us, again, it seems certain, that algebra can demonftrate its rules, juft as well as geometry. The fciences both reafon concerning quantity; the ideas, in both, are equally clear and well defined; they make ufe of the very fame axioms; and, therefore, that the conclufions of the one should be more certain than thofe of the other, what reafon can poffibly be affigned? Indeed, thofe mathematical reafonings, into which no idea of pofition is introduced, are not, ftrictly speaking, geometrical; they are mathematical; and if the arithmetic fymbols are ufed, which will in general contribute much to render them clearer and more concife, they become algebraic. The reproach, therefore, thrown against this fcience is ill-founded, and is injudicious; being calculated to diminish the attention paid to a part of mathematical learning that is of the very first importance. Farther, it is fo far from being abfurd to begin the study of the mathematics with algebra, rather than geometry, that it has been the practice to do fo with fome of the nations who have made the greatest progrefs in mathematical learning. One very great difadvantage that would neceffarily arife from forcing the ftudent of mathematics to read the feventh, &c. of the elements, is, that it would detain him long in the tudy of fynthetical reafonings, when he ought to be applying his mind to thofe that are analytical, and that lead to understand the methods of investigation. The fooner that the former method is abandoned for the latter, the fooner are the powers of invention called into action, and the more fpeedily do we acquire, not merely the knowledge of truth, but the capacity of difcovering it. As all the demonftrations in Euclid are fynthetical, the time fpent in the ftudy of thofe books we now fpeak of, would be far better beftowed in gaining a knowledge of the analytical inveftigations of algebra. It cannot indeed be denied, that many of the fundamental truths of algebra might be better proved than they are in fome of the books of that fcience; but this might certainly be done without abandoning the analytical methods, and without confuming time in the ftudy of demonftrations which, even when fully underflood, would not put the learner in poffeflion of the principle on which they were dif covered. Too great an attachment to fuch demonftrations is perhaps one of the chief reafons why the mathematical fciences have been for a long time fo ftationary in this country, compared with what they have been among our neighbours on the Continent. If there be any truth in this remark, the plan recommended by the Bishop of Rochester would tend greatly to retard the progress of fcience R 3 amongst amongst us, and to increase an evil, of which the magnitude is al ready fo much to be regretted. It is to be hoped, therefore, that they who have the care of the ftudies of the young men at the universities, will not haftily fuffer themfelves to be led away by the confidence with which Dr Horfley delivers his opinion on this fubject. The work, however, contains a fuller collection than ufual of the books of Euclid; and will, for that reafon, be very agreeable to thofe who are already verfed in mathematical ftudies, though, we apprehend, not very ufeful to thote who are only beginning them. But, waving the confideration of the purpofe, we are now to examine the execution of this work, and in what refpects the editor has improved on thofe who went before him. He profefles to have taken no aflittance from them, more efpecially from Simfon. • Quæcunque autem fint ea, vel qualefquales, quæ in editione hâc noftrâ fecimus emendationes, ducem in plerifque eorum SIMSONUM certiffime non fecuti fumus. Illud uobis propofitum fuit unice in Euclide emendando, Euclide ipfo duntaxat magiftro uti, per omnia intueri eum, et ad illius mentem quantum fieri potuit omnia componere.--Immo hoc ipfum erat ut rem non diffiteamur quod primo omnium ad Opus hoc noftrum excitavit nos, certa nimirum, et nunquam immutata opinio, Euclidem a Simfono fermone Anglico donatum juventutis academicæ inftitutioni non fufficere, aut fatis fideliter veterum geometrarum methodum, quæ nunquam non exißárn eft, iis in confpectu ponere. (Præf. ad fin.) The maxim, of employing only Euclid for the purpose of elucidating Euclid, feems at firft fight to be highly commendable, and to promife fomething very genuine and unfophifticated. This, however, is a hollow and deceitful appearance; for, in fact, no rule of criticism can be more injudicious and unfound. It is one which, if uniformly purfued, muft prevent the accumulation of learning and knowledge; and, instead of placing every fcholiaft on the thoulders of the preceding, would oblige him to begin his work anew, and execute the whole for himfelf. Had all men been vain enough to follow this maxim, the remains of antiquity, dug out from under the ruins of the barbarous ages, would not have gradually affumed all the perfection and elegance of the original compofitions; and the clallics in the days of Heyné would have been in no refpect better than in thofe of Chryfoloras. A few giants in hiterature may have been entitled to guide themfelves by this rule; but even they would have done more honour to themselves by the breach of it, than the obfervance. Such pretentions are much more likely to attend want of industry and patience in refearch, or an exceffive felf-confidence, than to accompany the poffeffion of real talents. But we must not cenfure Dr Dr Horley too feverely on this ground; for it will perhaps, appear that he has adhered lefs fcrupulously to his rule than the preceding paffage might lead us to imagine. As to what particularly regards Simfon in the above paffage, we acknowledge that the conftant attacks made by the learned Bishop on that excellent geometer has excited our surprise, and often our indignation. As an adept in the ancient gecinetry, a commentator on Euclid, and the reitorer of Apollonius, Simfon has merited the highest praife. The fpirit of the ancient geometry was known to him in its full extent; he ftudied it with industry and zeal; and poffelfed more power over it, as an inftrument for the discovery of truth, than any man of the prefent age, if we except his pupil and friend, the late Dr Mathew Stewart. Of this, his restoration of the Loci Plani, the Problems in his Conic Sections, and his restoration of the Porifms of Euclid, bear ample teftimony. His Euclid, though not admitting, like the works juft named, the fame exertion of original and inventive powers, is a model for the accuracy of its reafonings. What Dr Horfley refers to, therefore, when he speaks of it as giving but an imperfect idea of the extreme accuracy of the ancient geometry, we are unable to comprehend. Had he contented himself with faying that Simfon is now and then prolix, and that his notes are fometimes unneceffary, we could have feen reason for what he said, at least in a few cafes; but of this we cannot find a single inftance to justify the remark. As he has not specified what he meant particularly to fpeak of as deftitute of geometric agißa in Simfon, we cannot know precisely at what point the defence should be made; but we fhall proceed to confider on what his own pretenfions to fuperior accuracy are founded. For that purpose we muft look particularly into thofe parts. where the elements of geometry involve fome difficulty in them; and if Dr Horley has got over thofe in a more masterly way than any other editor, the oftentatious display in his preface will more easily be forgiven One of the firft quetions that has ufually exercised the ingenuity of the editors of Luclid, and the writers on elementary geometry in general, relates to parallel lines. It is eafy to fhow, that two lines having certain relations in their pofition with refpect to another line, will never meet; but it is very difficult, from the mere negative conficeration of two lines not meeting, to fhow what relation of poftion they must neceffarily have to a third line. Euclid himfelf coull find no other method of doing this, than by introducing an axiom, which almost every body has objected to as wanting on very effential property of an axiom, that of felf-evidence. Mathematicians. have therefore exerted themfelves, R 4 felves, in a variety of ways, to remove this difficulty, fome with more, and fome with lefs fuccefs; but none in a manner that has given entire fatisfaction. It has, however, we think, fared worfe with nobody in this matter than our author. Euclid had laid it down as an axiom, that lines which make with a third line the two interior angles lefs than two right angles, muft meet, if produced; and this propofition Dr Horfley endeavours to demonfirate; but he does fo by a procefs of reafoning which involves another axiom taken for granted without being expreffed; this is, that lines which incline toward one another, or have, as he calls it, their directions ad fe invicem, muft meet, if produced; where not only a new axiom, but alfo a new definition (that of the words inclined ad fe invicem, or toward one another) is implied. Now, if this definition be fupplied, the axiom juft mentioned will be found the very fame with that of Euclid, that is, with the propofition which it was Dr Horfley's purpose to demonstrate: His demonstration is therefore nothing more than a begging of the queftion, concealed under the obfcurity of a new and undefined expreffion. Such is the first example which he gives of geometrical precifion, when he is fairly left to himself, and has not Euclid for his guide. Dr Simfon has treated of this fame fubject, with confiderable prolixity, we will acknowledge, and without any thing remarkably happy or ingenious in his demonftration; but in a manner perfectly logical and accurate. Indeed, we are fully perfuaded, that if it had been proposed to that geometer to commit to the flames all that he had ever written concerning Euclid, or to infert the demonftration which Dr Horfley has given of this propofition, he would have fubmitted much more readily to the former than the latter mortification. The reader who will perufe with attention the corollary which Dr Horfley has annexed to the 28th of the firft of Euclid, will not think that in these remarks we have done him any injuftice. In the beginning of the third book of Euclid, it is stated as a definition, that equal circles are thofe of which the diameters are equal. This, however, is evidently not a definition, but a theorem; and is very improperly given as a definition by Euclid, or, as is more probable, by fome of his editors. Dr Horley has made an axiom of it, and this alfo feens not very agreeable to strict logic; for, as it is capable of being proved, by laying the one circle on the other, and thewing that they may wholly coincide, fo it ought to be proved in that nanner, becaufe the notion of equality has been before laid down as founded on the coincidence of magnitude; and no other idea of equality, but what is founded on this definition, and on the application to it of the other two axioms, that if equals are added to equals, or taken |