have of late fallen. We agree with his Lordship, that the 10th Book does not poffefs an equal degree of importance. The fubject of it, though wonderful in every part, and though it perhaps affords the beft fpecimen of the comprehenfive and difcerning mind of Euclid, is far more abftrufe and difficult than thofe of the other Books, and at the fame time lefs fubfervient to the exigencies and ufes of common life: and the Bishop evidently, from the various algebraical illuftrations, which he has given of fome of the propofitions in this Book, fuppofes his readers to have made fome proficiency in the ftudy of algebraic furds. But it will be neceffary for the learner to be acquainted with, at leaft, the two firft propofitions of this Book, or, as the Bishop with propriety recommends, the 20 firft, before he proceeds to the 11th and 12th. In the fix first Books, as well as in the 11th and 12th, the text of Commandine has been followed, and Gregory's tranflation of the Greek edition, Oxōn. 1703, in those between the 6th and 11th. But in both cafes the learned editor has made fuch corrections in the ftyle and emendations in the proof, as well as innumerable additions scattered through every part of the work, as feemed neceffary or convenient. Thefe alterations, we are informed in the Preface (which is written with great variety and depth of learning) chiefly occur in the 5th, 10th, and 11th Books; we have, however, obferved many other important emendations in every other part, but particularly in the 3d Book. We have obferved alfo, in almoft every page, fome alteration or omiffion, and scarcely ever without great advantage to the fubject. The change in the order of the definitions and propofitions, which often occurs, appears to be made, in every cafe, with judgment; and fometimes, when the demonftration adopted by the Bishop abfolutely requires it, with the utmost evidence of fcience. We are informed alfo, in the Preface, that the editor has feldom followed the plan of Simfon; and, with reafon, he paffes fome cenfures on that author's edition of Euclid. But we cannot help regretting, that his Lordship has not followed him in one very material point, which has rendered that book fo useful, and has been the means of giving it fo extenfive a circulation, that of its being compofed in the English language. We fear, indeed, that from this ingle circumftance, Simfon's edition will unavoidably continue in more general use than this, certainly more mathematical edition, by the learned prelate. In his Preface to his Elementary Treatifes, which volume is itfelf written in English, and which we noticed, with due commendation, in vol. xxi. p. 272 of our Review, Bishop Horfley fays, that works of fcience ought to be compofed in the Latin language. To this, as a general allertion, affertion, we cannot object; but, when we reflect that Euclid, the key to the science of geometry, is, with much propriety, put into the hands of our youth, at an early period of their education; when, if they attempt to perufe any Latin books, it ought to be rather for the fake of their Latinity; and, when it occurs further, that the teachers of our youth muft, at present, neceffarily explain and illuftrate the propofitions to their pupils in their vernacular language, we are inclined to think, that Euclid ought to make an exception to the rule. Who, indeed, that knows the value of this present edition of Euclid will not regret, that it is placed above the comprehenfion of industrious ftudents in mathematics, who have not had opportunity to acquire a fufficient knowledge of Latin for perufing it*? The first improvement of any confequence that we have obferved is, the new and much-wanted illuftration which the Bishop has given of the twelfth axiom of the firft Book, and has properly placed as a corollary to the 28th propofition; by means of which, and of the nature of a ftraight line, the truth of the axiom is established, without having recourse to the tedious method used by Simfon to illuftrate it, in his notes on the 28th propofition of the firft Book. In the fecond Book, we difcovered no material alteration, The two first definitions, which are the fecond and third of Commandine (for the firft definition of Commandine is properly placed here as an axiom) are, we think, rendered much more clear. In Commandine there are no definitions; for, according to him, it is only afferted, that things which touch one another do touch one another. In p. 65, we have a new propofition, which comprehends the fubftance of the 5th and 6th propofitions: it is neatly introduced, and we were glad to find it annexed to thefe propofitions without fuperfeding them, as it cannot fail to make an impreflion on the mind. Those to whom the education of our youth is entrufted muft have peculiar fatisfaction in perufing the 10th propofition, and the fcholion annexed to it. The diagram itfelf, to fay nothing of the demonftration, in Commandine as well as in Simfon, is calculated to confuse the learner refpecting the intersection of a circle and an elliptical figure. But here a diagram is used, in which two circles interfect each other, and are not therefore expofed to the fame confufion. A knowledge of the scholion annexed to this proposition is neceffary, in order to understand *It must be acknowledged, on the other hand, that the Latin language will facilitate its circulation in foreign countries. the the demonftration of the propofition itfelf. The 11th propofition (13th of Commandine) is here in its proper place, for the reafon given in the fhort fcholion which is placed after the 13th propofition of this edition. The diagram of this propofition is alfo happily improved: for even greater objections may be made to the diagram ufed by Commandine and Simfon for this propofition, than those which have been made against the diagram of the 10th propofition in thofe editions. The demonftration is also much more convincing and accurate. The remaining improvements in this Book confift chiefly of feveral neat and useful corollaries, many of which are taken from Clavius. They are well chofen, not only as they, for the most part, immediately arife from the propofitions to which they belong; but alfo, as they are well calculated to make the learner better acquainted with the properties of tangent lines. The improvements in the fourth Book confift entirely of a few fcattered infertions or omiffions. We are now arrived at that Book, which we are informed in the Preface required the greateft correction; and, in truth, we have found it in a state very different from that in which we have hitherto feen it. The change in the order of the definitions feems to be made with much propriety and judg ment; for, in the old order, there are two definitions (5 and 7) which entirely regard analogy, before it had been defined what analogy is, which is not furely much according to fcience: but, as they are here placed, they fucceed each other in a plain and natural order, In the 9th definition (the 7th of Commandine) a moft important alteration has been made; and his Lordship mentions it in his Preface, in a very learned and mafterly manner; and, independent of other confiderations, which we might cafily fuggeft, we are inclined to think it very good, because it is more adapted to the comprehenfion of the learner, who is ufually much puzzled at his entrance on this Book, and because it is also much more easily retained in his memory; for, in our reviews of elementary works, we feel ourselves obliged to confider them in the particular view of their utility to the perfons who may be entering upon the fubject of which they treat. The alterations alfo in the 7th, and three following propafitions, cannot fail to meet with univerfal approbation. The parts of each propofition, as they ftand here, are better and more ftrikingly oppofed to each other. The enunciations also are certainly more neatly expreffed, and more easily remembered. The demonftrations, likewife, of thefe four propofitions have been fo much altered, that they may not improperly be deemed new. new. The former part of the 9th propofition (the 8th of Commandine) is indeed quite new, in fubftance as well as in expreffion. The clearness and concifenefs of it cannot fail to give delight, as it forms fuch a contrast with the long and tirefome demonftration which is ufed by Commandine and Simfon. It is very fatisfactorily deduced from the new 9th definition, and is itfelf a fufficient reafon for the alteration made in that definition. The two cafes of the first part of the 3d propofition of the 6th Book (one only of which is given in Commandine) are well united in one demonstration, which is adapted to the two diagrams, as indeed are the two cafes of the fecond part also. Simfon has given not only two demonftrations of each of the various cafes of this propofition, but he has alfo given two enunciations. But the circumftance of the exterior angle not being particularly specified in the enunciation is not a fufficient reafon for an additional theorem; for, in the Greek editions, as the Bishop has proved, it is evidently included, though not particularly mentioned. No other very important difference from Commandine occurs, till we come to the 24th propofition, which is made much more general. In the 7th, 8th, and 9th Books, little important matter has been added, with the exception of a few corollaries; with most of which, the reader cannot fail to be pleased, and will generally find himself inftructed. The flyle and mode of proof is confiderably altered, in order to make it correfpond with the technical phrafeology of Commandine. The next Book abounds with fuch a variety of emendations of all kinds, that it ceafes to be Gregory's tranflation of the Greek, and affords us ample room for enlarging our criticifms; but, as we have been induced to fay fo much of fome of the former Books, from their fuperior importance and utility, we are obliged to pafs this over in a very curfory manner. We believe that there is not a fingle propofition in the Book that has escaped the fagacious eye of the prefent editor. The new and neat corollaries annexed to the improved 10th and 11th definitions of the 11th Book are of great utility, as they are well adapted to give the reader a proper idea of a folid angle, before he enters upon the doctrine of folids. The fub. ject alfo of the two corollaries to the 2d propofition, as well as of thofe to the 11th, give them confiderable importance; and the very fatisfactory manner in which they are demonftrated renders them interefting. The demonftration of the 21ft propofition is certainly more general than that in Conimandine, which is confined to a folid angle contained by three plane plane angles; and alfo much preferable to that of Simfon, who, after his ufual manner, and under the miftaken notion of perfpicuity, makes two cafes of it, when one will answer every purpose. We cannot alfo refrain from giving a decided preference to the diagrams and demonftrations of the 26th propofition, not only on account of their being more general, which is in itfelf a fufficient reafon, but becaufe they are much more clear and fatisfactory. The fame observations may be made with great truth respecting the 29th and 30th propofitions. The new fcholion, in p. 456, we by no means think unneceffary; but almoft regret that it is not placed. earlier in the work. The improved and unencumbered diagrams of the 35th propofition afforded us much fatisfaction, as well as the altered proof which is applied to it. This is the laft alteration of any importance that occurs in the volume. The remaining emendations confifl of verbal alterations, or of the addition or omiffion of a few fentences in the demonftrations, and these occur chiefly in the 17th propofition of the 12th Book. There are two pages of Addenda, which contain fome useful corollaries which were omitted in the body of the work. It has not occurred to us to remark many typographical errors of any great magnitude, which are not noticed in the lift at the end of the volume. Two or three we obferved amidit the marginal references, which may perhaps millead the inattentive reader; but none of them are of fufficient importance to be noted here. We have prefented our readers with a general view of what has been done in this edition of Euclid. Much, indeed, we have fhown has been done in it, to excite curiofity, and gratify the enquiring mind; and we do not hesitate to affert, that the learned editor has conferred a lafting obligation on the public, in having made fo valuable an addition to mathematical literature; and he demands a particular tribute of gratitude and honour from thofe who fuperintend the education of our youth, both in the Universities, and in all public or private feminaries. ART. VII. Pearfon on the Theory of Morals. IN the author's fecond chapter, where he treats of the rule of virtue, are many excellent remarks, and fome which we think liable to objection. When he contends, that there is no one |