taken from them, the refults are equal, can ever be admitted into geometry.

The fifth book of Euclid, which treats of the fubtle and difficult fubject of proportion, is the part of the elements which has most exercised the skill and ingenuity of commentators, and has given rife to much dispute, not concerning the conclufions, but concerning the mode of reafoning which the Greek geometer has employed. In this part Dr Horfley confiders himself as having made great improvements, though, when we compare his edition with Simfon's, except in one particular, we are quite at a lofs to perceive in what they confift. Yet, to hear him fpeak of them, one would imagine that, before his time, the fifth of Euclid was quite unintelligible: Siquidem omnia,' fays he, a nobis ita difpofita funt ut tandem aliquando, urá τινος ἐδῶ καὶ τάξεως explicetur hac definitionum feries, impedita antea, et mire interturbata. Fac enim in iifdem periculum, prout apud alias elementorum editiones extant, et nihil inveneris, quod aut perfpicuum, aut ad doctrinam utile, aut denique iis quibus interponitur fatis confonum eft. Rem ipfam deinde perpendito et fubductis rationibus, quomodo ex falebris hifce quis fe expediat aliter quam nos fecimus, ut opinor vix invenies:' (Præf. 7. ad fin.) Confidering what men they are who have undertaken to explain the matter in queftion before Dr Horfley, this may be confidered as one of the most ample panegyrics which any mathematician, fince the days of Cardan, has ventured to pronounce on his own performances. Yet we must acknowledge, that, after following the directions here given to his readers, fubductis rationibus,' the alterations he fpeaks of, feem all, except one, to be extremely immaterial.

This one, which feems of more importance than the reft, relates to the seventh definition, that of greater and less ratio, on which Dr Horley makes a remark, which we believe to be just, but by no means new. The remark is, that ratio being a relation, and not a quantity, greater or lefs, equal or unequal are not predicable of it; fo that to fpeak of one ratio being greater than another, is a catachreftic expreffion. When we fay, for inftance, that the ratio of A to B is greater than that of C to D, we mean that A is greater than that magnitude which has to B the fame ratio that C has to D. This is without doubt true in ftrictness; and the fame obfervation is made, and very well ilultrated, by Barrow in his Mathematical Lectures (lect. 20.), where he maintains against Gregory of St Vincents, Meibomius, Borelli, and others, that ratio is not quantity, and not strictly fufceptible of greater and lefs; and he adds, that when one ratio is called greater than another, it is by a kind of catachrefis

or

-or metonimy, which is the fame language that Dr Horley has employed. Barrow, however, though he has faid every -thing on the fubject of this definition, and the others that relate to proportion, which could be expected from a man of profound learning and great acuteness, has not propofed to make any change on the definition itself, nor on the demonftrations founded on it. Dr Horley has changed the former to one which he thinks preferable to what is ufually given as Euclid's: feveral demonftrations are changed in confequence of this, and they are perhaps in fome refpects improved; but they are certainly very different from the demonftrations of Euclid, and employ a poftulatum which he has never admitted into the fifth book. This, however, is the only change of any importance, that Dr Horsley feems to have made in the doctrine of proportion; the advan-tage from it is at beft but inconfiderable, and, at the fame time, the alteration feems rather to exceed that which a commentator has a right to make on his author's text. *

In the fixth book nothing occurs that requires to be taken notice of. The four books that follow are given with very little change from Gregory's folio edition.

In the eleventh and twelfth, where folids are treated of, the books of Euclid have been thought to require fome alteration. In this part, the Elements have been much indebted to Simson, who firft fhewed that Euclid's idea of equal and fimilar folids - was not accurate. Euclid holds thofe folids to be equal which are contained by the fame number of fimilar and equal plane figures; and yet it can be fhewn, that folids may be unequal in any proportion, though contained by fuch planes. This error was firit pointed out by Simfon; and Dr Horfley, without tak ing any notice of that circumstance, corrects Euclid's idea nearly as he had done. The great accuracy of Simfon was eminently fhewn in this part of the Elements; and he was the first who delivered the method of comparing folids with ftrict geometric accuracy. It is curious that this honour fhould have remained - for a geometer who wrote fo late as Simfon; and it is not a lit tle extraordinary, that any one should now treat of the fame fubject, and avail himself of all his improvements, without taking any notice of the perfon by whom they were firft fuggefted. This may be what Dr Horley means, when he fays, • Simfonum

Euclid gave no definition of compound ratio, though he uses the expreflion, and though it is certainly one that required to be explained. Dr Horley fallows Euclid in this, which is furely a defect; but, to Lave done otherwife, he mull have followed Simfon.

• Simfonum ducem minimè fecuti fumus. Not to acknowledge a leader, may certainly be faid, not to follow him...

In the twenty-fixth of the eleventh it is propofed to make a felid angle equal to a given folid angle, at a point in a given line. Of this problem, Euclid himself has given a very im perfect, and indeed a faulty folution, for which simfon fubftituted another, quite accurate, but not very happily conceived, nor fo extenfive as the nature of the thing requires: Dr Horfley has been more fuccessful in correcting this error; he has given a very fimple and general folution of the problem; and this fuperiority, he does not leave the reader to discover, but announces it with no fmall exultation. Problematis de quo agit propofitio libri XI. vicefima fexta, folutionem adjecimus uberiorem multo, quam quæ ex anguftis fuis principiis a Simfono prolata eft.'

Now, though it is true that Dr Horley's folution is more elegant and more general than Simfon's, this fuperiority might have been announced in lefs offensive terms. The problem is by no means of great difficulty; it admits of feveral folutions, fome of them even more fimple than that of Dr Horfley; but nothing that relates to fo eafy an investigation can decifively mark the genius of the inventor. A geometer, becaufe his folution was not the beft or most elegant, fhould not be charged with a limited and imperfect knowledge of the principles of his own fcience. Indeed, we are at a lofs to know what is here meant by the angufta principia of Simfon. His notions with regard to mathematics in general, might in fome refpects be accounted narrow and confined: he entertained ftrong and unreasonable prejudices against the algebraic methods of investigation, and feemed continually jealous of the encroachments which a barbarous rival (as he thought it) was every day making on his favourite fcience. This is confeffed on all hands; and to fuch prejudices the phrafe above quoted might not improperly be applied. But here the queflion is only concerning a matter of pure geometry, in which the extent and fertility of his genius were never before queftioned. The truth feems to be, that his excellence in this fcience was too great, to allow his defects to be eafily paffed over.

On the fubject of the eleventh book, we muft alfo remark, that Euclid, contrary to his cuftom, and not very confiftently with the rules of found logic, has given two definitions of a folid angle, of which one only is retained by Simfon. The definition retained is, that a folid angle is that which is formed by the meeting in a point of feveral plane angles which are not in the fame plane. The other definition is, that a folid angle is the mutual inclination of more than two ftraight lines which

meet,

meet, but are not in the fame plane. Dr Horley, in the fpirit of which we have seen so many examples, remarks,

Infcitè admodum Simfonus definitionum anguli folidi, quas duas Euclidis pofuit, alterâ repudiatâ alteram illam retinere maluit quæ vel minus univerfalis eft, vel fi aliter, ca faltem de quâ univerfalem effe, non equè manifeftum eft. '

This, we will not hesitate to fay, is a very uncandid criticism. There could be no reafon for retaining both definitions, as they either meant the fame thing, or they did not: If they meant the fame thing, one of them might be rejected; if they meant different things, one of them must be rejected, otherwife we must call different things by the fame name. Simfon, finding himself in this dilemma, retained the definition which most readily prefents to the mind that idea of a folid angle, which is the fubject of inveftigation in the Elements. Dr Horfley alleges that the other definition is more general, and that Euclid may have meant to include the vertex of a cone, or of any furface that terminates in a point, under the notion of a folid angle. But of this we have no proof; for nothing is more certain, than that he never takes the words folid angle in fuch a sense, in any part of the Elements. Indeed, to have done fo, was quite unfuitable to the ufual accuracy of his language. If he had ever called a cone by the name of a pyramid; if he had faid that the circumference of a circle was but a polygon of an infinite number of fides; if ever he had made any fuch deviation from the rigour of geometrical language-he might also have faid that a conical furface is made up of an infinite number of infinitely fmall plane angles. As he has never fpoken in this manner, we have no reason to think that he ever meant to do fo, nor would Dr Horfley, we believe, have afcribed to him that intention, but for the fake of accufing Simfon of ignorance, Infcitè admodum Simfonus. Our belief, therefore, in the ignorance of the latter, and the candour of the former, feems to rest on a very flight foundation.

The other work announced at the beginning of this article, is the book of Euclid's Data, from the fame editor, and with the addition of fome mathematical tracts of his own. This book, as being the foundation of the geometric analyfis, certainly deferves that the greatest attention fhould be exerted to give it to the public in the most perfect ftate. Some few inaccuracies feem to have entered originally into the compofition of it. In the fourth definition, for example, as it ftands in the Greek, and as it is given in the edition before us, there is without doubt an error; for it is there faid that lines, points and spaces are given in pofition which preserve always the fame fitúation. Now, if the word given' were really taken in fuch latitude as this, (fynonymous

(fynonymous with conftant or fixt) it would follow, as Simfon has juftly remarked, that a straight line dividing any given angle, in any given ratio, must be given in pofition, which is not true, because that pofition, though a thing determined in itself, cannot be found, except in a few cafes, by plane geometry. This limit therefore, is evidently implied, that the things proved to be given, must be found by the rules of plane geometry, that is, by constructions formed on the three poftulates prefixed. to the Elements. Dr Simfon, therefore, expreffed this definition differently from what it is in the Greek; and faid that. points, lines and spaces are given in pofition, which have always the fame fituation, and which are either actually exhibited, or can be found. Even the addition thus made, is not fufficiently. precife; for by being actually exhibited or found, is understood that they are found by the principles explained in the Elements.. Dr Horsley has paid no attention to thefe circumstances, but has followed exactly the Greek text, and has thus difcharged one part of the duty of a commentator at the expence of another. A fimilar remark may be made on his demonstration of the second propofition, where, by leaving out a limitation which Simfon had introduced, he has preferved the text, to the great prejudice of the fenfe.

In the general conduct of the book, however, little occurs to be cenfured, and not much to be praised, if we confider what others had done before. Simfon's edition of the Data always appeared to us to be excellent, and to admit of very little improvement; and in this opinion we are confirmed by the work before us. Dr Horfley, indeed, has added a fecond book to the Data, and has given, in a feparate tract, a felection of problems refolved by the geometric analyfis. We doubt, however, whether the first of these is a work of real utility; not that we doubt at all that new geometrical truths have their value, in whatever shape they appear, but becaufe they cannot always be proper for elementary inftruction. Propofitions of this nature maybe multiplied without end; and it is neceffary to make a felection of those that are of moft extenfive application, and are most frequently referred to, in order that the young geometer may. retain them in his mind, and have them always ready to be applied. The great fecret for preparing a young man to exert his talents in investigation, as well as in any thing elfe, is to fend him out furnished with all the principles neceffary to be known, but loaded with as few as poflible of thofe that are not neceffary,. or that may be easily supplied by his own ingenuity. The truths or principles that are not every day called for, had better be fupplied by the invention than the memory.

The