" case happen, A knowledges, this blemish in the demonftration of the 18th and Book v. that he may remove it, and render the demonstration we now have of it legitimate, he endeavours to demonstrate the follow: ing proposition, which is in page 115. of bis book, viz. “Let A, B, C, D be four magnitudes, of which the two “ first are of the one kind, and also the two others either of the “ same kind with the two first, or of some other the famo “ kind with one another. I say the ratio of the third Cro the “ fourth D, is either equal to, or greater, or less than the ratio " of the first A to the second B.” And after two propositions premised as Lemmas, he proceeds thus. " Either among all the poffible equimultiples of the first " A, and of the third C, and, at the same time, among all " the poffible equimultiples of the second B, and of the « fourth D, there can be found some one multiple EF of the “ first A, and one IK of the second B, that are equal to one " another; and also in the same cafe) fome one multiple “ GH of the third C equal to LM the multiple of the fourth " D, or such equality is no where to be found. If the first [i. e. if such E T " equality is to “ be found) it is B I -K " manifest from " what is be: 0 G H " fore demon. " strated, that D L -M " A is to B, as "C to D: but if such simultaneous equality be not to be “ found upon both sides, it will be found either upon one " fide, as upon the side of A [and B;] or it will be found "upon neither side; if the first happen; thercrore (from “ Euclid's definition of greater and lefler ratio foregoing) “ A has to B, a greater or less ratio than C to D; accord. "ing as GH the multiple of the third C is less, or greater " than LM the multiple of the fourth D : But if the second "case happen; therefore upon the one side, as upon the side " of A the first and B the second, it may happen that the multiple EF, (viz. of the first] may be less than IK the “maltiple of the second, while, on the contrary, upon the o. “ther lide, (viz. of C and D] the multiple GH [of the third " CJ is greater than the other multiple LM [of the fourth "D:] And then (from the same definition of Euclid) the ra. X 2 ho to & O TE Book V. " tio of the first A to the second B, is less than the ratio of the " third C to the fourth D; or on the contrary. “Therefore the axiom, [i.e. the proposition before set down], " remains demonstrated," &c. Not in the least; but it remains still undemonstrated : For what he says may happen, may, in innumerable cafes, never happen; and therefore his demonstration does not hold: For example, if A be the fide, and B the diameter of a square ; and C the side, and D the diameter of another square ; there can in no case be any multiple of A equal to any of B; nor any one of C equal to one of D, as is well known; and yet it can never happen that when any multiple of A is greater than a multiple of B, the multiple of C can be less than the multiple of D, nor when the multiple of A is less than that of B, the multiple of C can be greater than that of D, viz. taking equimultiples of A and C, and equimultiples of B and D: For A, B, C, D are proportionals; and so if the multiple of A be greater, &c. than that of B, so must that of C be greater, &c. than that of D; by sth Def. b. 5. The fame objection holds good against the demonstration which some give of the ift prop. of the 6th book, which we have made against this of the 18th prop. because it depends upon the same insufficient foundation with the other. PRO P. XIX. B. V. A corollary is added to this, which is as frequently used as the proposition itself. The corollary which is subjoined to it in the Greek, plainly News that the 5th book has been vitiated by editors who were not geometers : For the conversion of ratios does not depend upon this 19th, and the demonstration which several of the commentators on Euclid give of conver. fion, is not legitimate, as Clavius has rightly observed, who has given a good demonstration of it which we have put in proposition E; but, he makes it a corollary from the 19th, and be. gins it with the words, “ Hence it casily follows," though it does not at all follow from it. PRO P. XX, XXI. XXII. XXIII. XXIV. B. V. The demonstrations of the 20th and 21st propofitions are shorter than those Euclid gives of calier propofitions, either in the preceding, or following books : Wherefore it was proper to make them more explicit, and the 22d and 23d propositions are, as they ought to be, extended to any number of magnitudes : magnitudes : And, in like manner, may the 24th be, as is taken Book V. notice of in a corollary; and another corollary is added, as useful as the proposition, and the words “ any whatever” are sup. plied near the end of prop. 23. which are wanting in the Greek text, and the translations from it. In a paper writ by Philippus Naudaeus, and published, after his death, in the history of the Royal Academy of Sciences of Berlin, ann. 1745, page 5o. the 23d prop, of the gih book is censured as being obscurely enunciated, and, because of this, prolixly demonstrated : The enunciation there given is bot Eu. clid's but Tacquet's, as he acknowledges, which, though not to well expreffed, is, upon the matter, the same with that which is now in the Elements. Nor is there any thing obscure in it, though the author of the paper has fet down the proportionals in a disadvantageous order, by which it appears to be obscure : But, no doubt, Euclid enunciated this 23d, as well as the 22d, so as to extend it to any number of magnitudes, whichi, taken two and two, are proportionals, and not of six only; and to this general case the enunciation which Naudacus gives, cannot be well applied, The demonstration which is given of this 23d, in chat paper, is quite wrong; because, if the proportional magnitudes be plane or folid figures, there can do rectangle (which he improperly calls a product) be conceived to be made by any two of them : And it it should be said, that in this case straight lines are to be taken which are proportional to the figures, the demonftration would this way become much longer than Eu. clid's: But even tho' his demonstration had been right, who does not see that it could not be made use of in the 5th book ? : PROP. F, G, H, K, B. V. These propofitions are annexed to the 5th book, because they are frequently made use of by both ancient and modern geometers : And in many cases compound ratios cannot be brought into demonstration, without making use of them. Whoever desires to see the doctrine of ralios delivered in this 5th book folidly defended, and the arguments brought a. gainst it by And. Tacquet, Alph. Borellus, and others, fully lefuicd, may read Dr Barrow's mathematical lectures, viz. the 7th and 8th of the year 1066. The 5th book being thus corrected, I most readily agree to what the learned Dr Barrow says, “ That there is nothing 'p. 336. * X3 Pook V.“ in the whole body of the elements, of a more subtile invention, "" nothing more folidly established and more accurately bandled, " than the doctrine of proportionals." And there is some ground to hope, that geometers will think that this could not have been said with as good reason, fince Theon's time till the present. THI DEF. II. and V. of B. VI. Book VI. HE ad definition does not seem to be Euclid's, but fome unskilful editor's : For there is no mention made by Eu. clid, nor, as far as I know, by any other geometer, of reciprocal figures : It is obscurely expressed, which made it proper to render it more diftinét : 'It would be better to put the follow ing definition in place of it, viz. D E F. II. Two magnitudes are said to be reciprocally proportional to two others, when one of the first is to one of the other magnitudes, as the remaining one of the last two is to the remaining one of the first. But the 5th definition, which, fince Theon's time, has been kept in the elements, to the great detriment of learners, is now julliy thrown out of them, for the reasons given in the notes on the 23d prop. of this book. PRO P. I. and II. B. VI. To the first of these a corollary is added which is often used: And the enunciation of the second is made more general. A fecond case of this, as useful as the first, is given in prop. A, viz. the case in which the exterior angle of a triangle is bifected by a straight line: The demonstration of it is very like to that of the first case, and upon this account may, probably, have been left out, as also the enunciation, by fome unikilful editor : At least, it is certain, that Pappus makes use of this case, as an elementary propofition, without a demonftration of it, in prop. 39. of his 7th book of Mathematical Collections. PROP. Book VI. PRO P. VII. B. VI. To this a case is added which occurs not unfrequently in de. monstrations. It seems plain, that some editor has changed the demonstration that Euclid gave of this proposition : For, after he has de. monstrated, that the triangles are equiangular to one another, he particularly thews that their lides about the equal angles are proportionals, as if this had not been done in the demonstration of the 4th prop. of this book : This fuperfluous part is not found in the translation from the Arabic, and is now left out. This is demonstrated in a particular case, viz. that in which the third part of a straight line is required to be cut off ; which is not at all like Euclid's manner : Besides, the author of the demonstration, from four magnitudes being proportionals, concludes that the third of them is the same multiple of the fourth, which the first is of the second ; now this is no where demonstrated in the 5th book, as we now have it : But the editor assumes it from the confuled notion which the vulgar have of proporcionals : On this account, it was necessary to give a general and legitimate demonstration of this propofition. : The demonstration of this seems to be vitiated : For the proposition is demonstrated only in the case of quadrilateral figures, without mentioning how it may be extended to figures of five or more fides : Belides, from two triangles being equiangular, it is inferred, that a fide of the one is to the honolo. gous fide of the olher, as another side of the first is to the lide homologous to it of the other, without permutation of the proportionals; which is contrary to Euclid's manner, as. is clear from the next proposition : And the famie fault occurs again in the conclufion, wliere the fides about the equal angles are not fhewn to be proportionals ; by reason of again ncglecting permutation : On these accounts, a demonstration is given in Euclid's manner, like to that he makes use of in the X4 2002 |