knowledges, this blemish in the demonftration of the 18th and Book V. that he may remove it, and render the demonftration we now have of it legitimate, he endeavours to demonftrate the following propofition, which is in page 115. of his book, viz. "Let A, B, C, D be four magnitudes, of which the two "first are of the one kind, and alfo the two others either of the "fame kind with the two firft, or of fome other the fame "kind with one another. I fay the ratio of the third C to the "fourth D, is either equal to, or greater, or less than the ratio "of the firft A to the fecond B." And after two propofitions premised as Lemmas, he proceeds thus. "Either among all the poffible equimultiples of the first "A, and of the third C, and, at the fame time, among all "the poffible equimultiples of the fecond B, and of the "fourth D, there can be found fome one multiple EF of the "first A, and one IK of the fecond B, that are equal to one "another; and alfo (in the fame cafe) fome one multiple "GH of the third C equal to LM the multiple of the fourth D, or fuch equality is no where to be found. If the first "cafe happen, A "[i. e. if fuch "equality is to "be found ] it is "manifeft from B— "what is be- C "ftrated, that D L -M "A is to B, as "fore demon. "C to D: but if fuch fimultaneous equality be not to be "found upon both fides, it will be found either upon one "fide, as upon the fide of A [and B;] or it will be found upon neither fide; if the first happen; therefore (from "Euclid's definition of greater and leffer ratio foregoing) "A has to B, a greater or lefs ratio than C to D; accord"ing as GH the multiple of the third C is lefs, or greater "than LM the multiple of the fourth D But if the fecond "cafe happen; therefore upon the one fide, as upon the fide "of A the firft and B the fecond, it may happen that the "multiple EF, [viz. of the firft] may be lefs than IK the multiple of the fecond, while, on the contrary, upon the o"ther fide, [viz. of C and D] the multiple GH [of the third C] is greater than the other multiple LM [of the fourth "D:] And then (from the fame definition of Euclid) the ra. X 2 'Book V. "tio of the firft A to the fecond B, is lefs than the ratio of the "third C to the fourth D; or on the contrary. "Therefore the axiom, [i. e. the propofition before fet down], "remains demonftrated," &c. Not in the leaft; but it remains ftill undemonstrated: For what he fays 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 fide, and D the diameter of another fquare; there can in no cafe 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 lefs than the mul tiple of D, nor when the multiple of A is lefs 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 fo 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 5th Def. b. 5. The fame objection holds good against the demonstration which fome give of the 1ft prop. of the 6th book, which we have made against this of the 18th prop. because it depends upon the fame infufficient foundation with the other. PROP. XIX. B. V. A corollary is added to this, which is as frequently used as the propofition itfeif. The corollary which is fubjoined to it in the Greek, plainly fhews that the 5th book has been vitiated by editors who were not geometers: For the converfion of ratios does not depend upon this 19th, and the demonftration which feveral of the commentators on Euclid give of conver fion, is not legitimate, as Clavius has rightly obferved, who has given a good demonstration of it which we have put in propofition E; but he makes it a corollary from the 19th, and begins it with the words, "Hence it eafily follows," though it does not at all follow from it. PRO P. XX, XXI. XXII. XXIII, XXIV. B. V. The demonftrations of the 20th and 21ft propofitions are fhorter than thofe Euclid gives of eafier propofitions, either in the preceding, or following books: Wherefore it was proper to make them more explicit, and the 22d and 23d propofitions 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. In a paper writ by Philippus Naudaeus, and published, after his death, in the hiftory of the Royal Academy of Sciences of Berlin, ann. 1745, page 50. the 23d prop, of the 5th book is cenfured as being obfcurely enunciated, and, because of this, prolixly demonftrated: The enunciation there given is not Euclid's but Tacquet's, as he acknowledges, which, though not fo well expreffed, is, upon the matter, the fame with that which is now in the Elements. Nor is there any thing obfcure in it, though the author of the paper has fet down the propor tionals in a difadvantageous order, by which it appears to be obfcure: But, no doubt, Euclid enunciated this 23d, as well as the 22d, fo as to extend it to any number of magnitudes, which, taken two and two, are proportionals, and not of fix only; and to this general cafe the enunciation which Naudacus gives, cannot be well applied, The demonstration which is given of this 23d, in that paper, is quite wrong; becaufe, if the proportional magnitudes be plane or folid figures, there can no rectangle (which he improperly calls a product) be conceived to be made by any two of them: And if it fhould be faid, that in this cafe ftraight lines are to be taken which are proportional to the figures, the demonftration would this way become much longer than Euclid's: But even tho' his demonftration 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 ufe of by both ancient and modern geome ters: And in many cafes compound ratios cannot be brought into demonftration, without making use of them. Whoever defires to fee the doctrine of ratios delivered in this 5th book folidly defended, and the arguments brought against it by And. Tacquet, Alph. Borellus, and others, fully efuted, may read Dr Barrow's mathematical lectures, viz. the 7th and 8th of the year 1666. The 5th book being thus corrected, I most readily agree to what the learned Dr Barrow fays, "That there is nothing p. 336 " in X 3 1 Book V." in the whole body of the elements, of a more subtile invention, "nothing more folidly established and more accurately handled, "than the doctrine of proportionals." And there is fome ground to hope, that geometers will think that this could not have been faid with as good reafon, fince Theon's time till the prefent. Book VI. DE F. II. and V. of B. VI. THE DE F. II. Two magnitudes are faid to be reciprocally proportional to two others, when one of the firft 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 justly thrown out of them, for the reasons given in the notes on the 23d prop. of this book. PROP. I. and II. B. VI. To the first of these a corollary is added which is often used: And the enunciation of the fecond is made more general. PROP. III. B. VI. A fecond cafe of this, as ufeful as the firft, is given in prop. A, viz. the cafe in which the exterior angle of a triangle is bifected by a ftraight line: The demonftration of it is very like to that of the first cafe, and upon this account may, probably, have been left out, as alfo the enunciation, by fome unfkilful editor: At least, it is certain, that Pappus makes ufe of this cafe, as an elementary propofition, without a demonftration of it, in prop. 39. of his 7th book of Mathematical Collections. PROP. To this a cafe is added which occurs not unfrequently in demonftrations. It feems plain, that fome editor has changed the demonftration that Euclid gave of this propofition: For, after he has demonftrated, that the triangles are equiangular to one another, he particularly fhews that their fides about the equal angles are proportionals, as if this had not been done in the demonftration of the 4th prop. of this book: This fuperfluous part is not found in the tranflation from the Arabic, and is now left out. This is demonftrated in a particular cafe, 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: Befides, the author of the demonftration, from four magnitudes being proportionals, concludes that the third of them is the fame multiple of the fourth, which the firft is of the fecond; now this is no where demonftrated in the 5th book, as we now have it: But the editor affumes it from the confufed notion which the vulgar have of proportionals: On this account, it was neceffary to give a general and legitimate demonftration of this propofition. PROP. XVIII. B. VI. For the The demonftration of this feems to be vitiated: propofition is demonftrated only in the cafe of quadrilateral figures, without mentioning how it may be extended to figures of five or more fides: Befides, from two triangles being equiangular, it is inferred, that a fide of the one is to the homologous fide of the other, as another fide of the firft is to the fide homologous to it of the other, without permutation of the proportionals; which is contrary to Euclid's manner, as is clear from the next propofition: And the fame fault occurs again in the conclufion, where the fides about the equal angles. are not fhewn to be proportionals; by reafon of again neglecting permutation: On thefe accounts, a demonstration is given in Euclid's manner, like to that he makes ufe of in the X 4 20th Book VI. |