Book V.

ratio of the first A to the second 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 still undemonftrated. for what he fays may happen, may in innumerable cafes never happen, and therefore his demonftration does not hold. for example, if A be the fide and B the diameter of a fquare; 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 or lefs than a multiple of B, the multiple of C can, upon the contrary, be lefs or greater than the multiple 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, fo muft that of C be greater &c. than that of D. by 5. Def. B. 5.

The fame objection holds good against the Demonstration which fome give of the 1. Prop. of the 6. Book, which we have made against this of the 18. Propofition, 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 itself. the Corollary which is fubjoined to it in the Greek, plainly fhews that the 5. Book has been vitiated by Editors who were not Geometers. for the converfion of ratios does not depend upon this 19. and the Demonftration which feveral of the Commentators on Euclid give of Converfion, 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 19. and begins it with the words, "Hence "it eafily follows," tho' it does not at all follow from it.

PROP. XX, XXI, XXII, XXIII, XXIV. B. V. The Demonftrations of the 20. and 21. Propofitions are shorter 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 22. and 23. Propofitions are, as they ought to be, extended to any number of magnitudes. and in like manner

may the 24. be, as is taken notice of in a Corollary; and another Book V. Corollary is added, as ufeful as the Propofition. and the words

any whatever" are fupplied 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 Hiftory of the Royal Academy of Sciences of Berlin, Ann. 1745. page 50. the 23. Prop. of the 5. Book, is cenfured as being obfcurely enuntiated, and, because of this, prolixly demonftrated. the Enuntiation there given is not Euclid's but Tacquet's, as he acknowledges, which, tho' 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, tho' the Author of the paper has fet down the proportionals in a difadvantageous order, by which it appears to be obfcure. but no doubt Euclid enuntiated this 23. as well as the 22. 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 Enuntiation which Naudaeus gives, cannot be well applied.

The Demonstration which is given of this 23. in that paper, is quite wrong; because 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 fee that it could not be made ufe of in the 5. Book?

PROP. F, G, H, K. B. V.

Thefe Propofitions are annexed to the 5. Book, because they are frequently made ufe of by both antient and modern Geometers. and in many cafes Compound ratios cannot be brought into Demonftrations, without making ufe of them.

Whoever defires to fee the doctrine of Ratios delivered in this 5. Book, folidly defended, and the arguments brought against it by And. Tacquet, Alph. Borellus and others, fully refuted, may read Dr. Barrow's Mathematical Lectures, viz. the 7. and 8. of the year 1666.

The 5. Book being thus corrected, I most readily agree to what the learned Dr. Barrow fays*, " That there is nothing in the whole Page 336.

*

Book V. "body of the Elements, of a more fubtile 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 present.

Book VI.

THE

DE F. II. and V. of B. VI.

HE 2. Definition does not feem to be Euclid's but fome unfkilful Editor's. for there is no mention made by Euclid, nor, as far as I know, by any other Geometer, of reciprocal figures. it is obfcurely expressed, which made it proper to render it more diftinct. it would be better to put the following Definition in place of it, viz.

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 laft two is to the remaining one of the firft.

But the 5. Definition, which fince Theon's time has been kept in the Elements, to the great detriment of learners, is now juftly thrown out of them, for the reafons given in the Notes on the 23. Prop. of this Book.

PROP. I. and II. B. VI.

To the firft of thefe a Corollary is added which is often ufed. and the Enuntiation 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 Enuntiation, by fome unfkilful Editor. at least it is certain that Pappus makes ufe of this cafe, as an Elementary Propofition, without a Demonstration of it, in Prop. 39. of his 7. Book of Mathem. Collections.

PROP. VII. B. VI.

To this a cafe is added which occurs not unfrequently in Demonftrations.

Book VI.

PROP. VIII. B. VI.

It seems plain that fome Editor has changed the Demonstration that Euclid gave of this Propofition. for after he has demonstrated 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 Demonstration of the 4. Prop. of this Book. this fuperfluous part is not found in the Tranflation from the Arabic, and is now left out.

PROP. IX. B. VI.

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 first is of the fecond; now this is no where demonftrated in the 5. Book, as we now have it. but the Editor affumes it from the confused 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.

The Demonftration of this feems to be vitiated. for the Propofition is demonftrated only in the cafe of qudrilateral 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 Demonftration is given in Euclid's manner, like to that he makes ufe of in the 20. Prop.

320

Book VI. of this Book; and it is extended to five fided figures, by which it may be seen how to extend it to figures of any number of fides.

PROP. XXIII. B. VI.

Nothing is ufually reckoned more difficult in the Élements of Geometry by learners, than the doctrine of Compound ratio, which Theon has rendered abfurd and ungeometrical, by fubftituting the 5. Definition of the 6. Book, in place of the right Definition which without doubt Eudoxus or Euclid gave, in its proper place, after Theon's the Definition of Triplicate ratio, &c. in the 5. Book. Definition is this; a Ratio is faid to be compounded of ratios orar αἱ τῶν λόγων πηλικότητες ἐφ' ἑαυτὰς πολλαπλασιασαι ποιῶσι τινά. which Commandine thus tranflates, "quando rationum quantitates "inter fe multiplicatae aliquam efficiunt rationem;" that is, when the quantities of the ratios being multiplied by one another make a certain ratio. Dr. Wallis tranflates the word anλnótutes, “ratio"num exponentes," the exponunts of the ratios. and Dr. Gregory renders the last words of the Definition by" illius facit quantitatem," makes the quantity of that ratio. but in whatever sense the "quan"tities" or "exponents of the ratios," and their "multiplication" be taken, the Definition will be ungeometrical and useless. for there can be no multiplication but by a number; now the quantity or exponent of aatio (according as Eutocius in his Comment. on Prop. 4. Book 2. of Arch. de Sph. et Cyl. and the moderns explain that term) is the number which multiplied into the confequent term of a ratio produces the antecedent, or, which is the fame thing, thể number which arifes by dividing the antecedent by the confequent; but there are many ratios fuch, that no number can arise from the divifion of the antecedent by the confequent; ex. gr. the ratio which the diameter of a fquare has to the fide of it; and the ratio which the circumference of a circle has to its diameter, and fuch like. Befides, there is not the leaft mention made of this Definition in the writings of Euclid, Archimedes, Apollonius, or other antients, tho' they frequently make ufe of Compound ratio. and in this 23. Prop. of the 6. Book, where Compound ratio is first mentioned, there is not one word which can relate to this Definition, tho' here, if in any place, it was neceffary to be brought in; but the right Definition is exprefsly cited in thefe words, " but the ratio of K to "M is compounded of the ratio of K to L, and of the ratio of L "to M." this Definition therefore of Theon is quite ufelefs and