Book V. " body of the Elements, of a more fubtile invention, nothing more "folidly established and more accurately handled, than the doc"trine of Proportionals." And there is some ground to hope that Geometers will think that this could not have been faid with as good reason, fince Theon's time till the prefent.

Book VI.

THE

DEF. II. and V. of B. VI.

HE 2. Definition does not feem to be Euclid's but some 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 expreffed, which made it proper to render it more distinct. it would be better to put the following Definition in place of it, viz.

DEF. II.

Two magnitudes are said to be reciprocally proportional to two others, when one of the first is to one of the other magnitudes, að the remaining one of the laft two is to the remaining one of the first.

But the 5. 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 23. 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 Enuntiation of the second is made more general.

PROP. III. B. VI.

A fecond cafe of this, as useful as the first, is given in Prop. A, viz. the cafe 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 Enuntiation, by fome unfkilful Editor. at leaft 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.

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 Demonftration 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 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. befides, the Author of the Demonfration, 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 Demonstration of this Propofition.

PROP. XVIII. B. VI.

The Demonstration of this feems to be vitiated. for the Proposition is demonstrated 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 shewn to be proportionals; by reason of again neglecting permutation. on these accounts a Demonstration is given in Euclid's manner, like to that he makes use of in the 20. Prop.

Book VI.

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

نہ

PROP. XXIII. B. VI.

Nothing is ufually reckoned more difficult in the Elements of Geometry by learners, than the doctrine of Compound ratio, which Theon has rendered absurd 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 the Definition of Triplicate ratio, &c. in the 5. Book. Theon's Definition is this; a Ratio is faid to be compounded of ratios ὅταν αἱ τῶν λόγων πηλικότητες ἐφ' ἑαυτὰς πολλαπλασιαῆσαι 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 notes, “ratio"num exponentes," the exponents 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 a ratio (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 consequent term of a ratio produces the antecedent, or, which is the fame thing, the number which arifes by dividing the antecedent by the confequent; but there are many ratios fuch, that no number can arife from the divifion of the antecedent by the confequent; ex. gr. the ratio which the diameter of a square 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 least 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 firft 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 expressly cited in these 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 useless and

abfurd. for that Theon brought it into the Elements can fcarce be Book VI. doubted, as it is to be found in his Commentary upon Ptolomy's Meydan Zurrağıs, page 62. where he also gives a childish explication of it, as agreeing only to fuch ratios as can be expreffed by numbers; and from this place the Definition and explication have been exactly copied and prefixed to the Definitions of the 6. Book, as appears from Hervagius's.edition. but Zambertus and Commandine in their Latin Translations subjoin the fame to these Definitions. Neither Campanus, nor, as it seems, the Arabic manuscripts from which he made his Translation, have this Definition. Clavius in his Obfervations upon it, rightly judges that the Definition of Compound ratio might have been made after the fame manner in which the Definitions of Duplicate and Triplicate ratio are given, viz. "that as in several magnitudes that are continual proportionals, Eu"clid named the ratio of the first to the third, the Duplicate ratio "of the first to the second; and the ratio of the first to the fourth, "the Triplicate ratio of the first to the fecond; that is, the ratio

compounded of two or three intermediate ratios that are equal "to one another, and fo on; fo in like manner if there be several

magnitudes of the fame kind, following one another, which are "not continual proportionals, the first is faid to have to the laft "the ratio compounded of all the intermediate ratios,only for "this reason, that these intermediate ratios are interpofed betwixt "the two extremes, viz. the first and last magnitudes; even as in "the 10. Definition of the 5. Book, the ratio of the first to the "third was called the Duplicate ratio, merely upon account of two "ratios being interpofed betwixt the extremes, that are equal to one ❝another. so that there is no difference betwixt this compounding "of ratios, and the duplication or triplication of them which are "defined in the 5. Book, but that in the duplication, triplication, "&c. of ratios, all the interpofed ratios are equal to one another; "whereas in the compounding of ratios, it is not neceffary that the "intermediate ratios fhould be equal to one another." Alfo Mr. Edmund Scarburgh, in his English translation of the first fix Books, page 238, 266. exprefsly affirms that the 5. Definition of the 6. Book, is fuppofititious, and that the true Definition of Compound ratio is contained in the ro. Definition of the 5. Book, viz. the Definition of Duplicate ratio, or to be understood from it, to wit, in the fame manner as Clavius has explained it in the preceding citation. Yet these, and the reft of the Moderns, do notwithstanding

Book VI. retain this 5. Def. of the 6. B. and illuftrate and explain it by long Commentaries, when they ought rather to have taken it quite away from the Elements.

For, by comparing Def. 5. B. 6. with Prop. 5. B. 8. it will clearly appear that this Definition has been put into the Elements in place of the right one which has been taken out of them. because in Prop. 5. B. 8. it is demonstrated that the plane number of which the fides are C, D has to the plane number of which the fides are E, Z (fee Hervagius's or Gregory's Edition) the ratio which is compounded of the ratios of their sides; that is, of the ratios of C to E, and D to Z. and by Def. 5. B. 6. and the explication given of it by all the Commentators, the ratio which is compounded of the ratios of C to E, and D to Z, is the ratio of the product made by the multiplication of the antecedents C, D, to the product of the confequents E, Z, that is the ratio of the plane number of which the fides are C, D to the plane number of which the fides are E, Z. wherefore the Propofition which is the 5. Def. of B. 6. is the very fame with the 5. Prop. of B. 8. and therefore it ought neceffarily to be cancelled in one of these places; because it is abfurd that the fame Propofition should stand as a Definition in one place of the Elements, and be demonstrated in another place of them. Now there is no doubt that Prop. 5. Book. 8. should have a place in the Elements, as the fame thing is demonftrated in it concerning plane numbers, which is demonstrated in Prop. 23. B. 6. of equiangular parallelograms; wherefore Def. 5. B. 6. ought not to be in the Elements. and from this it is evident that this Definition is not Euclid's but Theon's, or fome other unfkilful Geometer's.

1

But no body, as far as I know, has hitherto fhewn the true use of Compound ratio, or for what purpose it has been introduced into Geometry; for every Propofition in which Compound ratio is made use of, may without it be both enuntiated and demonftrated. Now the use of Compound ratio confifts wholly in this, that by means of it, circumlocutions may be avoided, and thereby Propofitions may be more briefly either enuntiated or demonftrated, or both may be done; for instance, if this 23. Propofition of the 6. Book were to be enuntiated, without mentioning Compound ratio, it might be done as follows; If two Parallelograms be equiangular, and if as a fide of the first to a fide of the fecond, fo any affumed ftraight line be made to a second straight line; and as the other fide of the first to the other fide of the fecond, fo the second straight line be made