Book V. « body of the Elements, of a more subtile 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 said with as good reason, fince Theon's time till the present,

Book VI.

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


HE 2. Definition does not seem to be Euclid's but some

unskilful 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 obscurely expressed, which made it proper to render it more distinct. it would be better to put the following Definition in place of it, viz.


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 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 useda and the Enuntiation of the second is made more general.


A second 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 Demonftration 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 unskilful Editor. at leaft it is certain that Pappus makes use of this case, as an Elementary Proposition, without a Demonftration of it, in Prop. 39. of his 7. Book of Mathem. Collections.

Book VI.

PROP. VII. B. VI. To this a case is added which occurs not unfrequently in Demonstrations.


It seems plain that some Editor has changed the Demonstration that Euclid gave of this Proposition. for after he has demonstrated that the triangles are equiangular to one another, he particularly shews 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 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 Demontration, from four magnitudes being proportionals, concludes that the third of them is the fame multiple of the fourth, which the first is of the second; now this is no where demonstrated in the 5. Book, as we now have it. but the Editor assumes it from the confused notion which the vulgar have of proportionals on this account it was necessary to give a general and legitimate Demonstration of this Proposition.


The Demonstration of this seems to be vitiated. for the Proposition is demonstrated only in the case of quadrilateral figures, with out mentioning how it may be extended to figures of five or more sides. besides, from two triangles being equiangular it is inferred that a fide of the one is to the homologous fide of the other, as ano ther side of the first is to the side 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 same fault occurs again in the conclusion, where the sides about the equal angles are not thewn 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. 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.

PROP. XXIII. B. VI. Nothing is usually reckoned more difficult in the Elements of Geometry by learners, than the doctrine of Compound ratio, which Theon has rendered absurd and ungeometrical, by substituting 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 said to be compounded of ratios όταν αι των λογων πηλικότητες εφ' εαυτας πολλαπλασιασθείσαι ποιώσι τινα. which Commandine thus tranfates, “quando rationum quantitates “ inter se multiplicatae aliquam efficiunt rationem;" that is, when the quantities of the ratios being multiplied by one another make a certain ratio. Dr. Wallis translates the word innebTITES, “ ratio

num exponentes,” the exponents of the ratios. and Dr. Gregory renders the last words of the Definition by « illius facit quantitatem, mases 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 same thing, the number which arises by dividing the antecedent by the consequent; but there are many ratios such, that no number can arise from the division of the antecedent by the consequent; ex. gr. the ratio which the diameter of a square has to the side of it; and the ratio which the circumference of a circle has to its diameter, and such like. Besides, there is not the least mention made of this Definition in the writings of Euclid, Archimedes, Apollonius, or other antients, tho' they frequently make use 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 necessary to be brought in; but the right Definition is expressly cited in these words, “but the ratio of K to S 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

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absurd. for that Theon brought it into the Elements can scarce be Bouk VI. doubted, as it is to be found in his Commentary upon Ptolomy's Meydan Eurdens, page 62. where he also gives a childish explication of it, as agreeing only to such ratios as can be expressed 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 1'ranslations subjoin the fame to these Definitions. Neither Campanus, nor, as it feeins, the Arabic manuscripts from which he made his Translation, have this Definition. Clavius in his Observations upon it, rightly judges that the Definition of Compound ratio might have been made after the same 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 second ; that is, the ratio

compounded of two or three intermediate ratios that are equal “ to one another, and so on; so in like manner if there be severa!

inagnitudes of the same kind, following one another, which are “ not continual proportionals, the first is said to have to the last “ the ratio compounded of all the intermediate ratios only for “ this reason, that these intermediate ratios are interposed 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 interposed betwixt the extremes, that are equal to one

another. so that there is no diiference 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 interposed ratios are equal to one another; " whereas in the compounding of ratios, it is not neceffary that the

intermediate ratios should be equal to one another.” Alfo Mr. Edmund Scarburgh, in his English translation of the first fix Books, page 238; 266. exprefsly aflirms that the 5. Definition of the 6. Book, is supposititious, 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 same manner as Clavius has explained it in the preceding citation. Yet these, and the rest of the Moderns, do not withstanding,


Book Vl. retain this 5. Def. of the 6. B. and illustrate 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 (see Hervagius's or Gregory's Edition) the ratio which is compounded of the ratios of their fides; 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 consequents E, Z, that is the ratio of the plane number of which the fides are C, D to the plane number of which the fide's are E, Z. wherefore the Proposition which is the 5. Def. of B. 6. is the very fame with the 5. Prop. of B. 8. and therefore it ought necessarily to be cancelled in one of these places; because it is absurd that the same Proposition 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 same thing is demonstrated 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 some other unskilful Geometer's.

But no body, as far as I know, has hitherto shewn the true use of Compound ratio, or for what purpose it has been introduced into Geometry; for every Proposition in which Compound ratio is made use of, may without it be both enuntiated and demonstrated. Now the use of Compound ratio consists wholly in this, that by means of it, circumlocutions may be avoided, and thereby Propofitions may be more briefly either enuntiated or demonstrated, or both may be done ; for instance, if this 23. Proposition 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 side of the first to a side of the second, fo any assumed straight line be made to a second straight line; and as the other side of the firft to the other side of the second, so the second straight line be made

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