the Circumference DAB, as does the

DCB whereever the Point C is taken in the Circumference DCB. ... this Theorem is true.

BOOK IV.

Of PROPORTION, containing what is most valuable in Euclid's 5th and 6th Books.

152. Def. 1.

A

Leffer Magnitude is faid to be an aliquot Part of a greater Magnitude, when the leffer meafures the greater, i. e. when the leffer is contained a certain Number of Times exactly in the greater. Thus, a is an aliquot Part of 3a, a being contained in 3a, 3 Times.

153. Def. 2. A greater Magnitude is faid to be a Multiple of a leffer, when the leffer measures the greater, i. e. when the leffer is contained in the greater, exactly a certain Number of Times. Thus, 3a is a Multiple of a, a being contained in 34, 3 Times, and X b is a Multiple of b, for b is contained n Times in n x b.

154. The 3d Definition, "That Ratio is a mutual Relation of two Magnitudes of the fame Kind to one another in Refpect of Quantity," is rather metaphyfical than mathematical, and has been the Caufe of much Controverfy amongst the modern Mathematicians. Indeed it might well be omitted without any Lofs, and, as Mr. Simfon juftly obferves, feems to be the Addition of fome unskilful Editor.

155. Def. 4. These Magnitudes are faid to have a Ratio to one another, the leffer of which can be multiplied fo as to exceed the other.

G 3

156.

156. Def. 5. The first of four Magnitudes is faid to have the fame Ratio to the fecond which the third has to the fourth, when any Equimultiples whatfoever of the firft and third being taken, and any Equimultiples whatfoever of the fecond and fourth: If the Multiple of the firft be lefs than that of the fecond, the Multiple of the third is alfo lefs than that of the fourth; or, if the Multiple of the first be equal to that of the fecond, the Multiple of the third is alfo equal to that of the fourth; or, if the Multiple of the first be greater than that of the fecond, the Multiple of the third is also greater than that of the fourth.*

157. Def. 6. Magnitudes which have the fame Ratio are called Proportionals.

158. Def. 7. Of a greater and lefs Ratio being not used in this Effay, we have thought proper to omit it, as the Theorems which are obfcurely, (if not inaccurately) demonftrated by it, may be demonstrated more clearly by other Means.

159. The fame may be faid of the 8th Definition, viz. "That Analogy or Proportion is a Similitude of Ratios," as was remarked concerning the 3d. Definition.

160. Def. 9. Proportion confifts in three Terms · at least.

161.

* This is the true geometrical Definition of the Ancients however it may not be improper to give alfo the numerical or alge braical Definition, viz. Four Quantities are faid to be in direct Proportion to each other, when the Quotient of the first by the fecond is equal to the Quotient of the third by the fourth: Which Quotients are called Ratios. Or in other Words, Ratio is the Number expreffing how often the Confequent is contained in the Antecedent; or by which the Confequent being multiplied, the Product will be equal to the Antecedent.

161. Def. 10. When three Magnitudes are Proportionals, the firft is faid to have to the third the duplicate Ratio of that which it has to the fecond.

162. Def. 11. When four Magnitudes are continual Proportionals, the first is faid to have to the fourth the triplicate Ratio of that which it has to the fecond, and so forward quadruplicate, &c. increafing the Denomination each Time by Unity, in any Number of Proportionals.

163. Def. 12, of Compound Ratio. When there are any Number of Magnitudes of the fame Kind, the first is faid to have to the laft of them the Ratio compounded of the Ratio which the firft has to the fecond, and of the Ratio which the fecond has to the third, and of the Ratio which the third has to the fourth, and fo forward unto the laft Magnitude.

For Example, if A, B, C, D, be four Magnitudes of the fame Kind, the first A is faid to have to the laft D the Ratio compounded of the Ratio of A to B, and of the Ratio of B to C, and of the Ratio of C to D; or, the Ratio of A to D is faid to be compounded of the Ratios of A to B, B to C, and C to D.

And if A has to B the fame Ratio which E has to F; and B to C the fame Ratio which G has to H; and C to D the fame Ratio that K has to L; then, by this Definition, A is faid to have to D the Ratio compounded of Ratios which are the fame with the Ratios of E to F, G to H, and K to L. And the fame Thing is to be understood when it is more briefly expreffed by faying A has to D the Ratio compounded of the Ratios of E to F, G to H, and K to L.

In like Manner, the fame Things being fuppofed, if M has to N the fame Ratio which A has to D, then, for Shortnefs Sake, M is faid to have to N the

Ratio compounded of the Ratios of E to F, G to H, and K to L.

164. Scholium. Nothing has been thought more difficult, or caufed greater Difputes in Geometry, than the Definition of compound Ratio given by Theon in the 5th Def. of the 6th Book of Euclid, viz. "A Ratio is faid to be compounded of Ratios when the Quantities of Ratios being multiplied by one another, makes a certain Ratio ;" which is faulty, for there can be no Multiplication but by a Number, and fo the Definition will be ungeometrical. As to the Ufe of compound Ratio, it is introduced into Geometry only to render fome Demonftrations more concife, or that fome Propofitions may be more compendioufly enunciated. It may be further obferved, "that in any Magnitudes whatever, "of the fame Kind, A, B, C, D, &c. the Ratio "compounded of the Ratios of the firft to the fe"cond, of the fecond to the third, and fo on to "the laft, is only a Name or Expreffion, by which "the Ratio which the firit A has to the laft D is

fignified, and by which at the fame Time the "Ratios of all the Magnitudes, A to B, B to C, "C to D, from the first to the laft, to one another, "whether they be the fame or be not the fame, are "indicated, as in Magnitudes which are continual "Proportionals, A, B, C, D, &c. the duplicate "Ratio of the first to the second is only a Name, or "Expreffion, by which the Ratio of the firft A to "the third C is fignified, and by which, at the "fame Time, is fhewn that there are two Ratios. "of the Magnitudes from the firft to the laft, viz. "of the first A to the fecond B, and of the second "B to the third or laft C, which are the fame to "one another; and the triplicate Ratio of the firft "to the second is a Name, or Expreffion, by which "the Ratio of the first A to the fourth D is fignified, "and by which, at the fame Time, is fhewn that "there are three Ratios of the Magnitudes from the

"first to the laft, viz. of the firft A to the second "B, and of B to the third C, and of C to the fourth "or laft D, which are all the fame to one another; "and fo in the Cafe of any other multiplicate Ra<tios." 99, And this is the right Meaning of these Ratios as used by Euclid, Archimedes, Apollonius, and other ancient Geometricians. However, that of Theon abovementioned is ftill retained by most Moderns, and indeed may be made Ufe of when we are difcourfing of Numbers, or of Magnitudes as expreffed by Numbers, though it would be improper in this on Geometry.

165. Def. 13. In Proportionals the antecedent Terms are called homologous to one another, as alfo the Confequents to one another.

Note. Geometricians make Ufe of the following technical Words to fignify certain Ways of changing either the Order or Magnitude of Proportionals, fo as to continue ftill to be Proportionals.

166. Def. 14. Permutando, or alternando, by Permutation, or alternately: This Word is ufed when there are four Proportionals, and it is inferred, that the first has the fame Ratio to the third which the fecond has to the fourth; or that the first is unto the third, as the fecond to the fourth

fhewn in Article 199.

167. Def. 15. Invertendo, by Inverfion ; when there are four Proportionals, and it is inferred, that the second is unto the firft as the fourth to the third. See Art. 198.

168. Def. 16. Componendo, by Compofition ; when there are four Proportionals, and it is inferred, that the first, together with the fecond, is to the fecond, as the third, together with the fourth, is to the fourth. See Art. 200.