That is, if there be four Magnitudes, and you take any Equimultiples of the firft and third, and alfo any Equimultiples of the fecond and fourth; and if the Multiple of the firft be greater than the Multiple of the fecond; and alfo the Multiple of the third greater than the Multiple of the fourth; or, if the Multiple of the first be equal to the Multiple of the fecond; and alfo the Multiple of the third equal to the Multiple of the fourth; or, laftly, if the Multiple of the firft be less than the Multiple of the second; and also that of the third less than that of the fourth, and these Things happen according to every Multiplication whatfoever: Then the four Magnitudes are in the fame Ratio; the first to the second, as the third to the fourth.

VI. Magnitudes that have the fame Proportion, are called Proportionals.

Expounders ufually lay down here that Definition, for Magnitudes, which Euclid has given for Numbers, only, in his Seventh Book, viz. That

Numbers are proportional, when the first is either the fame Multiple of the fecond, as the third is of the fourth, or elfe the fame Part, or Parts.

But this Definition appertains only to Numbers, and commenfurable Quantities; and fo fince it is not univerfal, Euclid did well to reject it in this Element, which treats of the Properties of all Proportionals; and to fubftitute another general one, agreeing to all Kinds of Magnitudes. In the mean Time, Expounders very much endeavour to demonftrate the Definition here laid down by Euclid, by the ufual received Definition of proportional Numbers; but this much easier flows from that, than that from this; which may be thus demonstrated:

First, Let A, B, C, D, be four Magnitudes, which are in the fame Ratio, according to the Conditions that Magnitudes in the fame Ratio muft have according to the fifth Definition; and let the firft be a Multiple of the fecond: I fay, the third is alfo the fame Multiple of the fourth. For Example: Let A be equal to 5B: Then C fhall be equal to 5D. Take any

14.

Num

A B

:

C D

:

Number, for Example, 2, by which Iet 5 be multiplied, and the Product will be 10: And let 2A, 2C, be Equimultiples of the first and third Magnitudes A and C: Alfo, 2A, 10B, 2C, 10D let OB and 10D be Equimultiples of the second and fourth Magnitudes B and D. Then (by Def. 5.) if 2A be equal to 10B, 2C fhall be equal to 10D. But fince A (from the Hypothefts) is five Times B, 2A fhall be equal to 10 B; and fo 2C equal to 10D, and C equal to 5D; that is, C will be five Times D. W. W. D.

Secondly, Let A be any Part of B; theu C will be the fame Part of D. For, becaufe A is to B, as C is to D; and fince A is fome Part of B; then B will be a Multiple of A: And fo by (Cafe 1.) D will be the fame Multiple of C; and accordingly Chall be the fame Part of the Magnitude D, as A is of B.W.W.D.

Thirdly, Let A be equal to any Number of whatfoever Parts of B. I fay, C is equal to the fame Number of the like Parts of D. For Example: Let A be a fourth Part of five Times B; that is, let A be equal to B. I fay, C is alfo equal to D. For, because A is equal B, each of them being multiplied by 4, then 4A will be equal to 5B. And fo, if the Equimultiples of the first

A: B:: C: D

and third, viz. 4A, 4C, be affumed; as alfo the Equimultiples of the fecond and fourth, 4A, 5B, 4C, 5D viz. 5B, 5D; and (by the Definition) if 4A is equal to 5B; then 4C is equal to 5D. But 4A has been proved equal to 5B, and fo 4C fhall be equal to 5D, and C equal to D. W. W. D.

And univerfally, if A be equal

n

equal to D. For let A and C

m

be multiplied by m, and B and D by n. And because A is equal

n

to-B; mA fhall be equal to

m

n

to-B, C will be

m

ABC: D

mA, nB, mC, nD

nB; wherefore (by Def. 5.) mC will be equal fo-✯D,

VII. When, of Equimultiples, the Multiple of the firft exceeds the Multiple of the fecond, but the Multiple of the third does not exceed the Multiple of the fourth then the first to the fecond is faid to have a greater Proportion than the third to the fourth.

VIII. Analogy is a Similitude of Proportions.
IX. Analogy at least confifts of three Terms.
X. When three Magnitudes are Proportionals, the
firft is faid to have, to the third, a duplicate
Ratio to what it has to the fecond.

XI. But when four Magnitudes are continued
Proportionals, the first shall have a triplicate
Ratio to the fourth of what it has to the fe-
cond; and fo always one more in Order, as the
Proportionals fhall be extended.

XII. Homologous Magnitudes, or Magnitudes of a like Ratio, are faid to be fuch whofe Antecedents are to the Antecedents, and Confequents to the Confequents.

XIII. Alternate Ratio is the comparing of the Antecedent with the Antecedent, and the Consequent with the Confequent.

XIV. Inverse Ratio is, when the Confequent is taken as the Antecedent, and fo compared with the Antecedent as a Confequent.

XV. Compounded Ratio is, when the Antecedent and Confequent, taken both as one, is compared to the Confequent itself,

XVI. Divided Ratio is, when the Excess, whereby the Antecedent exceeds the Confequent, is compared with the Confequent.

XVII. Converse Ratio is, when the Antecedent is compared with the Excess, by which the Antecedent exceeds the Confequent.

XVIII. Ratio of Equality is, where there are taken more than two Magnitudes in one Order, and a

like

like Number of Magnitudes in another Order, comparing two to two being in the Jame Proportion; and it fhall be in the firft Order of Magnitudes, as the first is to the last, fo in the fecond Order of Magnitudes is the first to the laft: Or otherwife, it is the Comparison of the Extremes together, the Means being omited. XIX. Ordinate Proportion is, when as the Antecedent is to the Confequent, fo is the Antecedent to the Confequent; and as the Confequent is to any other, fo is the Confequent to any other. XX. Perturbate Proportion is, when there are three or more Magnitudes, and others also, that are equal to thefe in Multitude, as in the first Magnitudes the Antecedent is to the Confequent; fo in the fecond Magnitudes is the Antecedent to the Confequent: And as in the first Magnitudes the Confequent is to fome other, fo in the fecond Magnitudes is fome other, to the Antecedent.

AXIOM S.

Quimultiples of the fame, or of equal Magnitudes, are equal to each other.

II. Thofe Magnitudes that have the fame Equimultiple, or whofe Equimultiples are equal, are equal to each other.

PRO

X I. V.

PROPOSITION I.

THEOREM.

If there be any Number of Magnitudes Equimultiples of a like Number of Magnitudes, each of each; whatsoever Multiple any one of the former Magnitudes is of its correfpondent one, the fame Multiple are all the former Magnitudes of all the latter.

L

A

G+

B

C

E

E T there be any Number of Magnitudes A B, CD Equimultiples of a like Number of Magnitudes E F, each of each. I fay, what Multiple the Magnitude A B is of E, the fame Multiple A B and CD, together, is of E and F together. For, because A B and C D are Equimultiples of E and F, as many Magnitudes equal to E, that are in AB, fo many shall be equal to F in CD. Now, divide A B into Parts equal to E, which let be AG, GB; and CD into Parts equal to F, viz. CH, H D. Then the Multitude of Parts, CH, HD, fhall be equal to the Multitude of Parts, A G, G B. And fince A G is equal to E, and C H to F; AG and CH, together, fhall be equal to E and F together. By the fame Reafon, because G B is equal to E, and HD to F, GB and H D, together, will be equal to E and F together. Therefore as often as E is contained in AB, so often is E and F, together, contained in A B and CD, totogether. And fo as often as F is contained in CD, fo often are E and F, together, contained in A B and C D together. Therefore, if there are any Number of Magnitudes Equimultiples of a like Number of Magnitudes, each of each; whatsoever Multiple any one of the former Magnitudes is of its correfpondent one, the fame Multiple are all the former Magnitudes of all the latter; which was to be demonftrated.

H+

D

F

PRO