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That is, if there be four Magnitudes, and you take any Equimultiples of the firft and third, and also any Equimultiples of the second and fourth; and if the Multiple of the first be greater than the Multiple of the second ; and also 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 second ; and also the Multiple of the third equal to the Multiple of the fourth ; or, Jaftly, if the Multiple of the firtt 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 whatsoever: Then the four Magnitudes are in the lame Ratio ; the first to the second, as the third to the fourth.

VI. Magnitudes that have the same Proportion,

are called Proportionals. Expounders usually lay down here that Definition, for Magnitudes, which Éuclid has given for Numbers, only, in his Seventh Book, viz. That

Numbers are proportional, when the first is either the fame Multiple of the second, as the third is of the fourth, or else the same Part, or Parts.

But this Definition appertains only to Numbers, and commensurable Quantities, and so fince it is not universal, Euclid did well to reject it in this Element, which treats of the Properties of all Proportionals ; and to substitute another general one, agreeing to all Kinds of Magnitudes. In the mean Time, Expounders very much endeavour to demonstrate the Definition here laid down by Euclid, by the usual 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 same Ratio, according to the Conditions that Magnitudes in the fame Ratio must have according to the fifth Definitions and let the first be a Multiple of the second : I say, the third is also the fame Multiple of the fourth. For Example : Let A be equal to 5B: Then C hhall be equal to 5D. Take any

Num:

I A

Number, for Example, 2, by which let 5 be multiplied, and the Product will be

A:B::C:D 10: And let 2A, 2C, be Equimultiples of the first and third Magnitudes A and C: Alfo, 2A, 10B, 2C, 10D let 10B and 10D be Equimultiples of the second and fourth Magnitudes B and D. Then (by Def. 5.) if 2A be equal to 10B, 2C shall be equal to 10 D. But since A (from the Hypothesis) is five Times B, 2A shall be equal to 10B; and to 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 same Part of D. For, becaufe A is to B, as C is to D; and since 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 C shall be the fame Part of the Magnitude D, as A is of B.W.W.D.

Thirdiy, Let A be equal to any Number of whatsoever Parts of B. I fay, C is equal to the same 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 say, C is also equal to D. For, because A is equal B, each of them being multiplied by 4, then 4A will be equal to 5B. And so, if the Equimultiples of the first and third, viz. 4A, 4C, be al

A: B::C:D sumed ; as also the Equimultiples of the second and fourth, 4A, 5B, 4C, 3D viz. 5B, 5D ; and (by the Definition) if 4A is equal to 5B ; then 4C is equal to 5D. But 4 A has been proved equal to 5B, and to 4C shall be equal to 5D, and C equal to D. W. W. D.

And universally, if A be equal to-B, C will be

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nB; wherefore (by Def. 5.) mC will be equal to-nD,

and C equal 10-D. W.W. D.

VII. When

VII. Wben, of Equimultiples, the Multiple of the

first exceeds tbe Multiple of the second, but the Multiple of the third does not exceed the Mul. tiple of the fourth i then the first to the second is said to have a greater Proportion than the

tbird to the fourth. VIII. Analogy is a Similitude of Proportions. IX. Analogy at least consists of three Terms. X. When three Magnitudes are Proportionals, the

first is said to have, to the third, a duplicate

Ralio to what it bas to the second. XI. But when four Magnitudes are continued

Proportionals, the first shall have a triplicate Ratio to the fourth of what it has to the second; and so always one more in Order, as ibe

Proportionals fall be extended. XII. Homologous Magnitudes, or Magnitudes of

a like Ratio, are said to be such whole Antecedents are to the Antecedents, and Confequents to

the Consequents. XIII. Alternate Ratio is the comparing of the

Antecedent with the Antecedent, and the Cone

sequent with the Consequent. XIV. Inverse Ratio is, when the Consequent is

taken as ihe Antecedent, and so compared with

the Antecedent as a Consequent. XV. Compounded Ratio is, when the Antecedent

and Confequent, taken both as one, is compared

to the Consequent itself, XVI. Divided Ratio is, wben the Excess, where

by the Antecedent exceeds ibe Confequent, is com

pared with the Consequent. XVII. Converse Ratio is, when tbe Antecedent is

compared will tbe Excess, by which the Ante

cedent exceeds the Consequent. XVIII. Ratio of Equality is, where there are taken more iban two Magnitudes in one Order, and a

like Number of Magnitudes in another Order, comparing two 10 two being in the same Proportion, and it shall be in tbe first Order of Magnitudes, as the first is to the last, so in the second Order of Magnitudes is the first to the last : Or otherwise, it is the Comparison of the Ex

tremes together, the Means being omited. XIX. Ordinate Proportion is, wben as the Ante

cedent is to the Consequent, so is tbe Antecedent to the Consequent; and as the Consequent is to

any other, so is the Consequent to any other. XX. Perturbate Proportion is, when there are

three or more Magnitudes, and others also, that are equal to these in Multitude, as in the first Magnitudes the Antecedent is to the Consequent ; so in the second Magnitudes is the Antecedent to tbe Consequent: And as in the first Magnitudes the Consequent is to some other, fo in the fecond Magnitudes is fome other, to ibe Ante. cedent.

A XIOM Ş.

1. E Quimultiples of the fame, or of equal Magni

tudes, are equal to each other. II. Those Magnitudes that have the same Equi

multiple, or wbose Equimultiples are equal, are equal to each other.

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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 correspondent one, the same Multiple are all tbe former Magnitudes of all the latter.

L

G

E T there be any Number of Magnitudes A B,
CD Equimultiples of a like Number of Mag-

nitudes E F, each of each. I say, what Multiple the Magnitude A B is of E, the same Multiple A B and C D, 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 A B, so many shall be A equal to F in CD. Now, divide AB into Parts equal to E, which let be AG, GB; and C D into Parts equal to F,

+ viz, CH, HD. Then the Multitude of

E Parts, CH, HD, shall be equal to the B Multitude of Parts, AG,G B. And since

с
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 same Reason,
because G B is equal to E, and H D to F,

H+
G B and HD, together, will be equal to
E and F together. Therefore as often as
Eis contained in AB, so often is E and F,
together, contained in A B and CD, to-

DF together. And so as often as F is contained in CD, To 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 correspondent one, the same Multiple are all the former Magnitudes of all the latter ; which was to be demonstrated.

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