1

VII. When of Equimultiples, the Multiple of the First exceeds the Multiple of the Second, but the Multiple of the Third does not exceed the Multiple of the Fourth; then the First to the Second is faid to have a greater Proportion than the Third 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 Ratio to what - it has to the Second.

XI. But when four Magnitudes are Proportionals, the First shall have a Trilicate Ratio to the Fourth of what it has to the Second; and fo always one more in Order, as the Proportionals fhall be extended.

XII. Homologous Magnitudes, or Magnitudes of a like Ratio, are said to be fuch whose Antecedents are to the Antecedents, and Consequents to the Confe

quents.

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

XIV. Inverse Ratio, is when the Confequent is taken as the Antecedent, and so compared with the Antece dent 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 when the Excess wherein the Antecedent exceeds the Consequent, is compared with the Confequent.

XVII. Converse Ratio, is when the Antecedent is com pared 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 Number of Magnitudes in another Order, comparing two to two being in the fame Proportion; and it shall be in the first Order of Magnitude, 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 Extremes together, the Means being omitted,

XIX. Or

are

XIX. Ordinate Proportion, is when, as the Antecedent is to the Consequent, so is the Antecedent to the Consequent; and as the Consequent is to any other, So is the Confequent to any other. XX. Perturbate Proportion, is when there are three Magnitudes, and others also, that equal to these in as in the first Magnitudes the Antecedent is to the Consequent; so in the second Magnitudes is the Antecedent, to the Consequent: And as in the first Magnitudes the Consequent is to some other, so in the Second Magnitudes, is some other to the Antecedent.

I.

E

ΑΧΙΟMS.

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

II. Those Magnitudes that have the Same
Equimultiple, or whose Equimultiples

are equal, are equal to each other.

FH

PRO

1

PROPOSITION I.

THEOREM.

If there be any Number of Magnitudes Equimultiples of a like Number of Magnitudes, each to each; whatSoever Multiple any one of the former Magnitudes is of its Correspondent one, the Same Multiple is all the former Magnitudes of, all the latter.

L

A

E

ET there be any Number of Magnitudes AB, CD, Equimultiples of a like Number of Magnitudes E, F, each of each. I fay, what Multiple the Magnitude AB is of E, the fame Multiple A B, and CD, together, is of E and F together. For because AB and CD are Equimultiples of E and F, as many Magnitudes equal to E, that are in AB, so many shall be equal to Fin CD. Now divide AB into Parts equal to E, which let be AG, GB, and CD into Parts equal to F, viz. G CH, HD. Then the Multitude of Parts, CH, HD, shall be equal to the B Multitude of Parts AG, GB. And fince AG is equal to E, and CH to F; C AG and CH, together, shall be equal to E and F together. By the same Reafon, because GB is equal to E, and HD to F, GB and HD will be equal to E and F together. Therefore, as often as E is contain'd in A B, so often is E and F contain'd in AB and CD. D And so, as often as E is contain'd in AB, so often are E and F, together, contain'd in AB and CD together. Therefore, if there are any Number of Magnitudes Equimultiples of a like Number of Magnitudes, each to each; whatsoever Multiple any one of the former Magnitudes is of its Correspondent one, the same Multiple is all the former Magni

H

F

tudes of, all the latter; which was to be demonftrated.

PRO

PROPOSITION II.

THEOREM.

If the First be the Same Multiple of the Second, as the Third is of the Fourth, and if the Fifth be the same Multiple of the Second, as the Sixth is of the Fourth; then jhall the First, added to the Fifth, be the same Multiple of the Second, as the Third, added to the Sixth, is of the Fourth.

L ET the first AB be the same Multiple of the second C, as the third DE is of the fourth F; and

let the fifth BG be the fame

Multiple of the second C, as A
the fixth EH is of the fourth F.
I say the first added to the fifth,
viz. AG, is the same Multiple
of the second C, as the third ad-
ded to the fixth, viz. DH, is of
the fourth F.

B

D

E

GCHF

For because AB is the fame Multiple of C, as DE is of F, there are as many Magnitudes equal to Cin AB, as there are Magnitudes equal to Fin DE. And for the fame Reason there are as many Magnitudes equal to Cin BG, as there are Magnitudes equal to Fin EH. Therefore there are as many Magnitudes equal to C, in the whole A G, as there are Magnitudes equal to Fin DH. Wherefore A G is the same Multiple of C, as DH is of F. And so the first added to the fifth AG, is the fame Multiple of the second C, as the third, added to the fixth DH, is of the fourth F. Therefore, if the First be the Same Multiple of the Second, as the Third is of the Fourth; and if the Fifth be the same Multiple of the Second, as the Sixth is of the Fourth, then shall the First, added to the Fifth, be the Same Multiple of the Second, as the Third, added to the Sixth, is of the Fourth; which was to be demonftrated.

RO

PROPOSITION. III.

THEOREM.

If the First be the same Multiple of the Second, as the
Third is of the Fourth, and there be taken Equimul-
tiples of the First and Third; then will each of the
Magnitudes taken be Equimultiples of the Second and
Fourth.

IET the first A be the fame Multiple of the fe
cond B, as the third C is of the fourth D; and

let EF, GH, be Equimultiples of A and C. I say EF is the same Multiple of B, as GH is of D.

F

EAB

L

H

GCD

For because EF is the fame Multiple of A, K as GH is of C, there are as many Magnitudes equal to A in EF, as there are Magnitudes equal to C in GH. Now divide EF into the Magnitudes EK, KF, equal to A, and GH into the Magnitudes GL, LH, equal to C. Then the Number of the Magnitudes EK, KF, will be equal to the Number of the Magnitudes GL, LH. And because A is the fame Multiple of B, as C is of D, and EK is equal to A, and GL to C; EK will be the fame Multiple of B, as GL is of D. For the fame Reason, KF shall be the fame Multiple of B, as LH is of D. Therefore because the first EK is the fame Multiple of the second B, as the third GL is of the fourth D, and KF, LH are Equimultiples of the fecond B and fixth D. The first

2 of this. added to the fifth, EF, shall be the fame Multiple of the second B, as the third added to the fixth GH is of the fourth D. If, therefore, the First be the Same Multiple of the Second, as the Third is of the Fourth, and there be taken Equimultiples of the First and Third, then will each of the Magnitudes taken be Equimultiples of the Second and Fourth; which was to be demonftrated.