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 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 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 Confequent with the Confequent.

XIV. Inverfe Ratio is, when the Confequent is taken as the Antecedent, and so 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 Excefs, 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 Magnitudes, as the first is to the laft, fo in the fecond Order of Magnitudes is the first to the laft: Or otherwise, it is the Comparison of the Extremes together, the Means being omitted. 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.

AXIO M S.

I. EQuimultiples 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

"

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

ET 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 AB, and CD, together, is of E and F together.

A

G+

B

E

For, because A B and CD are Equimultiples of E and F, as many Magnitudes equal to E, that are in AB, fo many fhall be equal to F in CD. Now, divide A B into Parts equal to E, which let be A G, GB; and CD into Parts equal to F, viz. CH, HD. Then the Multitude of Parts, CH, HD, fhall be equal to the Multitude of Parts AG, GB. And fince AG is equal to E, and CH to F; AG and CH, together, fhall be equal to E and F together. By the fame Reafon, because GB is equal to E, and HD to F, GB and HD, 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, together. And fo as often as F is contained in CD, fo often are E and F, together, contained in A B, and CD 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+

DF

PRO

PROPOSITION II.

THEOREM.

If the first be the fame Multiple of the fecond, as the third is of the fourth; and if the fifth be the fame Multiple of the fecond, as the fixth is of the fourth; then shall the first, added to the fifth, be the fame Multiple of the second, as the third, added to the fixth, is of the fourth.

L

A

D

ET the firft A B be the fame Multiple of the fecond C, as the third DE is of the fourth F; and let the fifth BG be the fame Multiple of the second C, as the fixth EH is of the fourth F. I fay, the first added to the fifth, viz. AG, is the fame Multiple of the fecond C, as the third added to the fixth, viz. DH, is of the fourth F.

B+

E+

G C HF

For, because AB is the fame Multiple of C, as DE is of F; there are as many Magnitudes equal to C in AB, as there are Magnitudes equal to F in DE. And, for the fame Reason, there are as many Magnitudes equal to C in 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 F in DH. Wherefore A G is the fame Multiple of C, as DH is of F. And fo the firft, added to the fifth," AG, is the fame Multiple of the fecond C, as the third, added to the fixth, DH, is of the fourth F. Therefore, if the firfl be the fame Multiple of the fecond, as the third is of the fourth; and if the fifth be the fame Multiple of the fecond, as the fixth is of the fourth ; then fhall the firft, added to the fifth, be the fame Multiple of the fecond, as the third, added to the fixth, is of the fourth; which was to be demonftrated.

PRO

PROPOSITION III.

THEOREM.

If the first be the fame Multiple of the fecond, as
the third is of the fourth, and there be taken
Equimultiples of the first and third; then will
the Magnitudes fo taken be Equimultiples of
the fecond and fourth.

LET the firft A be the fame Multiple of the second
B, as the third C is of the fourth D; and let E F,

GH, be Equimultiples of A
and C. I fay, EF, is the fame
Multiple of B. as GH is of D.

For, becaufe EF is the fame Multiple of A, as GH is of C, there are as many Magnitudes equal to A in EF, as there are Magnitudes equal to Cin GH. Now divide E F into the Magnitudes EK, KF, each equal to A, and G H into the Magnitudes GL, LH, each equal

F

H

K+

L+

EAB GCD

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 E K is equal to A, and GL to C; EK will be the fame Multiple of B, as GL is of D. For the fame Reafon, KF hall be the fame Multiple of B, as L His of D. Therefore because the firft EK is the fame Multiple of the second B, as the third GL is of the fourth D, and KF the fifth, is the fame Multiple of B, the fecond, that LH, the fixth, is of D the fourth: Therefore the first added to the fifth, EF, fhall be the fame Multiple of the fecond B, as* 2 of this. the third added to the fixth, GH, is of the fourth D. If, therefore, the first be the fame Multiple of the fecond, as the third is of the fourth, and there be taken Equimultiples of the firft and third; then will the Magnitudes fa taken, be Equimultiples of the fecond and fourth; which was to be demonftrated.

PRO