G be equal to L, K will be equal to N; and if lefs, lefs. But G, K are Equimultiples of A, E, and L, N, are Equimultiples of B, F. Confequently, as A is to B, fo is E to F. Therefore, Proportions that are one and the fame to any third, are also the fame to one another; which was to be demonftrated.

PROPOSITION XII.

THEORE M.

If any Number of Magnitudes be proportional, as one of the Antecedents is to one of the Confequents, fo is all the Antecedents to all the Confequents.

G

A.

B.

ET there be any Number of proportional Magnitudes, A, B, C, D, E, F ; whereof as A is to B,

fo C is to D, and fo E to F. I fay, as A is to B, fo are all the Antecedents A, C, E, to all the Confequents B, D, F.

For let G, H, K, be Equimultiples of A, C, E; and L, M, N, any Equimultiples of B, D, F.

Then because as A is to B, fo is C to D, and fo E to F; and G, H, K, are Equimultiples of A, C, E, and L, M, N, Equimultiples of B, D, F ; if G exceeds L, H* will also exceed M, and K * Def. 5. of will exceed N; if G be equal to L, H will be equal this. to M, and K to N; and if lefs, lefs. Wherefore alfo, if G exceeds L, then G, H, K, together, will likewife exceed L, M, N, together; and if G be equal to L, then G, H, K, together, will be equal to L, M, N, together; and if lefs, lefs: But G, and G, H, K, are Equimultiples of A; and A, C, E; because, if there are any Number of Magnitudes Equimultiples to a like Number of Magnitudes, each to the other, the fame Multiple that one Magnitude is of one, fo fhall + all the Magnitudes be of all. +x of this, And for the fame Reason, L, and L, M, N, are

K 2

Equi

Equimultiples of B, and B, D, F. Therefore, as +5 Def. of A is to B, fot is A, C, E, to B, D, F. Wherefore, if there be any Number of Magnitudes proportional, as one of the Antecedents is to one of the Confe quents, fo are all the Antecedents to all the Confequents; which was to be demonftrated.

this.

PROPOSITION XIII.

THEOREM.

If the firft has the fame Proportion to the fecond, as the third to the fourth, and if the third bas a greater Proportion to the fourth than the fifth to the fixth, then also shall the first have a greater Proportion to the fecond, than the fifth bas to the fixth..

ET the firft A have the fame Proportion to the fecond B, as the third C has to the fourth D; and let the third C have a greater Proportion to the fourth D, than the fifth E to the fixth F. I fay,

this.

likewife, that the firft A to the fecond B has a greater Proportion than the fifth E to the fixth F.

For becaufe C has a greater Proportion to D, 7 Def of than E has to F; there are certain Equimultiples of C and E, and others of D and F, fuch that the Multiple of C may exceed the Multiple of D; but the Multiple of E not that of F. Now let thefe Equimultiples of C and E, be G and H; and K and L, thofe of D and F; fo that G exceeds K, and H not L: Make M the fame Multiple of A, as G is of C; and N the fame of B, as K is of D.

† 5 Def

Then, because A is to B as C is to D, and M and G are Equimultiples of A, C; and N, K, of B, D: If M exceeds N; then + G will exceed K; and if M be equal to N; G will be equal to K; and if lefs, lefs. But G does exceed K. Therefore M will al

*

fo exceed N. But H does not exceed L. And M, H, are Equimultiples of A, E; and N, L, any others of B, F. Therefore A has a greater Proportion to 7 Def. of B than E has to F. Wherefore, if the first has thethis. fame Proportion to the fecond, as the third to the fourth; and if the third has a greater Proportion to the fourth than the fifth to the fixth; then alfo fhall the first have a greater Proportion to the fecond, than the fifth has to the fixth; which was to be demonftrated.

PROPOSITION XIV.

THEOREM.

If the first has the fame Proportion to the fecond, as the third has to the fourth; and if the first be greater than the third; then will the fecond be greater than the fourth. But if the first be equal to the third, then the fecond fhall be equal to the fourth; and if the first be less than the third, then the fecond will be less than the fourth.

ET the firft A have the fame Proportion to the

And let A be greater than C. I fay, B is alfo greater than D.

For because A is greater than C, and B is any other Magnitude: A will have a greater Proportion to B than C has to B; but as A is to B, fo is C to D; therefore, alfo, fhall † have a greater Proportion to D than Chath to B. But that Magnitude to which the fame bears a greater Proportion, is the leffer Magnitude: Wherefore D is lefs than B; and confequently B will be greater than D. In A B C D like Manner we demonftrate, if A bę

equal to C, that B will be equal to D; and if A be lefs than C, that B will be less than D. Therefore, if the first has the fame Proportion to the fecond, as the third has to the fourth, and if the first be greater than the third, then will the fecond be greater than

K 3

the

*8 of this.

† 13 of this.

10 of this

the fourth. But if the first be equal to the third, then the fecond fhall be equal to the fourth; and if the first be less than the third, then the fecond will be less than the fourth; which was to be demonstrated.

PROPOSITION XV.

THEOREM.

Parts have the fame Proportion as their like Multiples, if taken correfpondently.

ET AB be the fame Multiple of C, as DE is

LETAB be

G

H

A

For because AB and D E are Equimultiples of C and F, there fhall be as many Magnitudes equal to C in AB, as there are Magnitudes equal to F in DE. Now, let AB be divided into the Magnitudes AG, GH, HB, each equal to C; and ED into the Magnitudes DK, KL, LE, each equal to F. Then the Number of the Magnitudes AG, GH, HB, will be equal to the Number of the Magnitudes DK, KL, LE. Now, because AG, GH, HB, are equal, as 7 of this. likewife DK, KL, LE, it fhall be as AG is to DK: So is GH to KL, and fo is HB to LE. But

D

K

L

B C E F

*

as one of the Antecedents is to one of the Confe

+12 of this. quents, fot all the Antecedents to all the Confequents. Therefore, as AG is to DK, fo is AB to DE. But AG is equal to C, and DK to F. Whence, as C is to F, fo fhall A B be to DE. Therefore, Parts have the fame Proportion as their like Multiples, if taken correfpendently; which was to be demonftrated.

PRO

PROPOSITION XVI.

THEORE M.

If four Magnitudes of the fame Kind are proportional, they fhall also be alternately proportional.

ET four Magnitudes ABCD, be proportional; whereof A is to B as C is to D. I fay likewife, that they will be alternately proportional, viz. as A is to C, fo is B to D; for take E, F, Equimultiples of A and B, and G,

H, any Equimulti- E

ples of C, D.

A.

B

Then because E is the fame Multiple of F

A, as F is of B, and

G

C.

D

H.

Parts have the fame Proportion * to their like Mul-* 15 of this. tiples, if taken correfpondently; it shall be as A is

to B, fo is E to F. But as A is to B, fo is C to

D. Therefore alfo as C is to D, fot is E to F. † 11 of this. Again, because G, H, are Equimultiples of C and D, and Parts have the fame Proportion with their like Multiples, if taken correfpondently, it will be as C is to D, fo is G to H; but as C is to D, so is E to F. Therefore alfo as E is to F, fo is G to H; and if four Magnitudes be proportional, and the first greater than the third, then the fecond will be ‡‡ 14 of this greater than the fourth; and if the first be equal to the third, the fecond will be equal to the fourth; and if lefs, lefs. Therefore, if E exceeds G, F will ex-. ceed H; and if E be equal to G, F will be equal to H; and if lefs, lefs. But E, F are any Equimultiples of A, B; and G, H any Equimultiples of C, D. Whence, as A is to C, fo fhall B be to + Def. 5. D. Therefore, if four Magnitudes of the fame Kind are proportional, they also shall be alternately propor tional.