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LK will be equal to N; and iflefs, lefs. But G and K are Equimultiples of A and E, and Land N are Equimultiples of B and F. Confequently, as A is to Def. 5. of B, fo is E to F. Therefore, Proportions that are one and the jame to any third, are alfo the fame to one another; which was to be demonftrated.

this.

PROPOSITION XII.

THEOREM.

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

LET there be any Number of Proportional Magnitudes, A, B, C, D, E, F, whereof as A is to B,

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this.

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

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

Then, because as A is to B, fo is C to D, and fo E to F; and G, H, and K, are Equimultiples of A, C, and E, and L, M, and N, Equimultiples of B, D, *Def. 5. and F ; if G exceeds L, H* will alfo exceed M, and K will exceed N; if G be equal to L, H will be equal to M, and K to N; and if lefs, lefs. Wherefore, also, if G exceeds L, then G, H, and K, together, will 1.kewife exceed L, M, and N together; and if G be equal to L, then G, H, and K, together, will be equal to L, M, and N, together; and if lef, lefs: But G, and G, H, and K, are Equimultiples of A, and A, C, and E; becaufe, if there are any Number of Magnitudes Equimultiples to a like Number of Magnitudes, each to the other, the fame Multiple that one Magni1 of this. tude is of one, fo fhall + all the Magnitudes be of all. And,

And for the fame Reafon, L, and L, M, and N, are.
Equimultiples of B, and B, D, and F. Therefore, as
A is to B, fo is A, C, and E, together, to B, D, Def. 5.
and F, together. Wherefore, if there be any Number this.
of Magnitudes proportional, as one of the Antecedents is to
one of the Confequents, fo are all the Antecedents to all
the Confequents; which was to be demonstrated.

PROPOSITION XIII.

THEOREM.

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

LET the first A have the fame Proportion to the fe

cond 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, likewife, that the

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

this.

For, because C has a greater Proportion to D, than E has to F; there * are certain Equinultiples of C* Def. 7. af 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 fate of B, as K is of D.

Then, because A is to B, as C is to D; and M and G are Equimultiples of A and C; and N and K of B and D: If M exceeds N, then + G will exceed K; and † Def. 5° if M be equal to N, G will be equal to K ; and if lefs,

K 4

lefs.

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* 8 of this.

*

lefs. But G does exceed K: Therefore M will also exceed N. But H does not exceed L. And M and H are Equimultiples of A and E; and N and L ́any others of B and F. Therefore A has a greater Proportion to B than E has to F. Wherefore, if the first has the 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 demonstrated.

9

PROPOSITION XIV.
THEOREM.

If the firft 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 shall be equal
to the fourth; and if the first be less than the
third, then the fecond will be less than the fourth.

L

ET the first A have the fame Proportion to the second B, as the third C has to the fourth D; and let A be greater than C. I fay, B is also 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; there13 of this. fore, alfo, C fhall ‡ have a greater Proportion to D, than C hath to B. But that Magnitude to which the fame bears +10 of this, a greater Proportion, is the leffer Magnitude. Wherefore D is lefs than B; and confequently B will be greater than D. In like manner we demonftrate, if A be 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 firft has the fame Proportion to the Jecond, as the third has to the fourth; and if the firft be greater than the third; then will the fecond be greater than the fourth. But if the first be equal to the third,

A B C D

then

then the fecond fhall be equal to the fourth; and if the firft be less than the third, then the fecond will be less than the fourth; which was to be demonftrated.

PROPOSITION XV.

THEOREM.

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

LE

A

G+

D

ET A B be the fame Multiple of C, as D E is of F. I fay, as C is to F, fo is A B to DE. For, because A B and D E are Equimultiples of C and F, there fhall be as many Magnitudes equal to Cin A B, as there are Magnitudes equal to F in D E. Now, let A B be divided into the Magnitudes AG, GH, HB, each equal to C; and ED into the Magnitudes D K, 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 A G,

K+

H+

L+

BC EF

7

of this.

GH, H B, are equal, as likewife DK, KL, LE; it fhall be, as AG is to DK, fo is GH to KL, and fo is HB to L E. But as one of the Antecedents is to one of the Confequents, fo + all the Antecedents to + 21 of this. all the Confequents. Therefore, as A G is to DK, fo is A B to D E. But A G is equal to C, and D K to F. Whence, as C is to F, fo fhall A B be to D E. Therefore, Parts have the fame Proportion as their like Multiples, if taken correfpondently; which was to be demonstrated.

PRO

PROPOSITION XVI.

THEOREM.

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

ET four Magnitudes A, B, C, D, 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 and F, Equimultiples of A and

B; and G and H, E-
any Equimultiples A
of C and D.

G--

C

B

D

H

Then, becaufe F.

E is the fame Mul

*

tiple of A, as F is of B, and Parts have the fame Pre• 15 of this. portion to their like Multiples, if taken correfpondently; it fhall 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 Cisto +11 of this. D, fot is E to F. Again, becaufe G and 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 Cis to D, fo 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 14 of this, will be 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 exceed H; and if E be equal to G, F will be equal to H; and if lefs, lefs. But E and F are any Equimultiples of A and B ; and G and H, any Equimultiples of C and D. Whence as A is to C, fo fhall B be to D. Therefore, if four Magnitudes of the fame Kind are proportional, they shall also be alternately proportional; which was to be demonftrated.

* Def 5.

*

PRO

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