II.

Those magnitudes of which the same, or equal magnitudes are equimultiples, are equal to one another.

III.

A multiple of a greater magnitude, is greater than the same multiple of a less.

IV.

That magnitude of which a multiple is greater than the same multiple of another, is greater than that other magnitude.

PROP. I. THEOR.

If any number of magnitudes be equimultiples of as many others, each of each; what multiple soever any one of the former magnitudes is of its part, the same multiple will all the former magnitudes be of all the latter.

Let any number of magnitudes AB, CD be equimultiples of as many others, E, F, each of each; whatsoever multiple AB is of E, the same multiple will AB and CD together, be of E and F together.

B

E

Because AB is the same multiple of E that CD is of F, as many magnitudes as are in AB equal to E, so many are there in CD equal to F. Divide AB into magnitudes equal to E, viz. AG, GB; and CD into CH, A HD, equal each of them to F: The number therefore of magnitudes CH, HD, in the one, will be G equal to the number of magnitudes, AG, GB, in the other: And because AG is equal to E, and CH to F, therefore AG and CH together are equal to a E and F together: For the same reason, because GB is equal to E, and HD to F; GB and HD together are equal to E and F together. Wherefore, as many magnitudes as there are in AB equal to E, so many are there in AB and CD together, equal to E and F together. Therefore, whatsoever multiple AB is of E, the same multiple D is AB and CD together of E and F together.

C

H

F

Therefore, "If any magnitudes, how many soever, be equimultiples of as many, each of each, whatsoever multiple any one of the former magnitudes is of its part, the same multiple will all the former magnitudes be of all the latter." For the same de

H

Book V.

a Ax. 2.5.

Book V.monstration holds in any number of magnitudes, which was here applied to two.' Q. E. D.

PROP. II. THEOR.

If the first magnitude be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth; then shall the first together with the fifth be the same multiple of the second, that the third together with the sixth is of the fourth.

Let AB the first, be the same multiple of C the second, that DE the third is of F the fourth; and BG the fifth, the same multiple of C the second, that EH the sixth is of F the fourth: Then is AG the first, together with the fifth, the same multiple of C the second, that DH the third, together with the sixth, is of F the fourth.

B

A

Because AB is the same multiple of
C, that DE is of F; there are as many
magnitudes in AB equal to C, as there
are in DE equal to F: In like manner, G C
as many magnitudes as there are in

D

E

H F

BG equal to C, so many are there in EH equal to F: As
many magnitudes, then, as are in the whole AG equal to C,
so many are there in the whole DH equal to F: therefore
AG is the same multiple of C, that DH is of F; that is, AG
the first and fifth together, is the same multiple of the second
C, that DH the third and sixth together
is of the fourth F. "If, therefore, the
"first be the same multiple," &c. Q. E. D. A

6

COR. From this it is plain, that if B

'any number of magnitudes AB, BG,
GH, be multiples of another magnitude
'C; and as many others DE, EK, KL G
be the same multiples of another magni-
'tude F, each of each; the whole of the
'former, viz. AH, will be the same mul-

D

E

K

tiple of C, that the whole of the latter, H C

L F

viz. DL is of F.'

PROP. III. THEOR.

If the first of four magnitudes be the same multiple of the second, as the third is of the fourth; and if of the first and third there be taken equimultiples, these will also be equimultiples, the one of the second, and the other of the fourth.

Let A the first, be the same multiple of B the second, that C the third is of D the fourth; and of A, C let the equimultiples EF, GH be taken; then will EF be the same multiple of B, that GH is of D.

Because EF is the same multiple of A, that GH is of C, there will be as many magnitudes in EF equal to A, as are in GH equal to C: Let EF be

divided into the magnitudes F EK, KF, each equal to A, and GH into the magnitudes GL, LH, each equal to C: Then will the number of magnitudes EK, KF, in the one, K be equal to the number of magnitudes GL, LH in the other: And because A is the same multiple of B, that C is

L

H

of D, and EK is equal to A,

and GL to C; therefore, EK E A B is the same multiple of B, that

Book V.

GL is of D: For the same reason, KF is the same multiple of B, that LH is of D; and so, if there be more points in EF, GH equal to A, C: Because, therefore, the first EK is the same multiple of the second B, as the third GL is of the fourth D, and the fifth KF is the same multiple of the second B, as the sixth LH is of the fourth D: EF the first, together with the fifth, is the same multiple of the second B, that GH the a 2. 5. third, together with the sixth, is of the fourth D. "If, therefore, the first," &c. Q. E. D.

H 2

Book V.

See N.

a 3. 5.

PROP. IV. THEOR.

If the first of four magnitudes has the same ratio to the second which the third has to the fourth; then any equimultiples whatever of the first and third shall have the same ratio to any equimultiples of the second and fourth, viz. the equimultiple of the first shall ' have the same ratio to that of the second, which the equimultiple of the third has to that of the fourth.

Let A the first, have to B the second, the same ratio which the third C has to the fourth D; and of A and C let there be taken any equimultiples whatever E, F; and of B and D any equimultiples whatever G, H: Then E will have the same ratio to G, that F has to H.

Take of E and F any equimultiples whatever K, L, and of G, H, any equimultiples whatever M, N: Then, because E is the same multiple of A, that F is of C; and of E and F have been taken the equimultiples K, L; therefore K

LK

L

FCDHN

is the same multiple of A, that L KE ABG M is of C: For the same reason, M is the same multiple of B, that N is of D: And because, as A is to b Hypoth. B, so is C to Db, and of A and C have been taken certain equimultiples K, L; And of B and D have been taken certain equimultiples M, N; if therefore K be greater than M, L will also be greater than N; and if equal, equal; c 5. def. 5. if less, less. And since K, L are any equimultiples whatever of E, F; and M, N any whatever of G, H: Therefore, as E is to G,

See N.

so is F to H. Therefore," if the first," &c. Q. E. D.
COR. Likewise, if the first has the same ratio to the second,
which the third has to the fourth, then also any equimultiples

whatever of the first and third have the same ratio to the second Book V. and fourth: And in like manner, the first and the third have the same ratio to any equimultiples whatever of the second and fourih.

Let A the first, have to B the second, the same ratio which the third C has to the fourth D, and of A and C let E and F be any equimultiples whatever; then E is to B as F to D.

Take of E, F, any equimultiples whatever K, L, and of B, D any equimultiples whatever G, H; then it may be demonstrated, as before, that K is the same multiple of A, that L is of C: And because A is to B, as C is to D, and of A and C certain equimultiples have been taken, viz. K and L; and of B and D certain equimultiples G, H; therefore, if K be greater than G, L will also be greater than H; and if equal, equal; if less, less c. And since K, L are any equimultiples c 5. def. 5. of E, F, and G, H any whatever of B, D; as therefore E is to B, so is F to D; and in the same way may the other case be demonstrated.

PROP. V. THEOR.

If one magnitude be the same multiple of another, See N. which a magnitude taken from the first is of a magnitude taken from the other; the remainder will be the same multiple of the remainder, that the whole is of the whole.

Let the magnitude AB be the same multiple of CD, that AE taken from the first, is of CF taken from the other; the remainder EB will be the same mul- G tiple of the remainder FD, that the whole AB is of the whole CD.

E

C

Take AG the same multiple of FD, that AE A is of CF: Therefore AE is a the same multiple of CF, that EG is of CD: But AE, by the hypothesis, is the same multiple of CF, that AB is of CD: Therefore EG is the same multiple of CD that AB is of CD; wherefore EG is equal to AB. Take from them the common magnitude AE; then will the remainder AG be equal to the remainder EB. Wherefore, since AE is the B same multiple of CF, that AG is FD, and AG is equal to EB: therefore AE is the same multiple of CF,

a 1. 5.

Fb 1. Ax. 5.

D