Book V. 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 the fourth; and BG the fifth, the same

the second, that EH

F the fourth: Len
st, together with the

fifth, the same multiple of C the se-
cond, that DH the third, together with
the sixth, is of F the fourth.

B

A

D

E

cl

F

H

C

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, as many as there are in BG equal to C, so many are there in EH equal to F: as many, 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 A fourth F. If, therefore, the first be the same multiple, &c. Q. E. D.

D

B

ET

COR. From this it is plain, that, if any
'number of magnitudes AB, BG, GH,
'be multiples of another C; and as many
'DE, EK, KL be the same multiples of GT
'F, each of each; the whole of the first,
'viz. AH, is the same multiple of C,

K

that the whole of the last, viz. DL, is ' of F.'

H

PROP. III. THEOR.

IF the first be the same multiple of the second, whic', the third is of the fourth; and if of the first and third there be taken equimultiples, these shall 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 EF is the same multiple of B, that GH is of D.

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

vided into the agnitudes EK, KF, each equal to A, and GH into GL, LH, each equal to C: the number therefore of the magnitudes EK, KF, shall be equal to the number of the others GL, LH: and because A is the same multiple of B, that C is of D, and that EK is equal to A, and GL to C; therefore EK is the same multiple of B, that GL is of D: for

E

Book V.

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

Book V.

See Note.

a 3. 5.

PROP. IV. THEOR.

IF the first of four magnitudes have the same ratio to the second which the third hath 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 equimultipies whatever

E, F; and of B and D any equi-
multiples whatever G, H: then
E has the same ratio to G, which
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 equimultiples K, L; therefore K is the same multiple of A, that L is of Ca: 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 is greater than N; and if equal, equal; if less, e 5. def. 5. less. And K, L are any equimultiples whatever of E, F; and M, N any whatever of G, H: as therefore E is to G, so ise F to H. Therefore, if the first, &c. Q. E. D.

K

LF

M

C

D

HN

COR. Likewise, if the first have 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 and fourth: and in like manner, the first and the third have the same ratio to any equimultiples whatever of the second and 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 E and F be any equimultiples whatever; then E is to B, as F to D.

Book V.

Take of E, F any equimultiples whatever K, L, and of B, Dany 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 is greater than H; and if equal, equal; if less, lesse: c 3. def, 5. and, K, L are any equimultiples 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 the other case is demonstrated.

PROP. V. THEOR.

IF one magnitude be the same multiple of another, See Note. which a magnitude taken from the first is of a magnitude taken from the other; the remainder shall 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 shall be the same multiple of the remainder FD, that the whole AB is of the whole CD.

G

E

C

Take AG the same multiple of FD, that AE 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; the remainder AG is equal to the remainder EB. Wherefore, since AE is the same multiple of CF, that AG is of FD, and that AG is equal to EB; therefore AE is the same multiple of CF, that EB is of FD: but AE is the same multiple of CF,

B

a 1. 5.

b 1. Ax. 5.

Book V. that AB is of C

that AB is of CD.

therefore EB is the same multiple of FD, Therefore, if any magnitude, &c. Q. E. D.

See Note.

PROP. VI. THEOR.

IF two magnitudes be equimultiples of two others, and if equimultiples of these be taken from the first two, the remainders are either equal to these others, or equimultiples of them.

Let the two magnitudes AB, CD be equimultiples of the two E, F, and AG, CH taken from the first two be equimultiples of the same E, F; the remainders GB, HD are either equal to E, F, or equimultiples of them.

First, Let GB be equal to E; HD is equal to F: make CK equal to F; and because AG is the same multiple of E, that CH is of F, and that GB is equal to E, and CK to F; therefore AB is the same multiple of E, that KH is of F. But AB, by the hypothesis, is the same multiple of E that CD is of F; therefore KH is the same multiple of F, that CD is of F; a 1. Ax. 5. wherefore KH is equal to CD: a take

b 2.5.

G

A

K

H

B

DE F

AK

away the common magnitude CH, then the
remainder KC is equal to the remainder
HD: but KC is equal to F; HD therefore is equal to F.
But let GB be a multiple of E; then
HD is the same multiple of F: make CK
the same multiple of F, that GB is of E:
and because AG is the same multiple of E,
that CH is of F; and GB the same multi-
ple of E, that CK is of F: therefore AB is
the same multiple of E, that KH is of Fb:
but AB is the same multiple of E, that CD
is of F; therefore KH is the same multi-
ple of F, that CD is of it; wherefore KH is
equal to CD: take away CH from both;
therefore the remainder KC is equal to the
remainder HD: and because GB is the

G

H

DEF

same multiple of E, that KC is of F, and that KC is equal to HD; therefore HD is the same multiple of F, that GB is of E. If therefore two magnitudes, &c. Q. E. D.