Book v.

PROP. II. THEOR.
IF
F 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 fixth 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 fixth is of ihe 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 fame multiple of C the second, that DH

D

А.
the third together with the sixth is of
F the fourth.
Because AB is the fame multiple of

B
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, G H F
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 fame multiple of the second C, that DH the third and fixth
together is of the fourth F. If therefore the
first be the same multiple, &c. Q. E. D.

D

A
Cor. From this it is plain, that if any
number of magnitudes AB, BG, GH be

B
* multiples of another C; and as many DE,
EK, KL be the same multiples of F, each

K
• of each; the whole of the first, viz. AH is GI
• the fame multiple of C, that the whole of

the last, viz. DL is of F.

E

PROP. III.

Bock V.

THEOR.

IF

.
F the first be the same multiple of the second, which

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 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 divided into the magnitudes EK, KF, each equal F

H 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

Kt

L+ 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 fame E Á B G C D inultiple of B, that 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 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. 2. 5. of the second B, which GH the third together with the fixth is of the fourth D. If therefore the first, &c. Q. E. D.

Book V.

PRO P. IV.

THE OR.

See N.

IF
F 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 fall have the same ratio to any equimultiples of the second and fourth, viz. · the equimultiple of the first shall have the fame 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 has the fame 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 fame multiple of A, that F is of C; and of E and F have been taken equimultiples K,

L; therefore K is the same mula. 3. s. tiple of A, that L is of Ca. for the

fame reason M is the same multiple

of B, that N is of D, and because K E A BGM b. Hypoth. 23 A is to B, so is C to Db, and of

L F F C DH N 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 e. s. Def. s. N; and if equal, equal; if less, lesse.

And K, L are any equimultiples
whatever of E, F, and M, N any
whatever of G, H. as therefore E
is to G, fo is F to H. Therefore
if the 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

the first and third have the same ratio to the second and fourth. and Book v. 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 fame ratio which the third Chas 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 is greater than H; and if equal, equal; if less, less. and K, L are any equimultiples of E, F,c. s. Def. s. 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 demonftrated,

PROP. V. THEOR. IF F one magnitude be the same multiple of another, Scc N,

which a magnitude taken from the first is of a magni-, tude taken from the other; the remainder shall be the same multiple of the remainder, that the whole is of the whole.

a. 1. S

Let the magnitude AB be the same multiple of CD, that AE
taken from the first, is of CF taken from the o- Gi
ther; the remainder EB shall be the same mul-
tiple of the remainder FD, that the whole AB is
of the whole CD.

Α.
Take AG the same multiple of FD, that AE is
of CF. therefore AE is the same multiple of
CF, that EG is of CD. but AE, by the hypothe-

С sis, is the same multiple of CF, that AB is of CD.

E therefore EG is the same multiple of CD that AB is of CD; wherefore EG is equal to AB. take F

b. 1. Ax s. from them the common magnitude AE; the remainder AG is equal to the remainder EB. B D Wherefore fince AE is the fame 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 fame

multiple

H 4

IF.

Book V. multiple of CF, that AB is of CD; therefore EB is the same mulwtiple of FD, that AB is of CD. Therefore if one magnitude, &c. Q.E.D.

PROP. VI. THEOR. F two magnitudes be equimultiples of two others, and See N.

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

Le the two magnitudes AB, CD be equimultiples of the two E, F, and AG, CH taken from the first two be equimultiples of the fame 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 fame multiple of
E, that CH is of F, and that GB is equal A K
to E, and CK to F; therefore AB is the fame
multiple of E, that KH is of F. But AB, by

C
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; wherefore

G

H a. 1. Ax. 5. KH is equal to CD. take away the common magnitude CH, then the remainder KC

B D E F
is equal to the remainder HD. but KC is e-
qual 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 fame multiple of F, that
GB is of E. and because AG is the same

K;
multiple of E, that CH is of F, and GB the A
same multiple of E, that CK is of F, there-

CH
fore AB is the same multiple of E, that KH
b. 2. 5. is of Fb. but AB is the same multiple of
E, that CD is of F; therefore KH is the G

H
fame multiple of F, that CD is of it; where-
fore K is equal to CD*. take away CH from
hoth, therefore the remainder KC is equal to
the remainder HD. and because GB is the

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

PROP.