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

IF

PROP. II. THEOR

(F the first magnitude be the fame multiple of the fecond that the third is of the fourth, and the fifth the fame multiple of the fecond that the fixth is of the fourth; then shall the first together with the fifth be the fame multiple of the fecond, that the third together with the fixth is of the fourth.

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

Because AB is the fame 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

B

D

E

H'

equal to C, fo many are there in EH equal to F: As many,
then, as are in the whole AG equal to C, fo many are there in
the whole DH equal to F: Therefore AG is the fame multi-
ple of C, that DH is of F; that is, AG the first and fifth to-
gether, is the fame multiple of the fecond
C, that DH the third and fixth together is
of the fourth F. If, therefore, the first be
the fame multiple, &c. Q. E. D.

D

A

E

K

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

H

CLF

PROP.

Book V.

IF

PROP. III. THEOR.

F the first be the fame multiple of the fecond, which the third is of the fourth; and if of the first and third there be taken equimultiples, thefe fhall be equimultiples, the one of the fecond, and the other of the fourth.

Let A the first, be the fame 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 fame multiple of B, that GH is of D.

Because EF is the fame 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 magnitudes F
EK, KF, each equal to A,
and GH into GL, LH,
each equal to C: The num-
ber therefore of the magni-
tudes EK, KF, fhall be e-
qual to the number of the
others GL, LH: And be-
caufe A is the fame multi-
ple of B, that C is of D,
and that EK is equal to A,
and GL to C; therefore

K

H

EK is the fame multiple of E ABG C D

B, that GL is of D: For

the fame reason, KF is the fame multiple of B, that LH is of D; and fo, if there be more parts in EF, GH equal to A, C: Because, therefore, the firft EK is the fame multiple of the fecond B, which the third GL is of the fourth D, and that the fifth KF is the fame multiple of the fecond B, which the fixth LH is of the fourth D; EF the first, together with the fifth, is the fame multiple of the fecond B, which GH the third, to- a 2. si gether with the fixth, is of the fourth D. If, therefore, the firft, &c. Q. E. d.

a

PROP.

Book V

See N.

a 3. 5.

I'

PROP. IV. THEOR.

F the first of four magnitudes has the fame ratio to the fecond which the third has to the fourth; then any equimultiples whatever of the firft and third fhall have the fame ratio to any equimultiples of the fecond and fourth, viz. the equimultiple of the first shall have the fame ratio to that of the fecond, which the equi⚫ multiple of the third has to that of the fourth.'

Let A the first, have to B the fecond, the fame 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 equi-
multiples whatever G, H: Then
E has the fame ratio to G, which
F has to H.

Take of E and F any equimul-
tiples 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
equinultiples K, L; therefore K
is the fame multiple of A, that L
is of C: For the fame reafon, MKE
is the fame multiple of B, that N

ABG M

is of D: And becaufe, as A is to L F C D H N

b Hypoth. By fo'is C to Db, and of A and B,

Chave 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, cs. def. 5. lefs 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 firft, &c. Q. E. D.

COR. Likewife, if the first has the fame ratio to the fecond, which the third has to the fourth, then alfo any equimulti

ples

ples whatever of the firft and third have the fame ratio to the fecond and fourth: And in like manner, the first and the third have the fame ratio to any equimultiples whatever of the second and fourth.

Let A the firft, have to B the fecond, the fame 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, D any equimultiples whatever G, H; then it may be demonftrated, as before, that K is the fame multiple of A, that L is of C: And because A is to B, as C is te 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 lefs, lefs : c s. def. s. And, K, L are any equimultiples of E, F, and G, H any whatever of B, D; as therefore E is to B, fo is F to D: And in the fame way the other cafe is demonstrated.

I

PROP. V. THEOR.

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

Let the magnitude AB be the fame multiple of CD, that AE taken from the firft, is of CF taken from the other; the remainder EB fhall be the fame multiple of the remainder FD, that the whole AB is of the whole CD.

E

Take AG the fame multiple of FD, that AE is of CF: Therefore AE is the fame multiple of CF, that EG is of CD: But AE, by the hypothefis, is the fame multiple of CF, that AB is of CD: Therefore EG is the fame 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, fince AE is the fame multiple of CF, that AG is of FD, and that AG is equal to EB; therefore AE is the fame multiple of CF, that EB is of FD: But AE is the fame multiple of CF,

a x. 5.

F

b 1. Ax. St

B D

that

Book V. that AB is of CD; therefore EB is the fame multiple of FD, that AB is of CD. Therefore, if any magnitude, &c. Q. E. D.

See N.

PROP. VI. THEOR.

F two magnitudes be equimultiples of two others, and if equimultiples of these be taken from the first two, the remainders are either equal to thefe 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 fame E, F; the remainders GB, HD are either equal to E, F, or equimultiples of them.

A K

C

First, Let GB be equal to E; HD is equal to F: Make CK equal to F; and becaufe AG is the fame multiple of E, that CH is of F, and that GB is equal to E, and CK to F; therefore AB is the fame multiple of E, that KH is of F. But AB, by the hypothefis, is the fame multiple of E that CD is of F; therefore KH is the fame multiple of F, that CD is of F; 1. Az. 5. wherefore KH is equal to CD: Take 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.

12.5.

B

H

A

C

c+

H

DEF

But let GB be a multiple of E; then HD is the fame multiple of F: Make CK the fame inultiple of F, that GB is of E: And because AG is the fame multiple of E, that CH is of F; and GB the fame multiple of E, that CK is of F; therefore AB is the fame multiple of E, that KH is of Fb: But AB is the fame G multiple of E, that CD is of F; therefore KH is the fame multiple of F, that CD is of its wherefore KH is equal to CD': Take away CH from both; therefore the remainder KC is equal to the remainder HD: And becaufe GB is the fame multiple of E, that KC is of F, and that KC is equal to HD; therefore HD is the fame multiple of F, that GB is of E: If therefore two magnitudes, &c. Q. E. D.

B DEF

PROP.