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Book V.

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PROP. II. THEOR.
F the first magnitude be the same multiple of the

second that the third is of the fourth, and the fifth the fame 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 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 fixth is of F the fourth. Then is AG the first together with the fifth the

D
fame multiple of C the second, that A
DH the third together with the sixth
is of F the fourth.

E
; Because AB is the same 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

G

HIF
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 to
gether, is the same 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.

D
Q. E. D

А

E
Cor. From this it is plain, that any B
number of magnitudes AB, BG, GH be
multiples of another C; and as many DE, Kt
‘EK, KL be the same multiples of F, each

GH
• of each; the whole of the first, viz. AH
is the same multiple of C, that the whole
of the last, viz. DL is of F.

H ČLF

d

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Book v. PRO P. III. THEO R. 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.

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Let A the first be the fame 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

F magnitudes EK, KF, each e

H qual to A, and GH into GL, LH, each equal to C. the number therefore of the magnitudes EK, KF, shall be e K qual to the number of the

L 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 fame E A B G G C D multiple 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 firft 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 fixth LH is of the fourth D; EF the first together with the fifth is the same mul. tiple • of the second B, which GH the third together with the 7, 2052 Sixth is of the fourth D. If therefore the first, &c. Q. E. D.

IF

Book V.

PROP. IV. THEOR. SE N.

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 shall have the same ratio to any equimultiples of the second and fourth, viz. ' the equiinultiple of the first shall - have the fame ratio to that of the second, which the

equimultiple of the third has to that of the fourthi’

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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
same 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 becaute 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 multiple
of A, that L is of C: for the same
reason M is the same multiple of B,

that N is of D. and because as A KE A B GM Þ. Hypoth. is to B, fo is C to D5, and of ALF CD HN

and C have been taken certain equi-
multiples K, L; and of B and D
have been taken certain equimul-
tiples M,N; if therefore K be great-

er than M, L is greater than N; c. $.Def s. and if equal, equal; if lefs, 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 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

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the first and third have the same ratio to the second and fourth. Book v.
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 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.

Take of E, F any equimultiples whatever K, L, and of B, D any equimultiples whatever G, H; then it may be demonstrated, as be-. fore, that Ķ 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, 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 demonstrated,

C.

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IF

PROP. V. THE OR
F one magnitude be the same multiple of another, See N.

which a magnitude taken from the first is of a mag-
nitude 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

G
other; the remainder EB shall be the same
multiple of the remainder FD, that the whole
AB is of the whole CD.

A
Take AG the same multiple of FD, that AE
is of CF. therefore AE is the same multiple of

2. Bol
CF, that EG is of CD. but AE, by the hypothe-
fis, is the fame multiple of CF, that AB is of CD.
therefore EG is the same multiple of CD that AB

E

F is of CD; wherefore EG is equal to AB6. take

b. 1. AX. from them the common magnitude AE; the remainder AG is equal to the remainder EB, Wherefore since AEis the same multiple of CF,

B D 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

HA

IF

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Book V. multiple of CF, that AB is of CD; therefore EB is the same mul

tiple of FD, that AB is of CD. Therefore if one magnitude, &c. Q. E. D.

PROP. VI. THEOR. See N. 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 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,

CH
by the hypothesis, is the fame multiple of E
that CD is of F; therefore KH is the same

multiple of F, that CD is of F; wherefore Gt at 2. 1. Ax. 5. 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 B

B DEF to F, HD therefore is equal to F.

But let GB be a multiple 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 fame mul

K
tiple of E, that CH is of F, and GB thé

A
fame multiple of E, that CK is of F, there с

fore AB is the same multiple of E, that KH
b. 2. s. is of Fb. but AB is the same multiple of

E, that CD is of F; therefore KH is the G
fame multiple of F, that CD is of it; where H
fore KH is equal to CD's take away CH
from both, therefore the remainder KC is
equal to the remainder HD. and because GB
is the same multiple of E, that KC is of F,

B DEF 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.

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