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PROP. II. THEOR.

IF the firft 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 fhall 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 sixth
is of F the fourth.

D

A

E

B

G

d

HF

. 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 equal to
C, fo 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 fame
multiple of C, that DH is of F; that is, AG the first and fifth to-
gether, 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 fame multiple, &c.
Q. E. D

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D

A

K

H CL

CLF

Book V.

IF

PRO P. III. THEOR.

F the first be the fame multiple of the second, 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 fecond, 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.

F

H

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 divided into the magnitudes 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 becaufe A is the fame multiple of B, that Cis of D, and that EK

K+

C; therefore EK is the fame E AB

L

is equal to A, and GL to

AB

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multiple of B, that GL is of

D. for the fame reafon KF is the fame 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 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 mul

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tiple of the second B, which GH the third together with the 2, 2, 5, fixth is of the fourth D. If therefore the first, &c. Q. E. D.

H 3

Book V,

See N.

8. 3. $•

IF

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 equimultiple 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 equimultiples 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 equimultiples K, L; therefore K is the fame multiple of A, that L is of C. for the fame reafon M is the fame multiple of B,

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

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; c. s. Def s. and if equal, equal; if less, less c. And K, L are any equimultiples whatever of E, F; and M, N any whatever of G, H. as therefore E

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COR. Likewife if the firft has the fame ratio to the fecond, which the third has to the fourth, then also any equimultiples whatever of

the firft and third have the fame ratio to the fecond and fourth. Book V. and in like manner the first and the third have the fame ratio to

any equimultiples whatever of the fecond and 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 E and F be any equimultiples whatever; then E is to B, as F to D.

Take of E, Fany equimultiples whatever K, L, and of B, D any equimultiples whatever G, H; then it may be demonstrated, as before, that K is the fame 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 lefs, lefs . and K, L are any equimultiples of c. 5. Def. s. 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,

IF

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F 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 shall be the fame multiple of the remainder, that the whole is of the whole.

G

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.

a

A

C

2. 1.6.

F

b. 1. Ax. j

E

Take AG the fame multiple of FD, that AE is of CF. therefore AE is the fame multiple of CF, that EGis 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. takę 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

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

See N.

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

IF

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F two magnitudes be equimultiples of two others, and if equimultiples of thefe 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 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 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

АК
C+

multiple of F, that CD is of F; wherefore Gf H 2. 1. Ax. 5. KH is equal to CD1. take away the common

magnitude CH, then the remainder KC is

equal to the remainder HD. but KC is equal B DEF

to F, HD therefore is equal to F.

But let GB be a multiple E; then HD is the fame multiple of F.

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Make CK the fame multiple of F, that GB
is of E. and because AG is the fame mul-

K

A

C

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H

tiple of E, that CH is of F, and GB the
fame multiple of E, that CK is of F, there-
fore AB is the fame multiple of E, that KH
is of Fb. but AB is the fame multiple of
E, that CD is of F; therefore KH is the G
fame multiple of F, that CD is of it; where-
fore 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 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

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