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Book V.

IF

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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 second 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.

Then is

Let AB the firft 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. AG the firft together with the fifth the fame multiple of C the fecond, that DH the third together with the fixth is of F the fourth.

B

D

A

E

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, G
fo many are there in EH equal to F. as

H F

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 multiple of C, that DH is of F; that is, AG the firft and fifth together, is the fame multiple of the fecond C, that DH the third and fixth together is of the fourth F. If therefore the

firft be the fame multiple, &c. Q. E. D.

D

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OF EUCLID.

PROP. III. THEOR.

IF 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, these shall 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.

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 mag

nitudes EK, KF, each equal F

to A, and GH into GL, LH,
each equal to C. the number
therefore of the magnitudes
EK, KF, fhall be equal to
the number of the others K

GL, LH. and because A is
the fame multiple of B, that
C is of D, and that EK is
equal to A, and GL to C;

therefore EK is the fame multiple of B, that GL is of

E

L

H

E ABG C D

117

Bock V.

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

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Book V.

See N.

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 first and third fhall have the fame ratio to any equimultiples of the second and fourth, viz. the equimultiple of the firft fhall have the fame ratio to that of the fecond, which the equimultiple of the third has to that of the fourth.'

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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 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 fame mul. 3. 5. tiple of A, that L is of C. for the fame reafon M is the fame multiple

of B, that N is of D. and becaufe K E ABG M

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

A and C have been taken certain e-
quimultiples K, L; and of B and
D have been taken certain equi-
multiples M, N; if therefore K be

greater than M, L is greater than

e. 5. Def. 5. N; and if equal, equal; if lefs, lefs. And K, L are any equimultiples whatever of E, F; and M, N any whatever of G, H.

is to G, fo is F to H.

as therefore E

Therefore

if the the firft, &c. Q. E. D.

LF

CDHN

COR. Likewife if the firft has the fame ratio to the fecond, which

the third has to the fourth, then alfo any equimultiples whatever of

the

the first and third have the fame ratio to the fecond and fourth. and Book V. in like manner the firft and the third have the fame ratio to any equimultiples whatever of the second 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, 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 fame multiple of A, that L is of C. and becaufe 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 E, F, c. 5. Def. §. 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 demonftrated.

IF

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 first, is of CF taken from the o- G ther; the remainder E B fhall be the fame multiple of the remainder FD, that the whole AB is of the whole CD.

A

E

a. 1. S.

C

F

b. 1. Ax s.

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

H 4

B D

multiple

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

See N.

IF

PROP. VI. THEOR.

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 firft two be equimultiples of the fame E, F; the remainders GB, HD are either equal to E, F, or equimultiples of them.

A K
АК
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; wherefore a. 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 to F, HD therefore is equal to F.

GH

B DEF

But let GB be a multiple of E; then HD is the fame multiple of
F. Make CK the fame multiple of F, that
GB is of E. and becaufe AG is the fame

A

Ki

C

H+

multiple 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 KB is equal to CDa. take away CH from
hoth, 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

PROP.

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