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Book v.

PROP. IV, THEOR.

See N.

TF 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 fhall have the same ratio to any equimultiples of the second and fourth, viz. the equimultiple of the first shall ¢ have the same 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 same 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 same 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 å. 3. 5.

of A, that L is of Ca. for the same
reason M is the fame multiple of B,

that N is of D. and because as A KE A B GM 6. Hypoth. is to B, fo is C to Db, and of A L F C D 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. 5. Def. 5. and if equal, equal; if less, less e,

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 C 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 af

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 same 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 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 c. and K, L are any equimultiples of c. 5. Def. 5. 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 is demonstrated.

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IF

F one magnitude be the same 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 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 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

"С is of CF. therefore AE is a the same multiple of

8. 1. S. CF, that EG is of CD. but AE, by the hypothefis, is the same multiple of CF, that AB is of CD.

E therefore EG is the same multipleof CD that AB

F is of CD; wherefore EG is equal to AB b. take

b. I. Axo 5 from them the common magnitude AE; the remainder AG is equal to the remainder EB, Wherefore since AEisthe 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 same

See N,

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1

c!

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. IV. 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 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

K
of E, that CH is of F, and that GB is equal A
to E, and CK to F; therefore AB is the same
multiple of E, that KH is of F. But AB, C
by the hypothesis, is the fame multiple of E
that CD is of F; therefore KH is the famę

multiple of F, that CD is of F; wherefore G 3. Ļ. Ax. 5. KH is equal to CD a. 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,
Make CK the same multiple of F, that GB K
is of E. and because AG is the same mul.
tiple of E, that CH is of F, and GB the

A
same multiple of E, that CK is of F, there,

C.
fore AB is the same multiple of E, that KH
is of Fb. but AB is the same multiple of
E, that CD is of F; therefore KH is the
same multiple of F, that CD is of it; where H Н
fore KH is equal to CD a. 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 D E F
and that KC is equal to HD; therefore HD is the same multiple of
7, that GB is of L. If thereforę two magnitudes, &c. Q.E. D.

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6. 2. 5.

Book v.

PROP. A.

THEOR.

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F the first of four magnitudes has to the second, the See N.

same ratio which the third has to the fourth; then if the first be greater than the second, the third is also greater than the fourth; and if equal, equal; if less, less.

Take any equimultiples of each of them, as the doubles of each. then by Def. 5th of this Book, if the double of the first be greater than the double of the second, the double of the third is greater than the double of the fourth. but if the first be greater than the second, the double of the first is greater than the double of the second, wherefore also the double of the third is greater than the double of the fourth; therefore the third is greater than the fourth. in like manner, if the first be equal to the second, or less than it, the third can be proved to be equal to the fourth, or less than ito Therefore if the first, &c. Q. E. D.

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F four magnitudes are proportionals, they are pro. See N.

portionals also when taken inversely.

If the magnitude A be to B, as Ç is to D, then also inversely B is to A, as D to C.

Take of B and D any equimultiples whatever E and F; and of A and C any equimultiples whatever G and H. First, Let E be greater than G, then G is less than E; and because A is to B, as C is to D, and of A and C the first and third, G and H are equimultiples; and of B and D the second and fourth, GA B E E and F are equimultiples; and that G is less

HCDF than E, H is also a less than F; that is, F is

a. 5. Def. 5: greater than H. if therefore E be greater than G, F is greater than H. in like manner, if E be equal to G, F may be fhewn to be equal to H; and if less, less. and E, F are any cquimultiples whatever of B and D, and G, H

Book V. any whatever of A and C. Therefore as B is to A, fo is D to C.

If then four magnitudes, &c. Q. E. D.

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2. 3. S.

See N. F the first be the same multiple of the second, or the

fame
part

of it, that the third is of the fourth; the
first is to the second, as the third is to the fourth.
· Let the first A be the same multiple of B the second, that C
the third is of the fourth D. A is to B, as c
is to D.

Take of A and Cany equimultiples whatever E and F; and of B and D any equimultiples whatever G and H. then because A is the fame multiple of B that C is of D; and that E is the fame multiple of A, that

A B C D F is of C; E is the same multiple of B, that F is of Da; therefore E and F are the same E G F H multiples of B and D. but G and H are equimultiples of B and D; therefore if E be a greater multiple of B, than G is; F is a greater multiple of D, than H is of D; that is, if E be greater than G, F is greater than H. in like manner, if E be equal to G, or less; F is equal to H, or less than it. But E, F are equimultiples, any whatever, of A,

C, and G, H any equimultiples whatever of b. 5. Def. 5. B, D. Therefore A is to B, as C is to Do.

Next, Let the first A be the same part of the second B, that the third Cis of the fourth D. A is to B, as c is to D. for B is the same multiple of A, that D is of C; wherefore by the

preceding case B is to A, as D is to C. B. 5. C; and inversely c A is to B, as c

is to D. Therefore if the first be A B C D the same multiple, &c. Q. E. D.

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