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BOOK V.

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PROP. IV. THEOR.

See N. IF 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 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.'

1

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 equimul

tiples whatever E, F; and of B
and D any equimultiples what-
ever G, H: Then E has the same
ratio to G, which F has to H.

Take of E and F any equi-
multiples whatever K, L, and of
G, H, any equimultiples what-
ever 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 K E A

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3. 5. of A, that L is of Ca: For the same reason, M is the same multiple of B, that N is of D: And because, as A is to B, so is b Hypoth. C to Db, and of A 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 and if equal, equal; * 5 Def. 5. if less, less. And K, L are any equimultiples whatever of E,

See N.

F; and M, N any whatever of

B G M

L F C DH N

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 equi

multiples whatever of the first and third have the same ra- Book V. tio to the second and fourth: 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 equimúltiples 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. Kand 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 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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PROP. V. THEOR.

If 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 G 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, A that the whole AB is of the whole CD.

F

CA

Take AG the same multiple of FD, that AE is of CF: therefore AE is a the same multiple of CF, that EG is of CD: But AE, by the hypothesis, is the same multiple of CF, that E AB is of CD: Therefore EG is the same multiple of CD that AB is of CD; wherefore EG is equal to ABb. Take from them the common magnitude AE; the remainder AG is equal to the remainder EB. Wherefore, since AE is the same multiple of CF, that AG is of FD, and that AG is equal to EB; therefore AE is the same multiple of CF, that EB is of FD: But AE is the same multiple of CF that

B

a 1. 5.

bi Ax. 5.

D

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

PROP. VI. THEOR.

See N. IF 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 A Ķ
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, by the hypothesis, is the same G
multiple of E that CD is of F; therefore
KH is the same multiple of F, that CD

H

1 Ax. 5. is of F; wherefore KH is equal to CD: B D E F 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.

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K

But let GB be a multiple of E; then HD is the same multiple of F: Make CK the same multiple of F, that GB is of E: And because AG is the same mul- A tiple of E, that CH is of F; and GB the same multiple of E, that CK is of F; therefore AB is the same multiple of E, 2. 5. that KH is of Fb: But AB is the same G multiple of E, that CD is of F; therefore KH is the same multiple of F, that CD is of it; Wherefore KH is equal to CDa: 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, 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.

B

DE F

BOOK V.

PROP. A. THEOR.

IF the first of four magnitudes has to the second See N. the 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 it. Therefore, if the first, &c. Q.E. D.

PROP. B. THEOR.

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IF four magnitudes are proportionals, they are pro- See N. portionals also when taken inversely.

If the magnitude A be to B, as C 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, E and F are equimul-G A BE
tiples; and that G is less than E, H is

also a less than F; that is, F is greater H C D F
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 shown to be
equal to H; and if less, less; and E, F,
are any equimultiples whatever of B and
D, and G, H any whatever of A and C ;

a 5 Def. 5.

Book V. therefore, as B is to A, so is D to C. If then four magnitudes, &c. Q.E.D.

PROP. C. THEOR.

See N. Ir the first be the same multiple of the second, or the same 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 C any equimultiples whatever E and F; and of B and D any equimultiples whatever G and H: Then, because A is the same multiple of B that C is of D; and that E is the same multiple A of A, that F is of C: E is the same multi- E G F H 3. 5. ple of B, that Fis of Da; therefore E and

F are the same 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, D. 5 Def. 5. Therefore A is to B, as C is to Db.

Next, Let the first A be the same part of the second B, that the third C is 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; and inB. 5. versely A is to B, as C is to D: There

c

B C D

fore, if the first be the same multiple, A B C D

&c. Q.E.D.

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