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Book v. PROP. A. THEOR. F the first of four magnitudes has to the fecond, the Sce 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.

IF

Take any equimultiples of each of them, as the doubles of cach. 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 fee cond. 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. 0. E. D.

PROP. B. THEO R.

IF
F 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 Dthe second and fourth, G A B E E and F are equimultiples; and that G is less than E, H is also · less than F; that is, Fis H Ç D F

a. s. Def.s. 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 shewn to be equal to H; and if less, less. and E, F are any equimultiples 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. nIf then four magnitudes, &c. Q. E. D.

Sce N.

PROP. C. THEOR.
IF

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 what-
ever E and F; and of B and D any equi-
multiples whatever G and H. then because
A is the same multiple of B that Cis of D;
and that E is the same multiple of A, that A B C D

F is of C; E is the same multiple of B, that 2. 3. 5.

Fis of D'; therefore I and F are the fame E G F H
multiples of B and D. but G and H are e-
quimultiples 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 bis. Def. 5. B, D. 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 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 q. B. s: C; and inversely " A is to B, as c is to D. Therefore if the first be

À B C D the same multiple, &c. Q. E, D,

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Book V.

PROP. D. THEO R.

If the first be to the second as the third to the fourth, See N.

and if the first be a multiple, or part of the second; the third is the same multiple, or the same part of the fourth.

Let A be to B, as C is to D; and firft let A be a multiple of B; C is the same multiple of D.

Take E equal to A, and whatever multiple A or E is of B, make F the same multiple of D. then because A is to B, as C is to D; and of B the second and D the fourth equimultiples have been taken E and F; A is to E, as C to F•. but A is equal to E, therefore

a. Cor.4.S. C is equal to Fb, and F is the same multiple

b. A.S. of D, that A is of B. Wherefore C is the A B C D lame multiple of D, that A is of B.

E F Seethe FiNext, Let the first A be a part of the lecond B; C the third is the same part of the

foot of the fourth D.

preceding Because A is to B, as C is to D; then,

page. inversely B is to A, as D to C. but A is

c. B. s. a part of B, therefore B is a multiple of A, and, by the preceding cafe, D is the same multiple of C; that is, C is the same part of D, that A is of B. Therefore if the first, &ç. Q. E. D.

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PROP. VII. THEO R.

QUAL magnitudes have the same ratio to the

fame magnitude; and the same has the same ratio to equal magnitudes.

Let A and B be equal magnitudes, and C any other. A and B have each of them the same ratio to C. and C has the fame ratio to each of the magnitudes A and B.

Take of A and B any equimultiples whatever D and E, and of

Book V. C any multiple whatever F. then because D is the same multiple

w of A, that E is of B, and that A is equal to a. 1. Ax. s. B; D is equal to E. thereforeif D be greater

than F, E is greater than F; and if equal, e-
qual ; if less, less. and D, E are any equimul-

tiples of A, B, and F is any multiple of C. b.s. Def.s. Therefore b as A is to C, so is B to C. Likewise C has the same ratio to A that

D A
it has to B. for, having made the fame con-
struction, D may in like manner be shewn e- E B
qual to E. therefore if :F be greater than D,

CF
it is likewise greater than E; and if equal,
equal; if less, less. and F is any multiple what-
ever of C, and D, E, are any equimultiples
whatever of A, B. Therefore C is to A, as
Cis to B b. Therefore equal magnitudes, &c.
Q. E. D.

See N.

PROP. VIII. THEOR.
Funequal magnitudes the greater has a greater

ratio to the fame than the less has. and same magnitude has a greater ratio to the less than it has to the greater:

Let AB, BC be unequal magnitudes of which AB is the greater, and let D be any magnitude whatever AB

E
has a greater ratio to D than BC to D. and
D has a greater ratio to BC than unto AB.

A
If the magnitude which is not the great-

F
er of the two AC, CB, be not less than D,
take EF, FG the doubles of AC, CB, as C
in Fig. 1. but if that which is not the
greater of the two AC, CB be less than D
(as in Fig. 2. and 3.) this magnitude can

G B
be multiplied so as to become greater than L K HD
D, whether it be AC or CB. Let it be
multiplied until it become greater than D,
and let the other be multiplied as often ;
and let EF be the multiple thus taken of
AC, and FG the same multiple of CB.

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F

therefore EF and FG are each of them greater than D. and in Book V. every one of the cases take H the double of D, K its triple, and so on, till the multiple of D be that which first becomes greater than FG. let L be that multiple of D which is first greater than FG, and K the multiple of D which is next less than L.

Then because L is the multiple of D which is the first that be. comes greater than FG, the next preceding multiple K is not greater than FG ; that is, FG is not less than K. and since EF is the same multiple of AC, that FG is of CB; FG is the fame multiple of CB, that EG is of AB'; wherefore EG and FG are equi- 2. 1. S. multiples of AB and CB. and it was thewn that FGE

E was not less than K, and, by the construction, EF is

A greater than D; therefore the whole EG is greater than K and D together.

F! A but K together with D is equal to L; therefore EG

С is greater than L; but FG

GB is not greater than L; and EG, FG are equimultiples. Ļ Ķ HD G B of AB, BC, and L is a multiple of D; therefore b AB has to D a greater ratio than BC has to D.

Also D has to BC a greater ratio than it has to AB. for, having made the same construction, it may be shewn, in like manner, that L is greater than FG, but that it is not greater than EG. and L is a multiple of D; and FG, EG are equimultiples of CB, AB. Therefore D has to CB a greater ratio than it has to AB. Wherefore of unequal magnitudes, &c. Q. E. D.

L K D b.y. Dcf.s.

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