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PROP. A. THEOR.

121

Book V.

F the first of four magnitudes has to the fecond, the see N.

fame ratio which the third has to the fourth; then if the first be greater than the fecond, the third is alfo greater than the fourth; and if equal, equal; if lefs, lefs."

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 fecond, the double of the third is greater than the double of the fourth. but if the first be greater than the fecond, the double of the firft is greater than the double of the fecond. wherefore alfo 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 firft be equal to the fecond, 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.

IF four magnitudes are proportionals, they are pro- see N.

portionals alfo when taken inverfely.

If the magnitude A be to B, as C is to D, then alfo 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 lefs 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 equimul

tiples; and of B and D the second and fourth, GA BE
E and F are equimultiples; and that G is lefs.

than E, H is alfo lefs than F; that is, F is H C D F
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 e-
qual to H; and if lefs, lefs. and E, F are any
equimultiples whatever of B and D, and G, H

a. 5. Def. 5.

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.

Sce N.

2.3.5.

PROP. C. THEO R.

IF the first be the fame multiple of the fecond, or the

fame part of it, that the third is of the fourth; the firft is to the second, as the third is to the fourth.

Let the first A be the fame multiple of B the fecond, 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 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 fame multiple of B, that

Fis of D; therefore E and F are the fame 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 lefs; F is equal to H, or lefs than it. But E, F are equimultiples, any whatever, of A, C, and G, H any equimultiples whatever of b.'s. Def. 5. B, D. Therefore A is to B, as C is to D. Next, Let the first A be the fame

4. B. 5.

part of the fecond B, that the third
Cis of the fourth D. A is to B, as C
is to D. for B is the fame multiple of
A, that D is of C; wherefore by the
preceding cafe B is to A, as D is to
C; and inverfely A is to B, as C
is to D. Therefore if the first be

c

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

A B C D

PROP. D. THEOR.

Book V.

F 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 fecond;

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the third is the fame multiple, or the fame 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 fame multiple of D.

Take E equal to A, and whatever multiple A or E is of B, make F the fame 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

C is equal to F. and F is the fame multiple
of D, that A is of B. Wherefore C is the A B
fame multiple of D, that A is of B.

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E

a. Cor. 4. 5. b. A. 5.

F See the Fi

gure at the foot of the preceding

page.

c. B. $.

Next, Let the first A be a part of the fecond B; C the third is the fame part of the fourth D.

Because A is to B, as C is to D; then, inversely B is to A, as D to C. but A is a part of B, therefore B is a multiple of A, and, by the preceding cafe, D is the fame

multiple of C; that is, C is the fame part of D, that A is of B. Therefore if the first, &c. Q. E. D.

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QUAL magnitudes have the fame ratio to the fame magnitude; and the fame has the fame ratio to equal magnitudes.

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

124

Book V. C any multiple whatever F. then because D is the fame multiple of A, that E is of B, and that A is equal to

a. 1. Ax. 5. B; D is equal to E. therefore if D be

greater than F, E is greater than F; and if equal, equal; if lefs, lefs. and D, E are any equimultiples of A, B, and F is any multiple of C. b.5.Def.s. Therefore bas A is to C, fo is B to C.

Likewife C has the fame ratio to A that it has to B. for, having made the fame conftruction, D may in like manner be shewn equal to E. therefore if F be greater than D, it is likewife greater than E; and if equal, equal; if lefs, lefs. and F is any multiple whatever of C, and D, E, are any equimultiples whatever of A, B. Therefore C is to A, as Cis to Bb. Therefore equal magnitudes, &c. Q. E. D.

DA

E B

CF

See N.

PROP. VIII. THEOR.

F unequal magnitudes the greater has a greater ratio to the fame than the lefs has. and the fame magnitude has a greater ratio to the lefs than it has to the greater.

E

A

C

B

KHD

Let AB, BC be unequal magnitudes of which AB is the greater, and let D be any magnitude whatever AB has a greater ratio to D than BC to D. and D has a greater ratio to BC than unto AB. If the magnitude which is not the great- F er of the two AC, CB, be not lefs than D, take EF, FG the doubles of AC, CB, as in Fig. 1. but if that which is not the greater of the two AC, CB be lefs than D (as in Fig. 2. and 3.) this magnitude can be multiplied fo as to become greater than 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 fame multiple of CB.

;

G
L

therefore EF and FG are each of them greater than D. and in Book V. every one of the cafes take H the double of D, K its triple, and fo 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 lefs than L.

Then because L is the multiple of D which is the first that becomes greater than FG, the next preceding multiple K is not greater than FG; that is, FG is not lefs than K. and fince EF is the fame 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- a. 1. s. multiples of AB and CB. and it was fhewn that FG E

was not less than K, and, by the construction, EF is greater than D; therefore the whole EG is greater than K and D together. but K together with D is equal to L; therefore EG

is greater than L; but FG

is not greater than L; and

F

E

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EG, FG are equimultiples L KHD

of AB, BC, and L is a

multiple of D; therefore b

AB has to D a greater ratio than BC has to D.

Alfo D has to BC a greater ratio than it has to AB. for, having made

the fame conftruction, 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.

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