Book V,

PROP. A. THEOR.

IF the first of four magnitudes have to the second See Note. the same ratio which the third has to the fourth; then, if the first be greater then the second, the third is also greater than the fourth; and if. equal, équal; 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 then the double of the fourth; but, if the first be greater then the second, the double of the first is greater then the double of the second; wherefore also the double of the third is greater then 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.

IF four magnitudes be proportionals, they are pro- See Note. 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 Erst and third, G, and H are equimultiples; and of B and D, the second and fourth, E and Fare equimultiples; and that G is less than È, I is also G a less than F; that is, F is 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 shown to be equal to H; and, if less, less; and E, F are any equimultiples whateyer of B and D, and G, H any whatever of A and C; therefore, as B is

R

H

E

C

.D

F

a 5. def. 5.

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

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 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 of A, that F is of C; E is the same multiple 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 any equimuhiples whatever, of A, C, and G, H any equimultiples whatever of 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 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 inversely A is to B, as C is to D. Therefore, if the first be the same multiple, &c. Q. E. D.

A B
EGF H

Book V.

PROP. D. THEOR.

IF the first be to the second as the third to the See Note. fourth, 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 first let A be a multiple of B; C is the same multiple of D.

Take E equal to A, and whatever muliple 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 Fa: but A is equal to E, therefore C is equal to Fb: and F is the same multiple of D, that A is of A B. Wherefore C is the same multiple of D, that A is of B.

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

Because A is to B, as C is to D; then, inversely, B is e to A, as D to C: but A is a part of B, therefore B is a multiple of A; and, by the preceding case, D is thǝ same multiple of C, that is, C is the same part of D, that A is of B: therefore, if the first, &c. Q. E. D.

E

a Cor. 4. 5.

b A. 5.

See the figure at the foot of the preceding page.

c B. 5,

PROP. VII. THEOR.

EQUAL magnitudes have the same ratio to the same 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 same ratio to each of the magnitudes A and B.

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

Book V.

of C any multiple whatever F: then, because D is the same multiple of A, that E is of B, and that A is

:

a 1. Ax. 5. equal to B; D is equal to E: therefore, if D be greater than F, E is greater than F; and if equal, equal; if less, less and D, E are any equimultiples of A, B, and F is any b 5. def. 5. multiple of C. Therefore, as A is to C, so is B to C.

See note.

Likewise C has the same ratio to A, that it has to B: for, having made the same construction, D may in like manner be shown equal to E: therefore, if F be greater than D, it is likewise greater than E; and if equal, equal; if less, less: and F is any multiple whatever of C, and D, E are any equimultiples whatever of A, B. Therefore C is to A, as C is to B. Therefore equal magnitudes, &c. Q. E. D.

57th L

PROP. VIII. THEOR.

OF unequal magnitudes, the greater has a greater ratio to the same than the less has; and the same magnitude has a greater ratio to the less, than it has to the greater.

Fig. 1.

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 to AB.

If the magnitude which is not the greater of the two AC, CB, be not less 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 less than D (as in Fig. 2. and 3.) this magnitude can be multiplied, so 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 same multiple of CB: therefore EF and FG are each of them greater than

E

F

L

A

K

H D

D.: and in every one of the cases, take H the, double of D, K, Book V. 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 becomes 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 same multiple of CB, that EG is of AB a; wherefore EG and FG are equimultiples of AB and CB: and it was shown, that FG was Fig. 2.

E

not less than K, and,
by the construction,
EF is greater than D;
therefore the whole
EG is greater than K F-
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 EG, FG
are equimultiples of
AB, BC, and L is a
multiple of De there-
foreb AB has to Da
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 shown, in like manner, that L is greater than

Fig. 3.

E

A

a 1.5.

K

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 ratiob than it has to AB. Wherefore, of unequal magnitudes, &c. Q. E. D.