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

See N.

a 3. 5.

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 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 of A that F is of C; E is the same multiple of B that F is 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 Gis, 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, b 5.def. 5. 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 inCB 5. versely, A is to B as C is to D. Therefore, if the first be the same multiple, &c. Q. E. D.

Book V.

PROP. D. THEOR.

IF the first be to the second as to the third to the See N. 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 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 four h 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 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 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 the same multiple of C, that is, C is the same part of

A

E

F

D, that A is of B. Therefore, if the first, &c. Q. E. D.

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 a 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 b, as A is to C,

so is B to C.

Likewise C has the same ratio to A, that
it has to B: for, having made the same con-
struction, 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 equimulti-
ples whatever of A, B. Therefore C is to A

as C is to Bb. Therefore, equal magnitudes,
&c. Q. E. D.

See N.

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.

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 ra- E tio to BC than unto 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

Fig. 1.

A

F

L

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.

E

F

Fig. 2.

Fig. 3.

E

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 ABa; wherefore EG and a 1. 5. FG are equimultiples of AB and CB: and it was shown, that FG 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 EG, FG are equimultiples of AB, BC, and L is a multiple of D; therefore 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 shown, in like manner, that L is greater

L

A

c!

F

A

K

H D

L

K

D

b 7. def. 5.

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

Book V.

See N.

PROP. IX. THEOR.

MAGNITUDES which have the same ratio to the same magnitude are equal to one another; and those to which the same magnitude has the same ratio are equal to one another.

Let A, B have each of them the same ratio to C: A is equal to B: for, if they are not equal, one of them is greater than the other; let A be the greater; then, by what was shown in the preceding proposition, there are some equimultiples of A and B, and some multiple of C such, that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than that of C. Let such multiples be taken, and let D, E be the equimultiples of A, B, and F the multiple of C, so that D may be greater than F, and E not greater than F: but, because A is to C as B is to C, and of A, B are taken equimultiples D, E, and of C is taken a multiple F; and that D is greater than F; a 5. def. E shall also be greater than Fa; but E is not greater than F, which is impossible; A therefore and B are not unequal; that is, they are equal.

5.

D

A

F

B

E

Next, Let C have the same ratio to each of the magnitudes A and B; A is equal to B: for, if they are not, one of them is greater than the other; let A be the greater; therefore, as was shown in Prop. 8th, there is some multiple F of C, and some equimultiples E and D, of B and A such, that F is greater than E, and not greater than D; but because C is to B, as C is to A, and that F, the multiple of the first, is greater than E, the multiple of the second; F, the multiple of the third, is greater than D, the multiple of the fourth*: but F is not greater than D, which is impossible. Therefore A is equal to B. Wherefore, magnitudes which, &c. Q. E. D.