See N.

+ Hyp.

+ Constr.

* 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 the second B, that the third C is of the fourth D: A shall be to B as C is to D.

ABCD

Take of A and C any equimultiples whatever, E and F; and of B and D any equi- EG FII multiples 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; therefore E is the same multiple of B*, that F is of D; that is, E and F are equimultiples of B and D: but G and H are + Constr. equimultiples+ of B and D; therefore, if E be a greater multiple of B than G is of B, 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 than it, F may be shewn to be equal + Constr. to H, or less than it: but E, F are equimultiples +, any whatever, of A, C; and G, H any equimultiples what† 5 Def. 5. ever of B, D; therefore + A is to B, as C is to D. Next, let the first A be the same part of the second B, that the third C is of the fourth D: A shall be to B, as C is to D.

* B. 5.

See N.

A B C D

For since A is the same part of B that C is of D, therefore 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 therefore inversely* A is to B, as C is to D. Therefore, if the first be the same multiple, &c.

Q. E. D.

PROP. D. THEOR.

If the first be to the second as the third to the fourth, and if the first be a multiple, or a 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 shall be 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*; therefore A is to E, as C to F: but A is equal to E, therefore C is equal * to F 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.

A B C D
E F

Next let A be a part of B: C shall be the same part

of 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, that is, B is a multiple of A; therefore, by the preceding case, 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, &c. Q. E. D.

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 shall each of them have the same ratio to C: and C shall have the same ratio to each of the magnitudes A and B.

*

Take of A and B any equimultiples whatever D and E, and of C any multiple whatever F: then, because D is the same+ multiple of A, that E is of B, and that A is equal+ to B: therefore D is equal to E: therefore, if D be greater than F, E is greater than F; and if equal, equal; if less, less; but D, E are any equimultiples of A, B, and F is any multiple of C; therefore, as A is to C, so is B to C.

CF

Likewise C shall have the same ratio to A, that it has to B. For, having made the same construction, D may in like manner be shewn to be equal to E: therefore, if F be greater than D, it is likewise greater than E; and if equal, equal; if less, less: but F is any multiple whatever of Ĉ, and D, E, are any equimultiples whatever of A, B; therefore, C is to A5 Def. 5. as C is to B. Therefore, equal magnitudes, &c.

QE. D.

I

See N.

+ Constr.

*1.5.

PROP. VIII. THEOR.

Of two unequal magnitudes the greater has a greater ratio to any other magnitude than the less has: and the same magnitude has a greater ratio to the less of two other magnitudes, than it has to the greater.

Fig. 1.

Ε

F A

G B

L KHD

Let AB, BC, be two unequal magnitudes, of which AB is the greater, and let D be any other magnitude. AB shall have a greater ratio to D than BC has to D: and D shall have a greater ratio to BC than it has 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 D: and in 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.

E

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 ACt, that FG is of CB; therefore FG is the same multiple of CB*, that EG is of AB; that is, EG and FG are equimultiples of AB and CB: and since it was shewn, that FG is 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 were proved to be equimultiples of AB, BC; and L is a multiple of D; there*7 Def. 5. fore* AB has to D a greater ratio than BC has to D.

+ Constr.

+ Constr.

+ Constr.

F

G B
LKHD

FA

G B

L KD

Also, D shall have 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 Constr. multiple of D; and FG, EG were proved to be equimultiples of CB, AB; therefore D has to CB a greater

ratio* than it has to AB. Wherefore, of two unequal *7 Def. 5. magnitudes, &c.

Q. E. D.

PROP. IX. THEOR.

Magnitudes which have the same ratio to the same mag- See N. nitude 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 shall be equal to B.

For, if they are not equal, one of them must be greater than the other: let A be the greater: then, by what was shewn 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 these multiples be taken; and let D, E be the equimultiples of A, B, and F the multiple of C, such that D may be greater than F, but E not greater than F: then, 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; therefore E is also greater than F: but E is not+ greater than F; which is impossible: therefore A and B B are not unequal; that is, they are equal.

+ Hyp.

5 Def. 5.

+ Constr.

Next, let C have the same ratio to each of the magnitudes A and B: A shall be equal to B. For, if they are not equal, one of them must be greater then the other: let A be the greater: therefore, as was shewn 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, but not greater than D: and because C is to B†, as C is to A, and that F the multi- † Hyp. ple of the first is greater than E the multiple of the second *; therefore F the multiple of the third is greater than D the multiple of the fourth: but F is

5 Def. 5.

+ Constr. not greater than D; which is impossible. Therefore A is equal to B. Wherefore, magnitudes which, &c.

See N.

Q. E. D.

PROP. X. THEOR.

That magnitude which has a greater ratio than another has unto the same magnitude, is the greater of the two: and that magnitude to which the same has a greater ratio than it has unto another magnitude, is the lesser of the two.

Let A have to C a greater ratio than B has to C: A shall be greater than B.

*

For, because A has a greater ratio to C, than B has *7 Def. 5. to C, 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 it: let them be taken; and let D, E be the equimultiples of A, B, and F the multiple of C such, that D is greater than F, but E is not greater than F: therefore. D is greater than E: and, because D and E are equimultiples of A and B, and that D * 4 Ax. 5. is greater than E; therefore A is* greater than B.

* 7 Def. 5.

Next, let C have a greater ratio to B than

it has to A: B shall be less than A.

c

For* there is some multiple F of C, and some equimultiples E and D of B and A such, that F is greater than E, but not greater than D: therefore E is less than D: and because E and D are equimultiples of B * 4 Ax. 5, and A, and that E is less than D, therefore B is less than A. Therefore, that magnitude, &c.

*

Q. E. D.

Ratios that are the same to the same ratio, are the same to one another.

Let A be to B as C is to D; and as C to D, so let E be to F: A shall be to B, as E to F.

Take of A, C, E, any equimultiples whatever G, H, K; and of B, D, F, any equimultiples whatever L, M, N. Therefore, since A is to B as C to D, and G, H are taken equimultiples of A, C, and L, M, of B, D;