The same by Algebra.

Let m, n, be two magnitudes, and p some other magnitude, and let m have a greater ratio to p, than ʼn has

m

to p; then m>n, for >, .. m>n. Again, if p have a greater ratio to n than it has to m; n<m. For if not, it would be either equal or greater; but n does not equal m; for then p:n :: p:m; but it is not, neither is it greater; for then p would have a greater ratio to m than to n, but it has not: whence n is not greater than m, nor does n = m; .. n < m.

PROPOSITION XI.

THEOREM.

Ratios which are the same to the same ratio are the same to one another.

For let a be to в as c is to D, and as c is to D so is E to F: then as A is to B so is E to f.

For take G, H, K, equimultiples of A, C, E; also any other L, M, N, equimultiples of B, D, f.

Therefore, because it is as A to B so is c to D, and G, H, are taken equimultiples of A, C, also L, M, any other equimultiples of B, D: if G exceed L, H will also • 5 Def. 5. exceed м; if equal, equal; and if less, less. Again, because it is as c is to D, so is E to F, and н, к, are taken equimultiples of c, E; also M, N, any other equimultiples of D, F: if H exceed м, K will also exceed N; if equal, equal; and if less, less. But if H exceed M, G will also exceed L; if equal, equal; and if less, less. Wherefore, if G exceed L, K will also exceed N; and if equal, equal; and if less, less. And G, K, are equimultiples of A, E, also L, N, any other equimultiples of B, F: therefore as A is to B, so will E be to F. Wherefore ratios which are the same to the same ratio, &c. Q. E. D.

The same by Algebra.

Let a bef; and c:d::e:f; then a: b:: c: d. Take g, h, i, equimultiples of a, c, e; and k, l, m, equimultiples of b, d, f. Because a : b :: e: f,* if g`<,

=, > k,† then also i <, =, > m.

And likewise be- † Def. 5.

cause e: ƒ :: c : d,‡ if h <, =, > l,† then is i also ‡ Hyp. <, =, > m:§ whence a: b:: c : d. Q. E. D.

PROPOSITION XII.
THEOREM.

If there be any number of magnitudes proportional; as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.

Let any number of magnitudes A, B, C, D, E, F, be proportional; that is, as A is to B, so is c to D, and E to F as A: B, so are A, C, E, to в, d, f.

For take G, H, K, equimultiples of A, C, E; also G L, M, N, any other equi- H multiples of B, D, F. Therefore, because as A is to K B so is c to D and E to F; and G, H, K, have been taken equimultiples of a, c, E; also L, M, N, any E other equimultiples of B,

A

C

L

M

N

B

D

F

D, F: if G exceed L, H will also exceed м, and 5 Def. 5. K exceed N; if equal, equal; and if less, less. Wherefore, if G exceed L; G, H, K, will also exceed L, M, N ; if equal, equal; and if less, less: G, and G, H, K, are equimultiples of a, and A, C, E. Because if there be 1.5.

b

any number of magnitudes equimultiples of as many magnitudes, each to each, whatsoever multiple any one of them is of its part, the same multiple shall all the first magnitudes be of all the others; for the same reason, L, and L, M, N, are equimultiples of B, and B, D, F: therefore it is as A is to B, so are A, C, E, to 5 Def. 5. B, D, F. Wherefore, if there be any number, &c.§

Q. E. D.

The same by Algebra.

Let s tms: mt::ns: nt, &c.; then will st::s +ms+ns: t + m t + nt, &c.

If

any

s + m s + n s

number of equal ratios be each greater than

§ This is the same as the twelfth proposition of the seventh book with regard to numbers.

a given ratio, the ratio of the sum of their antecedents, to the sum of their consequents, shall be greater than that given ratio.

PROPOSITION XIII.

THEOREM.

If the first magnitude have the same ratio to the second, as the third has to the fourth; but the third have a greater ratio to the fourth, than the fifth has to the sixth: the first magnitude shall also have a greater ratio to the second than the fifth has to the sixth.

For let the first magnitude, a, have the same ratio to the second, B, as the third, c, to the fourth, D; but let the third, c, have a greater ratio to the fourth, D, than the fifth, E, to the sixth, F: then shall the first magnitude, A, also have a greater ratio to the second, в, than the fifth, E, to the sixth, F.

For because c has a greater ratio to D than E to F, there are some equimultiples of c, E, and some other

equimultiples of D, F, such, that the multiple of c is greater than the multiple of D, but the multiple of E a7 Def. 5. is not greater than the multiple of F. And take G, H, equimultiples of C, E, also K, L, some other equimultiples of D, F, so that G shall exceed к, but й shall not exceed L; and whatever multiple G is of c, the same let м be of a; also whatever multiple к is of d, the same multiple let N be of B.

b

And because it is as A is to в so is c to D, and м, G, are taken equimultiples of A, C, also N, K, other equi5 Def. 5. multiples of B, D: if м exceed N, G, will also exceed K; if equal, equal; and if less, less. But G exceeds K; therefore м also will exceed N. But н does not exceed L; and м, н, are equimultiples of A, E, and N, L, some other equimultiples of B, F: wherefore A shall have a greater ratio to в than E to F. If therefore the first have the same ratio, &c. Q. E. D.

1. If the first of four magnitudes have a greater ratio to the second than the third has to the fourth; then shall the second have a less ratio to the first than the fourth to the third.

2. If any number of ratios be greater to the same ratio, then shall their sum be greater than that ratio multiplied by the number of the first ratios.

PROPOSITION XIV.

THEOREM.

If the first magnitude have the same ratio to the second, as the third has to the fourth, if the first be greater than the third, the second will also be greater than the fourth; if equal, equal; and if less, less.

For let A, the first magnitude, have to B, the second, the same ratio which c, the third, has to D, the fourth, and let A be greater than c; B will

also be greater than D.

A

C

For because A is greater than c, B and в some other magnitude; A shall c have a greater ratio to B, than c to D B. But as A is to B, so is c to D; wherefore also c will have a greater ratio to D than c has to в. But to that which the same magnitude has a greater ratio is the less: whence D is less than B: and, consequently, в will be greater than D.

In like manner we demonstrate, that if a be equal to C, B will also be equal to D; and if A be less than C, B will also be less than D. If, therefore, the first magnitude have the same ratio, &c. Q. E. D.

The same by Algebra.

a

с

a

b

с

Let a:b::c:d; or == ; then if a <, =, or > c, b shall also be <, then, but = ; whence,+<, therefore, † 13.5. b < d. In the same manner, if a > c,‡ then is b > d. But if a = c, then ab :: c: b. So that b

or > than d. Let ac, **8.5.

Q. E. D.

= d.

10. 5.

. 7.5.

b 12.5.

PROPOSITION XV.

THEOREM.

Magnitudes, when compared to one another, have the same ratio as their equimultiples have to one another.

For let AB be the same multiple of c, as DE is of F, then as c is to F, so is AB to DE.

E

LI

For because AB is the same multiple of c as DE is of F; as many magnitudes as are in AB equal to c, so many will there be in DE equal to F. Divide AB into magnitudes each equal to c, which let be AG, GH, HB; and DE into magnitudes B each equal to F, viz. in DK, KL, LE. Therefore the multitude in AG, GH, HB, HI will be equal to the multitude of DK, KL, G LE. And because AG, GH, HB, are equal to one another, also DK, KL, LE, are equal to one another, they will be as AG is to DK, so is GH to KL, and HB to LE. And it will be as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents: therefore it is as AG is to DK So is AB to DE. But AG is equal to c, and DK to F: therefore as c is to F so will AB be to DE. Therefore magnitudes, when compared to one another, &c. Q. E. D.

The same by Algebra.

A

Let a and b be any two quantities, and m a, mb, any equimultiples of them, m being any number whatever: then will a : b ::ma:mb. For =

ma
mb

PROPOSITION XVI.

THEOREM.

a

Q. E. D.

If four magnitudes of the same kind be proportional, they shall also be alternately proportional.

Let the four magnitudes A, B, C, D, be proportionals, viz. as a is to B so is c to D. They shall also be alternately proportional; viz, as a is to c so is B to D.

For take E, F, equimultiples of A, B; also G, H, any other equimultiples of c, D.

And because E is the same multiple of A as F is of B; and magnitudes when compared to one another have the same ratio as their equimultiples have to one another; it will be as A is to в so is E to F. But as

A is to в so is c to D; therefore as c is to D so is E to