Otherwise.

Hyp.. If A: B = C: D, and C: DE: F; then A: B = E: F; or, if 3 : 6 = 1 : 2; and 1 : 2 = 4 : 8, then 3 : 6 = 4 : 8.

Alg.

Arith.

Now m A, m E equims. of A, E ; n B, n F, of B, F;
.. by Def. 5, V A:B = = E: F.

Take the ms 2 X3, 2×1, 2×4;

and 2 × 6, 2 x 2, 2 × 8;

'.' 3 : 6 = 1 : 2, if 2 × 3 <2 × 6,

.. 2×1 < 2×2.

Again, . 1 : 2 = 4 : 8, if 2 × 1 < 2 × 2,

.. 2×4 < 2×8.

Now 2 × 3, 2 × 4 are equims; also 2 × 6 and 2 × 8;

..3:64: 8.

Scr. This Proposition is to Ratios. what Prop. 30, Bk. 1, is to Parallel lines; Ax. 1, Bk. I, to Magnitudes; and Ax. 1, Bk. V. to Equimultiples.

:

COR. 1. If A B= C: D, but C: D> or < E: F, then A: B> or <E: F.

For, whatever part of D be contained in C, a greater or less number of times than the like part of F is contained in E, the like part of B must be contained in A the same greater or less number of times.

COR. 2 Thus also, if A: B> or < C: D, and C: D = E: F, then A: B> or < E : F.

PROP. XII.-THEOR.

If any number of magnitudes be proportionals, as one of the antecedents is to its consequent, so shall all the antecedents taken together be to all the consequents.

CON. Pst. 1, V. DEM. Def. 5, V. Definition of Proportion. 1, V. If any no. of Ms be equims of as many, each of each, what m soever any one of them is of its pt., the same m shall all the first Ms be of all the others.

D. 1 Hyp.

3

6

V.

D

F

12 N.

18

Of A,C,E, take equims. of G,H,K;
and of B,D,F, equims. L,M,N.
then A: B = C: D

E: F,

and G,H,K, are equims of A,C,E,

and L,M,N, equims. of B,D,F;

.. G > = or < L, H also > = or < M, and K or > N;

wherefore if G > = or < L,

then G+H+K > = or < L+ M + N. But G, and G + H+ K, are equims. of A, and A+ C + E.

for whatever m of A, G is, the same m are all, G+H+K of A + C + E;

i. e. G, and G + H + K are equims. of A, and
A+ C + E.

So L, and L + M + N are equims. of B, and
B+ D + F.

.. A: B A+ C+E: B + D + F. Wherefore, if any number of magnitudes, &c.

Q. E. D.

Otherwise. Hyp. A : B= C:D=E:F; then A: B= A + C + E : B+ D + F; or 1:32

63: 9, then 1 : 3 = 1 + 2 + 3:3+ 6+ 9,

Of the Antecs. take mA, mC, mE,
Of the Conseqs.

A: BC: D, ..

• C:D = E: F, ..

nB, nD, nF;

if m A >=orn B, m C, >= or < n D; if m C> or < n D, mE >= or < n F: If m A> or < n B, m A + mC+mE >= or < n B+nD + n F.

=

Now (1, V.) m A + mC + mE = m (A + C + E),

so that m A, mA + mC+m E are equims. of A, and A + C + E; and nB, nB + nD,+ n F are equims. of B, and B + D + F;

.. A: B = A + C+E: C + D : F.

Arith. Of Antecs. take 3 x 1, 3 x 2, 3 x 3,

Of Conseqs.,, 2 X 3, 2 x 6, 2 × 9;

'.' 1 : 3 = 2 : 6, .. if 3 × 1>

2 X 6;

or < 2 × 3, 3 X

= or <

and.. 2 6 = 3 : 9,.. if 3 × 2 > or > 2 × 6, 3 X
> 2 X 9.

Now (3 x 1) + (3 × 2) + (3 × 3) = 3 (1 + 2 + 3),

or

so that 3 x 1, (3 × 1) + (3 × 2) + (3 × 3) are equims. of 1 and (1 + 2 + 3)

and 2 × 3, (2 × 3) + (2 × 6) + (2 X 9) equims. of 3 and (3 + 6 + 9);

.. 1 : 3 = (1 + 2 + 3) : (3 + 6 + 9), or 1 : 36: 18. Alg. II. For Antecs. take a, and a, b, c, d, e, &c.

For Conseqs." a', and a', b', c', d', e', &c.

с

d

By given Hyp.

== ==/ = &c.
e'

ab' a'b, a c'=

=

Take product of extremes and means;

a e' = a' e, &c.

Add Antecs. for numerator, Conseqs. for denominator,

Divide by

a a' + ab + a'c + a' d + a'e &c. and we have, aa' + ab + ac' + a' d + a'e &c. a a' + ab + a'c + a' d + a'e &c. Now = 1, & a a' + a b' + ac' + ad' + a e' &c.

a

α

And the quotients in each case being unity,

= 1.

.. a : a' = a +b+c+d+e &c. : a' + b' + c' + d' + e' &c.

PROP. XIII.-THEOR.

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

CON. Pst. 1, V. DEM. Def. 7, V.

Definition of greater and less ratio

Def. 5, V. Definition of Proportion.

2 Def. 7, V... some equims of C the 1st and E the third,
and some of D the 2nd and F the 4th;

3 Pst. 1,V.

4 Pst. 1, V.

so that m of C > m of D. but mof Em of F; Take such equims. G & H of C & E, K& Lof D & F, so that G > K, but HL;

Whatever m, G is of C, take M of A;

and whatever m, K is of C, take N of B.

D. 1 H&C.3,4. A: BC: D, and M, G, equims. of

of A, C; N, K of B, D;

2 Def. 5, V... if M > = or < N, G > = or < K.

4 C. 3 & 4. ButHL, and M, H equims of A, E;— N, L

of B, F;

5 Def, 7, V... A : B > E : F.

6 Rec.

Wherefore if the first has to the second &c. Q. E. D.

COR. If A: B> or < B: E, but C: D = E: F, it may also be demonstrated that A: B> or < E F.

Alg. & Arith Hyp. Let A 5: B6= C 10 : D 12; but C 10 : D 12 > E5: F9; then A 5: B6 > E 5: F 9. Let m3, & n = 2.

Now. A: BC: D,-if mC > n D, m A > nB;
.. mA > n B, and m E < nF,

.. A: B>E: F.

10 12 > 5:9,

Arith.

Assume 3 x 102 x 12, but 3 × 5 < 2 × 9.

Now 5 6

10: 12, if 3 × 10 > 2×12; and 3 × 5 > 2× 6 .. 3 × 5 > 2 x 6, and 3 x 5 < 2 × 9;

SCH.

.. 565 9.

"This proposition is equivalent to stating; 10. that if any ratio be greater than another, every ratio which is equal to the former will also be greater than the latter; 2°. Also, that if one ratio be greater than another, every ratio which is greater than the former is also greater than the latter." LARDNER.

PROP. XIV.-THEOR.

If the first has the same ratio to the second which the third has to the fourth; then, if the first be greater than the third, the second shall be greater than the fourth; and if equal, equal; and if less, less.

DEM. 8, V. The gr. M. a gr. ratio. 13, V.-10, V. one M with a gr. ratio to a third. 9, V. Two Ms. each with the same ratio to a third.