Since then C contains B oftener than C contains A, it is manifest that B must be less than A.

108.

Q. E. D.

// PROP. XI. THEOR. Ratios that are the same to the same ratio, are the same to one another.

DEMONSTRATION.

Let A be to Bas C to D, and also E to Fas C to D; it is to be shown that A is to Bas E is to F.

AC

Because A is to B as C to D, therefore B-D

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.

DEMONSTRATION.

By Cor. 2. Def. 5. any number of proportionals may be expressed by rA, A; B, B; rC, C;

Where rA, B, C, are the antecedents, and A, B, C, the consequents; and we are to prove that as rA is to A, so is rA+B+C to A+B+C.

rA

The ratio of rA to A is expressed by=r, and the ratio of rA+B+C to A+B+C, by rA+B+C

A+B+C=r; and therefore

A:A::rA+B+C:A+B+C.

13.

PROP. XIII. THEOR.

Q. E. D.

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.

DEMONSTRATION.

Let A, B, C, D, E, F be the first, second, third, fourth, fifth, and sixth magnitudes respectively. The ratios of A to B, of C to D, and of E to F ACE

COR. And if the first have a greater ratio to the second than the third has to the fourth, but the third the same ratio to the fourth which the fifth has to the sixth; it may be demonstrated, in like manner, that the first has a greater ratio to the second than the fifth has to the sixth.

PROP. XIV. THEOR. 14

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

DEMONSTRATION.

Let rA, A, B, B be any four proportionals.

1. Suppose rArB,

then by division AB:

next, suppose rA=rB,

then by division A= B:

lastly, suppose rA ZrB,

then by division A< B.

Q. E. D.

PROP. XV. THEOR. 15

Magnitudes have the same ratio to one another

which their equimultiples have.

DEMONSTRATION.

Let A, B be any two magnitudes of the same kind; and m being any integer greater than unity, let mA,

:

Since then C contains B oftener than C contains A, it is manifest that B must be less than A.

Ratios that are the same to the same ratio, are the

same to one another.

DEMONSTRATION.

Let A be to B as C to D, and also E to Fas C to D; it is to be shown that A is to Bas E is to F. AC

Because A is to B as C to D, therefore-B-D

E

for the same reason

therefore, that is, A : B :: E: F.

12. PROP. ΧΙΙ.

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 con

sequents.

DEMONSTRATION.

By Cor. 2. Def. 5. any number of proportionals may be expressed by A, A; B, B; C, C; Where A, B, C, are the antecedents, and A, B, C, the consequents; and we are to prove that as rå is to A, so is A+B+C to A+B+C.

A

The ratio of rA to A is expressed by and the ratio of rA+B+C to A+B+C, by

+A+B+C

A+B+C=r; and therefore

A:A::A+B+C:A+B+C.

first shall also have to the second a greater ratio than
the fifth has to the sixth.

DEMONSTRATION.

Let A, B, C. D. E. F tre an ars: весе fourth, fifth, and sixth magnitude respetarse The ratios of Au Lou

mB, be equimultiples of A, B; it is to be proved

that

A, B, MA, MB are proportionals.

The ratio of A to B is the numerical quotient

and the ratio of mA to mB is which is reducible

MA

mB'

If four magnitudes of the same kind be proportionals, they shall also be proportionals when taken

alternately.

rA

The ratio of rA to B is, which, because the factor r is in both numerator and denominator, is

A

evidently reducible to : again the ratio of the

A

third A to the fourth Bis also; therefore, the two ratios, viz. of rA to rB, and of A to B, being equal, we have

17

rA:rB::A: B.

PROP. XVII. THEOR.

E. D.

If magnitudes taken jointly be proportionals, they shall also be proportionals when taken separately; that is, if two magnitudes together have to one of them the same ratio which two others have to one of these, the remaining one of the first two shall have