Moreover, fince aN is fuppofed <4P (Prep.4), & aM >4P (Arg.6). 8. It is evident that the mgn. 4 P is > a N, & the fame mgn. 4 PM. But 4 4 P and 4 P being equimultiples of the antecedents P and P, and a N, a M'equimultiples of the confequents N and M, 9. It follows that P: N>P: M.

Which was to be demonstrated. 11.

II. Preparation.

If R be fuppofed > I N or N.

5. Take 1 a N a multiple of 1 N > P.

D. 7. B. 5.

6. Take a R & aN the fame multiples of R & of N that 1aN is of 1N. Pos.1. B. 5. 7. Let 4P be the firft multiple of P> a R; confequently the next preceding multiple 3 P will not be a R, or a R will not be <3 P. DEMONSTRATION.

IT

T may be proved as before (Arg. 1. 2 & 3), that

1. The mgns. a M and a N are equimultiples of the mgns. M & N. Moreover, a R & aN being equimultiples of R & of N (Prep. 6), and R being > N (Sup.),

2. It follows that a R is a N

But

a R not being < 3 P (Prep. 7), And the mgn. IaN being> P (Prep. 5), 3. Then by adding, a R+1 a N, or a M> 4 P.

But aR being 4P (Prep.7), & this fame aR being > aN (Arg.2), 4. Much more then a N is < 4 P.

But aM & aN are equimultiples of the antecedents M & N (Arg. 1) and 4 P, 4 P equimultiples of the confequents P & P, & moreover a M > 4 P & a N < 4 P (Arg. 3 & 4). 5. Confequently M: PN: P.

Which was to be demonstrated. 1.

Moreover, without changing the Preparation, it may be demonstrated as in the precedent cafe (Arg. 89), that

6. The ratio of P: N is the ratio of P: M.

Which was to be demonftrated. 11.

III.

And applying the fame preparation and fame reasoning to the laft
cafe when R= 1 N,

The demonstration will be completed as in the two precedent cafes.
Which was to be demonftrated. 1 & 1.

D. 7. B. 5.

PROPOSITION IX.

THEOREM IX.

MAGNITUDES (M & 1 M) which have the fame ratio to the fame

magnitude (N): are equal to one another. And those (M & 1 M) to which the fame magnitude (N) has the fame ratio, are equal to one another.

Hypothefis.

M: NIM: N.

TH

If not, the two mgns. M & 1 M are unequal.

I HEN the two mgns. M & 1 M have not the fame ratio to the fame mgn. N

But they have the fame ratio to this fame mgn. N (Hyp.) ; 2. Therefore the mgn. M is to the mgn. 1 M.

P. 8. B. 5.

Thefis.

1.

T

If not, the two mgns. M & 1 M are unequal.

HEN the fame mgn. N has not the fame ratio to the two mgns. M & 1 M.

But it has the fame ratio to those two mgns. (Hyp.).

2. Therefore the mgn. M is to the

mgn. I M.

Which was to be demonftrated.

P. 8. B. 5.

PROPOSITION X. THEOREM. X.

THAT magnitude (M) which has a greater ratio than another (P) has

unto the fame magnitude (N) is the greater of the two, and that magnitude (P) to which the fame (N) has a greater ratio than it has unto another magnitude (M) is the leffer of the two.

Hypothefis.

M. Nis > P: N.

Thefis.
The mgn. Mis > P.

1.

THE

CASE I. If M be = P.

HEN the mgns. M & P have the fame ratio to the fame mgn.N. P. But they have not the fame ratio to the fame mgn. N (Hyp.); 2. Therefore the mgn. M is not

to the mgn. P.

But the ratio M : N is not < the ratio P: N (Hyp.);

4. Therefore the mgn. M is not < the mgn. P.

But neither is the mgn. MP (Arg. 2),

5. It remains then that M be > P.

Hypothefis.

N: PN: M.

DEMONSTRATION.
II.

The

Thefis. mgn.

P is < M.

THE

HE ratio N: M would be to the ratio of N: P

2. Which being contrary to the Hypothefis, P cannot be = M.

3.

ČASE II. If P be > M.
>

THE ratio N: M would be the ratio N: P.

4. Which being alfo contrary to the Hypothefis, P cannot be > M. But neither is PM. (Arg. 2.);

3. Therefore P is < M.

Which was to be demonftrated.

P. 7. B. 5.

P. 8. B. 5.

RATIOS

ATIOS (A: B & E: F) that are equal to a fame third ratio (C: D), are equal to one another.

Hypothefis.
SA: B

The ratios

E: F

& are to the fame ratio C: D.

Preparation.

1. Take any equimultiples a A, a C, a E of the three ante-
cedents A, C, E.

2. And any equimultiples c B, c D, c F of the three confe-
quents B, D, F.

BECAUS

DEMONSTRATION.

ECAUSE A B C D (Hyp.),

Thefis.

A: BE: F.

1. If the multiple a A be >, or the multiple c B, the equimultiple a C is likewife >, = or < the equimultiple C D

In like manner fince C D E F (Hyp.)

:

2. If the multiple a C be >, or < the multiple c D, the equimultiple a E will be likewife >, or<the equimultiple c F.

3. Confequently if the multiple a A be >, or<, the multiple c B; the equimultiple a E is likewife >, = or < the equimultiple c F. 4. Confequently, A BE F

Which was to be demonftrated.

Pof. 1. B. 5.

D. 5. B. 5.

D. 5. B. 5.

IF

PROPOSITION XII. THEOREM XII.

any number of magnitudes (A, B, C, D, E, F, &c) be proportionals. The fum of all the antecedents (A+C+E &c) is to the fum of all the confequents (B+D+F &c), as one of the antecedents is to its confequent.

Hypothefis.

The mgns. A, B, C, D, E, F are proportionals

:

or A B C D E F &c.

Preparation.

'Thefis.

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

1. Take of the antecedents A, C, E the equimultiples m A,"
m C, mE

2. And of the confequents B, D, F the equimultiples # B,
n D, n F

Pof. 1. B. 5.

DEMONSTRATION,

SINCE

INCE then ABC: D=E:F (Hyp.) ;

1. If m A be >, = or < n B, likewife m C is >, = < n D; & m E

is, or nF

=

Therefore adding on both fides the mgns. >,, or <.

2. The mgns. A + mC+m E will be conftantly >,, or < the
mgns. B+D+n F according as m A is >,, or < n B.
But the mgns. m A+ mC+mE & m A are equimultiples of the
mgns. AC+ E & A (Prep. 1 & P. 1. B. 5.); alfo the mgns.
n B + n D + n F & n B are equimultiples of the mgns. B+D+
F & B (Prep. 2 & P. 1. B. 5.);

3. Confequently A+ C+E:B+D+FA: B

Which was to be demonftrated.

D. 5. B. 5.

D. 5. B. 5.