POSTULATES.

I.

ET it be granted, that any magnitude may be doubled, tripled, qua drupled, or in general, that any multiple of it may be taken.

II.

That from a greater magnitude, there may be taken one or several parts equal to a lets magnitude of the fame kind.

PROPOSITION I. THEOREM I

F any number of magnitudes (a M, a N, a O &c) be equimultiples of as many (M, N, O &c) each of each, the fum (a M+aN+a O&c) of all the first is the fame multiple of the fum (M+N+O &c) of all the second, as any one of the first fa M) is of its part (M).

Hypothefis.

are

equimultiples N of
of to each

Thefis.

a MaNaO is the fame multiple of M+N+O that a M is of M, or a N of N-&c.

Preparation,

The mgn. a M being the fame multiple of M, that a N is of N (Hyp), as many magnitudes A, B, C, &c. as can be taken out of a M each equal to M, fo many X,Y,Z, &c. can be taken out of a N, each equal to N.

A be each

Let then Bequal to M &

BECAUSE

C

DEMONSTRATION.

ECAUSE a M is the fame multiple of M, that a N is of N (Hyp), 1. As many magnitudes X, Y, Z, &c. as are in a N each equal to N, so many A, B, C, &c. are there in a M each equal to M.

But

A=M & X=N (Prep.),

A+ XM+N

B being M& Y=N (Prep.),

B+Y=M+N

C = M&Z=N (Prep.),
C+ZM+N

Confequently there is in a M as many Magnitudes

are in a Ma NM+N.

Pof. 2. B.5.

Ax. 2. B. 1.

Ax. 2. B. 1.

Ax. 2. B. 1.

M, as there

5. From whence it follows that a MaN is the fame multiple of

M+N, that a M is of M, or that a N is of N, & likewise a M+aN

a

O is the fame multiple of M+N+O, that a M is of M or aN of N, &c.

Which was to be demonstrated

PROPOSITION II. THEOREM. II.

IF the firft. magnitude (a M) be the fame multiple of the fecond (M), that

the third (a N) is of the fourth (N); & the fifth (cM) the fame multiple of the fecond (M), that the fixth (CN) is of the fourth (N); then shall the first together with the fifth (a McM) be the fame multiple of the second (M), that the third together with the sixth (a N+cN) is of the fourth (N),

BECAUSE

ECAUSE M is the fame multiple of M, that N is of N (Hyp.),

1. There are as many magnitudes in a M to M as there are in a N = to N.

In like manner, because c M is the fame multiple of M, that eN is of N (Hyp.),

2. There are as many magnitudes in Mto M as there are in N = to N.

3. Confequently, as many as are in the whole a M + c M equal to M, fo many are there in the whole a N +cN to N.

4. Therefore a McM is the fame multiple of M that @N+cN is of N.

Which was to be demonstrated.

Ax. 2. B. 1.

IF

PROPOSITION III.

THEOREM III.

F the firft magnitude (a M) be the fame multiple of the second M, that the third (a N) is of the fourth ( N), and if of the firft (a M) and third (a N) there be taken equimultiples (e a M, e a N); thefe (e a M, e a N) fhall be equimultiples, the one of the fecond (M) and the other of the fourth (N).

1.

Hypothefis.

11. ea M ・e

equimultiples
of

are two Sa M each equimultiples & of

fa N

of

a N each

Preparation.

Thefis.

ea M is the fame _multiple of M that e a N is of N.

Divide a M into its parts 1 a M, a 1 M, &c. each a M,
And e a N into its parts 1 a N, a 1 N, &c. each = a N.
DEMONSTRATION,

BECAUSE

ECAUSE ea M is the fame multiple of a M, that e a N is of aN (Hyp. 2.).

1. There are as many magnitudes in e a M

are

in e a N

to a M as there

to a N 2. Therefore the number of parts 1 a M, a 1 M, &c. in e a M, is = the number of parts 1 a N, a 1N, &c. in ea N.

to

But because a M is the fame multiple of M, that a N is of N, and
that i a Ma M, 1 a Na N.

3. The magnitude 1 a M is the fame multiple of M, that 1 a N is of N.
4. In like manner a 1 M is the fame multiple of M, that a 1 N is of N.
Since then I mgn. I a M is the fame multiple of the II mgn. M.
that the III mgn. I a N is
of the IV mgn. N

& that the V mgn. a 1 M is the fame multiple of the II mgn. M
that the VI mgn. a 1 N is
of the IV mgn. N.
5. It follows that the magnitude ea M, compofed of the I & V mgn.
I a M+ai M, is the fame multiple of the II mgn. M, that the mgn.
e a N, compofed of the III & VI mgn. 1 a N+a 1 N is of the IV
mign. N.

Which was to be demonftræted

P. 2. B.

PROPOSITION IV.

THEOREM IV.

IF four magnitudes (M, N, O, P,) are proportional: then any equimultiples

(a M, a O) of the first (M) and third (O), fhall have the fame ratio to any equimultiples (N, c P) of the fecond (N) and fourth (P).

Hypothefis.

C

I. M: NO: P.

(a M) are

11. & equimult. & alfo & O of Ο c P

I. Take

SM CN

}

are (N

equimult. &.

of Preparation.

Thefis.

aMcNaO : c P.

of a M & of a O any equimult. Ra M, Ra O

2. Likewife of c N & of c P any equimult. Sc N, Sc P

DEMONSTRATION.

} Pof. 1. B. 5.

BECAUSE & M is the fame mult. of M, that a Ois of O (Hyp. 2),

a

& the mgns. Ra M, Ra O are equimult, of the mgns. aM, aO (Prep. 1). 1. The magnitude Ra M is the fame multiple of M, that the magnitude Ra O is of O.

2. In like manner, the magnitude S c N is the fame multiple of N that S c P is of P.

P. 3. B. 5.

And as M: NO:P (Hyp. 1.) & Ra M, Ra O are any equimultiples of the I term M and of the III O; and Se N, SCP any equimultiples of the II term N and of the IV P (Arg. 1 & 2), 3. If Ra M be >, or < Sc N, Ra O will be >, or <S cP. D. 5. But the magnitudes Ra M & R a O are any equimultiples of the magnitudes a M & a O, and the magnitudes Sc N, SCP are any equimultiples of the magnitudes c N& cP (Prep. i & 2).

4. Confequently, the ratio, of a M to c N is to the ratio of a O
to c P; or a McNa 0: c P

Which was to be demonftrated.
COROLLARY

B. S

D. 5. B. 5.

IT SNR,LLARAM; be>,

Tis manifeft that if Sc N be> or <Ra M; likewife Sc P will be >, = or < Ra O (Arg. 3.); hence c Na Mc PaQ (D. 5. B. 5.). Therefore, if four magnitudes be proportional, they are also by inverfion or invertendo.