II.

Those magnitudes of which the fame, or equal magnitudes, are equimultiples, are equal to one another.

III.

A multiple of a greater magnitude is greater than the fame multiple of a less.

IV.

That magnitude of which a multiple is greater than the fame multiple of another, is greater than that other magnitude.

PROP. I. THEOR.

If any number of magnitudes be equimultiples of as many, each of each; what multiple foever any one of them is of its part, the same multiple shall all the first magnitudes be of all the other.

Let any number of magnitudes AB, CD be equimultiples of as many others E, F, each of each; whatsoever multiple AB is of E, the same multiple shall AB and CD together be of E and F together.

Becaute AB is the fame multiple of E that CD is of F, as many magnitudes as are in AB equal to E, so many are there in CD equal to F. Divide AB into magnitudes equal to E, viz. AG, GB; and CD in- A to CH, HD equal each of them to F: The number therefore of the magnitudes CH, HD shall

be equal to the number of the others AG, G

GB: And because AG is equal to E, and CH to F, therefore AG and CH together are equal to E and F together: For the fame reafon, because GB is equal to E, and HD to F; GB and HD together are equal to E and F together. Wherefore, as many magnitudes as

B

C

E

are in AB equal to E, so many are there in H. F

AB, CD together equal to E and F together.
Therefore, whatsoever multiple AB is of E,
the fame multiple is AB and CD together of D
E and F together.

Therefore, if any magnitudes, how many foever, be equimultiples of as many, each of each, whatsoever multiple any one of them is of its part, the fame multiple shall all the firft magnitudes be of all the other: For the fame demonftration

۱

holds

Book. V.

a Ax. 2. 1.

Book V. holds in any number of magnitudes, which was here applied ' to two.' Q. E. D.

PROP. II. THEOR.

IF the first magnitude be the fame multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the fixth is of the fourth; then shall the first together with the fifth be the fame multiple of the second, that the third together with the fixth is of the fourth,

Let AB the first be the same multiple of C the second, that
DE the third is of F the fourth; and BG the fifth the fame
multiple of C the second, that EH
the fixth is of F the fourth: Then

is AG the first together with the fifth
the fame multiple of C the second,
that DH the third together with the
fixth is of F the fourth.

Because AB is the fame multiple
of C, that DE is of F; there are as
many magnitudes in AB equal to C,

B

A

D

E

GCH

1

F

manner, as many as there are in BG

equal to C, so many are there țin EH equal to F: As many then as are in the whole AG equal to C, so many are there in the whole DH equal to F: Therefore AG is the same multiple of C, that DH is of F; that is, AG the first and fifth tom

gether, is the fame multiple of the fecond

C. that DH the third and fixth together is

D

of the fourth F. If, therefore, the first be

the same multiple, &c. Q. E. D.

PROP. III. THEOR.

If the first be the same multiple of the second, which
the third is of the fourth; and if of the first and
third there be taken equimultiples, these shall be equi-
multiples the one of the second, and the other of the
fourth.

Let A the first be the same multiple of B the second, that C the third is of D the fourth; and of A, C let the equimultiples EF, GH be taken: Then EF is the fame multiple of B, that GH is of D.

Because EF is the fame multiple of A, that GH is of C, there are as many magnitudes in EF equal to A, as are in GH

equal to C: Let EF be di

vided into the magnitudes F

H

1

EK, KF, each equal to A,

and GH into GL, LH,

Book V.

cause A is the same multi

ple of B, that C is of D,

and that EK is equal to A,

and GL to C; therefore

EK is the fame multiple of E ABGCD

B, that GL is of D: For

the fame reason, KF is the fame multiple of B, that LH is of
D; and so, if there be more parts in FF, GH equal to A, C:
Because, therefore, the first EK is the fame multiple of the fe-
cond B, which the third GL is of the fourth D, and that the fifth
KF is the fame multiple of the second B, which the fixth LH
is of the fourth D; EF the first together with the fifth is the
same multiple of the second B, which GH the third together 2 2. S.
with the fixth is of the fourth D. If, therefore, the first, &c.
Q. E. D.

PROP.

Book V.

Sec N.

PROP. IV. THEOR.

F the first of four magnitudes has the same ratio to the fecond which the third has to the fourth; then any equimultiples whatever of the first and third shall have the fame ratio to any equimultiples of the second and fourth, viz. ' the equimultiple of the first shall have ⚫ the fame ratio to that of the fecond, which the equi• multiple of the third has to that of the fourth."

Let A the first have to B the second, the same ratio which the third C has to the fourth D; and of A and C let there be taken any equimultiples whatever E, F; and of B and D any equimultiples whatever G, H: Then E has the fame ratio to G, which F has to H.

Take of E and F any equimultiples whatever K, L, and of G, H, any equimultiples whatever M, N: Then, because E is the fame multiple of A, that F is of C; and of E and F have been taken

23.5.

equimultiples K, L; therefore K

is the fame multiple of A, that LKEABGM

is of C: For the fame reason, M

is the fame multiple of B, that N

is of D: And because as A is to b Hypoth. B, so is C to Db, and of A and C have been taken certain equimultiples K, L; and of Band D have been taken certain equimultiples M, N; if therefore K be greater than M, L is greater than N; and if equal, equal; if less, cs. def. 5. less. And K, L are any equimultiples whatever of E, F; and M, Nan any whatever of G, H: As therefore E is to G, so is F to H. Therefore, if the first, &c. Q. E. D.

LFCDHN

Cor. Likewise, if the first has the same ratio to the second, which the third has to the fourth, then also any equimulti

ples

ples whatever of the first and third have the fame ratio to the Book V. fecond and fourth: And in like manner, the first and the third have the fame ratio to any equimultiples whatever of the second and fourth.

Let A the first have to B the second, the fame ratio which the third C has to the fourth D, and of A and C let E and F be any equimultiples whatever; then E is to B, as F to D.

Take of E, F any equimultiples whatever K, L, and of B, | D any equimultiples whatever G, H; then it may be demonftrated, as before, that K is the same multiple of A, that L is of C: And because A is to B, as C is to D, and of A and C certain equimultiples have been taken, viz. K and L; and of B and D certain equimultiples G, H; therefore, if K be greater than G, L is greater than H; and if equal, equal; if less, less : cs. def. sa And K, L are any equimultiples of E, F, and G, H any whatever of B, D; as therefore E is to B, so is F to D: And in the fame way the other cafe is demonstrated.

PROP. V. THEOR.

F one magnitude be the fame multiple of another, see N. which a magnitude taken from the first is of a magnitude taken from the other; the remainder shall be the fame multiple of the remainder, that the whole is of the whole.

Let the magnitude AB be the fame multiple G

of CD, that AE taken from the first, is CF taken from the other; the remainder EB shall be the fame multiple of the remainder FD, that the whole AB is of the whole CD.

A

Take AG the fame multiple of FD, that AE is of CF: Therefore AE is the same multiple of CF, that EG is of CD: But AE, by the hypothesis, is the same multiple of CF, that AB is of CD: Therefore EG is the fame mul. E

tiple of CD that AB is of CD; wherefore EG is equal to AB: Take from them the common

magnitude AE; the remainder AG is equal to the remainder EB. Wherefore, since AE is

BD

the fame multiple of CF, that AG is of FD,

and that AG is equal to EB; therefore AE is the fame multiple

of CF, that EB is of FD: But AL is the fame multiple of CF,