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Book. V.

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

III.
A multiple of a greater magnitude is greater than the same
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.

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IF any number of magnitudes be equimultiples of as

many, each of each ; what multiple soever any one of them is of its part, the same multiple shall all the first magnitudes be of all the other.

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a Ax. 2. 1.

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.

Because AB is the fame multiple of Ethat 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 20 CH, HD equal each of them to F: The number therefore of the magnitudes CH, HD shall E 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 same rea-
fon, 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
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 same multiple is AB and CD together of D
E and F together.

Therefore, if any magnitudes, how many soever, be equimultiples of as many, each of each, whatsoever multiple any one of them is of its part, the same multiple hall all the first magnitudes be of all the other : · For the same demonstration

· holds

Book V.

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

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IF
F the first magnitude be the same multiple of the se-

cond that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth ; then shall the first together with the fifth be the same 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 same multiple of the second, that EH

D
the sixth is of F the fourth : Then A
is AG the first together with the fifth
the same multiple of C the second,
that DH the third together with the

E
fixth is of F the fourth.

B
Because AB is the same multiple
of C, that DE is of F; there are as
many magnitudes in AB equal to C,
as there are in DE equal to F : In like G
manner, as many as there are in BG
equal to C, so many are there fin EH equal to F: As many
then as are in the whole AG equal to C, lo many are there in
the whole DH equal to F: Therefore AG is the same multi-
ple of C, that DH is of F; that is, AG the first and fifth toa
gether, is the same multiple of the second
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.

A.

E

dH

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Book v.

PROP. III.

THEOR.

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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 equimaltiples the one of the second, and the other of the fourth.

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

Because EF is the same 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 EK, KF, each equal to A, and GH into GL, LH, each equal to C:The number therefore of the magnisudes EK, KF, shall be equal to the number of the

L+ others GL, LH: And becaufe A is the same multiple of B, that C is of D, and that EK is equal to A, and GL to C; therefore EK is the same multiple of

E A B G C D B, ihat GL is of D: For the same reason, KF is the same 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 same multiple of the fee cond B, which the third GL is of the fourth D, and that the fifth KF is the same multiple of the second B, which the fixth LH is of the fourth D; EF the first together with the fifth is the fame multiple of the second B, which GH the third together a 2. s. with the fixth is of the fourth D. If, therefore, the first, &c. Q. E. D.

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Book V.

PRO P. IV.

THE O R.

Sec N.

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F the first of four magnitudes has the same ratio to

the second which the third has to the fourth ; then any equimultiples whatever of the first and third fhall 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 second, which the equi.

multiple of the third has to that of the fourth.'

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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 same 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 same multiple of A, that F is of C; and of E and F have been taken equimultiples K, L; therefore K

is the same multiple of A, that L K E A B GM a 3.5.

is of Ca: For the same reason, M
is the same multiple of B, that N

L F C DH N is of D: And because as A is to b Hypoth. B, fo is C to Db, and of A and

C have been taken certain equimultiples K, L; and of B and D have been taken certain equimultiples M, N; if therefore K be greater than M, L is greater than

Ñ; and if equal, equal; if less,
E S. def. 5. less. And K, L fare any equi-

multiples whatever of E, É; and
M, N any whatever of G, H:
As therefore E is to G, fo is CF
to H. Therefore, if the first, &c.
Q. E. D.

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

ples

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

Le A the first have to B the second, the same ratio which the third C bas 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 demonstrated, 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 same way the other case is demonstrated.

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F one magnitude be the fame multiple of another, scc 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.

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

A Take AG the same multiple of FD, that

3 I. S. AE is of CF: Therefore AE is a the same mul. tiple 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 same mul.

E tiple of CD that AB is of CD; wherefore EG

F is equal to AB6: Take from them the common

br. AX. so magnitude AE ; the remainder AG is equal to the remainder EB. Wherefore, fince AE is B D the same multiple of CF, that AG is of FD,

and that AG is equal to EB; therefore AE is the same multiple E of CF, that EB is of FD ; But AL is the same multiple of CF

that

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