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II.

Thofe 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 lefs.

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 fame 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 fame multiple shall AB and CD together be of E and F together.

G

E

Becaule AB is the fame multiple of E that CD is of F, as many magnitudes as are in AB equal to E, fo 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 fhall be equal to the number of the others AG, GB: And because AG is equal to E, and CH to F, therefore AG and CH together are B 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 are in AB equal to E, fo many are there in H 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 E and F together.

D

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

' holds

Book. V.

a Ax. 2. I.

Book V.

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

PRO P. II. THE OR.

IF the first magnitude be the fame multiple of the se

cond that the third is of the fourth, and the fifth the fame multiple of the fecond that the fixth is of the fourth; then fhall the first together with the fifth be the fame multiple of the fecond, that the third together with the fixth is of the fourth,

A

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

:

manner, as many as there are in BG

B

G

D

E

H

equal to C, fo many are there in EH equal to F: As many
then as are in the whole AG equal to C, fo many are there in
the whole DH equal to F: Therefore AG is the fame multi-
ple of C, that DH is of F; that is, AG the firft and fifth to-
gether, is the fame multiple of the fecond
C. that DH the third and fixth together is
of the fourth F. If, therefore, the first be
the fame multiple, &c. Q. E. D.

COR. From this it is plain, that if any number of magnitudes AB, BG, GH, be multiples of another C; and as many DE, EK, KL be the fame multiples of F, each of each; the whole of the first, viz. AH, is the fame multiple of C, that the whole of the laft, viz. DL, is of F.'

D

A.

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Book V.

I'

PROP. III. THEOR.

[F the first be the fame multiple of the second, which the third is of the fourth; and if of the first and third there be taken equimultiples, thefe fhall be equimultiples the one of the fecond, and the other of the fourth.

Let A the firft be the fame multiple of B the fecond, 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

K

vided into the magnitudes F
EK, KF, each equal to A,
and GH into GL, LH,
each equal to C: The num-
ber therefore of the magni-
tudes EK, KF, fhall be e-
qual to the number of the
others GL, LH: And be-
caufe A is the fame 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

B, that GL is of D: For

L

H

E

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

a

PROP.

Book V.

Sec N.

a 3.5.

IF

PROP. IV. THEOR.

F the first of four magnitudes has the fame ratio to the fecond 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 firft fhall 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 fecond, the fame 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 equi-
multiples whatever G, H: Then
E has the fame ratio to G, which
F has to H.

Take of E and F any equimul-
tiples 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
equimultiples K, L; therefore K
is the fame multiple of A, that LKE
is of C: For the fame reason, M

ABG M

is the fame multiple of B, that NL F C D H N

is of D: And because as A is to

b Hypoth. B, fo is C to Db, and of A and

Chave 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 N; and if equal, equal; if lefs, es. def. 5. lefs. And K, L fare any equimultiples whatever of E, F, and M, N any whatever of G, H: As therefore E is to G, fo is F to H. Therefore, if the firft, &c. Q. E. D.

COR. Likewife, if the firft has the fame ratio to the fecond, which the third has to the fourth, then alfo any equimulti

ples

ples whatever of the firft 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 fecond, 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 fame 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 lefs, lefs: e s. def. 5. And K, L are any equimultiples of E, F, and G, H any whatever of B, D; as therefore E is to B, fo is F to D: And in the fame way the other cafe is demonstrated.

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c

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

Let the magnitude AB be the fame multiple of CD, that AE taken from the first, is of 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.

G

A

E

Take AG the fame multiple of FD, that AE is of CF: Therefore AE is the fame multiple of CF, that EG is of CD: But AE, by the hypothefis, is the fame multiple of CF, that AB is of CD: Therefore EG is the fame multiple of CD that AB is of CD; wherefore EG is equal to ABb: Take from them the common magnitude AE; the remainder AG is equal to the remainder EB. Wherefore, fince AE is 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 AE is the fame multiple of CF,

a 1. S.

F

br. Ax. 5,

B D

that

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