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II.

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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 same multiple of another, is greater than that other magnitude.

PROP. I. THEOR.

If any number of magnitudes be equimultiples of as IF 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.

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 same 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 magni

tudes equal to E, viz. AG, GB; and CD into A
CH, HD equal each of them to F: the num-
ber therefore of the magnitudes CH, HD shall

be equal to the number of the others, AG, GB; GE
and because AG is equal to E, and CH to F,
therefore AG and CH together are equal tó B
(Ax. 2.5.) E and F together: for the same rea-
son, because GB is equal to E: and HD to F;
GB and HD together are equal to E and F to- C
gether. Wherefore, as many magnitudes as

are in AB equal to E, so many are there in HT 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 shall all the first

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magnitudes be of all the other: 'For the same demonstration holds in any number of magnitudes, which was here applied 'to two.' Q. E. D.

PROP. II. THEOR.

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

A

DI

E

cl

F

H

G

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 C the second, that EH the sixth is of F the fourth: then is AG the first, together with the fifth, the same multiple of C the second, that DH the third, together with the sixth., 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 manner, as many as there are in BG equal to C, so many are there in 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 together, is the same multiple of the second C, that DH the third and sixth A| together, is of the fourth F. If, therefore the first be the same multiple, &c. Q. E. D.

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COR. From this it is plain, that, ' if any number of magnitudes AB,

B

' BG, GH, be multiples of another C;
' and as many DE, EK, KL, be the G
'same multiples of F, each of each;
'the whole of the first, viz. AH, is
'the same multiple of C, that the
'whole of the last, viz. DL, is of F.'

H

D

E

K

F

PROP. III. THEOR.

Ir 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 equimultiples, 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 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

H

vided into the magnitudes F EK, KF, each equal to A, and GH into GL, LH, each equal to C: the number therefore of the magnitudes EK, KF, shall be equal to the number of the others K GL, LH: and because 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 B, that 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 EF, GH equal to A, C: because, therefore, the first EK is the same multiple of the second 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 sixth LH is of the fourth D; EF the first together with the fifth, is the same multiple (2. 5.) of the second B, which GH the third, together with the sixth, is of the fourth D. If, therefore, the first, &c. Q. E. D.

E A

B

G

D

PROP. IV. THEOR.

Ir 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 shall have the same ratio to any equimultiples of the second and fourth, viz. the equimultiple of the first shall have the 'same ratio to that of the second, which the equimultiple ' 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 is of C (3.5.); for the same reason, M is the same multiple of B, that N is of D. and because, as A is to B, so is C to D (Hypoth.), 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 then N; and if equal, equal; if less, less (5. def. 5.). And K, L are any equimultiples whatever of E, F; and M, N any whatever of G, H: as therefore E is to G, so is (5. def. .) F to H. Therefore, if the first, &c. Q. F. D.

K E A B G M

L F C DH N

COR. Likewise, if the first have the same ratio to the second, which the third has to the fourth, then also any equimultiples

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whatever of the first and third have the same ratio to the second and fourth: and in like manner, the first and the third have the same ratio to any equimultiples whatever of the second and 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 E and F be any equimultiples whatever; then E is to B, as F to D.

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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 (5. 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, so is F to D: and in the same way the other case is demonstrated.

PROP. V. THEOR.

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If one magnitude be the same multiple of another, which a magnitude taken from the first is of a magnitude taken from the other; the remainder shall be the same multiple of the remainder, that the whole is of the whole.

Let the magnitude AB be the same multi- G ple 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

ET

C

FT

Take AG the same multiple of FD, that AE is of CF: therefore AE is (1. 5.) 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 same multiple of CD that AB is of CD; wherefore EG is equal to AB (1. Ax. 5.). Take from them the common magnitude AE; the remainder AG is equal to the remainder EB. Wherefore, since AE is the same multiple of CF, that AG is of FD, and that AG is equal to EB; therefore AE is the same multiple of CF, that EB is of FD: but

B

D

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