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Book V.

b 5. def. 5.

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D, and of A and C, M and G are equimultiples: and of B and D, N and K are equimultiples; if M be greater than N, G is greater than K, and if equal, equal; and if less, less; but G is greater than K, therefore M is greater than N: But H is not greater than L; and M, H are equimultiples of A, E; and N, equimultiples of B, F: Therefore, A has a greater ratio to c 7. def. 5. B than E has to F. Wherefore, " if the first, &c. Q. E. D. COR. And if the first have a greater ratio to the second than the third has to the fourth, but the third the same ratio to the fourth which the fifth has to the sixth; it may be demonstrated in like manner, that the first has a greater ratio to the second than the fifth has to the sixth.

See N.

a 8. 5.

PROP. XIV. THEOR.

If the first has to the second, the same ratio which the third has to the fourth; then, if the first be greater than the third, the second will be greater than the fourth; and if equal, equal; and if less, less.

Let the first A, have to the second B, the same ratio which the third C has to the fourth D; then, if A be greater than C, B will also be greater than D.

Because A is greater than C, and B is any other magnitude, A has to B a greater ratio than C to Ba But as A is to B, so

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b 13. 5. c 10. 5.

d 9. 5.

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is C to D: therefore also C has to D a greater ratio than C has to B. But of two magnitudes, that to which the same has the greater ratio is the less. Wherefore, D is less than B; that is, B is greater than D.

Secondly, If A be equal to C, B will be equal to D: For A is to B as C (that is A) to D; B therefore is equal to Da. Thirdly, If A be less than C, B will be less than D: For C is greater than A, and because C is to D, as A is B, D is greater than B, by the first case; wherefore, B is less than D. Therefore, "if the first," &c. Q. E. D.

Book V.

PROP. XV. THEOR.

Magnitudes have the same ratio to one another which their equimultiples have.

Let AB be the same multiple of C, that DE is of F; C is to F, as AB to DE.

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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 magnitudes in DE equal to F: Let AB be divided into magnitudes, each equal to C, viz. AG, GH, HB; and DE into magnitudes each equal to F, viz. DK, KL, LE: Then the number of the first AG, GH, HB will be equal to the number of the last DK, KL, LE: And because AG, GH, HB are all equal, and DK, KL, LE, are also equal to one another; therefore, AG is to DK, as GH to

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KL, and as HB to LE: And as one of the antecedents to a 7. 5. its consequent, so are all the antecedents together to all the

consequents together; wherefore, as AG is to DK, so is AB b 12. 5. to DE: But AG is equal to C, and DK to F: Therefore, as C is to F, so is AB to DE. Therefore, "magnitudes," &c. Q. E. D.

PROP. XVI. THEOR.

If four magnitudes of the same kind be proportionals, they will also be proportionals when taken alternately.

Let the four magnitudes A, B, C, D, be proportionals, viz. as A is to B, so is C to D: They shall also be proportionals when taken alternately; that is, A is to C, as B to D.

For, of A and B take any equimultiples whatever E and F; and of C and D take any equimultiples whatever G and H: and

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Book V. because E is the same multiple of A, that F is of B, and because magnitudes have the same ratio to one another which their equimultiples have ; therefore, A is to B, as E is to F: But

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as A is to B, so is C

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D, so is E to F.

Wherefore, as

E to F: Again, because G, H are equimultiples of C, D, as C is to D, so is G to H; but as C is to E is to F, so is G to Hb. But when four magnitudes are proportionals, if the first be greater than the third, the second will be greater than the fourth; and if equal, equal; if c 14. 5. less, less. Wherefore, if E be greater than G, F likewise

is greater than H; and if equal, equal; and if less, less; and E, F are any equimultiples whatever of A, B; and G, H any d def. 5. whatever of C, D. Therefore, A is to C as B to Dd. • If, then, four magnitudes," &c. Q. E. D.

See N.

a 1. 5.

PROP. XVII. THEOR.

If magnitudes, taken jointly, be proportionals, they will also be proportionals when taken separately; that is, if two magnitudes together have to one of them the same ratio which two others have to one of these, the remaining one of the first two will have to the other the same ratio which the remaining one of the latter two has to the other of these.

Let AB, BE, CD, DF be the magnitudes taken jointly which are proportionals; that is, as AB to BE, so is CD to DF; they will also be proportionals taken separately, viz. as AE to EB, so is CF to FD.

For, of AE, EB, CF, FD take any equimultiples whatever GH, HK, LM, MN, and again of EB, FD take any equimultiples whatever KX, NP: And because GH is the same multiple of AE, that HK is of EB, therefore GH is the same multiple of AE, that GK is of AB: But GH is the same multiple of AE that LM is of CF; wherefore, GK is the same multiple of AB, that LM is of CF: Again, because LM is the same multiple of CF, that MN is of FD; therefore, LM

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is the same multiple of CF, that LN is of CD: But LM was Book V. shown to be the same multiple of CF, that GK is of AB; GK, therefore, is the same multiple of AB, that LN is of a 1. 5. CD; that is, GK, LN are equimultiples of AB, CD. Next, because HK is the same multiple of EB, that MN is of FD; and KX is also the same multiple of EB, that NP is of FD; therefore, HX is the same multiple of EB, that MP is of FD. And because AB is to BE, as CD is to DF, and because of AB and CD, GK and LN are equimultiples, and of EB and FD, HX and MP are equi multiples; if GK be greater than HX, then LN is greater than MP; and if equal, equal; and if less, less: But if GH be greater than KX, by adding the common part HK to both, GK is greater than HX; wherefore also, LN is greater than MP; and by taking away MN from both, LM is greater than NP: Therefore, if GH be greater than KX, LM is greater than NP. In like manner, it may be demonstrated, that if GH be equal to KX, LM likewise is equal to NP; and if less, less: And GH, LM are any equimultiples whatever of AE, CF, and KX, NP are any equimultiples whatever of EB, FD. Therefore, as AE is to EB, so is CF to FD. If, then, magnitudes," &c. Q. E. D.

GAC C L

PROP. XVIII. THEOR.

If magnitudes, taken separately, be proportionals, See N. they will also be proportionals when taken jointly; that is, if the first be to the second, as the third to the fourth, the first and second together will be to the second, as the third and fourth together to the fourth.

Let AE, EB, CF, FD be proportionals; that is, as AE to EB, so is CF to FD: they will also be proportionals, when taken jointly; that is, as AB to BE, so is CD to DF.

For, of AB, BE, CD, DF take any equimultiples whatever GH, HK, LM, MN: and again of BE, DF take any equimultiples whatever KO, NP: And because KO, NP are equi

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Book V. multiples of BE, DF; and KH, NM are also equimultiples
of BE, DF; if KO, the multiple of BE, be greater than KH,
which is a multiple of the same BE, NP, likewise the multiple
of DF, will be greater than MN, a
multiple of the same DF; and if KO

be equal to KH, NP will be equal to
NM; and if less, less.

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First, Let KO not be greater than KH, therefore, NP is not greater than NM: And because GH, HK are equimultiples of AB, BE, and AB is greater than BE; therefore, a 3. Ax.5. GH is greater a than HK ; but KO is not greater than KH, wherefore, GH is greater than KO. In like manner, it may be shown, that LM is greater than NP. Therefore, if KO be not greater than KH, then GH, the multiple of AB, is always greater than KO, the multiple of BE; and likewise LM, the multiple of CD, is greater than NP, the multiple of DF.

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c 6. 5.

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Next, Let KO be greater than KH: therefore, as has been shown, NP is greater than NM: and because the whole GH is the same multiple of the whole AB, that HK is of BE, the remainder GK is the same multiple

of the remainder AE that GH is of

ABb: which is the same that LM is

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of CD. In like manner, because LM H
is the same multiple of CD, that MN
is of DF, the remainder LN is the
same multiple of the remainder CF,
that the whole LM is of the whole CD: K
But it was shown, that LM is the same
multiple of CD, that GK is of AE;
therefore GK is the same multiple of
AE, that LN is of CF; that is, GK,
LN are equimultiples of AE, CF:
And because KO, NP are equimulti-
ples of BE, DF, if from KO, NP there
be taken KH, NM, which are likewise equimultiples of BE,
DF, the remainders HO, MP are either equal to BE, DF,
or equimultiples of them. First, let HO, MP, be equal to
BE, DF; and because AE is to EB, as CF to FD, and that

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