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K, but H not greater than L: Also whatever multiple G is of C, take M the same multiple of A; and whatever

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multiple K is of D, take N the same multiple of B: Then, because A is to B as C to D, and M and G are equimultiples of A and C, and N and K of B and D, therefore 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: And H is not greater than L; and M, H are equimultiples of A, E, and N, L equimultiples of B, F; therefore (5. Def. 7) A has a greater ratio to B than E has to F.

Wherefore, If the first &c. Q.E.D.

COR. Also, 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 shewn, in like manner, that the first has a greater ratio to the second than the fifth has to the sixth.

PROP. XIV. THEOR.

If the first has the same ratio to the second which the third

has to the fourth, then, if the first be greater than the third, the second shall be greater than the fourth, and if equal, equal, and if less, less.

Let the first A have the same ratio to the second B which the third C has to the fourth D: then, if A be greater than C, B shall be greater than D, if equal, equal, and if less, less.

First, let A be greater than C: Then, since A is

greater than C, and B any other magnitude, therefore A has to B a greater ratio than C to B (5.8): But, as A is to B so is C to D; therefore also C has to Da greater ratio than C has to B (5. 13): But, of two magnitudes, that to which the same has the greater ratio is the lesser (5. 10); therefore D is less than B, that is, B is greater than D.

Next, let A be equal

to C: Then, A is to B as C, that is, A, is to D; therefore (5. 9) B is equal to D.

Lastly, let A be less

than C: Then C is

A B C D A B C D

A B C D

greater than A; and because C is to D as A is to B, and that C is greater than A, therefore, by the first D is greater than B, that is, B is less than D. Wherefore, If the first &c. Q. E.D.

case,

PROP. XV. THEOR.

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

Let AB and DE be equimultiples of C and F: C shall be to F as AB to DE.

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For, 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: Divide AB 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 magnitudes AG, GH, HB, will be equal to the number of the magnitudes, DK, KL, LE: And because AG,

GH, HB are all equal, and that DK, KL, LE are also equal to one another, therefore AG is to DK as GH to KL, and as HB to LE: But as one of the antecedents is to its consequent, so are all the antecedents together to all the consequents together (5. 12); therefore, as AG is to DK so is AB to DE: But AG is equal to C, and DK to F; therefore, as C is to F so is AB to DE.

Wherefore, Magnitudes &c.

Q. E. D.

PROP. XVI. THEOR.

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

Let the four magnitudes A, B, C, D be proportionals, that is, let A be to B as C to D: they shall also be proportionals when taken alternately, that is, A shall be to C as B to D.

Take of A, B, any equimultiples whatever E, F, and of C, D, any equimultiples whatever G, H: Then, because E is the same multiple of A that F is of B, and that magnitudes have the same ratio to one another which their equimultiples have (5. 15) therefore A is to B as E is to F: But A is to B as C is to D; therefore C is to D as E is to F (5. 11): Again, because G, H are equimultiples of B C, D, therefore C is to D

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A

F

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C

D

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as G is to H: But C is to D as E is to F; therefore E is to F as G to H: But when four magnitudes are proportionals, if the first be greater than the third, the second shall be greater than the fourth, and if equal, equal, and if less, less (5. 14); therefore, if E be greater than G, F is greater than H, and if equal, equal, and if less, less: But E, F, are any equimultiples whatever

of A, B, and G, H any whatever of C, D; therefore A is to C as B is to D.

Wherefore, If four magnitudes &c. Q. E. D.

COR. Hence, by the help of (5. 14), it follows that, if four magnitudes be proportionals, then if the first be greater than the second, the third will be greater than the fourth, and if equal, equal, and if less, less.

PROP. XVII. THEOR.

If four magnitudes, taken jointly, be proportionals, they shall 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 shall have to the other the same ratio which the remaining one of the last two has to the other of these.

Let AB, BE, CD, DF be four magnitudes which, taken jointly, are proportionals, that is, let AB be to BE as CD to DF: they shall also be proportionals when taken separately, that is, AE shall be to EB as CF to FD.

Take of AE, EB, CF, FD, any equimultiples whatever GH, HK, LM, MN, and, again, of EB, FD, take any equimultiples whatever KX, NP: Then, 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 (5.1): But GH is the same multiple of AE that LM is of CF; therefore 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 is the same multiple of CF that LN is of CD: But LM was shewn to be the same multiple of CF that GK is of AB; therefore GK is the same multiple of AB that LN is of CD, that

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is, GK, LN, are equimultiples of AB, CD: Again, because HK is the same multiple of EB that MN is of FD, and that 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 (5.2), that is, HX, MP, are equimultiples of EB, FD: And because AB is to BE as CD is to DF, and that GK, LN, are equimultiples of AB, CD, and HX, MP, are equimultiples of EB, FD, therefore, if GK be greater than HX, LN is greater than MP, and if equal, equal, and if less, less: But if GH be greater than KX, then, by adding HK to both, GK is greater than HX; therefore also LN is greater than MP, and, by taking away MN from both, LM is greater than NP; that is, if GH be greater than KX, LM is greater than NP: And, in like manner, it may be shewn that, if GH be equal to KX, LM is equal to NP, and if less, less: But GH, LM, are any equimultiples whatever of AE, CF, and KX, NP are any whatever of EB, FD; therefore AE is to EB as CF to FD.

Wherefore, If four magnitudes &c. Q. E.D.

PROP. XVIII. THEOR.

If four magnitudes, taken separately, be proportionals, they shall 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 shall be to the second, as the third and fourth together to the fourth.

Let the four magnitudes AE, EB, CF, FD be proportionals, that is, let AE be to EB as CF to FD: they shall also be proportionals, when taken jointly, that is, AB shall be to BE as CD to DF.

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