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As many magnitudes, therefore, as there are in the whole ae equal to c, so many will there be in the whole dh equal to F.

The whole Aē, therefore, is the same multiple of c, as the whole Dh is of F.

Q. E, D,

PRO P. III. THEOREM.

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If the first of four magnitudes be the same multiple of the second as the third is of the fourth ; and if of the first and third there be taken equimultiples, these will also be equimultiples, the one of the second, and the other of the fourth.

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Let A the first, be the same multiple of B the second, as c the third, is of the fourth ; and let EF and Gh be equimultiples of A and c; then will EF be the same multiple of B, that Gh is of D.

For fince EF is the same multiple of A that Gh is of c (by Hyp.), there will be as many magnitudes in Ef equal to A, as there are in gh equal to c.

Divide EF into the magnitudes EK, KF each equal to A (I. 35.); and Gh into the magnitudes GL, LH, each equal to c.

Then

Then will the number of magnitudes EK, KF in the one, be equal to the number of magnitudes GL, Lh in the other.

And because a is the same multiple of B that c is of D (by Hyp.), and Ek is equal to A, and GL to c (by Confi.), EK will be the same multiple of B that GL is of D.

In like manner, since it is equal to A, and lh to co KF will be the same multiple of B, that LĦ is of D.

And since EK, KF are each multiples of B, and GL, LH are each the same multiples of D, the whole ef will be the fame multiple of B, as the whole GH is of D (V. 2.)

Q. E. D.

PRO P. IV. THEORE M.

If the first of three magnitudes be greater than the second, and the third be any magnitude whatever, some equimultiples of the first and second may be taken, and some multiple of the third such, that the former shall be greater than that of the third, but the latter not greater.

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Let AB, Bc be two unequal magnitudes, and D any other magnitude whatever ; then there may be taken fome equimultiples of AB, BC, and some multiple of D such,

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that the multiple of AB shall be greater than that of D, but the multiple of bc not greater.

For of BC, ca take any equimultiples GF, fe such, that they may be each greater than D; and of D take the multiples K and I such, that I may be that which is first greater than GF, and k that which is next less than 1.

Then, because is that multiple of D which is the first that becomes greater than GF, the next preceding multiple K will not be greater than GF; that is GF will not be less than k.

And, since FE is the same multiple of ac that GF is of BC (by Conft.), GF will also be the same multiple of BC that eg is of AB (V. 1.)

The magnitudes Eg and GF are, therefore, equimultiples of the magnitudes AB and BC, and 1 is a multiple of D.

And, since GF is not less than K, and Er is greater than D (by Conft.), the whole eg will be greater than K and D taken together.

But K and D, taken together, are equal to L (by Conft.); therefore EG will be greater than 1, and FG not greater than L, as was to be shewn.

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PROP

PRO P. V. THEOREM.

If four magnitudes be proportional, any equimultiples whatever of the antecedents will be proportional to any equimultiples whatever of the consequents.

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Let a be to B as c is to D, and of A and c take ang equimultiples EK, FL; and of B and D any equimultiples GM, HN; then will Ek be to GM, as FL is to hn. For of EK and FL take any equimultiples whatever EP,

and of GM and an any equimultiples whatever GS, HT:

Then, fince EK is the same multiple of A, that FL is of c (by Conft.), and of EK, FL have been taken the equimultiples EP, FR, EP will be the same multiple of A, that FR is of c (V. 3.)

And, in the same manner, it may be shewn, that as is the same multiple of B, that he is of D.

But A has the same ratio to B that c has to D (by Hyp.); and of A and c have been taken the equimultiples EP, FR; and of B and D the equimultiples Gs, AT.

If, therefore, EP be greater than GS, FR will also be greater than ht; and if equal, equal; and if lefs, less (V. Def. 5.)

And, since EP, FR are any equimultiples whatever of EK, FL; and'os, ut are any equimultiples whatever of GM, HN; EK will have the same ratio to GM, that FL has to HN (V. Def. 5.)

Q. E. D.

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If four magnitudes be proportional, and the first be greater than the second, the third will also be greater than the fourth; and if equal, equal; and if less, less.

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Let A have to B the same ratio that c has to D; then if A be greater than B, c will also be greater than D; and if equal, equal; and if less, less.

For, of A and c take any equimultiples e and g, and of B and d the same equimultiples F and H.

Then, because a is to B, as c is to d (by Hyp.), if E be greater than F, G will also be greater than H; and if equal, equal; and if less, less (V. Def. 5.)

And, since É, F, G, H are the same multiples of A, B, C, D, each of each, these last magnitudes will also observe the same agreement of equality, excess, or defect with their equimultiples.

If, therefore, a be greater than b, c will also be greater than D; and if equal, equal ; and if less, less.

Q. E. D. PROP

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