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Then, since e is the same multiple of a that G is of B, and that magnitudes have the same ratio as their equimultiples (V. 13.), A is to B as e is to g.

But a is to B'as c is to d, by the propofition; whence c is to D as e is to G (V.11.)

In like manner, because F is the fame multiple of c that ħ is of D, c will be to D as F is to H (V. 13.)

But c has been shewn to be to Das E is to G; consequently, E will be to G as F is to H (V. 11.)

Since, therefore, E has the same ratio to G that F has to H, if e be greater than F, G will also be greater than H; and if equal, equal ; and if less, less (V. Def. 5.)

But E and G are any equimultiples whatever of A and B; and F and H are any equimultiples whatever of c and D; therefore a is to c as B is to D (V. Def. 5.)

Q. E. D.

PRO P. XVI. THEOREM.

If four magnitudes be proportional, the sum of the first and second, will be to the first or second, as the sum of the third and fourth, is to the third or fourth.

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Let Ae be to EB as CF is to Fd; then will AB be to BE, Or AE, as cd is to DF, or cf.

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For, fince is to es as CF is to FD (by Hyp.); therefore, alternately, Ae will be to CF as EB to FD (V. 15.)

And, since the antecedent is to its consequent as all the antecedents are to all the consequents (V. 12.), AE will be to cf as AB is to cd.

But ratios which are the same to the fame ratio are the fame to each other (V. ID); whence AB will be to CD as' EB is to FD; and, alternately, AB to EB as CD to DF (V. 15.)

Again, since ae has been shewn to be to CF as AB is to CD, therefore, by alternation, AE will be to AB as CF is to cD (V. 15.)

But quantities which are dire@ly proportional are also proportional when taken inversely; whence AB will be to AE as cd is to ck (V.7.)

Q. E. D.

PROP. XVII. THEOR E M. If four magnitudes be proportional, the difference of the first and second, will be to the first or second, as the difference of the third and fourth, is to the third or fourth,

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Let AB be to BE as cd is to DF; then will AE be to AB, or EB, a$ CF is to CD, of FD.

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For,

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For, of AE, EB, CF, FD take any equimultiples whatever GH, HK, LM, MN; and of EB, FD any other equimultiples whatever KP, NR:

Then, because Gh is the same multiple of AE that HK is of EB (by Confl.), GH will be the same multiple of AE that GK is of AB (V. 1.)

But oh is the fame multiple of AE that LM is of cf (by Conf.); therefore GK is the same multiple of AB that LM is of cf.

In like manner, because LM is the same multiple of cf that mn is of FD (by Conf.), lm will be the same multiple of cf that en is of ¢D (V. 1.)

But lm has been shewn to be the same multiple of cf that GK is of AB; therefore GK is the fame multiple of AB that in is of cd.

Again, because HK, KP are the fame multiples of EB, that mN, NR are of FD (by Conft.), HP will be the same multiple of EB, that MR is of FD (V. 2.)

And, since AB is to Be as CD'to DF (by Hyp.), and GK, LN are equimultiples of AB, CD, and HP, MR of be, if

greater

than
HP, IN will be

greater than MR ; and if equal, equal; and if less, less (V. Def. 5.) From the two former of these take

away

the common part hk, and from the two latter, the common part MN; then if gh be greater than KP, LM will be greater than NR, and if equal, equal; and if lefs, less.

But GH, LM are any equimultiples whatever of AE, CF (by Const.), and KP, NR are any equimultiples whatever of EB, FD; whence he is to cF as EB to FD (V. Def. 5.); and, alternately, AE to EB as CF to FD,

DF,

GK be

And since AB is the sum of AE, EB, and cd of 'CF, FD, AB will be to AE Or EB, as cd is to cF or FD (V. 16.); and, inversely, AE or EB to AB, as ce or FD to CD.

Q. E, D, SCHOLIUM. When the consequents are greater than the antecedents, the same demonstration will hold, if the terms be taken inversely.

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If four magnitudes of the same kind be proportional, the greatest and least of them, taken together, will be greater than the other two.

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Let AB, CD, E, F be the four proportional magnitudes, of which al is the greatest and r the least, then will AB and F together, be greater than cd and E together.

For in AB take AG equal to E, and in co take ch equal to F (1. 3.)

Then, because AB is to cd as E to F (by Hyp.), and AG is equal to E and ch to F (by Confl.), AB will be to CD as AG to CH.

But magnitudes which are proportional, are also proportional when taken alternately (V. 15.); therefore AB will be to AG as co to ch.

And, since BG is the difference of AB and AG, and DH of cd and ch, BG will be to AB as DH to CD (V. 17.); and, inversely, AB to BG as CD to DH.

But

But AB is greater than cd (by Hyp.) ; whence GB is also greater than HD (V. 14).

And, because AG is equal to E, and ch to F, AG and F together, are equal to ch and e together. : To the first of these equals add GB, and to the second HD, then will AG, GB and F together, be - greater than CH, HD and E together.

But AG, GB are equal to AB, and cH, HD to CD; confequently AB and F together are greater than cd and E together.

Q. E, D, SCHOLIUM. That F muft be the least of the four magnitudes when A is the greatest, appears from propositions VI, and XIII.

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