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

multiples whatever KP, NR:

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

But GH is the fame multiple of AE that LM is of CF (by Conft.); therefore GK is the fame multiple of AB that LM is of CF.

In like manner, because LM is the fame multiple of CF that MN is of FD (by Conft.), LM will be the fame multiple of CF that LN is of CD (V. 1.)

But LM has been fhewn to be the fame multiple of cr that GK is of AB; therefore GK is the fame multiple of AB that LN 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.)

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And, fince AB is to BE as CD to DF (by Hyp.), and GK, LN are equimultiples of AB, CD, and HP, MR of BE, DF, if GK be greater than HP, LN 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, lefs.

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

And fince AB is the fum 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 CF or FD to CD.

Q. E. D, SCHOLIUM. When the confequents are greater than the antecedents, the fame demonftration will hold, if theterms be taken inversely.

PROP. XVIII. THEOREM,

If four magnitudes of the fame 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 AB is the greatest and F the leaft; then will AB and F together, be greater than CD and E together.

For in AB take AG equal to E, and in CD take Cн equal to F (I. 3.)

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Then, because AB is to CD as E to F (by Hyp.), and AG is equal to E and CH to F (by Conft.), 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 CD to CH.

And, fince 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.

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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 leaft of the four magnitudes when A is the greateft, appears from propofitions VI. and XIII.

воок VI.

DEFINITIONS.

1. Similar rectilineal figures, are those which are equiangular, and have the fides about the equal angles proportional.

2. The homologous, or like fides, of fimilar figures, are those which are oppofite to equal angles.

3. Two figures are faid to have their fides reciprocally proportional, when the first confequent, and fecond antecedent, of the four terms, are both fides of the fame figure.

4. Of three proportional quantities, the middle one is faid to be a mean proportional between the other two; and the last a third proportional to the first and second.

5. Of four proportional quantities, the laft is faid to be a fourth proportional to the other three, taken in order.

6. If any number of magnitudes be continually proportional, the ratio of the first and third is faid to be duplicate that of the first and fecond; and the ratio of the first and fourth, triplicate that of the first and second.

7. And of any number of magnitudes, of the fame kind, taken in order, the ratio of the first to the laft, is faid to be compounded of the ratio of the first to the second, of the fecond to the third, and so on, to the last.

8. A right line is faid to be divided in extreme and mean ratio, when the whole line is to the greater fegment, as the greater fegment is to the lefs.

PROP.

PRO P. I.

Triangles and parallelograms, having the fame altitude, are to each other as their bafes.

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Let the triangles ABC, ACD, and the parallelograms EC, CF have the fame altitude, or be between the fame parallels BD, EF; then will the bafe BC be to the base CD, as the triangle ABC is to the triangle ACD, or as the parallelogram EC is to the parallelogram CF.

For, in BD produced, take any number of parts whatever BG, GH, each equal to BC; and DK, KL, any number whatever, each equal to CD; and join AG, AH, AK and AL:

Then, because CB, BG, GH are all equal to each other, the triangles AHG, AGB, ABC will alfo be equal to each other (II. 5.); and whatever multiple the base HC is of the bafe BC, the fame multiple will the triangle AHC be of the triangle ABC.

And, for the fame reason, whatever multiple the base LC is of the bafe CD, the fame multiple will the triangle ALC be of the triangle ADC.

If, therefore, the bafe HC be equal to the base CL, the triangle AHC will be equal to the triangle ALC; and if greater, greater; and if lefs, less.

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