The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed

¶ The second Proposition.

If there be foure quantities, and if the proportion of the first to the second be greater then the pro∣portion of the third to the fourth, then alternately the proportion of the first to the third, shall be grea∣ter then the proportion of the second to the fourth.

Let A haue vnto B a greater proportion then hath C to D. Then I say alternately A hath to C a greater proportion then hath B to D. For one and the same proportion it can not haue:* 1.1 for then alternately A should be to B as C is to D, which is contrary to the suppositiō. But if it haue a lesse proportiō, let E be vnto C as B is to D. Now thē by the 13. of this booke

[illustration]

E hath vnto C a greater proportion then hath A to C. Wherefore (by the first part of the tenth of the same) E is greater thē A. Wherefore by the first part of the 8. of the same, E hath to B a greater pro¦portion then hath A to B. And forasmuch as by suppositiō E is vnto C, as B is to D, therfore alternately E is to B as C is to D. Wherfore by the 13. of the same, C hath to D a greater proportion then hath A to B, which is contrary to the suppositiō. Wherfore the proportion of A to C is neither one and the same with the proportion of B to D, no•• lesse then it, wherefore it is greater. Which was required to be proued.

* 1.2This may also be demonstrated affirmatiuely, let E be vnto B as C is to D. Now thē by the first part of the tenth of this booke, E is lesse then A: wherfore by the first parte of the 8. of the same, the proportion of A to C is greater then the proportion of E to C. But alternately E is to C as B is to D. Wherfore (by the 13. of the same) A hath to C a

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Notes

• * 1.1

  Demonstratiō leading to a•• impossibilitie.
• * 1.2

  An other de∣monstration of the same affirmatiuely.