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 13. Theoreme. The 13. Proposition. If the first haue vnto the second the self same proportion that the third hath to the fourth, and if the third haue vnto the fourth a greater proportiō thē the fifth hath to the sixth: thē shall the first also haue vnto the second a greater proportion then hath the fifth to the sixth.

SVppose that there be sixe magnitudes, of which let A be the first, B the second, C the third, D the fourth, E the fifth, and F the sixth. Suppose that A the first haue vnto B the second, the self same propor∣tion that C the third hath to D the fourth. And let C the third haue vnto D the fourth, a greater proportion then hath E the fifth to F the sixth.

Then I say that A the first hath to B the second a greater proportion, then hath E the fifth to F the sixt.* 1.1 For forasmuch as C hath to D a greater proportion then hath E to F, therfore there are certaine equemultiplices to C and E, and likewise any other equemultiplices whatsoeuer to D and F, which being compared toge∣ther, the multiplex to C shall exceede the multiplex to D, but the multiplex to E shall not exceede the multiplex to F (by the conuerse of the eight definition of

[illustration]

this booke). Let those multiplices be taken, and suppose that the equemultipli∣ces to C and E, be G and H: and likewise to D and F take any other equemulti∣plices whatsoeuer, and let the same be K and L, so that let G exceede K but let not H exceede L. And how multiplex G is to C, so multiplex let M be to A. And how multiplex K is to D, so multiplex also let N be to B. And because that as A is to B, so is C to D: and to A and C are taken equemultiplices M and G.* 1.2 And likewise to B and D are taken certayne other equemultiplices N & K: if ther∣fore M exceede N, G also excedeth K: and if it be equall it is equall, and if it be lesse it is lesse (by the conuersion of the sixt definition of the fifth.) But by con∣struction G excedet•• K, wherfore M also excedeth N, but H excedeth not L. But M & H are equemultiplices to A & E: and N & L are certaine other e∣quemultiplices•• whatsoeuer, to B and F. Wherfore A hath vnto B a greater pro∣portion then E hath to F (by the 8. definition.) If therefore the first haue vnto the second the selfe same proportion that the third hath to the fourth, and if the third haue vnto the fourth a greater proportion then the fifth hath to the sixth, then shall the firs•• also haue vnto the second a greater proportion then hath the 〈◊〉〈◊〉 to the sixth•• Which was required to be proued.

¶ An addition of Campane.

If there be foure quantities,* 1.3 and if the first haue vnto the second a greater proportion thē hath the third to the fourth: then shall there be some equemultiplices of the first and the third, which beyng compared to some equemultiplices of the second and the fourth, the mul∣tiplex of the first shall be greater then the multiplex of the second, but the multiplex of the third shall not be greater then the multiplex of the fourth.

Which is thus proued. Suppose that AB haue vnto C a greater proportion thē hath D to E. And let AF be to C as D is to E. Now then by this proposition & the tēth, AF is lesse then AB. Let it be lesse then AB by the quantitie FB. And multiply FB vntil there be produced a quantitie greater then C, which let be GH: which also must be such a multiplex, as D be¦yng

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so oftentymes multiplied, maye produce a quanti∣tie not lesse thē E•• whiche multiplex let be K. And let LG be so multiplex to AF, as GH is to FB, or K to D. Now then by the first of this booke LH is equemultiplex to AB as K is to D. And let M be to E the first multiplex greater then K•• & let N be equemultiplex to C as M is to E. Now then N is the first multiplex to C greater then LG: For for that as D is to E, so is AF to C, and K is equemultiplex to D as GL is to AF, also M is equemultiplex to E, as N is to C: therfore (by the 4. of this booke) as K is to M, so is GL to N•• but K is to M the first multiplex lesse then M: wherfore al∣so GL is the first multiplex lesse then N: and GL by supposition is not lesse thē C. Wher¦fore take the greatest multiplex of C vnder N: or a multiplex equall to N, if peraduen∣ture N be the first of the multiplices of C, which let be O. Now then then N shall consist of O and C. Wherfore forasmuch as LG is not lesse then O, and GH is greater then C, therfore LH shall be greater then N. And forasmuch as K is lesse then M, therfore that which was required to be proued, is manifest.

Although this proposition here put by Campane nedeth no demonstration for that it is but the conuerse of the 8. definition of this booke, yet thought I it not worthy to be omitted, for that it reacheth the way to finde out such equemultipli∣ces, that the multiplex of the first shall excede the multiplex of the second, but the multiplex of the third shall not exceede the multiplex of the fourth.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  An addition of Campane