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 17. Theoreme. The 19. Proposition. If there be foure numbers in proportion: the number produced of the first and the fourth, is equall to that number which is produced of the second and the third. And if the number which is produced of the first and the fourth be equall to that which is produced of the second & the third: those foure numbers shall be in proportion.

* 1.1SVppose that there be foure numbers in proportion A, B, C, D, as A is to B, so let C be to D. And let A multiplieng D produce E: and let B multiplieng C pro∣duce F. Then I say that the number E is equall vnto the number F. Let A mul∣tiplieng C produce G. Now forasmuch as A multiplieng C, produceth G, & mul∣tiplieng D it produceth E: it followeth that the number A

[illustration]

multiplieng two numbers C and D, produceth G and E. VVherfore by the 17 of the seuenth, as C is to D, so is G to E. But as C is to D, so is A to B, wherfore as A is to B, so is G to E.* 1.2 Againe, forasmuch as A multiplieng C produced G, and B multiplieng C produced F:* 1.3 therfore two numbers A and B, multiplieng one nūber C, do produce G & F. VVher∣fore by the 18. of the seuenth, as A is to B, so is G to F. But as A is to B, so is G to E: wherfore as G is to E, so is G to F. VVherfore G hath to either of these E and F one & the same proportion (But if one number haue to two numbers one and the same proportion, the said•• two numbers shall be equall). VVherfore E is equall vnto F.

But now againe, suppose that E be equall vnto F. Then I say that as A is to B, so is C to D.* 1.4 For the same order of construction remayning still, forasmuch as A multiplieng C & D produced G and E, therfore by the 17. of the seuenth, as C is to D so is G to E, but E is equall vnto F (But if two numbers be equall, one number shall haue vnto them on•• and the same proportion) wherfore as G is to E, so is G to F. But as G is to E, so is C to D. Wherefore as C is to D, so is G to F, but as G is to F, so is A to B by the 18. of the seuenth, wherfore as A is to B so is C to D:* 1.5 which was required to be proued.

* 1.6Here Campane addeth, that it is needeles to demonstrate, that if one number haue to two numbers one and the same proportion, the said two numbers shall be equall: or that if they be equal, one number hath to them one and the same proportion. For (saith he) if G haue vnto E and F one and the same proportion, thē either, what part or partes G is to E, the same part or parts is G also of F: or how multiplex G is to E, so multiplex is G to F (by the 21. definition) And therfore by the 2 and 3 common sentence, the said numbers shall be equall. And so conuersedly, if the two numbers E and F be equal, then shall the numbers E and F be either the selfe same parte or partes of the number G, or they shall be equemultiplices vnto it. And therfore by the same definition the number G shall haue to the numbers E and F one and the same proportion.

Notes

• * 1.1

  This proposi∣tion in num∣bers demon∣strateth that which the 16. of the sixth booke demon∣strateth in lines.
• * 1.2

  Construction.
• * 1.3

  Demonstra∣tion.
• * 1.4

  The second part of this proposition which is the conuerse of the first.
• * 1.5

  Demonstra∣tion.
• * 1.6

  An assumpt added by Campane.