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 16. Theoreme. The 18. Proposition. Betwene two like plaine or superficiall numbers there is one meane propor∣tionall number. And the one like plaine number is to the other like plaine number in double proportion of that which the side of like proportion, is to the side of like proportion.

SVppose that there be two like plaine or superficiall numbers A & B. And let the sides of A be the nūbers C, D: and the sides of B be the numbers E, F. And forasmuch as like plaine numbers are those which haue their sides proportionall (by the 22. defini∣tion of the seuenth) therefore as C is to D, so is E to F.* 1.1 Then I say that betwene A and B there is one meane proportionall number, and that

[illustration]

A is vnto B in double proportiō of that which C is vnto E, or of that which D is vnto F, that is, of that which side of like proportion is to side of like proportion. For for that as C is to D, so is E to ••, therefore alternately (by the 13. of the seuenth) as C is to E, so is D to F. And forasmuch as A is a plaine or superfi∣ciall number, and the sides thereof are C and D: therefore D multiplying C produced A. And by the same reason also E multiplying F produced B. Let D multiplying E produce G. And forasmuch as D multiplying C produced A, and multiplying E produced G, therefore (by the 17. of the seuenth) as C is to E, so is A to G. But as C is to E, so is D to F, wherefore as D is to F, so is A to G. Againe forasmuch as E multiplying D produced G, and multiplying •• produced B, therefore (by the 17. of the seuenth) as D is to F, so is G to B. But it is proued

that as D is to F, so is A to G: wherfore as A is to G, so is G to B. Wherefore these numbers A, G, B, are in continuall prorortion. Wherefore betwene A and B there is one meane proportio∣nall number.

* 1.2Now also I say that A is vnto B in dooble proportiō of that which side of like proporti∣on is to side of like proportion, that is, of that which C is vnto E, or of that which D is vnto F. For forasmuch as A, G, B, are in continuall proportion, therefore (by the 10. definition of the ••ift) A is vnto B in double proportion of that which A is vnto G. But as A is to G, so is C to E, and D to F: wherefore A is vnto B in double proportion of that which C is to E, or D to F: which was required to be demonstrated.

Notes

• * 1.1

  Demonstra∣tion of the fi••st part of this proposition.
• * 1.2

  Demonstra∣tion of the second part.