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 9. Proposition.

If a number be deuided into two partes: that which is produced of the whole number into him selfe together with that which is produced of one of the partes into him self, is equall to that which is produced of the whole into the sayd part twise together with that which is produced of the other part into him selfe.

This proueth in numbers that which the 7. of the second proued in lines. For let the number A be deuided into the numbers B and D.* 1.1 Then I say that the square of A together with the square of D is equall to that which is produced of A into D twise together with the

[illustration]

square of B. For it is manifest by the 6. of these propositions that the square of A is equall to the squares of B and D together with that which is produced of B into D twise. Wherefore the square of A together with the square of D, is equall to two squares of D••* 1.2 and to that which is produced of D into B twise together with the square of B. But by the first of these propositions two squares of D, and that which is produced of D into B twise is e∣quall to that which is produced of D into A twise. Wherfore that which is produced of D into A twise together with the square of B, is equall to the square of A together with the square of D: wherfore that is manifest which was required to be proued.

Notes

• * 1.1

  This answe∣reth to the 7. of the second.
• * 1.2

  Demonstra∣tion.