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 tenth Proposition.

If an euen nomber be deuided into two equall nombers, and vnto it be added any other nomber: the square nomber of the whole nomber composed of the nōber and of that which is added, and the square nomber of the nōber added: these two square nōbers (I say) added together, are double to these square nombers, namely, to the square of the halfe nomber, and to the square of the nomber composed of the halfe nomber and of the nomber added.

Suppose that the nomber AB being an euen nomber be deuided into two equall nombers AC and CB: and vnto it let be added an other nomber BD. Then I say, that the square nombers of the nombers AD and DB are double to the square nombers of AC and CD. For forasmuch as the nomber AD is deuided into the nombers AB and BD: therefore the square nombers of the nombers AD and DB are equall to the su∣perficiall nomber produced of the multiplication of the nombers AD and DB the on into the other twise, together with the square of the nomber AB (by the 7 propositiō) But the square of the nomber AB is equal to fower squares of either of the nombers AC or CB (for AC is equall to the nomber CB): wherfore also the squares of the nom∣bers AD and DB are equall to the superficiall nomber produced of the multiplication of the nombers AD and DB the one into the other twise, and to fower squares of the nomber BC or CA. And forasmuch as the superficiall nomber produced of the multi∣plication of the nombers AD and DB the one into the other, together with the square of the nomber CB, is equal to square of the nomber CD (by the 6 propositiō): therfore the nomber produced of the multiplication of the nomber•• AD and DB the one into the other twise together with two squares of the nomber CB, is equall to two squares of the nomber CD. Wherefore the squares of the nombers AD and DB are equall to

two squares of the nomber CD, and to two squares of the nomber AC. Where∣fore they are double to the squares of the numbers AC and CD. And the square of the nomber AD is the square of the whole and of the nomber added. And the square of DB is the square of the nomber added: the square also of the nomber CD is the square of the nomber composed of the halfe and of the nomber added: If there∣fore an euen nomber be deuided. &c. Which was required to be proued.

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