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 ninth Proposition.

If a number be deuided into two equall numbers, and againe be deuided into two inequall partes: the square numbers of the vnequall numbers, are double to the square which is made of the multiplicati∣on of the halfe number into it selfe, together with the square whiche is made of the number set be∣twene them.

For let the number AB being an euen number be deuided into two equall numbers AC & CB: & into two vnequall nūbers AD and DB. Then I say that the square num∣bers of AD and DB, are double to the squares which are made of the multiplication of the numbers AC and CD into themselues. For forasmuch as the number AB is an euen number, and is deuided also into two equal numbers AC and CB, and afterward into two vnequal nūbers AD and DB: therefore the superficial nūber produced of the multiplicatiō of the nūbers AD & DB, th'one into the other, together with the square of the number DC, is equal to the square of the number AC (by the fift proposition) Wherfore the superficiall number produced of the multiplication of the numbers AD and DB the one into the other twise, together with two squares of the number CD, is double to the square of the number AC. Forasmuch as also the number AB is deui∣ded into two equal numbers AC and CB, therfore the square number of AB is qua∣druple to the square number produced of the multiplication of the number AC into it selfe (by the 4. proposition). Moreouer forasmuch as the superficiall number produ∣ced of the multiplication of the numbers AD & DB the one into the other twise to∣gether, with two squares of the number DC, is double to the square number of CA•• &

forasmuch as there are two numbers, of whiche the one is quadruple to

[illustration]

one and the selfe same number, and the other is double to the same num¦ber: therefore that number whiche is quadruple shall be double to that number whiche is double. Wherefore the square of the number AB is double to the number produced of the multiplicatiō of the numbers AD and DB the one into the other twise together with the two squares of the number DC. Wherfore the number which is produced of the mul∣tiplication of the numbers AD and DB the one into the other twise, is lesse thē halfe of the square of the number AB by the two squares of the numbers DC. And forasmuch as the nūber produced of the multiplica∣tion of the nūbers AD & DB the one into the other twise, together with the nūber cōposed of the squares of the numbers AD and DB is (by the 4. proposition) equall to the square of the number AB•• therfore the nū∣ber composed of the squares of the numbers AD & DB is greater then the halfe of the square nūber of AB, by the two squares of the number DC. And the square of the number AB is quadruple to the square of the number AC. Wherfore the number composed of the squares of the numbers AD and DB is greater then the double of the square of the number AC by two squares of the number DC. Wherfore the said num∣ber is double to the squares of the numbers AC and CD. If therefore a number be deuided &c. which was required to be demonstrated.

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