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. Theoreme. The 9. Proposition. If a right line be deuided into two equall partes, and into two vnequall partes, the squares which are made of the vne∣quall partes of the whole, are double to the squares, which are made of the halfe lyne, and of that lyne which is betwene the sections.

SVppose that a certayne right line AB be deuided into two equall partes in the pointe C, and into two vnequall partes in the pointe D. Then I say that the squares which are made of the lines AD and DB, are double to the squares whiche are made of the lynes AC and CD.* 1.1 For (by the 11. of the

[illustration]

first) erect from y^e point C to the right line AB a perpendiculer line CE. And let CE (by the 3. of the first) be put equall vnto either of these lines AC & CB: and (by the first peticiō) draw lines from A to E, and from E to B. And (by the 31. of the first) by the point D draw vnto the line EC a parallel lyne, and let the same be DF: and (by the selfe same) by the point F draw vnto AB a line parallel, and let the same be FG. And (by the first peticion) draw a line from A to F.* 1.2 And forasmuch as AC is e∣quall vnto CE, therfore (by the 5. of the first) the angle EAC is equal vnto the angle CEA. And forasmuch as the angle at the point C is a right angle: therfore the angles remayning EAC, and AEC, are equall vnto one right angle, where∣fore eche of these angles EAC and AEC is the halfe of a right angle. And by the same reason also eche of these angles EBC and CEB is the halfe of a rig••t angle. VVherfore the whole angle AEB is a right angle. And forasmuch as the angle GEF is the halfe of a right angle, but EGF is a right angle. For (by the 29 of the first) it is equall vnto the inward and opposite angle, that is, vnto ECB: wherfore the angle remayning EFG is the halfe of a right angle. VVhere∣fore (by the 6. common sentence) the angle GEF is equall vnto the angle EFG. VVherfore also (by the 6. of the first) the side EG is equall vnto the side FG. A∣gaine ••orasmuch as the angle at the point B is the halfe of a right angle, but the angle FDB is a right angle, for it also (by the 29. of the first) is equall vnto the

inwarde and opposite angle ECB. VVherefore the angle remayning BFD is the halfe of a right angle. VVherfore the angle at the point B is equall vnto the angle DFB. VVherfore (by the 6. of the first) the side DF is equall vnto the side DB. And forasmuch as AC is equall vnto CE, therfore the square which is made of AC is equall vnto the square which is made of CE. VVherefore the squares which are made of CA and CE are double to the square which is made of AC. But (by the 47. of the first) the square which is made of EA is equall to the squares which are made of AC and CE (For the angle ACE is a right an∣gle) wherefore the square of AE is double to the square of AC. Agayne foras∣much as EG, is equall vnto GF, the square therfore which is made of EG is e∣qual to the square which is made of GF. VVherfore the squares which are made of GE and GF are double to the square which is made of GF. But (by the 47. of the first) the square which is made of EF is equall to the squares which are made of EG and GF. VVherfore the square which is made of EF is double to the square which is made of GF. But GF is equall vnto CD. VVherefore the square which is made of

[illustration]

EF is double to the square which is made of CD. And the square whiche is made of AE is double to the square which is made of AC. VVherefore the squares which are made of AE and EF are double to y^e squares which are made of AC and CD. But (by the 47. of the first) the square which is made of AF is equal to the squares which are made of AE and EF (For y^e angle AEF is a right angle). VVherfore the square which is made of AF is double to the squares which are made of AC & CD. But (by the 47. of the first) y^e squares which are made of AD and DF are e∣quall to y^e square which is made of AF. For y^e angle ot y^e point D is a right angle. VVherfore the squares which are made of AD and DF are double to y^e squares which are made of AC and CD. But DF is equall vnto DB. VVherfore the squares which are made of AD and DB, are double to the squares which are made of AC and CD. If therfore a right line be deuided into two equall partes and into two vnequall partes, the squares which are made of the vnequall partes of the whole, are double to the squares which are made of the halfe lyne, and of that lyne which is betwene the sections: which was required to be proued.

¶ An example of this proposition in numbers.

Take any euen number as 12. And deuide it first equally as into 6. and 6. & then vn∣equally as into 8. & 4. And take the difference of the halfe to one of the vnequal partes

which is 2. And take the square numbers of the vnequall partes 8, and 4, which are 6•••• and 16: and adde them together, which make 80. Then take the squares of the halfe 6. and of the differēce 2: which are 36, and 4: which added together make 40. Vnto which number, the number composed of the squares of the vnequall partes, whiche i•• 80, i•• double. As you see in the example.

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The demonstration wherof followeth in Barlaam.

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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Notes

• * 1.1

  Constr••ction.
• * 1.2

  Demonstra∣tion.