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

An example of this proposition in numbers.

Take any number as 20: and deuide it into two equall partes 10. and 10. and then into two vnequall partes as 13. and 7. And take the differēce of the halfe to one of the vnequall partes which is 3. And multiply the vnequall partes, that is, 13 and 7. the one into the other, which make 91. take also the square of 3. which is 9. and adde it to the foresayde number 91: and so shall there be made 100. Then multiply the halfe of the whole number into himself, that is, take the square of 10. which is 100. which is equal to the number before produced of the multiplication of the vnequal parts the one in∣to the other, & of the difference into it selfe which is also 100. As you se in the example.

〈 math 〉〈 math 〉

The demonstration wherof followeth in Barlaam.

The fifth proposition.

If an euen number be deuided into two equall partes, and againe also into two vnequall partes: the superfic••all number which is produced of the multiplication of the vnequall partes the one into the other, together with the square of the number set betwene the parts, is equal to the square of halfe the number.

Suppose that AB be an euen number: which let be

[illustration]

deuided into two equall numbers AC and CB, and into two vnequall numbers AD and DB. Then I say, that the square number which is produced of the multiplication of the halfe number CB into it selfe, is equall to the su∣perficiall number produced of the multiplication of the vnequall numbers AD and DB the one into the other, and to the square number produced of the number CD which is set betwene the sayde vnequall partes. Let the square number produced of the multiplication of the halfe number CB into it selfe be E. And let the superfi∣ciall number produced of the multiplication of the vne∣qual nūbers AD and DB the one into the other, be the number FG: and let the square of the number DC which is set betwene the partes be GH. Now forasmuch as the number BC is deuided into the numbers BD and DC, therfore the square of the number BC, that is, the num∣ber E, is equall to the squares of the numbers BD and DC, and to the superficiall number which is composed of the multiplication of the numbers BD and DC the one into the other twise, (by the 4. proposition of this boke) Let the square of the number BD be the number KL: & let NX be the square of the number DC: and finally of

the multiplication of the numbers BD and DC the one into the other twi••e, let be pro¦duced either of these numbers LM and MN. Wherefore the whole number KX is equall to the number E•• And forasmuch as the number BD multiplying it selfe produ∣ced the number KL, therefor it measureth it by the vnities which are in it selfe. More ouer forasmuch as the number CD multiplying the number BD produced the num∣ber LM, therefore also DB measureth LM by the vnities which are in the number CD: but it before measured the number KL by the vnities which are in it selfe. Where∣fore the number DB measureth the whole number KM by the vnities which are in CB. But the number CB is equall to the number CA. Wherefore the number DB mea∣sureth the number KM by the vnities which are in CA. Agayne forasmuch as the nū∣ber CD multipli••ng the number DB produced the number MN: therefore the num∣ber DB measureth the number MN by the vnities which are in the number CD: but it before measured the number KM by the vnities which are in the number AC. Wher¦fore the number BD measureth the whole number KN by the vnities which are in the number AD. Wherefore the number FG 〈◊〉〈◊〉 equall to the number KN. For numbers which are equemultiplices to one and the selfe same number, are equall the one to the other. But the number GH is equall to the number NX: for either of them is suppo∣sed to be the squ••re of the number CD. Wherefore the whole number KX is equall to the whole number FH. But the number KX is equall to the number E•• Wherefore also the number FH is equall to the number E. And the number FH is the superficial num∣ber produced of the multiplication of the numbers AD and DB the one into the o∣ther together with the square of the number DC. And the number E is the square of the number CB. Wherfore the superficiall number produced of the multiplication of the vnequal partes AD and DB the one into the other, together with the square of the nū∣ber DC which is set betwene those vnequall partes, is equall to the square of the num∣ber CB, which is the halfe of the whole number AB. If therfore an euen number be de∣uided into two equall partes, &c. which was required to be proued.