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 1. Theoreme. The 1. Proposition. If a right line be deuided by an extreme and meane proportion, and to the greater segment, be added the halfe of the whole line: the square made of those two lines added together shalbe quintuple to the square made of the halfe of the whole lyne.

SVppose that the right

[illustration]

line AB be deuided by an extreme and meane pro∣portiō in the point C. And let the greater segment therof, be AC. And vnto AC, adde directly a ryght line AD, and let AD be equall to the halfe of the line AB.* 1.1 Then I say that the square of the line CD is quintuple to the square of the line DA. Describe (by the 46. of the first) vpon the lines AB and DC squares, namely, AE & DF. And in the square DF, describe and make complete the figure. And ex∣tend the line FC, to the point G.* 1.2 And forasmuch as the line AB is deuided by an extreme and meane proportion in the point C, therefore that which is contayned vnder the lines AB and BC is equall to the square of the line AC. But that which is contayned vn∣der the lines AB and BC, is the pa∣rallelogramme CE, and the square of

the line AC is the square HF. Wherefore the parallelogramme CE is equall to the square HF. And forasmuch as the line BA, is double to the line AD, by construct••on: 〈◊〉〈◊〉 the lyne BA is equall to the line KA, and the line AD, to the lyne AH: therefore also, the lyne KA, is double to the line AH. But as the lyne KA is to the line AH, so is the parallelogramme C∣K to the parallelogramme CH: Wherefore the parallelogramme CK is double to the paralle∣logramme CH. And the parallelogrammes LH and CH are double to the parallelo∣gramme CH (for supplementes of parallelogrammes are b•• the 4••. of the first equall the one to the other). Wherefore the parallelogramme CK is equall to the parallelogrammes LH & CH. And it is proued that the parallelogramme CE is equall to the square FH. Wherefore the whole square AE is equall to the gn••mon MXN. And forasmuch as the line BA, i•• dou∣ble to the line AD, therefore the square of the line BA is, by the 20. of the sixth, quadruple to the square of the line DA, that is, the square AE to the square DH. But the square AE is equall to the gnomō MXN, wherefore the gnomō MXN, is also quadruple to the square DH. Wherefore the whole square DF is quintuple to the square DH. But the square DF, i•• the square of the line CD, and the square DH is the square of the line DA. Wherefore the square of the line CD, is quintuple to the square of the line DA. If therefore a right line be deuided by an extreame and meane proportion, and to the greater segment, be added the halfe of the whole line: the square made of those two lines added together shalbe quintuple to the square made of the halfe of the whole line: Which was required to be demonstrated.

Thys proposition is an other way demonstrated after the fiueth proposition of this booke.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.