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

* 1.1An addition of Pelitarius.

To reduce two vnequall squares to two equall squares.

Suppose that the squares of the lines AB and AC be vnequall. It is required to re∣duce them to two equall squares. Ioyne the two lines AB and AC at their endes in such sort that they make a right angle BAC. And draw a line from B to C. Then vppon the two endes B and C make two angles eche of which may be equal to halfe a right an∣gle (This is done by erecting vpon the line BC perpē∣diculer

[illustration]

lines, from the pointes B and C: and so (by the 9. proposition) deuiding ••che of the right angles into two equall partes): and let the angles BCD and CBD be either of thē halfe of a right angle. And let the lines BD and CD concurre in the point D. Then I say that the two squares of the sides BD and CD, are equall to the two squares of the sides AB and AC. For (by the 6 proposition) the two sides DB and DC are equall, and the angle at the pointe D is (by the 32. proposition) a right angle. Wherefore the square of the side BC is e∣qual to the squares of the two sides DB and DC (by the 47. proposition): but it is also equall to the squares of the two sides AB and AC (by the self same proposition) wher∣fore (by the common sentence) the squares of the two sides BD and DC are equall to the squares of the two sides AB and AC: which was required to be done.

Notes

• * 1.1

  An addition of P••l••tari••••.