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 6. Theoreme. The 13. Proposition. When a right line standing vpon a right line maketh any an∣gles: those angles shall be either two right angles, or equall to two right angles.

SVppose that the right line AB standing vppon the right line CD do make these angles CBA and ABD. Then I say, that the angles CBA and ABD are eyther two right angles, or its ••quall to two right angles. If the angle CBA be equall to the angle ABD: then are they two right angles (by the tenth difinition.* 1.1) But if not, raise vp (by the 11. proposition) vnto the right line CD, and from the pointe geuen in it, namely••* 1.2 B, a perpendiculer line BE•• VVherfore (by the x. definition) the angle CBE and EBD ar•• right angles. Now forasmuch as the angle CBE, is equall to these two angles CBA and ABE, put the angle EBD common to them both•• wherfore the angles CBE and EBD, are equal to thes•• three angles CBA, ABE, and EBD. Agayne forasmuch as the angle DBA is equall vnto these two angles DBE and EBA, put the angle ABC common to them both: wherfore the angles DBA

[illustration]

and ABC, are equal to these three angles, DBE, EBA, and ABC. And it is proued that the an∣gles CBE and EBD are equal to the selfe same three angles: but thinges equall to one & the self same thing•• are also (by the first commō sentence) equall the o••e to the othe. VVherfore the angles CBE and EBD are equall to the angles DBA & ABC. But the angles CBE and EBD are two right angles: wherfore also the angles DBA and ABC are equall to two right angles. VVherfore when a right line standing v∣pon a right line maketh any angles: those angles shalbe either two right angles, or equall to two right angles: which was required to be demonstrated.

An othe•• demonstration after Pelitarius.

Suppose that the right line AB do stand vpon the right line CD. Then I say, that the two angles ABC and ABD,* 1.3 are either two right angles, or equal to two right angles. For if AB be perpēdiculer vnto CD: thē is it manifest, that they are right angles (by the conuersion of the definition) But if it incline towardes the end C, then (by the 11. pro∣position) from the point B, erect vnto the line CD a perpendicul••r line BE. By which•• construction the propositiō is very manifest. For forasmuch as the angle ABD is gre••∣ter then the right angle DBE by the angle ABE•• and

[illustration]

the other angle ABC is lesse then the right angle CBE by the selfe same angle ABE: if from the great••r b••e taken away the excesse, and the same bee added to the lesse, they shall be made two right angles. That is, if from the obtuse angle ABD be taken away the angl•• ABE, there shal remayne the right angle DBE. And then if the same angle ABE be added to the ••cute an∣gle CBA, there shall bee made the right angle CBE. Wherefore it is manifest, that the two angles, namely, the obtuse angle ABD, & the acute angle ABC, are equall to the two right angles CBE and DBE: which was requ••red to be proued.

Notes

• * 1.1

  Construction,
• * 1.2

  Demonstration
• * 1.3

  An other de∣monstration af∣ter Peli••••riu••.