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

Correlary.

Hereby it is manifest,* 1.1 that if in a triangle one angle be equall to the two other angles remayning the same angle is a right

angle: for that the side angle to that one angle (namely, the angle which is made of the side produced without the trian∣gle) is equall to the same angles, but when the side angles are equall the one to the other, they are also right angles.

¶ An addition of Pelitarius.

If in a circle be inscribed a rectangle triangle, the side opposite vnto the right angle shall be the diameter of the circle.

* 1.2Suppose that in the circle ABC be inscribed a

[illustration]

rectangle triangle ABC, whose angle at the point B let be a right angle. Then I say, that the side AC is the diameter of the circle. For if not, then shall the centre be without the line AC, as in the point E.* 1.3 And draw a line from the poynt A to the point E, & produce it to the circumference to the point D: and let AED be the diameter: and draw a line from the point B to the point D. Now (by this 31. Propositiō) the angle ABD shall be a right angle, and therefore shall be equall to the right angle ABC, namely, the part to the whole: which is ab∣surde. Euen so may we proue, that the centre is in no other where but in the line AC. Wherfore AC is the diameter of the circle: which was required to be proued.

¶ An addition of Campane.

* 1.4By thys 31. Proposition, and by the 16. Proposition of thys booke, it is mani∣fest, that although in mixt angles, which are contayned vnder a right line and the circumference of a circle, there may be geuen an angle lesse & greater then a right angle, yet can there neuer be geuē an angle equall to a right angle. For euery secti∣on of a circle is eyther a semicircle, or greater then a semicircle, or lesse, but the an∣gle of a semicircle is by the 16. of thys booke, lesse then a right angle, and so also is the angle of a lesse section by thys 31. Proposition: Likewise the angle of a greater section, is greater then a right angle, as it hath in thys Proposition bene proued.

Notes

• * 1.1

  A Corollary.
• * 1.2

  An addition of P••litarius.
• * 1.3

  Demonstra∣tion lea••ing to an absurdi∣t••••.
• * 1.4

  An addition of Campane.