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 8. Theoreme. The 9. Proposition. If within a circle be taken a poynt, and from that poynt be drawen vnto the circumference moe then two equall right lines, the poynt taken is the centre of the circle.

SVppose that the circle be ABC, and within it let there be taken the poynt D. And from D let there be drawen vnto the circumference ABC moe then two equall right lines,* 1.1 that is, DA, DB, and DC. Then I say that the poynt D is the centre of the circle ABC. Draw (by the first petition) these right lines

[illustration]

AB and BC:* 1.2 and (by the 10. of the first) deuide thē into two equall partes in the poyntes E and F: namely, the line AB in the poynt E, and the line BC in the poynt F. And draw y^e lines ED and FD, and (by the second pe∣tition) extend the lines ED and FD on eche side to the poyntes K, G, and H, L. And for asmuch as the line AE is equall vnto the line EB, and the line ED is common to them both, there∣fore these two sides AE and ED are equall vnto these two sides BE, and ED: and (by supposition) the base DA is equall to the base DB. Wherfore (by the 8. of the first) the angle AED is equall to the angle BED. Wherfore eyther of these angles AED and BED is a right angle. Wherefore the line GK deuideth y^e line AB into two equall partes and maketh right angles. And for asmuch as, if in a circle a right line deuide an other right line into two equall partes in such sort that it maketh also right angles, in y^e line that deuideth is the centre of the circle (by the Correllary of the first of the third). Therfore (by the same Correllary) in the line GK is the centre of the circle ABC. And (by the same reason) may we proue that in y^e line HL is the centre of the circle ABC, and the right lines GK, and HL haue no other poynt common to them both besides the poynt D. Wherefore the poynt D is the centre of the circle ABC. If therefore within a circle be taken a poynt, and from that point be drawen vnto the circumference more then two equall right lines, the poynt taken is the centre of the circle: which was required to be proued.

¶ An other demonstration.

Let there be taken within the circle ABC the poynt D.* 1.3 And from the poynt D let there be drawen vnto the circumference more then two equall right lines, namely, DA, DB, and DC. Then I say that the poynt D is the centre of the circle. For if not, then if it be possible

[illustration]

let the point E be the centre: and draw a line from D to E, and extend DE to the poyntes F and G. Wherefore the line FG is the diameter of the circle ABC. And for asmuch as in FG the diameter of the circle ABC is taken a poynt, namely D, which is not the centre of that circle, therefore (by the 7. of the third) the line DG is y^e grea∣test, and the line DC is greater then the line DB, and the line DB is grea∣te•• then the line DA. But the lines DC, DB, DA, are also equall (by supposi∣tion): which is impossible. Wherefore the poynt E is not the centre of the circle ABC. And in like sort may we proue that no other poynt besides D. Wherefore the poynt D is the centre of the circle ABC: which was required to be proued.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  An other de∣monstration of the same leading also to an impossi∣bilitie.