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 31. Theoreme. The 37. Proposition. If without a circle be taken a certaine point, and from that point be drawen to the circle two right lines, of which, the one doth cut the circle and the other falleth vpon the circle, and that in such sort, that the rectangle parallelogramme which is cōtayned vnder the whole right line which cutteth the circle, and that portion of the same line that lieth betwene the point and the vtter circumferēce of the circle, is equall to the square made of the line that falleth vpon the circle: then that line that so falleth vpon the circle shall touch the circle.

LEt the circle be ABC: and with∣out

[illustration]

the same circle take a point, and let the same be D,* 1.1 & from the point D let there be drawen to the circle ABC two right lines DCA and DB: and let DCA cut the circle, and DB fall vpon the circle. And that in such sort, that that which is contayned vnder the lines AD and DC, be equall to the square of the line DB. Then I say, that y^e line DB toucheth the circle ABC. Drawe (by the 17. of the third) from the poynt D a right line touching the circle ABC, and let the same be DE.* 1.2 And (by the first of the same) let the point F be the centre of the circle ABC: and draw these right lines FE, FB, and FD. Wherfore the angle FED is a right angle.* 1.3 And for asmuch as the right line DE toucheth the circle ABC, and the right line DCA cutteth the same, therfore (by the Proposition going before) that which is contayned vnder the lines AD and DC, is equall to the square of the line DE. But that which is contayned

vnder the lines AD and DC, is supposed to be equall to the square of the line DB. Wherefore the square of the line DE is equall to the square of the line DB. Wherefore also the line DE is equall to the line DB. And the line FE is equall to the line FB, for they are drawen from the centre to y^e circumference. Now therefore these two lines DE and EF are equall to these two lines DB and BF, and FD is a common base to them both. Wherefore (by the 8. of the first) the angle DEF is equall to the angle DBF. But the angle DEF is a right angle. Wherefore also the angle DBF is a right angle. And y^e line FB being produced, shall be the diameter of the circle. But if from the end of the di∣ameter of a circle be drawen a right line making right angles, the right line so drawen toucheth the circle (by the Correllary of the 16. of the third). Wherfore the right line DB toucheth the circle ABC. And the like demonstration will serue if the centre be in the line AC. If therefore without a circle be taken a cer∣taine point, and from that poynt be drawen to the circle two right lines, of which the one doth cut the circle, and the other falleth vpon the circle, and that in such sort, that the rectangle parallelogramme which is contayned vnder the whole right line which cutteth the circle, and that portion of the same line that lieth be∣twene the poynt and the vtter circumference of the circle, is equall to the square made of the line that falleth vpon the circle: then the line that so falleth vpon the circle shall touch the circle: which was required to be proued.

¶ An other demonstration after Pelitarius.

Suppose that there be a circle BCD, whose

[illustration]

centre let be E:* 1.4 and take a point without it, name∣ly, A: And frō the poynt A drawe two right lines ABD, and AC: of which let ABD cut the circle in the poynt B, & let the other fall vpon it. And let that which is contained vnder the lines AD and AB, be equall to the square of the line AC. Then I say, that the line AC toucheth the circle. For first if the line ABD do passe by the centre, draw the right line CE. And (by the 6. of the second) that which is contayned vnder the lines AD and AB together with the square of the line EB, that is, with the square of the line EC (for the lines EB and EC are equall) is equall to the square of the line AE. But that which is contained vnder the lines AD and AB, is supposed to be equall to the square of the line AC: Wherefore the square of the line AC together with the square of the line CE, is equall to the square of the line AE. Wherefore (by the last of the first) the angle at the point C is a right angle. Wherfore (by the 18. of this boke) the line AC toucheth the circle.

But if the line ABD doo not passe by the centre, drawe from the point A the line AD, in which let be the centre E. And forasmuch as that which is contained vnder this whole line and his outward part, is equall to that which is contained vnder the lines AD and AB by the first Corollary before put, therefore the same is equall to the square of the line AC, wherefore the angle ECA is a right angle as hath before bene proued in the first part of this Proposition. And therfore the line AC toucheth the circle: Which was required to be proued.

Notes

• * 1.1

  This proposi∣tion is the cō∣uerse of the former.
• * 1.2

  Construction.
• * 1.3

  Demonstra∣tion.
• * 1.4

  An other de∣monstration after Pelita∣rius.