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 12. Theoreme. The 12. Proposition. If in a circle be described an equilater triangle: the square made of the side of

the triangle, is treble to the square made of the line, which is drawen from the centre of the circle to the circumference.

SVppose that ABC be a circle, and in it describe an equilater triangle ABC. Then I say that the square made of the side of the triangle ABC is treble to the square made of the line drawen from the center of the circle ABC to the circum∣ference.* 1.1 Take (by the 1. of the third) the centre of the circle, and let the same be D. A••d draw a right line from the point A to the poynt D, and extend it to the point E. And draw a right line from the point B to the poynt E.* 1.2 Now forasmuch as the triangle ABC is equilater, t••••refore eche of these three circumferences AB,

[illustration]

AC, & BEC is the third part of the whole circum∣ference of the circle ABC wherefore the circumfe∣rence BE is the sixth part of the circūference of the circle [for the circumferēce of the semicircle ABE is equall to the circumference of the semicircle A∣CE, from which taking away equal circumferences AB and AC, the circumference remayning BE shalbe equal to the circumference remayning EC]: wherefore the right line BE is the side of an equi∣later hexagon figure described in the circle. Where∣fore it is equall to the line drawen from the centre of the circle to the circumference, that is vnto the line DE (by the corollary of the 15. of the sixth). And forasmuch as the line AE is double to the line DE, therefore the square of the line AE is quadruple to the square of the line DE (by the 4. of the second): that is, to the square of the line BE. But the square of the line AE is equall to the squares of the lines AB, and BE (by the 47. of the first) for the angle ABE is (by the 31. of the third) a right angle. Wherfore the squares of the line AB & BE are qua∣druple to the square of the line BE. Wherefore taking away the square of the line BE, the squa••e of the line AB sh••lbe treble to the square of BE: but the line BE is equall to the line DE. Wh••r••fore the square of the line AB is treble to the square of the line DE. Wherefore the square made of the side of the triangle, is treble to the square made of the line drawen frō the centre of the circle to the circumference: which was required to be proued.

A Corollary added by Campane.

Hereby it is manifest, that the line BC, which is the side of the equilater triangle, diuideth the semi∣diameter DE into two equall parts. For let the poynt of the diuision be F. And suppose a line to be drawen from the poynt D to the B, and an other from the poynt D to the poynt C. Now it is manifest (by the 4. of the first) that the line BF is equall to the line FC, and therefore (by the 3. of the third) all the angles at the poynt F are ••ight angles. Wherefore (by the 47. of the first) the square of the line BD is equall to the squares o•• the line•• ••F and FD, and by the same the square of the line BE is equall to the squares of the lines BF, and FE: b••t the line BD is equall to the line BE (as hath before bene proued). Wherefore by the common sentence the twoo squares of the two lines BF and FD are equall to the two squares of the line•• BF, and FE. Wherefore taking away the square of the line B∣F which is cōmon to them both: the residue, namely, the square of the line DF shalbe equall to the re∣sidue, namely, to the square of the line FE. Wherfore also the line FD is equal to the line FE. Wher∣fore hereby it is manifest that a ••erpendicular line drawen from the centre of a circle to the side of an equilater triangle inscribed in it, is equall to halfe of the line drawen from the centre of the same circle, to the circumference thereof.

A Corollary added by Flussas.

The side of an equilater triangle is in power sesquitertia to the perpendicular line which is drawē from one of the angles to the opposite side. For of what parts the line AB contayneth in power 12. of

such parts the line BF which is the halfe of AB contayned in power 3.* 1.3 Wherefore the residue, namely, the perpendicular line AF contayneth in power of such parts 9. (for the squares of the lines AF, and BF are by the 47. of the first equall to the square of the line AB). Now 1••. to 9. is sesquitertia•• wherfore the power of the line AB is to the power of the line AF in sesquitertia proportion.

Moreouer the side of the triangle is the meane proportionall betwene the diameter and the per∣pendicular line: For (by the Corollary of the 8. of the sixth) the line AE is to the line AB as the line AB is to the line AF.

Farther the perpendicular line drawen from the angle diuideth the base into two equal parts and passeth by the center.* 1.4 For if there should be drawen any other right line frō the point A to the poynt F, thē that which is drawen by the point D, two right lines should include a superficies, which is impossi∣ble. Wherefore the contrary followeth, namely, that the line, which being drawen from the angle pas∣seth by the center, is a perpendicular line to the base (by the 3. of the third).

Notes

• * 1.1

  Constr••yction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  This Corollary is the 11. prop••∣sition of the 14. booke after Campane.
• * 1.4

  This Corollary is the 3. Corolla∣ry after the 17. proposition of the 14 booke af∣ter Campane.