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 9. Theoreme. The 9. Proposition. If the side of an equilater hexagon, and the side of an equilater decagon or ••••u••gled figure, which both are inscribed in one & the selfe same circle, be added together: the whole right line made of them is a line diuided by a•• extreame and meane proportion, and the greater segment of the same is

the side of the hexagon.

SVppose that there be a circle ABC. And let the side of a decagon or tenangled figure inscribed in the circle ABC, be BC, and let the side of an hexagon or sixe angled figure inscribed in the same circle, be CD. And let the lines BC and CD be so ioyned together directly that they both make one right line, namely, BD. Then I say that the line BD is diuided by an extreame

[illustration]

and meane proportion in the point C: and that the greater segmēt therof is the line CD.* 1.1 Take (by the 1. of the third) the centre of the circle. And let it be the point E: and draw these right lines EB, EC, and ED. And extend the line BE to the point A. Now forasmuch as BC is the side of an equilater decagon,* 1.2 therefore the circumference or semicir∣cle ACB is quintuple to the circumference CB. Where∣fore the circumference AC is quadruple to the circumfe∣rence CB. But as the circumference AC is to the circum∣ference CB, so is the angle AEC to the angle CEB, by the last of the sixth. Wherefore the angle AEC is quadruple to the angle CEB. And forasmuch as the angle EBC is equall to the angle ECB (by the 5. of the first,) for the line EB is equall to the line EC, by the diffinition of a circle, therefore the angle AEC is double to the angle ECB, by the 32. of the first. And forasmuch as the right line EC is equall to the right line CD, by the corollary of the 15. of the fourth (for either of them is equall to the side of the hexagon inscribed in the circle ABC) therefore the angle CED is equall to the angle CDE wherefore the angle ECB is double to the angle EDC, by the 32. of the first. But it is proued that the angle AEC is double to the angle ECB, wherefore the angle AEC is qua∣druple to the angle EDC. And it is proued that the angle AEC is quadruple to the angle BEC. Wherefore the angle EDC is equall to the angle BEC. And the angle EBD is com∣mon to the two triangles BEC and BED: wherefore the angle remayning BED is equall to the angle remayning ECB, by the corollary of the 32. of the first. Wherefore the triangle EBD is equiangle to the triangle EBC. Wherfore, by the 4. of the sixt, proportionally, as the line BD is to the line BE, so is the line BE to the line BC. But the line EB is equall to the line CD. Wherefore as the line BD is to the DC, so is the line DC to the line CB. But the line BD is greater then the line DC: wherefore also the line DC is greater then the line C∣B. Wherefore the right line BD is diuided by an extreame and meane proportion in the point C: and his greater segment is DC. If therefore the side of an equilater hexagon, and the side of an equilater decagon or tenangled figure, which both are inscribed in one and the selfe same circle, be added together•• the whole right line made of them, is a line diuided by an extreame and meane proportion, and the greater segment of the same is the side of the hexa∣gon: which was required to be proued.

A Corollary added by Flussas.

* 1.3Hereby it is manifest, that the side of an exagon inscribed in a circle being cut by an extreame and meane proportion, the greater segment thereof is the side of the decagon inscribed in the same circle. For if from the right line DC be cut of a right line equall to the line CB, we may thus reason, as the whole DB is to the whole DC, so is the part taken away DC to the part taken away CB: wherefore by the 19. of the fifth, the residue is to the residue as the whole is to the whole. Wherefore the line D∣C is cutte like vnto the line DB: and therefore is cut by an extreame and meane proportion.

Campane putteth the conuerse of this proposition after this maner.

If a line be diuided by an 〈◊〉〈◊〉 and ••••ane proportion, of 〈◊〉〈◊〉 circle the greater segment is the side of an equilater Hexagon, of the same shall the lesse segment be the side of an equilater Decagon. And of what circle the lesse segment is the side of an equilater Decagon, of the same is the greater seg∣ment the side of an equilater Hexagon.

For the former figure remayning, suppose that the line BD be diuided by an extreme and meane proportion in the point C: and let the greater segment therof be DC. Thē I say that of what circle the line DC is the side of an equilater Hexagon, of the same circle is the line CB the side of an equi∣later decagon: and of what circle, the line BC is the side of an equilater Decagon, of the same is the line DC the side of an equilater Hexagon.* 1.4 For if the line

[illustration]

DC be the side of an Hexagon inscribed in the circle, then by the corollary of the 15. of the fourth, the line DC is e∣qual to the line BE. And forasmuch as the proportiō of the line BD to the line DC is as the proportion of the line D∣C to the line CB, by supposition: therfore (by the 7. of the fifth) the proportion of the line BD to the line BE, is as the proportion of the line BE to the line BC. Wherefore (by the 6. of the sixth) the two triangles DEB, EBC are equiangle (for the angle B is common to eche triangle). Wherefore the angle D is equall to the angle CEB: for they are subtended of sides of like proportion. And foras∣much as the angle AEC is quadruple to the angle D (by the 32. of the first twise taken, and by the 5. of the same) therefore the same angle AEC is quadruple to the angle CEB. Wherefore (by the last of the sixth) the circumfe∣rence AC is quadruple to to the circumferēce CB. Wher∣fore the line BC is the side of a decagon inscribed in the circle ACB.

But now if the line BC be the side of a decagon in∣scribed in the circle ABC, the line CD shalbe the side of an Hexagon inscribed in the same circle.* 1.5 For let DC be the side of an Hexagon inscribed in the circle H. Now by the first part of this proposition the line BC shalbe the side of a decagon inscribed in the same circle. Suppose that in the two circles AC∣B and H be inscribed equilater decagons, all whose sides shalbe equall to the line CB. And forasmuch as euery equilater figure inscribed in a circle is also equiangle, therefore bothe the decagons are equi∣angle. And forasmuch as al the angles of the one taken to∣gether

[illustration]

are equall to al the angles of the other taken tog••∣ther, as it is easy to be proued by that which is added af∣ter the 32. of the first, therefore one of these decagons is equiangle to the other: and therfore the one is like to the other by the diffinition of like superficieces. And for that if there be two like rectiline figures inscribed in two cir∣cles, the proportion of the sides of like proportion of those figures, shalbe as the proportion of the Diameters of those circles, as it is easy to proue by the corollary of the 20. of the sixth, and first of this book••: but the sides of the like deca••ons inscribed in the two circles ABC and H are equall: therefore theyr Diameters also are equall. Wherefore also theyr semidiameters are equall. But the s••midi••meters and the side of the Hex••gon are equall, by the ••o••ollary of the 1••. of the fourth. Wherefore the line DC is the side of an hexagon in the circle ABC, as also it is the side of an hexagon inscribed in the circle F, which is equall to the ci••cle ABC: which was required to be proued.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  This Corollary is the 3. propo∣sition of the ••4. booke af∣ter Campane.
• * 1.4

  Demonstra∣tion of the first part.
• * 1.5

  Demonstrati∣on of the se∣cond part.