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 16. Probleme. The 16. Proposition. In a circle geuen to describe a quindecagon or figure of fiftene angles, equilater and equiangle.

SVppose that the circle geuen be ABCD. It is required in the circle ABCD to describe a figure of fiftene angles consisting of equall sides and of equall angles. Describe in the circle ABCD the sides of an e∣quilater triangle, and let the same be AC, and in y^e arke AC describe the side of an equilater pentagon and let the same be AB.* 1.1 Now then of such e∣quall partes wherof the whole circle ABCD containeth fiftene, of such partes I say, the circumference ABC being the third parte of the circle shall contayne fiue. And the

[illustration]

circumference AB being the fift part of a circle shall contain thre, wherefore the residue BC shall containe two. Deuide (by the 30. of the first) the arke BC into two equall partes in the point E.* 1.2 Wher¦fore either of these circum∣ferences BE & EC is the fiftene part of the circle ABCD. If therfore there be drawn right

lines from B to E, and from E to C, and then beginning at the point B or at the point C there be applied into the circle ABCD right lines equall vnto EB or EC, and so continuing till ye come to the point C if you began at B, or to y^e point B if you began at C, and there shall be described in the circle ABCD a figure of fiftene angles equilater and equiangle: which was required to be done. And in like sort as in a pentagon, if by the pointes where the circle is deuided, be drawen right lines touching the circle in the said pointes, there shall be described about y^e circle a figure of fiftene angles equilater & equiangle. And in like sort by y^e selfe same obseruations that were in Pentagons, we may in a figure of fiftene angles geuen being equilater and equiangle either inscribe, or circumscribe a circle.

* 1.3¶ An addition of Flussates to finde out infinite figures of many angles.

If into a circle from one poynt be applyed the sides of two* 1.4 Poligonon figures: the ex∣cesse of the greater arke aboue the lesse, shall comprehend an arke contayning so many sides of the Poligonon figure to be inscribed by how many vnities the denomination of the Poligonon figure of the lesse side excedeth the denomination of the Poligonon figure of the greater side: and the number of the sides of the Poligonon figure to be inscribed is produced of the multiplication of the denominations of the foresayd Poligonon figures the one into the other.

As for example. Suppose that into the circle ABE be applyed the side of an equi∣later and equiangle Hexagon figure (by the 15. of thys booke) which let be AB: and likewise the side of a Pentagon (by the 11. of this booke) which let be AC: and the side of a square (by the 6. of thys booke) which let be AD: and the side of an equilater tri∣angle (by the 2. of this booke) which let be AE. Then I say, that the excesse of the arke AD aboue the arke AB, which excesse is the arke BD, contayneth so many sides of the Poligonon figure to be inscribed, of how many vnities the denominator of the Hex∣agon AB, which is sixe, excedeth the denominator of the square AD, which is foure. And forasmuch as that excesse it two vni∣ties,

[illustration]

therfore in BD there shall be two sides. And the denominator of the Poligonon fi∣gure which is to be inscribed shall be pro∣duced of the multiplication of the deno∣minators of the foresayd Poligonon fi∣gures, namely, of the multiplication of 6. into 4. which maketh 24. which number is the denominator of the Poligonon fi∣gure, whose two sides shall subtend the arke BD. For of such equall partes wherof the whole circumference cōtayneth 24, of such partes I say, the circumference AB con∣tayneth 4, and the circumference AD contayneth 6. Wherefore if from AD which subtendeth 6. partes be taken away 4. which AB subtendeth, there shall re∣mayne vnto BD two of such partes of which the whole contayneth 24. Wherfore of an Hexagon and a square is made a Poligonon figure of 24. sides. Likewyse of the Hexagon AB and of the Pentagon AC shall be made a Poligonon figure of 30.

sides, one of whose sides shall subtend the arke BC. For the denomination of AB which is 6. excedeth the denomination of AC which is 5. onely by vnitie. So also forasmuch as the denomination of AB which is 6. excedeth the de∣nomination of AE which is 3. by 3. therefore the arke BE shall contayne 3. sides of a Poligonon figure of .18. sides. And obseruing thys selfe same methode and order, a man may finde out infinite sides of a Poligonon figure.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  An addition of Flussates.
• * 1.4

  A Poligonon figure is a fi∣gure consisting of many sides.