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 1. Theoreme. The 1. Proposition. * 1.1A perpendicular line drawen from the centre of a circle to the side of a Pentagon described in the same circle: is the halfe of these two lines, name∣ly, of the side of an hexagon figure, and of the side of a decagon figure be∣ing both described in the selfe same circle.

SVppose that the circle be ABC.* 1.2 And let the side of an equilater Pentagon described in the circle ABC, be BC. And (by the 1. of the third) take the centre of the circle, and let the same be D. And (by the 12. of the first) from the point D draw vnto the line BC a perpendicular line DE. And extend the right line DE direct∣ly to the point F. Then I say, that the line DE (which is drawen from the centre to BC the side of the pentagon) is the halfe of the sides of an hexagon and of a decagon taken together and descri∣bed in the same circle. Draw these right lines DC and CF. And vnto the line EF put an equall line GE. And draw a right

[illustration]

line from the point G to the point C.* 1.3 Now forasmuch as the circumference of the whole circle is quintuple to the circū∣ference BFC (which is subtended of the side of the penta∣gon) and the circumference ACF is the halfe of the cir∣cumference of the whole circle, and the circumference CF (which is subtended of the side of the decagon) is the halfe of the circumference BCF: therefore the circumference ACF is quintuple to the circumference CF (by the 15. of the ••i••t). Wherefore the circumference AC is qradruple to the circumference FC. But as the circumference AC is to the circumference FC, so is the angle ADC to the angle FDC, by the last of the sixt. Wherefore the angle ADC is quadruple to the angle FDC. But the angle ADC is double to the angle EFC, by the 20. of the third: Wherefore the angle EFC is double to the angle GDC. But the angle EFC is equall to the angle EGC, by the 4. of the first. Wherfore the angle EGC is double to the an∣gle EDC. Wherefore the line DG is equall to the line GC (by the 32. and 6. of the first). But the line GC is equall to the line CF, by the 4. of the first. Wherfore the line DG is equall to the line CF. And the line GE is equall to the line EF (by construction). Wherefore the line DE is equall to the lines EF and FC added together. Vnto the lines EF and FC adde the line DE. Wherefore the lines DF and FC added together, are double to the line DE. But the line DF is equall to the side of the hexagon: and FC to the side of the decagon.

Wherefore the line DE is the halfe of the side of the hexagon, and of the side of the decagon being both added together and described in one and the selfe same circle.

It is manifest * 1.4 by the Propositions of the thirtenth booke, that a perpendicular line drawen from the centre of a circle to the side of an equilater triangle described in the same circle; is halfe of the semidiameter of the circle. Wherefore by this Proposition, a perpendicular dra∣wen from the c••ntre of a circle to the side of a Pentag••n, is equall to the perpendicular drawen from the centre to the side of the triangle, ••nd to halfe of the side of the decagon described in the same circle.

Notes

• * 1.1

  First proposi∣tion after Flussas.
• * 1.2

  Construction.
• * 1.3

  Demonstra∣tion.
• * 1.4

  This is mani∣fest by the 12. propositiō of the thirtenh booke as Campane well gathereth in a Corollary of the same.