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. Probleme. The 16. Proposition. Two circles hauing both one and the selfe same centre being geuen, to in∣scribe in the greater circle a poligonon figure, which shall consist of equall and euen sides, and shall not touch the superficies of the lesse circle.

SVppose that there be two circles ABCD, and EFGH hauing one & the selfe same centre, namely, K. It is required in the greater circle which let be ABCD to inscribe a poligonon figure which shalbe of equal and euen sides and not touch the circle EFGH. Drawe by the centre K a right line BD.* 1.1 And (by the 11. of the first) from the point G rayse vp vnto the right line BD a perpendicular line AG, and ex∣tend it to the point C. Wherefore the line AC toucheth the circle EFGH (by the 15. of the third). Now therfore if (by the 30. of the third) we diuide the circumference BAD into two equall partes, and againe the halfe of that into

[illustration]

two equal partes, and thus do cōtinually, we shall (by the corollary of the 1. of the tenth) at the length leaue a certayne circumference lesse then the circumference AD. Let the circumference left be LD. And from the point L. Drawe (by the 12. of the first) vnto the line BD a perpendi∣culare line* 1.2 LM,* 1.3 and extende it to the point N. And draw these right lines LD and DN. And forasmuch as the angles DML, and DMN are right angles, therfore (by the 3. of the third) the right line BD diuideth the right line LN into two equall parts in the pointe M. Wherfore (by the 4. of the first) the rest of the sides of the triangles DML, and DMN, namely, the lines DL, and DN shalbe equall. And forasmuch as the line AC is a parallell to the LN (by the 28. of the first): But AC toucheth the circle EFGH, wherfore the line LM toucheth not the circle EFGH, and much lesse do the lines LD, and DN touch the circle EFGH. If therefore there be applied right lines equall to the line LD continually into the circle A∣BCD (by the 1. of the fourth) there shalbe described in the circle ABCD a poligonon figure which shalbe of equall, and* 1.4 euen sides, and shall not touch the lesse circle, namely, EFGH: (by the 14. of the third or by the 29.) which was required to be done.

¶ Corollary.

Hereby it is manifest that a perpendicular line drawen from the poynt L to the line BD toucheth not one of the circles.

¶An Assumpt added by Flussas.

If a Sphere be cut of a playne superficies, the common section of the superficieces, shall be the cir∣cumference of a circle.

Suppose that the sphere ABC be cut by the playne superficies AEB, and let the centre of the sphere be the poynt D.* 1.5 And from the poynt D, let there be drawne vnto the playne superficies AEB a perpendicular line (by the 11. of the eleuenth) which let be the line DE. And from the poynt E draw in the playne superficies AEB vnto the common section of the sayd superficies and the sphere, lines how many so euer, namely, EA and EB. And draw these lines DA and DB. Now forasmuch as the right angles DEA and DEB are equall (for the line DE is e∣rected

[illustration]

perpendicularly to the playne superficies).* 1.6 And the right lines DA & DB which subtend those angles are by the 12. defini••ion of the eleuenth equall: which right lines moreouer (by the 47. of the first) do contayne in power the squares of the lines DE, EA, and DE, EB: if therfore from the squares of the lines DE, EA, and DE, EB ye take away the square of the line DE which is cōmon vnto them, the residue namely the squares of the lines EA & EB shall be equal. Wherfore also the lines EA and EB are equall. And by the same reason may we proue, that all the right lines drawne from the poynt E to the line which is the cōmon section of the superficies of sphere and of the playne su∣perficies, are equall. Wherefore that line shall be the cir∣cumference of a circle, by the 15. definition of the first. * 1.7 But if it happen the plaine superficies which cutteth the sphere, to passe by the centre of the sphere, the right lines drawne from the centre of the sphere to their common section, shal•• be equall by the 12. definition of the eleuenth. For that common section is in the superficies of the sphere. Wherefore of neces••itie the playne superficies comprehended vnder that line of the common section shall be a circle, and his centre shall be one and the same with the centre of the sphere.

Iohn Dee.

Euclide hath among the definition of solides omitted certayne, which were easy to conceaue by a kinde of Analogie. As a segment of a sphere, a sector of a sphere, the vertex, or toppe of the segment of a sphere: with such like. But that (if nede be) some farthe•• light may be geuē, in this figure next before,* 1.8 vn∣ders••and a segment of the sphere ABC to be that part of the sphere contayned betwen the circle AB, (whose center is E) and the sphericall superficies AFB. To which (being a lesse segment) adde the cone ADB (whose base is the former circle: and toppe the center of the sphere) and you haue DAFB a sector of a sphere, or solide sector (as I call it). DE extended to F, sheweth the top or vertex of the seg∣ment, to be the poynt F and EF is the altitude of the segment sphericall. Of segmentes, some are grea∣ter thē the halfe sphere, some are lesse. As before ABF is lesse, the remanent, ABC is a segment greater then the halfe sphere.

¶A Corollary added by the same Flussas.

By the foresayd assumpt it is manifest, that if from the centre of a sphere the lines drawne per∣pendicularly vnto the circles which cutte the sphere, be equall: those circles are equall. And the perpendicular lines so drawne fall vpon the centres of the same circles.

For the line which is drawne frō the centre of the sphere to the circumference, containeth in pow¦er, the power of the perpendicular line, and the power of the line which ioyneth together the endes of those lines. Wherfore frō that square or power of the line from the center of the sphere to the circum∣ference or cōmon sectiō drawne, which is the semidiameter of the sphere, taking away the power of the perpendicular, which is cōmon to them ••ound about, it followeth, that the residues how many so euer they be, be equall powers, and therefore the lines are equall the one to the other. Wherefore they will describe equall circles, by the first definition of the third. And vpon their centers fall the perpendicular lines by the 9. of the third.

* 1.9And those circles vpon which falleth the greater perpendicular lines are the lesse circles. For the pow∣ers of the lines drawne from the centre of the sphere to the circumference being alwayes one and e∣quall, to the powers of the perpendicular lines and also to the powers of the lines drawne from the cen∣tres of the circles to their circumference, the greater that the powers of the perpendicular lines taken away from the power contayning them both are, the lesse are the powers and therefore the lines re∣mayning, which are the semidiameters of the circles and therefore the lesse are the circles which they describe. Wherefore if the circles be equall, the perpendicular lines falling from the centre of the sphere vpon thē, shall also be equall. For if they should be greater or lesse, the circles should be vnequall as it is before manifest. But we suppose the perpendiculars to be equall.* 1.10 Also the perpendicular lines falling vpon those bases are the least of all, that are drawne from the centre of the sphere: for the other drawne

from the centre of the sphere to the circumference of the circles, are in power equall both to the pow∣ers of the perpendiculars and to the powers of the lines ioyning these perpendiculars and these subten∣dent lines together: making triangles rectangleround about: as most easily you may conceaue of the figure here annexed.

A the Center of the Sphere.

[illustration]

AB the lines from the Center of the Sphere to the Circumfe∣rence of the Circles made by the Section.

BCB the Diameters of Circles made by the Sections.

AC the perpendiculars from the Center of the Sphere to the Circles•• whose diameters BC∣B are one both sides or in any situation els.

CB the Semidiameters of the Cir∣cles made by the Sections.

AO a perpendicular longe then AC: and therefore the Semi∣diameter OB is lesse.

ACB, & AOB triangles rectangle.

Notes

• * 1.1

  Construction.
• * 1.2

  Note this L∣M, because of KZ in the next proposition, and here the point M: for the point Z in the next demonstration.
• * 1.3

  I. Dee.
• * 1.4

  For that the sections were made by the number two: that is, by ta∣king halues, and of the resi∣due the hal••e•• and so, to LD, being an halfe, and a residue which shall be a cōmon measure backe againe, to make sides of the Poligonon figure.
• * 1.5

  Construction.
• * 1.6

  Demonstra∣tion.
• * 1.7

  The circles so made: or so con∣sidered in the sphere, are cal∣led the greatest circles: All o∣ther, not ha∣uing the center of the sphere, to be their center also•• are called lesse circles.
• * 1.8

  Note these de∣scriptions.
• * 1.9

  An other Co∣rollary.
• * 1.10

  An other Co∣rollary.