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

¶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.

* 1.1And 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.* 1.2 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

  An other Co∣rollary.
• * 1.2

  An other Co∣rollary.