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 Flussas.

Euery playne superficies extended by the center of a parallelipipedon, diuideth that solide into two equall partes: and so doth not any other playne superficies not extended by the center.

For euery playne extended by the center, cutteth the diameter of the parallelipipedon in the cen∣ter into two equall partes. For it is proued, that playne superficieces which cutte the solide into two equall partes, do cut the dimetient into two equall partes in the center. Wherefore all the lines drawen by the center in that playne superficies shall make angles with the dimetient. And forasmuch as the di∣ameter falleth vpon the parallel right lines of the solide, which describe the opposite sides of the sayde solide, or vpon the parallel playne superficieces of the solide, which make angels at the endes of the diameter: the triangles contayned vnder the diameter, and the right line extended in that playne by the center, and the right line, which being drawen in the opposite superficieces of the solide, ioyneth together the endes of the foresayde right lines, namely, the ende of the diameter, and the ende of the line drawen by the center in the superficies extended by the center, shall alwayes be equall, and equi∣angle, by the 26. of the first. For the opposite right lines drawen by the opposite playne superficieces of the solide do make equall angles with the diameter, forasmuch as they are parallel lines, by the 16. of this booke. But the angles at the cēter are equall, by the 15. of the first, for they are head angles: & one side is equall to one side, namely, halfe the dimetient. Wherefore the triangles contayned vnder e∣uery right line drawen by the center of the parallelipipedon in the superficies, which is extended also by the sayd center, and the diameter thereof, whose endes are the angles of the solide, are equall, equi∣later, & equiangle (by the 26. of the first). Wherfore it followeth that the playne superficies which cut∣teth the parallelipipedon, doth make the partes of the bases on the opposite side, equall, and equiangle, and therefore like, and equall both in multitude, and in magnitude: wherefore the two solide sections of that solide, shalbe equall and like, by the 8. diffinition of this booke. And now that no other playne superficies, besides that which is extended by the center, deuideth the parallelipipedon into two equall partes, it is manifest: if vnto the playne superficies which is not extended by the center, we extend by the center a parallel playne superficies (by the Corollary of the 15. of this booke). For forasmuch as that superficies which is extended by the center, doth deuide the parallelipipedō into two equall par••••: it is manifest, that the other playne superficies (which is parallel to the superficies which deuideth the solide into two equall partes) is in one of the equall partes of the solide: wherefore seing that the whole is euer greater then his partes, it must nedes be that one of these sections is lesse then the halfe of the solide, and therefore the other is greater.

For the better vnderstanding of this former proposition, & also of this Corollary added by Flussas, it shalbe very nedefull for you to describe of pasted paper or such like matter a parallelipipedō or a Cube, and to deuide all the parallelogrāmes therof into two equall parts, by drawing by the c••̄ters of the sayd parallelogrammes (which centers are the poynts made by the cutting of diagonall lines drawen frō th•• opposite angles of the sayd parallelogrāmes) lines parallels to the sides of the parallelogrāmes: as in the former figure described in a plaine ye may see, are the sixe parallelogrāmes DE, EH, HA, AD, DH, and CG, whom these parallel lines drawen by the cēters of the sayd parallelogrāmes, namely, XO, OR, PR, and PX, do deuide into two equall parts: by which fower lines ye must imagine a playne superfi∣cies to be extended, also these parallel lynes KL, LN, NM, and M••, by which fower lines likewise y•• must imagine a playne superficies to be extended ye: may if ye will put within your body made thus of pasted paper, two superficieces made also of the sayd paper, hauing to their limites lines equall to the foresayde parallel lines: which superficieces must also be deuided into two equall partes by parallel

lines drawen by their centers, and must cut the one the other by these parallel lines. And for the dia∣meter of this body, extēd a thred from one angle in the base of the solide to his opposite angle, which shall passe by the center of the parallelipipedon, as doth the line DG in the figure before described in the playne. And draw in the base and the opposite superficies vnto it, Diagonall lines, from the angles from which is extended the diameter of the solide: as in the former description are the lines BG and DE. And when you haue thus described this body, compare it with the former demonstration, and it will make it very playne vnto you, so your letters agree with the letters of the figure described in the booke. And this description will playnely set forth vnto you the corollary following that proposition. For where as to the vnderstanding of the demonstration of the proposition the superficieces put within the body were extended by parallel lynes drawen by the cēters of the bases of the parallelipipe∣don: to the vnderstanding of the sayd Corollary, ye may extende a superficies by any other lines dra∣wen in the sayd bases, so that yet it passe through the middest of the thred, which is supposed to be the center of the parallelipipedon.