As here thou seest.
22. If the sides of a cube be with right lines cut into two equall parts, and three bisegments of the bisecants in the abbuting plaines, neither meeting one the other, nor parallell one unto another, two of one, the third of that next unto the remainder, be so proportionally cut that the lesser segments doe bound the bisecant: three lines without the cube perpendicular unto the sayd
plaines from the points of the proportionall sections, equall to the greater segment knit together, two of the same bisecant, betweene themselves and with the next angles of cube; the third with the same angles, they shall comprehend a dodecaedrum. 17 p xiij.
Let there be two plaines for a cube for all, that one quinquangle for twelve may be described, and they abutting one upon another, aeio, and euyi, having their sides halfed by the bisecantes, sr, lm, rn, jv: And the three bisegments or portions of the bisegments lm, and rn, neither concurring or meeting, nor parallell one to another; two of the said lm, to wit, fl, and fm: The third next unto the remainder, that is lr. And let each bisegment be cut proportionally in the points d, c, g; so that the lesser segments doe bound the bisecant, to wit, dl, cm, and gr. Lastly let there be three perpendiculars from the points db, cg, to the said d, cp, gz: And the two first knit one to another, by bp: And againe with the angles of the cube, by be, and pi: The third knit with the same angles, by ze, and zi: And let all the plaines be made up. I say first, that the five sides bp, pi, iz, ze, and eb are equall; Because, every one of them severally are the doubles of the same greater segment. For in drawing the right lines de and eg, ig, it shall be plaine of two of them; And after the same manner of the rest. First therefore cd, and bp, are equall by the [6 e xxj], and by the [27 e v]. Therefore bp, is the double of the greater segment. Then the whole fl, cut proportionally, and the lesser segment dl, they are by the [7 e xiiij], of treble power to the greater fd, that is, by the fabricke db. Therefore le wich is equall to lf, the line cut, and ld, are of treble power to the same db: But by the [9 e xij], de is of as much power as le, and ld too.
Therefore de is of treble power to db. Therefore both ed, and db, are of quadruple power to db. But be, by the [9 e xij], is of as much power as ed, and db. And therefore be, is of quadruple value to db: And by the [14 e xij], it is the double of the said db. Therefore the two sides eb, and bp, are equall: And by the same argument pi, iz, and ze, are equall. Therefore the quinquangle is equilater.
I say also that it is a Plaine quinquangle: For it may be said to be an oblique quinquangle; and to be seated in two plaines. Let therefore fh be parallell to db, and cp: and be equall unto them. And let hz, be drawne: This hz shall be cut one line, by the [14 e vij]. For as the whole tr, that is rf, is unto the greater segment that is to fh: so fh, that is zg, is unto gr. And two paire of shankes fh, gr, fc, gz, by the [6 e xxj], are alternely or crosse-wise parallell. Therefore their bases are continuall.
Hitherto it hath beene prooved that the quinquangle made is an equilater and plaine: It remaineth that it bee prooved to be Equiangled. Let therefore the right lines ep, and ec, be drawne: I say that the angles, pbe, and ezi, are equall: Because they have by the construction, the bases of equall shankes equall, being to wit in value the quadruple of le. For the right line lf, cut proportionally, and increased with the greater segment df, that is fc, is cut also proportionally, by the [4 e xiiij], and by the [7 e xiiij], the whole line proportionally cut, and the lesser segment, that is cp, are of treble value to the greater fl, that is of the sayd le. Therefore el, and lc, that is ec, and cp, that is ep, is of quadruple power to el: And therefore by the [14 e xij], it is the double of it: And ei, it selfe in like manner, by the fabricke or construction, is the double of the same. Therefore the bases are equall. And after the same manner, by drawing the right lines id, and ib, the third angle bpi, shall be concluded to be equal to the angle ezi. Therefore by the [13 e xiiij], five angles are equall.