Seeing therefore a Mingled ordinate pyramidate is thus made or compounded of pyramides the geodesy of it shall be had from the Geodesy of the pyramides compounding it: And one Base multiplyed by the number of all the bases shall make the surface of the body. And one Pyramis by the number of all the pyramides; shall make the solidity. The base of the pyramis appeareth to the eye: The heighth lieth hidde within, but it is discovered by a right angle triangle, whose base is the semidiagony or halfe diagony, the shankes the ray of the circle, and the perpendicular of the heighth. Therefore subtracting the quadrate of the ray, from the quadrate of the halfe diagony the side of the remainder, by the 9 e xij. shall be the heighth. But the ray of the circle shall have a speciall invention, according to the kindes of the base, first of a triangular, and then next of a quinquangular. The division of a Polyhedron ariseth from the bases upon which it standeth. As is manifest by the 12 e. xviij. And this is the invention or way to finde out the circular ray for an octoedrum, and an icosoedrum. This division also ariseth from the bases of the figures. As here thou seest, in this Monogrammum and solidum, that is lines and solid octahedrum. Therefore And For foure angles of a Tetraedrum are equall to three angles of the Octoedrum: And therefore 12. are equall to And This construction is easie, as it is manifest in the example following: Where thou seest as it were two equilater and equall triangles of a double pentaedrum to cut one another. For the perpendicular yu, and su, with the semidiagoni's, ua, uo, ui, ue, shall be made equall by the 2 e vij, the eight sides ya, ye, yo, yi, se, si, sa, so; And also eight triangles. Therefore As is manifest by the 9 e xij. And As in the figure following, the side is 6. The quadrate is 36. the double is 72. whose side 8.8/17, is the diagony. And from hence doth arise the geodesy of the octaedrum. For the semidiagony is 4.4/17. whose quadrate is 17.171/289. And the quadrate of 6, the side of the equilater triangle, being of treble power to the ray, by the 12 e, xviij. is 36. And the side of 12. the third part 3.3/7 is the ray of the circle. Wherefore 8.8/17. that is 5.21/289. is the quadrate of the perpendicular, whose side 2.1/5 is the height of the same perpendicular: whose third part againe 11/25. multiplied by 15.18/31. the triangular base doe make 11.66/155 for one of the eight pyramides: Therefore the same 11.66/155 multiplied by eight, shall make 91.63/155 for the whole octoedrum. Therefore And This fabricke is ready end easie, as is to be seene in this example following. For there shall be made 20 triangles, both equilaters and equall. Let there be therefore two ordinate quinquangles, the first aeiou; The second ysrlm; each of whose sides let them subtend two sides of a decangle; to wit, utym, let it subtend ya, and am. Then let there be five perpendiculars from the angles of the second quinquangle yj, sy, rv, In like manner also shall it be proved of the five upper triangles, by drawing the right lines dy and cn which as afore (because they knit together equall parallells, to wit, dc, and yn) they shall be equall. But dy, is the side of a sexangle: Therefore cn, shall be also the side of a sexangle: And cg, is the side of a decangle: Therefore an, whose power is equall to both theirs by the 9 e xij. shall by the converse of the 15 e xviij, be the side of a quinquangle: And in like manner gt, shall be concluded to be the side of a quinquangle. Wherefore ngt, is an equilater: And the foure other shall likewise be equilaters. The other five triangles beneath shall after the like manner be concluded to be equilaters. Therefore one shall be for all, to wit, ibe, by drawing the raies di, and de. For ib, This is the fourth example of irrationality, or incommensurability. The first was of the Diagony and side of a square or quadrate. The second was of the segments of a line proportionally cut. The third of the Diameter of a circle and the side of a quinquangle. And For by the 13 e, xviij. the line continually made of the side of the sexangle and decangle is cut proportionally, and the greater segment is the side of the sexangle: As here. Let the perpendicular ae, be cut into two equall parts in i. Then eo, that is the lesser segment continued with the halfe of the greater, that is, with ie. it shall by the 6 e xiiij, be of power five times so great as is the power of the same halfe. Therefore seeing that io, the halfe of the diagony is of power fivefold to the halfe: the whole diagony shall be of power fivefold to the whole cut. And from hence also shall be the geodesy of the Icosaedrum. For the finding out of the heighth of the pyramis, there is the semidiagony of the side of the decangle and the halfe ray of the circle: But the side of the decangle is a right line subtending the halfe periphery of the side of the quinquangle, or else the greater segment of the ray Therefore And As here thou seest. 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. 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. This is the fifth example of irrationality and incommensurability. The first was of the diagony and side of a quadrate or square. The second was of a line proportionally cut and his segments: The third is of the diameter of a Circle and the side of an inscribed quinquangle. The fourth was of the diagony and side of an icosahedrum. The fifth now is of the diagony and side of a dodecahedrum. For that hath beene told you even now. But from hence also doth arise the geodesy or maner of measuring of a dodecahedrum. For if the quadrate of the line subtending the angle of a quinquangle be trebled, the half of the treble shall be the side of the semidiagony of the dodecahedrum: Because by the 6 e xxiiij, the diagony of the cube, that is of the dodecahedrum is of treble power to the side of the cube. But if the quadrate of the side of the decangle be taken out of the quadrate of the side of the quinquangle; The side of the remainder shall be the ray of the circle circumscribed about a quinquangle. Lastly if the quadrate of the ray, be taken of the quadrate of the half-diagony; the side of the remainder shall be the heighth of perpendicular. As if the side of the decangle be 7.3/5: The quadrate of that shall be 57.19/25: the treble of which is 173.7/25 whose side is about 13.107/131 for the side of the Dodecahedrum, therefore 6.119/131 the halfe shall be the semidiagony of the dodecahedrum. The ray of the The semidiagony and ray of the circle thus found, the altitude remaineth. Take out therefore the quadrate of the ray of the circle, 16.4/225 out of the quadrate of the semidiagony 47.12458/17161, the side of the remainder 31.2714406/3861225 is for the altitude or heighth: whose 1/3 is 5/3. The quinquangled base is almost 38. Which multiplied by 5/3 doth make 63.1/3 for the solidity of one Pyramis; which multiplied by 12, doth make 760. for the soliditie of the whole dodecaedrum. This appeareth plainely out of the nature of a solid angle, by the kindes of plaine figures. Of two plaine angles a solid angle cannot be comprehended. Of three angles of an ordinate triangle is the angle of a Tetrahedrum comprehended: Of foure, an Octahedrum: Of five, an Icosahedrum: Of sixe none can be comprehended: For sixe such like plaine angles, are equall to 12 thirds of one right angle, that is to foure right angles. But plaine angles making a solid angle, are lesser than foure right angles, by the 8 e xxij. Of seven therefore, and of more it is, much lesse possible. Of three quadrate angles the angle of a cube is comprehended: Of 4. such angles none may be comprehended for the same cause. Of three angles of an ordinate quinquangle, is made the angle of a Dodecahedrum. Of 4. none may possibly be made; For every such angle: For every one of them severally doe countervaile one right angle and 1/5 of the same, Therefore they would be foure, and three fifths. Of more therefore much lesse may it be possible. This demonstration doth indeed very accurately and manifestly appeare, Although there may be an innumerable sort of ordinate plaines, yet of the kindes of angles five onely ordinate bodies may be made; From whence the Tetrahedrum, Octahedrum, and Icosahedrum are made upon a triangular base: the Cube upon a quadrangular: And the Dodecahedrum, upon a quinquangular. |