The Way to Geometry, Book 25
Of Geometry the twenty fifth Booke; Of mingled ordinate Polyedra's.
1. A mingled ordinate polyedrum is a pyramidate, compounded of pyramides with their toppes meeting in the center, and their bases onelyonly outwardly appearing.
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.
2 The heighth of the compounding pyramis is found by the ray of the circle circumscribed about the base, and by the semidiagony of the polyedrum.
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 half… diagony the side of the remainder, by the 9 e x ij. 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.
3 A mingled ordinate polyedrum hath either a triangular, or a quinquangular base.
The division of a Polyhedron ariseth from the bases upon which it standeth.
4 If a quadrate of a triangular base be divided into three parts, the side of the third part shall be the ray of the circle circumscribed about the base.
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.
5 A mingled ordinate polyedrum of a triangular base, is either an Octoedrum, or an Icosoedrum.
This division also ariseth from the bases of the figures.
6 An octoedrum is a mingled ordinate polyedrum, which is comprehended of eight triangles. 27 d xj.
As here thou seest, in this Monogrammum and solidum, that is lined and solid octahedrum.
Therefore
7 The sides of an octoedrum are 12. the plaine angles 24, and the solid 6.
And
8 Nine octoedra's doe fill a solid place.
For foure angles of a Tetraedrum are equall to three angles of the Octoedrum: And therefore 12. are equall to nine. Therefore nine angles of an octaedrum doe countervaile eight solid right angles.
And
9 If eight triangles, equilaters and equall be joyned together by their edges; they shall comprehend an octaedrum.
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.
10 If a right line on each side perpendicular to the center of a quadrate and equall to the halfe diagony be tied together with the angles, it shall comprehend an octaedrum, 14. d xiij.
For the perpendicular y u, and s u, with the semidiagoni's, u a, u o, u i, u e, shall be made equall by the a e v ij, the eight sides y a, y e, y o, y i, s e, s i, s a, s o; And also eight triangles.
Therefore
11 The Diagony of an octaedrum is of double power to the side.
As is manifest by the 9 e xij.
And
12 If the quadrate of the side of an octaedrum, be doubled, the side of the double shall be the diagony.
As in the figure following, the side is 6. The quadrate 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 17 271/28…. And the quadrate of 6, the side of the equilater triangle, being of treble power to the ray, by the 6 e, xviiij. is 36. And the side of 12. the third part 3 3/7. is the ray of the circle. Wherefore 8 …/17. that is 5 21/289. is the quadrate of the perpendicular, whose side 21/5 is the height of the same perpendicular: whose third part againe 11/25. mulliplied by 15 18/31. the triangular base doe make 11 66/1… for one of the eight pyramides: Therefore the same 11 60/155 multiplied by eight, shall make 91 …/155 for the whole octoedrum.
13 An Icosaedrum is an ordinate polyedrum comprehended of 20 triangles 29 d xj.
Therefore
14 The sides of an Icosaedrum are 30. the plaine angles 60. the solid 12.
15 If twentie ordinate and equall triangles be joyned with solid angles, they shall comprehend an Icosaedrum.
This fabricke is ready end easie, as is to be seene in this example following.
16 If ordinate figures, to wit, a double quinquangle, and one decangle be so inscribed into the same circle, that the side of both the quinquangle doe subtend two sides of the decangle, sixe right lines perpendicular to the circle and equall to his ray, five from the angles of one of the quinquangles, knit together both betweene themselves, and with the angles of the other quinquangle; the sixth from the center on each side continued with the side of the decangle, and knit therewith the five perpendiculars, here with the angles of the second quinquangle, they shall comprehend an icosaedrum. è 15 p xiij.
For there shall be made 20 triangles, both equilaters and equall. Let there be therefore two ordinate quinquangles, the first a e i o u; The second y s r l m; each of whose sides let them subtend two sides of a decangle; to wit, u t y m, let it subtend y a, and a m. Then let there be five perpendidiculars from the angles of the second quinquangle y j, s y, r v, l f, m t. And let them be knit first one with another, by the lines n j, j v, v f, f t, t n. Secondarily, with the angles of the first quinquangle, by the lines n e, e j, j i, i v, o f, f u, u t, t a, a n. The sixth perpendicular from the center d, let it be b g, the ray d c, continued at each end with the side of the decangle, c g, and d b, tied together about with the perpendiculars, as by the lines n g, t g: Beneath with the angles of the first quinquangle, as by the lines b e, b i, and in other places in like manner, and let all the plaines be made up. This say I, is an Icosaedrum; And is comprehended of 20. triangles, both equilaters and equall. First, the tenne middle triangles, leaving out the perpendiculars, that they are equilaters and equall, one shall demonstrate, as n a t. For m t, and y u, because they are perpendiculars, they are also, by the 6 e x xj. parallells: And by the grant, equall. Therefore by the 27 e, v, n t, is equall to y m, the side of the quinquangle. Item n a, by the 6 e x i j. is of as great power, as both the shankes n y, and y a, that is, by the construction, as the sides of the sexangle aud decangle: And, by the converse of the 15. e xviij. it is the side of the quinquangle. The same shall fall out of o t. Wherefore n a t, is an equilater triangle. The same shall fall out of the other nine middle triangles, n a e, n e j, e j i, j i v, i v o, v o f, f o u, f u t, u t a, t a n.
In like manner also shall it be proved of the five upper triangles, by drawing the right lines d y, and c n, which as afore (because they knit together equal parallells, to wit, d c, and y n) they shall be equall. But d y, is the side of a sexangle: Therefore c n, shall be also the side of a sexangle: And c g, is the side of a decangle: Therefore a n, 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 g t, shall be concluded to be the side of a quinquangle. Wherefore n g t, 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, i b e, by drawing the raies d i, and d e. For i b, whose power, as afore, is asmuch as the sides of the sexangle, and decangle, shall be the side of the quinquangle: And in like sort b e, being of equall power with d e, and d o, the sides of the sexangle and decangle, shall be the side of the quinquangle. Wherefore the triangle e b i, is an equilater: And the foure other in like manner may be shewed to be equilaters. Therefore all the side of the twenty triangles, seeing they are equall, they shall be equilater triangles: And by the 8 e, vij. equall.
17 The diagony of an icosaedrū is irrational unto the side.
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
18 The power of the diagony of an icosaedrum is five times as much as the ray of the circle.
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 a e, be cut into two equall parts in i. Then e o, that is the lesser segment continued with the halfe of the greater, that is, with i e. 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 i o, 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 proportionally cut. For so it may be taken Geometrically, and reckoned for his measure. Therefore if the quadrate of the side of the decangle, be taken out of the quadrate of the side of the quinquangle, there shall by the 15 e xviij, remaine the quadrate of the sexangle, that is of the ray. The side of the decangle (because the side of the quinquangle here is 6) shall be 3 3/35 to wit a right line subtending the halfe periphery. Now the halfe ray shall thus be had. The quadrates of the quinquangle and decangle are 36, and 9 639/1225 And this being subducted frō that, the remaine 26.586/1225 by the 15 e xviij, shall be the quadrate or square of the sexangle: And the side of it, 5, and almost 5/7 shall be the ray: The halfe ray therefore shall be 2 6/7. To the side of the decangle 3 3/35 adde 2 6/7; the whole shal be 5 33/35 for the semidiagony of the Icosaedrum. The ray of the circle circumscribed about the triangle, is by the 12 e xv iij, the same which was before 3 3/7 to wit of the quadrate 12. Therefore if the quadrate of the circular ray, be taken out of the quadrate of the halfe diagony, there shall remaine the quadrate of the heighth and perpendicular: the quadrate of the halfe-diagony is 35 389/1225: the quadrate of the circular ray is 12. This taken out of that beneath 23 …89/1225: whose side is almost 5, for the perpendicular and heighth proposed: From whence now the Pyramis is esteemed. The base of a triangular pyramis is 15 18/31. The Plaine of this base and the third part of the heighth is 25 30/31 for the solidity of one Pyramis. This multiplyed by 20 maketh 519 11/31 for the summe or whole solidity of the Icosaedrum. And this is the geodesy or manner of measuring of an Icosaedrum.
19. A mingled ordinate polyedrum of a quinquangular base is that which is comprehended of 12 quinquangles, and it is called a Dodecaedrum.
Therefore
20. The sides of a Dodecaedrum are 30, the plaine angles 60. the solid 20.
And
21. If 12 ordinate equall quinquangles be joyned with solid angles, they shall comprehend a Dodecaedrum.
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, a e i o, and e u y i, having their sides halfed by the bisecantes, s r, l m, r n, j v: And the three bisegments or portions of the bisegments l m, and r n, neither concurring or meeting, nor parallell one to another; two of the said l m, to wit, f l, and f m: The third next unto the remainder, that is l r. And let each bisegment be cut proportionally in the points d, c, g; so that the lesser segments doe bound the bisecant, to wit, d l, c m, and g r. Lastly let there be three perpendiculars from the points d b, c g, to the said d, c p, g z: And the two first knit one to another, by b p: And againe with the angles of the cube, by b e, and p i: The third knit with the same angles, by z e, and z i: And let all the plaines be made up. I say first, that the five sides b p, p i, i z, z e and e b. are equall; Because every one of them severally are the doubles of the same greater segment. For in drawing the right lines d e and e g, i g, it shall be plaine of two of them… And after the same manner of the rest. First therefore o d, and b p, are equall by the 6 e xxj, and by the 27 e v. Therefore b p, is the double of the greater segment. Then the whole f l, cut proportionally, and the lesser segment d l, they are by the 7 e x iiij, of treble power to the greater f d, that is, by the fabricke d b. Therefore le wich is equall to l f, the line cut, and l d, are of treble power to the same d b: But by the 9 e xij, d e is of as much power as l e, and l d too. Therefore d e is of treble power to d b. Therefore both e d, and d b, are of quadruple power to d b. But b e, by the 9 e xij, is of as much power as e d, and d b. And therefore b e, is of quadruple value to e b: And by the 14 e x i j, it is the double of the said e b. Therefore the two sides e b, and b p, are equall: And by the same argument p i, i z, and z e, 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 f h be parallell to d b, and c p: and be equall unto them. And let h z, be drawne: This h z shall be cut one line, by the 14 e vij. For as the whole tr, that is r f, is unto the greater segment that is to f h: so f h, that is z g, is unto g r. And two paire of shankes f h, g r, f c, g z, 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 e p, and e c, be drawne: I say that the angles, p b e, and e z i, are equall: Because they have by the construction, the bases of equall shankes equall, being to wit in value the quadruple of l e. For the right line l f, cut proportionally, and increased with the greater segment d f, that is f c, 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 c p, are of treble value to the greater f l, that is of the sayd l e. Therefore e l, and l c, that is e c, and c p, that is e p, is of quadruple power to e l: And therefore by the 14 e xij, it is the double of it: And e i, 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 i d, and i b, the third angle b p i, shall be concluded to be equall to the angle e z i. Therefore by the 13 e xiiij, five angles are equall.
23. The Diagony is irrationall unto the side of the dodecahedrum.
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.
24 If the side of a cube be cut proportionally, the greater segment shall be the side of a dodecahedrum.
For that hath beene told you even now. But from hence also doth arise the geodesyor māner 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 Circle shall now thus be found. If the quadrate of the side of the decangle be taken out of the quadrate of the side of the sexangle; the side of the remainder, shall be the Ray of the Circle, by the .15 and 9 e xviij. As here the side of the Quinquangle is 4⅔. The side of the Decangle 2 2/5: And the quadrates therefore are 2… 7/9. and 5 19/25. This s…bducted from that leaveth 16 4/225 whose side is 4 2/55 for the Ray of the Circle.
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 3… 2…14406/3861225 is for the altitude or heighth: whose ⅓ is 5/3. The quinquangled base is almost 38. Which multiplied by 5/3 doth make 63 ⅓ for the solidity of one Pyramis; which multiplied by 12, doth make 760. for the soliditie of the whole dodetacedrum.
25 There are but five ordinate solid plaines.
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 compr…hended: 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 onelyonly 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.