This is to be noted, that if a man should demaund 〈◊〉〈◊〉 many sides an Icosahedron hath, we may thus answere: It is manifest that an Icosah••r••n is contayned vnder 20. triangles, and that euery triangle consisteth of three right lin••s. Now then multiply the 20. trian∣gles into the sides of one of the triangles, and so shall there be produced 6••. ••he halfe of which is 30. And so many sides hath an Icosahedron. And in like sort in a dodecahedron, foras∣much as 12. pentagons make a dodecahedron, and euery pentagon contayneth ••. right lines•• multiply •••• into 12. and there shall be produced 60. the halfe of which is 30. And so many are the sides of a dodecahedron. And the reason why we take the halfe, i••, for that euery side whe∣ther it be of a triangle or of a pentagon, or of a square as in a cube, ••s taken twise. And by the same reason may you finde out how many sides are in a cube, and in a pyramis, and in an octohedron.
But now agayne if ye will finde out the number of the angles of euery one of the solide figures, when ye haue done the same multiplication that ye did before, di••id•• the same sides, by the number of the plaine superficieces which comprehend one of the angles of the solides As for example, forasmuch as 5. triangles contayne the solide angle of an Icosahedron, diuide 60. by 5. and there will come forth 12. and so many solide angles hath an Icosahed••on. In a dodecahedron, forasmuch as three pentagons comprehend an angle, diuide 60. by 3. and there will come forth 20: and so many are the angles of a dodecahedron. And by the same reason may you finde out how many angles are in eche of the rest of the solide figures.
* 1.1If it be required to be knowne, how one of the plaines of any of the fiue solides being ge∣uen, there may be found out the inclination of the sayd plaines the one to the other, which con¦tayne eche of the solides. This (as sayth Isidorus our greate master) is fo••••d out after this maner. It is manifest that in a cube, the plaines which contayne i••, do•• 〈◊〉〈◊〉 the one the other by a right angle. But in a Tetrahedron, one of the triangles being geuen, let the endes of one of the sides of the sayd triangle be the centers, and let the space be the perpendicular line drawne from the toppe of the triangle to the base, and describe circumfer••nces of a circle, which shall cutte the one the other: and from the intersection to the centers draw right lines, which shall containe the inclination of the plaines cōtayning the Tetrahedron. In an Octo••e∣dron, take one of the sides of the triangle ther••of, and vpon it describe a square, and draw the diagonall line, and making the centres, the endes of the diagonall line, and the space likewise the perpendicular line drawne from the toppe of the triangle to the base, describe circumfe∣rences: and agayne from the common section to the centres draw right lines, and they shall contayne the inclination sought for. In an Icosahedron, vpon the side of one of the tri∣angles thereof, describe a pentagon, and draw the line which subtendeth one of the angles of the sayd pentagon, and making the centres the endes of that line, and the space the perpendi∣cular line of the triangle, describe circumferences: and draw from the common intersectio•• of the circumferences, vnto the centres right lines: and they shall contayne likewise the incli∣nation of the plaines of the icosahedron. In a dodecahedron, take one of the pentagons, and draw likewise the line which subtendeth one of the angles of the pentagon, and making the centres the endes of that line, and the space, the perpendicular line drawne from the section into two equall partes of that line to the side of the pentagon, which is parallel vnto it, de∣scribe circumferences: and from the point of the intersection of the circumferences draw vn∣to the centres right lines: and they shall also containe the inclination of the plaines of the do∣decahedron. Thus did this most singular learned man reason, thinking the de••onstration in euery one of them to be plaine and cleare. But to make the demonstration of them mani∣fest,