The Way to Geometry, Book 26
Of Geometry the twenty sixth Booke; Of a Spheare.
1 AN imbossed solid is that which is comprehended of an imbossed surface.
2. And it is either a spheare or a Mingled forme.
3. A spheare is a round imbossement.
It may also be defined to be that which is comprehended of a sphearicall surface. A sphearicall body in Greeke is called Sphaera, in Latine Globus, a Globe.
Therefore
4. A Spheare is made by the conversion of a semicircle, the diameter standing still. 14 d xj.
As here thou seest:
5. The greatest circle of a spheare, is that which cutteth the spheare into two equall parts.
Therefore
6. That circle which is neerest to the greatest, is greater than that which is farther off.
And
7. Those which are equally distant from the greatest are equall.
As in the example above written.
8. The plaine of the diameter and sixth part of the sphearicall is the solidity of the spheare.
As before there was an analogy betweene a Circle and a Sphericall: so now is there betweene a Cube and a spheare. A cubicall surface is comprehended of sixe quadrate or square and equall bases: And a spheare in like manner is comprehended of sixe equall sphearicall bases compassing the cubicall bases. A cube is made by the multiplication of the sixth part of the base, by the side: And a spheare likewise is made by multiplying the sixth part of the sphearicall by the diameter, as it were by the side: so the plaine of 616/6, and 1 4, the diameter is 1437 ⅓ for the solidity of the spheare.
Therefore
9. As 21 is unto 11, so is the cube of the diameter unto the spheare.
As here the Cube of 14 is 2744. For it was an easy matter for him that will compare the cube 2744, with the spheare, to finde that 2744 to be to 1437 ⅓ in the least boundes of the same reason, as 21 is unto 11.
Thus much therefore of the Geode…y of the spheare: The geodesy of the Setour and section of the spheare shall follow in the next place.
And
10. The plaine of the ray, and of the sixth part of the sphearicall is the hemispheare.
But it is more accurate and preciser cause to take the halfe of the spheare.
11. Spheares have a trebled reason of their diameters.
So before it was told you; That circles were one to another, as the squares of their diameters were one to another, because they were like plaines: And the diameters in circles were, as now they are in spheares, the homologall sides. Therefore seeing that spheres are figures alike, and of treble dimension, they have a trebled reason of their diameters.
12 The five ordinate bodies are inscribed into the same spheare, by the conversion of a semicircle having for the diameter, in a tetrahedrum, a right line of value sesquialter unto the side of the said tetrahedrum; in the other foure ordinate bodies, the diagony of the same ordinate.
The adscription of ordinate plainebodies is unto a spheare, as before the Adscription plaine surfaces was into a circle; of a triangle, I meane, and ordinate triangulate, as Quadrangle, Quinquangle, Sexangle, Decangle, and Quindecangle. But indeed the Geometer hath both inscribed and circumscribed those plaine figures within a circle. But these five ordinate bodies, and over and above the Polyhedrum the Stereometer hath onelyonly inscribed within the spheare. The Polyhedrum we have passed over, and we purpose onelyonly to touch the other ordinate bodies.
13 Out of the reason of the axeltree of the sphearicall the sides of the tetraedrum, cube, octahedrum and dodecahedrum are found out.
The axeltree in the three first bodies is rationall unto the side, as was manifested in the former. For it is of sesquialter valew unto the side of the tetrahed…um; of treble, to the side of the cube: Of double, to the side of the Octahedrum. Therefore if the axis a e, be cut by a double reason in i: And the perpendicular i o, be knit to a, and e, shall be the side of the tetrahedrum; and o e, of the cube, as was manifest by the 10 e viij, and 25 iiij: And the greater segment of the side of the cube proportionally cut, is by the 24 e, xxv.
If the same axis be cut into two halfes, as in u: And the perpendicular u y, be erected: And y, and a, be knit together, the same y a, thus knitting them, shall be the side of the Octahedrum, as is manifest in like manner, by the said 10 e, viij. and 25 e iiij.
The side of the Icosahedrum is had by this meanes.
14 If a right line equall to the axis of the sphearicall, and to it from the end of the perpendicular be knit unto the center, a right line drawne from the cutting of the periphery unto the said end shall be the side of the Icosahedrum.
As here let the Axis a e; be the diameter of the circle a u e… and a i, equall to the same axis, and perpendicular from the end, be knit unto the center, by the right line i o: A right drawne from the section u, unto a, shall be the side of the Icosahedrum. From u, let the perpendicular u y, be drawne: Here the two triangles i a o, & u y o, are equiangles by the 13 e, vij. Therefore by the 12 e, vij. as i a, is unto a o: so is u y, unto y o. But i a, is the double of the said a o: Therefore u y, is the double of the same y o: Therefore by the 14 e, xij. it is of quadruple power unto it: And therefore also u y, and y o, that is, by the .9 e xij, u o, that is againe by the 28 e, iiij. a o. is of quintuple power to y o. But y o, is lesser than a o, that is, than o e: Let therefore o s, be cut off equall to it. Now as the halfe of a o, is of quintuple valew to the halfe of y o: so the double a e, is of quintuple power to the double y s. Therefore, by the 18 e xxv. seeing that the diagony a e, is of quintuple power to y s; the said y s, shall be the side of the sexangle inscribed into a circle, circumscribing the quinquangle of the Icosahedrum. But the perpendicular u y, is equall to y s; because each of them is the double of y o. Wherefore u y, is the side of the sexangle. But a y, is the side of the Decangle: For it is equall to s e: Because if from equall rayes a o, and o e, you take equall portions o y, and o s: There shall remaine equall, y a, and s e. And the Diagony of an Icosahedrum by the 16 e xxv, is compounded of the side of the sexangle, continued at each end with the side of the decangle. Wherefore a y, is the side of the decangle. Lastly, u a, whose power is as much as the sides of the sexangle and decangle, by the 15. e, xviij. shall be the side of an Icosahedrum.
15 Of the five ordinate bodies inscribed into the same spheare, the tetrahedrum in respect of the greatnesse o… his side is first, the Octahedrum, the second; the Cube, the third; the Icosahedrum, the fourth; and the Dodecahedrum, the fifth.
As it will plainely appeare, if all of them be gathered into one, thus. For a i, the side of the Tetrahedrum, subtendeth a greater periphery than a o, the side of the Octahedrum; And a o, a greater than i e, the side of the Cube; because it subtendeth but the halfe: And i e, greater than u e, the side of the Icosahedrum: And u e, greater than y e, the side of Dodecahedrum.
The latter, Euclide doth demonstrate with a greater circumstance. Therefore out of the former figures and demonstrations, let here be repeated, The sections of the axis first into a double reason in s: And the side of the sexangle r l: And the side of the Decangle a r, inscribed into the same circle, circumscribing the quinquangle of an icosahedrum: And the perpendiculars i s, and u l.
Here the two triangles a i e, and i e s, are by the 8 e, viij. alike; And as s e, is unto e i: So is i e, unto e a: And by 25 e, iiij, as s e, is to e a: so is the quadrate of s e, to the quadrate of e i: And inversly or backward, as a e, is to s e: so is the quadrate of i e, to the quadrate of s e. But a e, is the triple of s e. Therefore the quadrate of i e, is the triple of s e. But the quadrate of a s, by the grant, and 14 e xij, is the quadruple of the quadrate of s e. Therefore also it is greater than the quadrate of i e: And the right line a s, is greater than i e, and a l, therefore is much greater. But a l, is by the grant compounded of the sides of the sexangle and decangle r l, and a r. Therefore by the 1 c. 5 e, 18. it is cut proportionally: And the greater segment is the side of the sexangle, to wit, r l: And the greater segment of i e, proportionally also cut, is y e. Therefore the said r l, is greeter than y e: And even now it was shewed u l, was equall to r l. Therefore u l. is greater than y e: But u e, the side of the Icosahedrum, by 22. e vj. is greater than u l. Therefore the side of the Icosahedrum is much greater, then the side of the dodecahedrum.