The Way to Geometry, Book 20
Of Geometry the twentieth Booke, Of a Bossed surface.
1. A bossed surface is a surface which lyeth unequally betweene his bounds.
IT is contrary unto a Plaine surface, as wee heard at the 4 e v.
2. A bossed surface is either a sphericall, or varium.
3. A sphericall surface is a bossed surface equally distant from the center of the space inclosed.
Therefore
4 It is made by the turning about of an halfe circumference the diameter standeth still. è 14 d x j.
As here if thou shalt conceive the space betweene the periphery and the diameter to be empty.
5. The greatest periphery in a sphericall surface is that which cutteth it into two equall parts.
Those things which were before spoken of a circle, the same almost are hither to bee referred. The greatest periphery of a sphericall doth answere unto the Diameter of a Circle.
Therefore
6. That periphery that is neerer to the greatest, is greater than that which is farther off: And on each side those two which are equally distant from the greatest, are equall.
The very like unto those which are taught at the 15,16, 17, 18. e. xv. may here againe be repeated: As here.
7 The plaine made of the greatest periphery and his diameter is the sphericall.
So the plaine made of the diameter 14. and of 44. the greatest periphery, which is 616. is the sphericall surface. So before the content of a circle was measured by a rectangle both of the halfe diameter, and periphery. But here, by the whole periphery and whole diameter, there is made a rectangle for the measure of the sphericall, foure times so great as was that other: Because by the 1 e vj. like plaines (such as here are conceived to be made of both halfe the diameter, and halfe the periphery, and both of the whole diameter and whole periphery) are in a doubled reason of their homologall sides.
Therefore
8 A plaine of the greatest circle and 4, is the sphericall.
This consectarium is manifest out of the former element.
And
9 As 7 is to 22. so is the quadrate of the diameter unto the sphericall.
For 7, and 22, are the two least bounds in the reason of the diameter unto the periphery: But in a circle, as 14, is to 11. so is the quadrate of the diameter unto the circle. The analogie doth answer fitly: Because here thou multipliest by the double, and dividest by the halfe: There contrariwise thou multipliest by the halfe, and dividest by the double. Therefore there one single circle is made, here the quadruple of that. This is, therefore the analogy of a circle and sphericall; from whence ariseth the hemispherical, the greater and the lesser section.
And
10 The plaine of the greatest periphery and the ray, is the hemisphericall.
As here, the greatest periphery is 44. the ray 7. The product therefore of 44. by 7. that is, 308. is the hemisphericall.
11 If looke what the part be of the ray perpendicular from the center unto the base of the greater section, so much the hemisphericall be increased, the whole shall be the greater section of the sphericall: But if it be so much decreased, the remainder shall be the lesser.
As in the example, the part of the third ray, that is, of 3/7, is from the center: such like part of the hemispherical 308, is 132. (For the 7, part of 308. is 44. And three times 44. is 132.) Therefore 132. added to 308. do make 440. for the greater section of the sphericall. And 132. taken from 308. doe leave 176. for the lesser section of the same.
12 The varium is a bossed surface, whose base is a periphery, the side a right line from the bound of the toppe, unto the bound of the base.
13 A varium is a conicall or a cylinder like forme.
14 A conicall surface is that which from the periphery beneath doth equally waxe lesse and lesse unto the very toppe.
Therefore
15 It is made by turning about of the side about the periphery beneath.
16 The plaine of the side and halfe the base is the conicall surface.
As in the example next afore going, the side is 13. The halfe periphery is 15 5/7: And the product of 15. 5/7. by 13. is 204. 2/7. for the conicall surface. To which if you shall adde the circle underneath, you shall have the whole surface.
17 A cylinder like forme is that which from the periphery underneath unto the upper one, equall and parallell unto it, is equally raised.
Therefore
18 It is made by the turning of the side about two equall and parallell peripheries.
19 The plaine of his side and heighth is the cylinder like surface.
As here the periphery is 22. as is gathered by the Diameter, which is 7. The heighth is 12. The base therefore is 38.½. And 38.½ by 12. are 462. for the cylinderlike surface. To which if you shall adde both the bases on each side, to wit, 38.½. twise, or 77. once, the whole surface shall be 539.