Sackes in which they carry corne, are for the most part of a cylinderlike forme. If an husbandman therefore shall lend unto his neighbour a sacke full or corne, and the base of the sacke be 4 foote over. And the neighbour afterward for that one sacke, shall pay him 4 sacke fulls, every sacke being as long as that was, yet but one foote over in the diameter, he may be thought peradventure to have repayed that which he borrowed in equall measure, to wit in heighth and base. But it shall be indeed farre otherwise: For there is a great difference betweene the quadrate of the foure severall diameters, 1. 1. 1. 1. that is 4: and 16, the quadrate of 4, the diameter of that sacke by which it was lent. For Circles are one unto another as the quadrates of their diameters are one to another, by the 2 e xv. Therefore he payd him but one fourth part of that which he borrowed of him.

And

16 Cylinders reciprocall in base and heighth are equall. 15 p xij.

Both these affections are in common attributed to the equally manifold of first figures.

And

17. If a cylinder be cut with a plaine surface parallell to his opposite bases, the segments are, as their axes are 13 p xij.

As here thou seest. For the axes are the altitudes or heights. It is likwise a consectary following upon that generall theoreme of first figure, but somewhat varyed from it. It doth answere unto the 10 e 23.

The unequall sections of a spheare we have reserved for this place: Because they are comprehended of a surface both sphearicall and conicall, as is the sectour. As also of a plaine and sphearicall, as is the section: And in both like as in a Circle, there is but a greater and lesser segment. And the sectour, as before, is considered in the center.

18. The sectour of a spheare is a segment of a spheare, which without is comprehended of a sphearicall within of a conicall bounded in the center, the greater of a concave, the lesser of a convex.

Archimedes, maketh mention of such kinde of Sectours, in his 1 booke of the Spheare. From hence also is the geodesy following drawne. And here also is there a certaine analogy with a circular sectour.

19. A plaine made of the diameter, and sixth part of the greater, or lesser sphearicall, is the greater or lesser sector.

As here of the Diameter 14, and of 73.1/3 and 4.2/3 (which is the one sixth part of the greater sphearicall) the plaine is 1026.2/3 for the solidity of the greater sectour, so of the same diameter 14, and 29.1/3 which is the 1/6 part of 176, the lesser sphæricall, the plaine is 410.2/3 for the solidity of the lesser sectour.

And from hence lastly doth arise the solidity of the section, by addition and subduction.

20. If the greater sectour be increased with the internall cone, the whole shall be the greater section: If the lesser be diminished by it, the remaine shall be the lesser section.

As here the inner cone measured is 126.4/63. The greater sectour, by the former was 1026.2/3. And 1026.2/3 + 126.4/63 doe make 1152.46/63.

Againe the lesser sectour, by the next precedent, was 410.2/3: And here the inner cone is 126.4/63 And therefore 410.2/3 – 126.4/63 that is 284.38/63 is the lesser section.

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