For here the base is to be added to the variable surface. It is a Consectary out of the 19 e, iiij. For here the axes and diameters are, as it were, the shankes of equall angles, to wit, of right angles in the base, and perpendicular axis. The cause of this division of a varied or mingled body, is to be conceived from the division of surfaces. Here the base is a circle. Therefore As it appeareth out of the definition of a variable body. And Here a threefold difference of the heighth of a Cone is professed, out of the threefold difference of the angles, whereby the toppe of the halfed cone is distinguished: Notwithstanding this consideration belongeth rather to the Optickes, than to Geometry. For a Cone a farre off seeme like triangle. Therefore according to the difference of the heighth, it And For a Cone is so the first in variable solids, as a triangle is in rectilineall plaines: As a Pyramis is in solid plaines: For neither may it indeed be divided into any other variable solids more simple. And As here you see. And These are consectaries drawne out of the 16 and 18 e. iiij. As here you see. For here two circles, parallell one to another are the bases of a Cylinder. Therefore As is apparant out the same definition of a varium. The geodesy here is fetch'd from the prisma: As if the base of the cylinder be 38.1/2: Of it and the heighth 12, the solidity of the cylinder is 462. This manner of measuring doth answeare, I say, to the manner of measuring of a prisma, and in all respects to the geodesy of a right angled parallelogramme. If the cylinder in the opposite bases be oblique, then if what thou cuttest off from one base thou doest adde unto the other, thou shalt have the measure of the whole; as here thou seest in these cylinders, a and b. From hence the capacity or content of cylinder-like As here the diameter of the inner Circle is 6 foote: The periphery is 18.6/7: Therefore the plot or content of the circle is 28.2/7. Of which, and the heighth 10, the plaine is 282.6/7 for the capacity of the vessell. Thus therefore shalt thou judge, as afore, how much liquor or any thing else conteined, a cubicall foote may hold. The demonstration of this proposition hath much troubled the interpreters. The reason of a Cylinder unto a Cone, may more easily be assumed from the reason of a Prisme unto a Pyramis: For a Cylinder doth as much resemble a Prisme, as the Cone doth a Pyramis: Yea and within the same sides may a Prisme and a Cylinder, a Pyramis and a Cone be conteined: And if a Prisme and a Pyramis have a very multangled base, the Prisme and Cylinder, as also the Pyramis and Cone, do seeme to be the same figure. Lastly within the same sides, as the Cones and Cylinders, so the Prisma and Pyramides, from their axeltrees and diameters may have the similitude of their bases. And with as great reason may the Geometer demand to have it granted him, That the Cylinder is the treble of a Cone: As it was demanded and granted him, That Cylinders and Cones are alike, whose axletees are proportionall to the diameters of their bases. Therefore The heighth is thus had. If the square of the ray of the base, be taken out of the square of the side, the side of the remainder shall bee the heighth, as is manifest by the 9 e xij. Here therefore the square of the ray 5, is 25. The square of 13, the side is 169. And 169 - 25, are 144; whose side is 12 for the heighth: The third part of which is 4. Now the circular base is 78.4/7: And the plaine of these is 314.2/7 for the solidity of the Cone. But the analogie of a conicall unto a Cylinder like surface doth not answeare, that the Conicall should be the subtriple of the Cylindricall, as the Cone is the subtriple of the Cylinder. Of two cones of one common base is made Archimede's Rhombus, as here, whose geodÆsy shall be cut of two cones. And Sackes in which they carry corne, are for the most part of And Both these affections are in common attributed to the equally manifold of first figures. And 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. 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. 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. 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. FINIS. |