The Way To Geometry
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15 Of the five ordinate bodies inscribed into the same spheare, the tetrahedrum in respect of the greatnesse of 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 ai, the side of the Tetrahedrum, subtendeth a greater periphery than ao, the side of the Octahedrum; And ao, a greater than ie, the side of the Cube; because it subtendeth but the halfe: And ie, greater than ue, the side of the Icosahedrum: And ue, greater than ye, 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 rl: And the side of the Decangle ar, inscribed into the same circle, circumscribing the quinquangle of an icosahedrum: And the perpendiculars is, and ul.
Here the two triangles aie, and ies, are by the 8 e, viij. alike; And as se, is unto ei: So is ie, unto ea: And by 25 e, iiij, as se, is to ea: so is the quadrate of se, to the quadrate of ei: And inversly or backward, as ae, is to se: so is the quadrate of ie, to the quadrate of se. But ae, is the triple of se. Therefore the quadrate of ie, is the triple of se. But the quadrate of as, by the grant, and 14 e xij, the quadruple of the quadrate of se. Therefore also it is greater than the quadrate of ie: And the right line as, is greater than ie, and al, therefore is much greater. But al, is by the grant compounded of the sides of the sexangle and decangle rl, and ar. Therefore by the 1 c. 5 e, 18. it is cut proportionally: And the greater segment is the side of the sexangle, to wit, rl: And the greater segment of ie, proportionally also cut, is ye. Therefore the said rl, is greeter than ye: And even now it was shewed ul, was equall to rl. Therefore ul, is greater than ye: But ue, the side of the Icosahedrum, by 22. e vj. is greater than ul. Therefore the side of the Icosahedrum is much greater, then the side of the dodecahedrum.
Of Geometry the twenty seventh Book; Of the Cone and Cylinder
1 A mingled solid is that which is comprehended of a variable surface and of a base.
For here the base is to be added to the variable surface.
2 If variable solids have their axes proportionall to their bases, they are alike. 24. d xj.
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.
3 A mingled body is a Cone or a Cylinder.
The cause of this division of a varied or mingled body, is to be conceived from the division of surfaces.
4 A Cone is that which is comprehended of a conicall and a base.
Here the base is a circle.
Therefore
5 It is made by the turning about of a right angled triangle, the one shanke standing still.
As it appeareth out of the definition of a variable body.
And
6 A Cone is rightangled, if the shanke standing still be equall to that turned about: It is Obtusangeld, if it be lesse: and acutangled, if it be greater. ê 18 d xj.
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 appeareth with a right angled, or obtusangled or acutangled toppe: As here the least Cone is obtusangled: the middle one rightangled: and the highest acutangled. But the cause of this threefold difference in the angles from of the difference of the shankes, is out of the consectaries of the threefold triangle of a right line cutting the base into two equall parts, as appeareth at the end of the viij Booke.
And
7 A Cone is the first of all variable.
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
8 Cones of equall heighth are as their bases are 11. p xij.
As here you see.
And
9 They which are reciprocall in base and heighth are equall, 15 p xij.
These are consectaries drawne out of the 16 and 18 e. iiij. As here you see.
10 A Cylinder is that which is comprehended of a cylindricall surface and the opposite bases.
For here two circles, parallell one to another are the bases of a Cylinder.
Therefore
11 It is made by the turning about of a right angled parallelogramme, the one side standing still. 21. d xj.
As is apparant out the same definition of a varium.
12. A plaine made of the base and heighth is the solidity of a Cylinder.
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 vessell or measure is esteemed and judged of. For the hollow or empty place is to be measured as if it were a solid body.
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.
13. A Cylinder is the triple of a cone equall to it in base and heighth. 10 p xij.
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
14. A plaine made of the base and thid part of the height, is the solidity of the cone of equall base & height;
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
15. Cylinder of equall heighth are as their bases are. 11 p xij.
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.
FINIS