752

THEORY AND PRACTICE OF ENGINEERING.

Book II

The Axis of the Ellipsis M is shown at KE. If amoving or generating circle roll along the concave circum-ference of a fixed circle in the same plane, and the radiusof the former be half that of the latter, any given pointin the plane of the generating circle within or without itwill describe an ellipse; such a curve has been provedto be an epicycloid ; when the circle revolves on the in-side of the circumference, the curve is sometimes calledthe hypocycloid. The revolving circle is the generatingcircle, the circle on which the revolution is performedthe fundamental circle, and the portion of the fundamentalcircle on which the epicycloid rests is the base.

The Axis of the Parabola P is the perpendicular lineN O, which falls upon the line QR.

The Axes of Spheroids are the two lines which cross atright angles, TV being that which is horizontal.

Solids have length, breadth, and thickness. The massA has length from B to C, breadth from B to D, andthickness from D to E.

The boundaries of solids are surfaces, and all the regularsolids are terminated by regular and equal planes; theyare five in number, as the tetraedron, the hexaedron, theoctaedron, the dodecacdron, and the icosaedron ; these arealso called Platonic bodies, on account of their beingtreated of and described by Plato : besides these five therecan be no other solids bounded by like equal and regularplane figures, and whose solid angles are all equal:three of these, as the tetraedron, octaedron, and icosaedron,are contained by equilateral triangles; one, viz. the cubeor hexaedron by squares, and the other, the dodecaedron,by pentagons. The sphere may be inscribed in either ofthese, as may also another around or circumscribing it,the common centre of which may be found by bisectingany three of the dihedral angles, or by bisecting any threeof the edges by planes at right angles with them.

The solid content of any regular polyedron is equalto one-third of the product of its convex surface and apo-them, which is the radius of the inscribed sphere.

The regular polyedronsof 6, 8, 12, and 20 faces, have forevery face a face opposite and parallel to it, and the oppo-site edges of those faces likewise parallel; also the straightline which joins two opposite angles passes through thecentre of the polyedron: any one of them may be in-scribed in a regular polyedron which has a greaternumber of faces by taking for its vertices certain of thevertices of the latter, or of the centres of its faces, or ofthe middle points of its edges.

Similar and equal bodies are those which have allthese dimensions of one size, the square OKL and PMNbeing the same in both, as well as the other sides.

In the Cube A the sides BCD are all equal. Thehexaedron or cube has six faces, eight solid angles, andtwelve edges; the centres of its faces are the vertices ofan inscribed regular octaedron; four of its vertices arethe vertices of an inscribed octaedron ; its adjoining facesare at right angles to each other, and the diameter of acircumscribed sphere is to the edge as the hypothenuseto the lesser side of a right-angled triangle whose sidesare as the side and diagonal of a square. The diameter ofthe inscribed sphere is equal to the edge of the cube.

The Sphere has its surface represented by F. As aline according to Euclid is generated by the motion of apoint, so a surface is generated by the motion of a line :if the generating line be a straight line, and move, subjectto the condition of having always two consecutive positionsin the same plane, the surface generated is developable,and can be stretched out on a plane, as that of a cylinder.

Fig. 728.

n

Fig. 720.

ig. 730.

E

Fig. 731.

1 /

Fig. 732.

W \

y

/

/

i.

733.

A

/ B

y

c

D

Fig. 734.

Fig. 73.%