OF SPACE. 



tiid the fluxion expressed by a cylindroid is equal to a pa- 

 rallelepiped on the same base and of the same height. 



183. Theorem. The curve surface of a 

 sphere is equal to the rectangle contained by 

 its versed sine and the sphere's circumference. 



The fluxion of the surface is obviously equal to the rect- 

 angle contained by the fluxion of the circumference and 

 the circumference of the circle of which the radius is the 

 sine ; it varies therefore as the sine ; but the fluxion of 

 the cosine or of the versed sine varies as the sine, conse- 

 quently the surface varies as the versed sine. Now where 

 the tangent becomes parallel to the axis, the fluxion of the 

 surface becomes equal to the rectangle contained by the 

 sphere's circumference, and the fluxion of the versed sine : 

 hence the whole surface of any segment is equal to the 

 whole rectangle contained by its versed sine and the 

 sphere's circumference ; and the surface of the whole 

 sphere is four times the area of a great circle. 



184. Theorem. The content of a sphere 

 is two thirds of that of the circumscribing cy- 

 linder. 



The fluxion of the sphere is to that of the cylinder as the 

 square of the sine to the square of the radius ; or if the 

 fluxion of the cylinder be aabi, that of the sphere will be 

 (aor — xx]bi, or labxx — hxxi, of which the fluent is ali* 

 — it'j' ; which, when x-zia, becomes \aH; while the con- 

 tent of the cylinder is a?b.- - 



185. Theorem. When a picture is pro- 

 jected on a plane, by right lines supposed to 

 be drawn from each point to the eye, the 

 whole image of every right line, produced 

 without limit, is a right line drawn from its 

 intersection with the plane of projection, to 

 its vanishing point, or the point where a line 

 drawn from the eye, parallel to the given 

 line, meets the plane of projection ; and this 

 image is divided by the image of any given 

 point in the ratio of the portion of the line 

 intercepted by that point and the picture, to 

 the line drawn from the eye to the vanishing 

 point ; so that if any two parallel lines be 

 drawn from the ends of the whole image, 

 and the distances of the eye and of the given 

 point be laid oft' on them respectively, the 

 line joining the points thus found, will deter- 



\\ 



mine the place of thg requued image of the 

 point. 



For A being the eye, and 

 B the vanishing point of r^ B I A. 



the line CD; AB and CD ~ 



being parallel, are in the 

 same plane, and AD is also 

 in that plane (62) ; and BC is the intersection of this plane 

 with that of the picture ; therefore E, the image of the 

 point D, is always in the line BC ; and AB : CD :: BE : EC ; 

 and taking the parallel lines BF, CG, in the same ratio, FG 

 will also cut BC in E. When AB is perpendicular to the 

 plane, B is called the point of sight, and is the vanishing 

 point of all lines perpendicular to the plane of the picture ; 

 and the vanishing point of any other line may be found by 

 setting off from B a line equal to the tangent of its inclina- 

 tion to the perpendicular line, the radius being AB. 



Scholium. When a line becomes parallel to the plane 

 of the picture, the distance of its vanishing point becomes 

 infinite, and the image is therefore parallel to the original. 

 In this case, the magnitude of the image may be deter- 

 mined by means of lines drawn in any other direction 

 through the extremities of the original line. In the ortho- 

 graphical projection, the images of all parallel lines what- 

 ever become parallel, the distance of the eye, and conse- 

 quently that of the vanishing point, becoming infinite. 



186. Definition. The subcontrary sec- 

 tion of a scalene cone is that which is per- 

 pendicular to the triangular section of the 

 cone passing through the axis, and perpen- 

 dicular to the base, and which cuts off" IVom 

 it a triangle similar to the whole, but in a 

 contrary position. 



187. Theorem. The subcontrary section 

 of a scalene cone is a circle. 



Through any point A of the 

 section, let a plane be drawn 

 parallel to the base ; then its 

 section will be a circle, as is 

 easily shown by the properties 

 of similar triangles j and the 

 common section of the planes 

 will be perpendicular to the 

 triangular section of the cone to which they are both per- 

 pendicular ; consequently, ABqirCB.BD ; but since the 

 triangles CBE, FBD are equiangular and similar, CB : BE 

 ::BF:BD, and CB.BD=BE.BF=ABq ; therefore EAF 

 is also a circle. 



