29 



Again, because solids may in like manner be con 

 sidered as composed of infinitely thin solids or plates, 

 one placed upon the other, their differential is the area of 

 the surface multiplied by the differential of the axis. Thus 

 the base of any solid generated by the revolution of a 

 surface rectilinear or curved must be a circle, and the 

 proportion of the radius to the circumference being taken 

 as r : c, y being the ordinate to the line bounding the 



o 



/2 



s ?y 



vertical section, the surface will be -~- and the differential 

 of the axis x being dx, the differential of the solid will be 

 -^r , in which y in terms of x being inserted from the 



boundary line s equation, the integral gives the solid con 

 tent. Thus if the line which bounds is straight and 

 parallel to the axis, or the solid is a cylinder, its content 

 is the circle multiplied by the axis ; and if the line is drawn 

 to a point in the axis, or the solid is a cone, then its content 

 is one-third of the same product, or one-third of the cy 

 linder well-known properties of those two figures, proved 

 by ordinary geometry. So in like manner we find the 

 sphere to be two-thirds of the circumscribing cylinder, 

 the celebrated discovery of Archimedes, of which he caused 

 the diagram to be inscribed on his tomb. 



Lastly, it may in like manner be shown that the radius 

 of the osculating circle at any point of any curve, that 

 is, the circle touching it at such point, and having 

 the same curvature with it at that point, is equal to 



where dy being found in terms of 



x, the differential of j| is to be taken, so that there 

 will in the result in each case be no differential at all. 



