1 10 



ELEMENTARY LESSONS ON [CHAP. n. 



as before (Fig. 63), but at the point O draw a straight line 

 touching the circle, the tangent line at O ; 

 let us also prolong C P until it meets the 

 tangent line at T. We may measure the 

 angle between O C and O P in terms of 

 the length of the tangent O T as compared 

 with the length of the radius. Since our 

 radius is I, this ratio is numerically the 

 length of O T, and we may therefore call 

 the length of O T the "tangent" of the 

 angle O C P. It is clear that smaller angles 

 will have smaller tangents, but that larger 

 angles may have very large tangents ; in 

 fact, the length of the tangent when P C was 

 moved round to a right angle would be 

 infinitely great. It can be shown that the 

 ratio between the lengths of the sine and 





C M 



Fig. 63. 



of the cosine of the angle is the same as the ratio between the 

 length of the tangent and that of the radius ; or the tangent of 

 an angle is equal to its sine divided by its cosine. The formula 

 for the tangent may be written : 



133. Solid Angles. When three or more surfaces intersect 

 at a point they form a solid angle: there is a solid angle, for 

 example, at the top of a pyramid, or of a cone, and one at every 

 corner of a diamond that has 

 been cut. If a surface of any 

 given shape be near a point, it 

 is said to subtend a certain solid 

 angle at that point, the solid 

 angle being mapped out by 

 drawing lines from all points 

 of the edge of this surface to the 

 point P (Fig. 64. ) An irregular 

 cone will thus be generated 

 whose solid angle is the solid 

 angle subtended at P by t the 

 surface E F. To reckon this 

 solid angle we adopt an expedient similar to that adopted when 

 we wished to reckon a plane angle in radians. About the point 

 P, with radius of I centimetre, describe a sphere t which will 

 intercept the cone over an area M N : the area thus intercepted 

 measures the solid angle. If the sphere have the radius I, its 

 total surface is 471-. The solid angle subtended at the centre by 

 a hemisphere would be 2?r. 



Fig. 64. 



