584 KEPOKT — 1893. 



for two of which the spiral of Archimedes is employed, and in the third 

 the hyperbohc spiral. 



Lastly, there is a method of rectifying the semicircle given by 

 Kochanski, about 1685 (see Cremona), in which an angle at 30° laid 

 off at the extremity of a vertical diameter meets the tangent to the latter 

 from the point of intersection ; along the tangent a distance equal to three 

 times that of the radius is taken, giving a point whose distance from the 

 extremity of the vertical diameter is 



Z=rN/'4+(3-tan30°)2 

 =3-14153r. 



Using one of the foregoing methods for finding geometrically the length 

 of an arc, the area of the sector of a circle can now be found by drawing 

 a tangent to the arc equal to it in length and joining the extremities of 

 the segment with the centre. The area of the triangle so found is equal 

 to the area of the sector. 



Cnlmann has compared the difference of area resulting from the as- 

 sumption of a parabolic arc or of a circular arc as the approximate form 

 of the bounding curve of an area. 



Let F, = the area of a circular segment. 

 F2 = ,, ,, parabolic „ 



r,=«/ {I 



1.3.5 3.5.7 5.7.9 



and 'P"=\fl 



(/, I, and t having the same values as previously), 



4 77 1 

 3 ^'^5 



the difference F, - F2 < ;^ Wi x 5 f^, 



or <^ «2xP,, 

 Therefore if /"Th ' 



the difference < ^^^p^ F 1 . 



10 

 •100 



In the reduction of volumes to segments the same principles and con- 

 structions are employed as in the reductions of areas. 

 Thus, if 



V=volume=area x I 

 =bhl, 



where h, h, and I are respectively the breadth, height, and length, 

 by taking two of these equal to a certain unit base, then for the third a 

 value can be obtained in terms of which volume can be measured, and 

 the volume represented by a segment. Many special problems occur 

 in connection with canals and earthwork which admit of graphical 

 solution by means of the foregoing principles. 



