CosC 



Log cos C 



1842.] Area, Spherical and Plane. 779 



C be less than this, the spherical area will be less than that of the plane 

 triangle. 



When cos C is a maximum C = O. In this case a 2 + W- = 3 a b, 



or 1 + r-i = 3 *-• ; which solved as a quadratic gives •— = ' = 



a 2 a * a 2 



2*618 nearly. This is the maximum inequality in the sides, so as to 



have equal areas. 



In like manner, the value of the angle may be found for any given 

 ratio of the containing sides within these limits ; or the angle being 

 given, the ratio of the sides may be found. To save the trouble of these 

 calculations, I have constructed the small table in the margin, which shews 



for given ratios of a and b the value of C 

 with which the spherical and plane tri- 

 angles have equal areas. If the sides 

 were so large in regard to the radius, 

 that the terms omitted could sensibly 

 affect these results, it would be neces- 

 sary to take into account those of the 

 next, and perhaps also of higher orders. 

 To ascertain the actual difference in 

 the areas of the spherical and plane tri- 

 angles in an extreme case, suppose an 

 equilateral with sides of 1^ degrees : 

 the direct formula gives the excess 

 = 61" '21 7; and the difference in the 

 areas of the two triangles will be 0*395 1 

 square miles, corresponding to an ex- 

 cess of 0"*005245 : One-third of this 

 would be the error on each angle, and, 

 were it ten times as great, it would still 

 be, in Troughton's phrase, a quantity 

 less than what is visible in the telescope. 



1*0 



1*1 



1*2 



1*3 



1*4 



1*5 



1*6 



1*7 



1*8 



1*9 



2*0 



2*1 



2*2 



2*3 



2*4 



2*5 



2*6 



300 

 221 

 330 

 244 



390 



325_ 



450 

 356 

 480 

 389 

 5l0 

 424 

 540~ 

 461 



570 

 500 

 600 

 541 



"630 

 584 



b90 

 676 

 720 

 725 



750 

 776 



780 



9*82391 



•82588 

 •83109 

 •83869 

 •84804 

 •85067 

 •87021 

 •88238 

 •89498 

 •90783 

 •92082 

 •93386 

 •94687 

 •95980 

 •97262 

 •98528 

 •99777 



48°- 11' 

 47-57 

 47-20 

 46*23 

 45-11 

 43-46 

 42-08 

 40-18 

 31-16 

 36-01 

 33-33 

 30-50 

 27-46 

 24-16 

 20*08 

 14-50 

 5-48 



It is almost needless to remark that the supposed triangle is larger 

 than any which has yet occurred in practice. The great triangle in the 

 French arc, (long supposed to be the largest in the world), has an 



5 K 



