1842.] Area, Spherical and Plane. 781 



maximum difference of areas. For any other ratio of sides the angle will 

 be smaller. For the ratio ~ — the angle is 1 20°. When the ratio 



A 



10 a* -f- W . 101 



is —■ the value of — - — — is -—-'the log of which 0*52720 corres- 

 1 o ao oil 



ponds to a value of C somewhat less than 150° or 140°.55 , .45 // ; and so in 



other cases. When the ratio of the sides becomes indefinitely great, 



the maximum difference angle approaches indefinitely near to 90°. 



In well chosen triangles there is not usually any very great differences 

 in the sides, and hence, practically, the greatest differences of area will 

 usually occur when C is not far from 120°. 



If, for example, we suppose a triangle with sides of a degree each, 

 and containing an angle of 120°, by the original formula, the excess is 

 27".210; and the difference in area between the spherical and plane 

 triangles is 0" 18214 square miles, the excess corresponding to which is 

 0*0024176. On a triangle with degree sides and the maximum angle 

 124°.02'.35", the excess is 26 // .035 : the difference of areas is 0*18320 

 square miles, the excess corresponding to which is // .0024318. Such 

 differences, though utterly insensible in the telescope, are still much 

 greater than have ever occurred in practice ; for though a single side of 

 more than a degree be nothing very extraordinary, it is but rarely that 

 two such sides can be found forming a triangle with a third side of from 

 118 to 120 miles. 



The difference here treated of is, in similar triangles, proportional to 

 the 4th powers of the homologous sides : Hence, in an equilateral with 

 half degree sides, this difference would be ^ of 0"* 005 245, or 

 0*00006475 ; and on the isosceles with half degree side containing 

 120°, the difference would be ~ of 0"*0024176, or 0"*00001511. Trian- 

 gles such as these are not very uncommon, but it is much more common 

 to have triangles with less than half of their area. 



It is thus fairly proved that the difference between the excess on a 

 spherical triangle computed rigorously and the excess deduced by reck- 

 oning its area as equal to that of a plane triangle with the same sides 

 and contained angle, is a quantity so small that, even in extreme cases, 

 the neglect of it will not induce any sensible error; and that, on 

 triangles such as usually occur in practice, the difference is so utterly 

 insignificant that to go much out of the usual way in order to take 

 account of it, would be a very needless refinement. 



