Interpretation of Formults in Spherical Trigonometry, 23 



able in addition to what has been pointed out in the other 

 cases. 



10. I shall conclude with some remarks which naturally 

 arise from what has been already said. 



(1.) Dr. Simson and others define a spherical triangle as 

 being " a figure upon the surface of a sphere comprehended 

 by three arcs of three great circles, each of which is less than 

 a semicircle" The words in italics should be omitted. Ana- 

 lysis, as we have seen above, detects the error, and shows that 

 a side may be of any magnitude not exceeding a complete 

 circle. An author, as the writers on trigonometry have done, 

 may properly confine his attention to triangles having each of 

 their sides less than a semicircle, but he ought not to state 

 that there can be no others. 



(2.) The proposition is not true, in which it is asserted, 

 that " the three sides of a spherical triangle are together less 

 than a circle." For a triangle understood in the ordinary 

 meaning, (as bounded by three arcs of great circles, without be- 

 ing intersected by them,) the limit is ^lijo circles, instead of one. 

 To illustrate this familiarly, let us take two points, A and B, 

 on the horizon of a globe, very near each other, and a third 

 C, nearly diametrically opposite to them, but above the hori- 

 zon ; then draw the smaller of the arcs joining A C and B C. 

 By these two arcs, and by the greater arc of the horizon, the 

 sphere will be divided into two triangles, each having one side 

 almost a circle, and each of the others nearly a semicircle. 



(3.) Neither is the proposition true, in which it is asserted, 

 that " any two sides of a spherical triangle taken together, are 

 greater than the third side." It is easy to see, by taking one 

 of the sides almost a circle, and the others small, that one 

 side may exceed the sum of the other two in any ratio what- 

 ever. 



(4.) The assertion is likewise erroneous, that " the sum of 

 the three angles of a spherical triangle cannot be less than 

 two right angles, nor greater than six." The true limits are two 

 right a?igles and ten right angles. To illustrate this, let the 

 surface of a sphere be divided into two triangles having ex- 

 tremely small sides. Then, the angles of the smaller triangle 

 being A, B, C, those of the other will be 2 tt— A, 2 ?r— B, and 

 2 TT— C. The sum of the former will be 7r4- E, where E may 

 be as small as we please; while, by adding the others, we 

 find for their sum 6 7r — (A + B + C), or 6 7r— -tt— E, or finally, 

 5 TT— E; that is, ten right angles wanting E. 



(5.) The area of a spherical triangle may be of any magni- 

 tude between zero and the surface of the sphere. 



(6.) The same principles explain some things regarding the 



