w Disquisitiones generales circa Superficies Curvas." 335 



angle formed by shortest lines above two right angles, is equal 

 to the entire curvature of the triangle. It is here supposed,, 

 that the unity of angles is the one whose corresponding arc 

 equals the radius (57° 17' 45"); and that for the entire curva- 

 ture, which is a portion of the surface of the auxiliary sphere, 

 the area of a square whose side is the radius of the auxiliary 

 sphere is considered as unity. This important theorem may 

 evidently be thus expressed : The excess of the angles of a 

 triangle formed by shortest lines above two right angles : eight 

 right angles : : the portion of the auxiliary sphere corre- 

 sponding to that triangle : the whole surface of the auxiliary 

 sphere. Generally the excess of the angles of a polygon of 

 n sides, all shortest lines, above 2n — 4 right angles, is equal 

 to the entire curvature of the polygon. 



The general investigations contained in the paper are finally 

 applied to the theory of triangles formed by shortest lines. 

 We shall mention only a few of the principal theorems of this 

 theory. If a, b, c be the sides of such a triangle (which are 

 considered as quantities of the first order), A, B, C the oppo- 

 site angles, a, (3, y the measures of curvature in the angular 

 points, <r the area of the triangle ; then \ (a -f /3 + y) <r will 

 be the excess of the sum A + B + C above two right angles 

 down to quantities of the fourth order. With the same ac- 

 curacy the angles of a plane rectilinear triangle, the sides of 

 which are a, b, c, will be 



A - T V (2a + J9 + y) or 



c- T V(« + /3 + 2 y )<r 



It is clear, that the latter theorem is a generalization of a well 

 known proposition first given by Legendre, by which, omit- 

 ting the quantities of the fourth order, the angles of the recti- 

 linear triangle are obtained by diminishing the angles of the 

 spherical one by one-third of the spherical excess. For sur- 

 faces which are not spherical, unequal reductions must be ap- 

 plied to the angles, and, generally speaking, the unequal part 

 is a quantity of the third order : if, however, the whole surface 

 deviates but little from the spherical form, it involves besides 

 a factor of the same order with the deviation from the spheri- 

 cal form. It is, undoubtedly, important for the theory of geo- 

 detical operations, to be able to calculate the inequalities of 

 these reductions, and to obtain thereby the full conviction that 

 they are to be considered as insensible for all measurable tri- 

 angles on the surface of the earth. Thus it is found, that in 

 the largest triangle in the measurement conducted by the au- 

 thor, 



