

Geodesic Lines on the Earth's Surface. 353 



gress is to be made in the investigation of the figure of the earth, 

 certainly no " small quantities" may be set aside without at least 

 a distinct idea of their magnitude and ultimate effect. For in- 

 stance, the course of the geodesic line joining two points on a 

 spheroid of small excentricity is probably not investigated in any 

 English work, and it is therefore not unnecessary to inquire how 

 far this line departs from any of the plane curves joining its ex- 

 tremities. 



Although the observed angles of a triangulation are not geo- 

 desic angles, yet in the calculation of the distance and reciprocal 

 bearings of two points far apart and connected by a long chain 

 of triangles we may fall upon the geodesic line in this way : — If 

 A, Z be the points, then to start the calculation from A we get 

 by some preliminary calculation the approximate azimuth of Z 

 at A, or the angle made by the direction of Z with either of the 

 sides AB, AC of the first triangle. Let P L be the point where 

 this line or direction intersects B C ; then to find P 2 , where the 

 line cuts the next triangle-side C D, we make the angle B Y^ P 2 

 such that B V i P 2 + B P t A= 180°. This fixes P 2 ; and P 3 is fixed 

 by a repetition of the same process ; so for P 4 , P 5 , . . . Now it is 

 clear that the points Pj, P 2 , P 3 , . . . so computed are those which 

 would be actually fixed by an observer with a theodolite, proceed- 

 ing in the following manner. Having set the instrument up at 

 A and turned the telescope in the direction of the computed 

 bearing, an assistant places a mark Pj on the line B C, adjusting- 

 it until it is bisected by the cross hairs of the telescope fixed at 

 A. The theodolite is then removed from A, placed over the 

 mark Pj, and the telescope turned to A ; the horizontal circle, or 

 merely the telescope itself on its axis, is then turned through 

 180° and the instrument clamped. The assistant then places a 

 mark P 2 on the line C D so as to be bisected by the cross hairs 

 of the telescope, which is then removed to P 2 ; and in the same 

 manner is P 3 fixed. Now it is clear that the string of points P 

 approximates to the geodesic line; for the plane of any two con- 

 secutive elements V n _ x P n , P«P W+1 contains the normal at the 

 common point V n . 



With the exception of the greater part of Norway, Sweden, 

 and Turkey, the whole of Europe may be considered to be co- 

 vered with chains of triangulation binding its countries together. 

 From the north-western extremity of the Hebrides to Dunkirk in 

 Prance there is a strong connexion by the British triangulation ; 

 this line may be continued through France to Berne, and by a 

 triangulation, possibly not quite complete, through Switzerland 

 and Italy to Corfu and Palermo. Again, a line may be drawn 

 from Cadiz to St. Petersburg : if the necessary triangulation is 

 not quite complete, it will doubtless be so before many years 



Phil. Mag. S. 4. Vol. 39. No. 262. May 1870. 2 A 



