Goniomefnc Prollems. 137 



each angle, which makes iheir sum, tluis corrected, equal to 

 1S0° exactly : with these llie uistances are conijiuled, con- 

 sidering the triangle merely as a plane one. 



When the sides of a spheric triangle are very small in 

 comparison of the radius, which is the case la most geodesic 

 operations, there are some convenient formu'.Le in Legen- 

 dre's Trigonometry, p. 463, for its sohuion. Another way 

 is to take the chords of the arcs and Had the angles formed 

 by those chords at each of the stations, and the triangle 

 then to be computed is actuallv a plane one. On this sub- 

 ject see the xviiith chapter of Cagnoli's Trigonometry*. 



As all substances expand by heat and contract by cold, 

 if the measurement of the base has been conducted under 

 different temperatures, it will require reducing to that length 

 which it would have been found had the temperature re- 

 mained uniform during the whole time of measuring. 



The most advantageous shape of a triangle for determining 

 the angles is, when the triangle is equilateral. (Cagnoli, 33t:.) 

 We must therefore in practice approach as near to that as 

 circumstances will admit. 



Persons unaccustomed to the practice would naturally 

 suppose that the small errors in a chain of triangles f would 

 accumulate to such a degree as to render the work very un- 

 certain after passing through the first six or eight triangles : 

 however, the contrary is known to be the fact. The small 

 errors compensate each other, and the work may be depended 

 upon with as much certainty after twenty triangles as after 

 two. Colonel JNIudge, the present able conductor of the tri- 

 gonometrical survey in England, from the base measured on 

 llounslovv Heath, carried on a series of 24 triangles over a 

 distance of 60 miles; and then, having measured a base of 

 verilication, and compared it with the computed triangles, 

 had the satisfaction to find a difference of no more ilian 4-| 

 inches between them. This remarkably near agrcenjcnt, 

 whilst it proves that the errors compensate each other, 



• This excellent work lias been translated into English by tJic autlior of 

 thii memoir, and is now ready for the press. 



+ Snell appears to have been the first who measured an arc of the meridian 

 by a chain of ttianj;U-». 



shows 



