the 'Trigonometrical Operation . 169 
three angles of each of the triangles, into whatever number, 
great or fmall, it might be divided, would conftantly amount 
to 180 0 . But the earth being a fphere or fpheroid, it follows, 
that the fame inftrument, fucceffively adjufted at each of the 
Rations, will have its axis perpendicular, on a fphere, to an 
equally curved furface ; on a fpheroid, to one unequally curved, 
in either cafe forming the horizon of the ftation ; and the fum 
of the three angles of fuch a fpherical or fpheroidical triangle 
nuift, as is known, always exceed 180 0 , lefs or more, iti pro- 
portion to the lengths of the fides. When the triangles are 
very fmall, the excefs being of courfe fmall cannot poffibly 
be difcernible by common inftruments. Even the fineft, fup- 
pofing them free from error of divifion, will fcarcely render it 
perceptible, without the utmoft care in making the obferva- 
tions. This will be fufficiently exemplified in the following 
calculations, where a column is inferted containing the fpherical 
excefs ; and another for the difference or error between that and 
the excefs of the fum of the obferved angles above 180 0 . From 
thefe it will appear, that, notwithftanding the goodnefs of our 
inftrument, and the pains taken in ufing it, we have frequently 
failed in bringing out an excefs ; and indeed the refults have 
even fometimes been in a fmall degree defe£tive. 
It had been at firft propofed to multiply the obfervations as 
much as poffible, and particularly by fucceffively changing the 
zero of the inftrument to new points (PhiL Tranf. 17 87? 
p. 219.), to meafure the fame angles on different parts of the 
circle, fo as to fubdivide any errors that might arife from in- 
accuracy of divifion, or fihake at the center. This principle, 
perfeftly good in theory, and which was adhered to as far as 
the circumftances would permit, was neverthelefs found, on 
many occafions, to be impoffible in pradtice, without facri- 
Vol. LXXX. Z ' firing 
