Measuring Surface and Solidity. 



And in the same manner, if all the errors were placed on B 

 we would have 



25 2S = m (_ 2 A 4- 3B + 3C 2D 2E) + n(2a 3b 3 c 4- 2d + 2e\ 



Similar formula? are easily deduced from this for other poly- 

 gons. Supposing all the error to be on A, and using the same 

 notation, we have 



inSgon 2( S) = w(A + B 2C) ...... +n ( a 5 + 2c) 



in4gon =ro(2A + 2B - 2 C _ 2 D) . . . + 



\- n( 2a 



in5gon = m(3 A + 3B 2C 2D 2E) -f 



in6gon = w(4A + 4B 2C 2D 2E 2F) + 



+ w(K-4a_4&_2c+2d +2e+ 2/) 

 &c. &c. 



In order to investigate a formula for solidity, we must sup- 

 pose that the co-ordinates are determined of so many points 

 in the surface, that the spaces included by joining them in 

 threes are planes, and that the field-book indicates which of the 

 points are to be thus connected. If two of these adjacent tri- 

 gons are in different planes, their common side, in strict lan_ 

 guage, is a side of the solid ; but, for the sake of convenience, 

 I shall define the term, side of the solid, to mean the line form- 

 ing the common base of a pair of trigons, although they be in 

 the same plane ; every one of the lines joining the points indi- 

 cated in the field-book, will thus be a side of the solid. A 

 point will also be understood to be in the upper or under sur- 

 face of the solid, according as a perpendicular from it to the 

 horizon passes through the solid, or does not pass through it. 



It is evident, that the lines ordinating 

 the apices of any solid to the horizontal 

 plane, must divide the space between the 

 upper surface and that plane into as many 

 triangular prismoids as there are trigons 

 in that surface ; and the space between 

 the under surface and the same plane in- 

 to as many triangular prismoids as there 

 are trigons in the under surface; and 

 that the capacity of the solid is equi- 

 valent to the sum of one set of these 



