Measuring Surface and Solidity. 243 



linear mass is equivalent to the altitude of each apex, multi- 

 plied respectively by the product of the latitude of each apex 

 connected with it by a side, into the longitude of the apex be- 

 fore minus, the longitude of the apex after it in the order of 

 their connexion with the first by a side. 



The co-ordinates of all the points in the survey (except points 

 determined by offsets) are thus calculated, in order to examine 

 the accuracy of the work, to perform the necessary corrections, 

 and to calculate the area, but they present another important 

 advantage in the facility which they afford by protracting either 

 plans or models. This is performed by means of two graduat- 

 ed slips of ivory or brass, the one adjusted so as to slide on the 

 other at right angles, and called T scales, or offset scales ; the 

 lines which determine every point are previously corrected, and 

 they cease, with regard to the drawing or model, to have any 

 dependence on one another, so that in the representation there 

 can be no accumulation of errors. 



The formula for the area of a polygon, given above, differs 

 in a practical point of view from that of M. Lexel of Peters- 

 burgh, which is in this shape. 



, j 



A . B . sin (480 _ a) 

 A . C . sin (360 a b) 

 A . D . sin (540 _ a b e) 

 2 S = < A . E sin (720 a bc d) 

 B. C sin (180 I) 

 B . D sin (360 b c) 

 &c. &c. &c. 



in so far, that the number of members may be always one less 

 than the number of sides, while in this the members are more 

 complex, and their number increases with the number of ways in 

 which the sides, except one, can be collated together in pairs. 



The full advantages of the system are not apparent until its 

 application to the field operations is examined. In a polygon 

 determined by the measurement of all its sides and their bear- 

 ings, there is absolutely no form which produces, from a given 

 error in measurement, a minimum alteration of shape ; the 

 square of the error in position being in every case equivalent to 

 the square of the error in length, plus the square of the length 

 into the square of the error in bearing. And the error in area 

 is a minimum only when the area itself is a maximum, the peri- 



