Measuring Surface and Solidity. 239 



Referring to the same figure, and conceiving it now to be 

 entirely in the horizontal plane, we perceive that its area is 

 equivalent to the sum of the trapezoids BC, CD diminished by 

 the sum of the trapezoids BA, AE, ED ; or if we designate 

 the latitudes of the several points by the letters A, B, C, &c 

 and the longitudes by the small letters abc 9 &c., and the sur- 

 face by S, we obtain this equation, 



D) (e + d)(E D) (a + *)(A E 

 _(6 + a)(B A) 



which being simplified, becomes 



2S = A(6 e)+ B(c a) + C(rf 6) + D(e c) + E(a rf) 



Or, in words, twice the area of a polygon is equivalent to the 

 latitude of each point, multiplied by the longitude of the point 

 before, minus the longitude of the point after it. And the area 

 will be a positive or negative quantity, according to the direc- 

 tion in which we proceed. To determine the area of any poly- 

 gon, we thus require no more multiplications than the number 

 of its sides ; or, since the differences, not the absolute lengths 

 of the co-ordinates, are considered, we can subtract from all the 

 latitudes the latitude of the most southerly point, and thus have 

 one fewer multiplication. It may also be noticed, that the for- 

 mula simplifies itself when applied to the tetragon, so as to 

 require only two multiplications. It is true that these formulae 

 are not well adapted for logarithmic calculation ; but the whole 

 sum of the products is readily obtained by the use of a table 

 of quarter squares. For example, let the co-ordinates of the 

 points of a polygon be as follows : 



Lat. Long. 



A 3089 14566 



B 3527 14846 



C 2682 15793 



D 2478 15660 



1753 15156 



F 2997 14643 



Subtracting 1753 from the latitudes, to save a multiplication, 

 we have twice, the area, equal to 



