VI. 



TABLE FOR COMPUTING TERRESTRIAL SURFACES. 



THE tables under No. VI. were published by Delcros in the Annuaire Meteoro- 

 logique de la France pour 1850, p. 65 et seq. 



The formula from which they have been computed reads as follows : 



s = at 



I sin (f> cos (L + i <) 

 I" / 7i\ / 7\ 2T 



iI 3 v|-r-J + (- rJ sin (4>-r-<Z>) cos 

 \a.+bj \a + &/J 



+ i [ 3 (-^!)' + (^j)l sin (S H- i *) cos [5 L + (2 , + 0)] 



90 



[etc.; 



in which a = J great axis of the globe ; I = small axis ; L = the latitude of the 

 lower limit of a quadrilateral surface ; L' = the latitude of the upper limit of the 

 same ; < = L' L ; S = the area of a quadrilateral surface of one degree in lon- 

 gitude ; TT = the ratio of the circumference to the diameter. 



Substituting the numerical values, the quarter of the meridian being = 10,000,724 

 legal metres ; the ^ great axis, or , = 6,376,989 metres ; the small axis, or J, 

 6,356,323 metres ; the ratio of the axis ^i.^* 5 an ^ ma king ^ = 1 nonagesimal, the 

 formula becomes, 



224.996360 cos ( L + 30') 



0.730851 cos (3 L + 1 30') 

 S= { + 0.001784 cos (5 L + 2 30') 



0.000004 cos (7 L + 3 30') 

 + etc. 



The first three terms of the formula give the results with sufficient accuracy. 



In order to avoid too large a number of figures, the results are given in square 

 miles, the linear base of which is a mile equal to T ^ of the mean degree of the 



meridian. That mile is thus = ( 90 x f 5 ) = 7407.942 metres. In order to con- 

 vert the results into new geographical miles, of which 60 = 1, multiply by 16, 

 log = 1.2041200 ; into common French leagues, 25 = 1, multiply by 2.777778, 

 log = 0.4436975 ; into nautical leagues, 20= 1, multiply by 1.777778, log = 

 0.2498775 ; into English statute miles, 69.163 = 1, by 21.711034, log = 1.3366868. 



USE OF THE TABLES. 



Table L, which gives the number of square miles contained in the quadrilateral 

 surfaces of one degree in latitude and longitude, successively from the equator to the 

 pole, will be more frequently used. Table II. has been computed for maps on a 

 smaller scale ; and Tables III. and IV. for maps of very small scale, covering large 

 areas, in which surfaces of one degree could not be estimated with sufficient accuracy. 

 If the scale is large enough to have the minutes traced on, then Table V. is to be used. 



For computing a surface by Table I., which may serve as an example for all the 

 others, find first the lowest parallel circle which crosses, on the map, the surface to 

 be estimated; suppose it is 40 lat. N., and the zone within 40 and 41 lat. N. 

 contains four integral degrees of longitude, that is, four surfaces of one degree each 

 way ; then in the first column of the table, on the line beginning with latitude 40, 

 and in the vertical column headed 4, take the value of these four surfaces, viz. 

 685.88. Then take likewise the value of the number of surfaces between 41 and 

 42 lat. N., artd so on. The fractional parts left outside of the integral degrees are best 

 estimated, with the compass, in decimals, the values of which can be found in the 

 columns of the multiples, by properly moving the decimal point to the left. Having 

 taken them in that way, and summing them up with all the integral surfaces, we ob- 

 tain the total surface required. 



F 11 



