te Radius of the Earth. 
r * 
at once the value of 4, corresponding to a latitude not expressed in 
column A of the table. 
Radius of the Earth. 
1. Knowing the Reduction of Latitude, we can readily obtain the 
Radius of the Earth, corresponding to any given latitude . Thus— 
let the radius Cm be designated by p ; and we shall have, from the 
triangle ee =p*sin.2PCm, and 7? =p?cos.2PCm; but PCm= 
L—s, hence, y?=p?sin.2({—6), and w*=p*cos.?(.)—é)=p? X 
Ferd oe ; these values of y? and «? being placed in equa- 
tion (1) will give the relation 
p*sin.?()—6)= —_ (a? —p2[1—sin.?()—3)]), whence we ob- 
1 
ate Se Xsin. ates -»}? 
for a, reduces to p= 
7 which, by substituting ap) 301 
(7 0.00667Sin. ey et 
2, Formula (7) can be easily reduced to numbers, for we shall 
have the value of 4, given by the table for the reduction of latitude, 
when J is itera: Regarding a, the earth’s equatorial radius, 
unity, we have qakcleted from formula (7) the table below, in which 
columns A contain the given latitudes differing by 1°, and columns 
B, the corresponding values of the terrestrial radius. 
Barth. fe ies he enna , i7, , 
Values of the E adius—the equatorial radius being 1.000000, 
B B yA; 8 Ay B 
9999: ale 0.999614 ilo 0.999175) ‘010; 5 
9938 21/0.999577)31 0.991241 1 0.998577 110.99 
).999857}22|0.999537)32\0.999073142\0.99861 9}52 0.997944 
2 999496 0.9978 
When it is required to find the length of the roils by means of 
the table, for a place ee latitade falls between any two consecu- 
tive ones of the table, we in a manner entirely similar to 
that explained for the reduetio of latitude. 
~ 3. Since ais equal to 3963 English miles, we have only to mul- 
tiply this number by the decimal given by the table, and the product 
will be the number of miles in the terrestrial radius, at the place 
whose latitude is in column A, on the left of the decimal used. 
