Reduction of Latitude. 231 
If the length of a degree be required, whose middle point falls 
between any two consecutive latitudes expressed in the table, it 
would be sufficiently accurate to proceed as follows: Suppose we 
wish the length of the degree whose middle point is at the latitude 
41° 24’ 10’—we perceive from the table, that the difference be- 
tween L answering to 42°, and L answering to 41° of latitude, is 
00.011920 miles; then we say, as ] is to 00.011920, so is 0° 24’ 10” 
is x.01 zs 
1c 
to a fourth term , which being added to the tab- 
ular length of the degree whose middle point corresponds to the lat- 
itude 41°, will give the length of the degree sought. ‘This method, 
of course is but an approximation ; but if the utmost accuracy be de- 
sired, we should place 41° 24’ 10” for J in formula (5), and de- 
duce from it the corresponding value for L. 
Reduction of Latitude. 
1. It is evident that the vertical and radius at every point on the 
earth’s surface will make an angle with each other, excepting at the 
equator and the poles. This angle is called the reduction of lati- 
tude ; and for the purpose of determining its value, let Cz’ be the 
radius of the earth produced through any place m; and designate 
by 6 the required angle zmz’ or its equal CmN ; and the angle 
PCm by =e ss = triangle PNm, _—— at P, we have 
tang. emerge ia x2 7” since PN is equal to =; 2. We also have 
tang. c=", whence, by combining this with the value of tang. L, we 
2 
obtain tang. Cn: x tang. 1; butd=L—C, therefore, we shall have 
tang. 1 — tang. C (a? —b*) tang.) 
I-+tang) Xtang.C ~ a? +btang.2 ” 
which, by substituting in it a(1 —~) for b, is reduced to tang. d= 
[l—(1—<a)?] tang. 
1+(1—a)*tang.*y ’ 
value of 6, at any given latitude J 
2. In proceeding from the equator towards either pole, and vice 
versa, the angle 0 increases to a certain value depending upon the 
oblateness, and then begins to decrease ; hence, at that certain lati- 
tude 6 has a true maximum value ; and watt is a maximum when its 
tang. d= tang. ()—C)= 
(6), which is the formula for estimating the 
