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 1 is to 00.011920, so is 0° 24' 10'' 



/24'10"X.01192\ 

 to a fourth term I tq I , 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 -l 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 5 the required angle zmz' or its equal CmN ; and the angle 

 PCm by C. From the triangle PNm, right-angled at P, we have 



Pm a^ 2/ . h^ 



tang. 4'=pi\r~X2 '^~' ^'"^^ -^^ '^ equal to — x. We also have 



y 



tang. C=-3 whence, by combining this with the value of tang. ^|^, we 



obtain tang. C = — 2 Xtang. 4^; but 6 =4^ — C, therefore, we shall have 



tang. ■\. - ta ng. C (o^—S 2 ) tang.4. 



tang. 0= tang. (nL' — L' )=:;—;"; rz7. r^ = ' — ttToZ ^, — ' 



o & \r / l+tang.-^Xtang.C a^+^^tang.^^, 



which, by substituting in it a(\ — ■«) for h, is reduced to tang. ^= 



[1 — (1— a)2] tang.-l' 

 , ■ /, _ X 2 — ^T^TT' (^)' ^^^J<^^ i^ ^^^6 formula for estimating the 



value of^, at any given latitude -\>. 



2. In proceeding from the equator towards either pole, and vice 

 versa, the angle (5 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 as (5 is a maximum when its 



