DEGREE, 
fi OOM Si 
Hf in the expreffion # sx j fin, oY, 
fathoms for the lat. 522 2’ 20”, as found by the meafure- 
ment of the arc between Cito and Dunnofe, the ellipti- 
we take g’ = 60820 
city, a, will be. found = aoe 
For pe — 60494. $= 335 =e —% 
I 
3x ei 8 fin.* 52.2. 20 == 1 SOLO 336 5- 
Log. ——22: 
Method of M. Legendre for calculating the Latitude, er 
tude, and Amis of the Stations on a Meridian Line 
ig. 65.) be the pole of the earth, PA, PB 
two elliptic meridians, let L be the known latitude of the 
peint A, it is required to find that of the point B, fituated 
on the arc , perpendicular to PA, its longitude, and 
the azimuth of A as feen from B. 
pines A to = — to the points A and 
B; make A M A= Be 
The rie ll are C4, having r tee its ais of curvature, it 
follows that a fimilar arc 9, whofe radius =1, will have for 
2 
its value 
If pon: the point M as a centre, and with the radius 
aight geal then iy ae try, 
cof. (p 5)=fin. L cof. 0, 
tang. P= tang. 
cof, L. 
tang. b= cof. L 
fin. @ 
plb=oe°—-L4+ ie +08. L 
@ ‘ae oe 
as cof, L aa ae L’. 
b=go0°—9 tang. Lig? tang. L (3+tang.? L). 
From the value 4 may be deduced the approximate value 
of ce latitude B. 
g0°—p b= L—5 ¢ tang. L. 
The angle P is the difference of longitude between A and 
B, and the angle J the required azimuth P BA. 
To have more exatly the latitude of the point B, it may 
be dais that it is a to the complement of the angle 
of the angle P +N b+N 
But as he angle NAM is very nearly equal to the angle 
MN col, L_ 
NBM, we fhall have NBM= 
N may be tap by aed aa (8), for if we oe 
w/=r, then CM= n. L, fame manner fo 
point B, ye aa is Lee Noe fin. L’; 
nearly we hav 
; thus aves 
MN=¢’r (fin. L—fin, L’). 
It is eafy to fee that this value is always pofitive, that is 
CM7CN, for the latitude of Ab 
Sin, L—fin, L’=za fin, (=) cof, (=) ; hence 
MN=20r fin, (=~) eon (+, 
Upon the fuppofition that L—L’ is very fmall, the are 
. Uy 
may be taken for its fine, and cof. L for cof. (= 5 ) ; 
then MN=er (L—L’) cof. L, 
But the approximate latitude of B or L’ =L— 
tang. L; hence MN=Ze*r @ tang. L oof. L=ie io 
fin. 
It may — defirable to have a more exaét expreffion 
fer NS d , which may be obtained thus: fince 
3; r may betaken without fenfible er. 
—e oe L 
ror =1; then calling L—L’/=dL, L+L’=2L—dL, and 
: —L’ L+L’.. 
the equation M N=2¢'r fin. cof, 7 will be- 
come M N=2¢? fin. 3 dL cof, ete 
Expanding the factor cof. —d L being fuppofed very 
{mall, M N=2¢? fin. dL cof. L+3¢ fin. dL fin. Le - 
But when the arc A is very fmall, fin. A=2 fin, (=), 
and M N=e*'fin. dL cof. L+ de" fin. dL, cof. L. 
MN fin. 
? 
Since fn. NBM=fin. J BNM _ MN cof. Ls 
we fhali have ee Ay preceding values of MN a 
fin. be fin. dLoof.Lc Tr de ant d Lefin. Lcot, i 
But cof; aa L-+fin. L fin. dL; 
Hence be’? dL cof L432dL fin. dL fin. L cof. Le 
Hence it follows that the angle NBM=2¢? 2 fin. L 
ecf.. L, and Bos yaa! ies true latitude of B= a 
(ed, 1? ¢ L—#e? @ fin. L cof. L. 
Jé L be ce of the point A, L’ the latitude of 
B, y the perpendic dle diftance of B ieee 
A, and r the radi f the ee or no 
; taking R” to exprefs the number of fesonds i in this rae 
due. we fhall obtain the following equatio 
LV’=L~— gRY(Z ; tang. L—iR”: (4) fin. L cof. L (a). 
reciprocally L= L’+ aR(Z :) tang. L—ZR” (5) fin. L? 
r 
col. L’. (4). 
It is evident that ¢ reprefenting the a the fe- 
cond term, in moft cafes may be —— The difference of 
longitude of A and B= P = ——+— 
r col, pp (4-8 Stang L), 
_() 
ue i azimuth of aa are BA or angle PBA= x= 
2 tang. L+3 R"Z J tang L (3+ tang,* L). (d). 
If L be only known, the laft se may, by a trigos 
go° — 
nometrical formula, be changed into z =90°—R"2 ~ tang, 
3 
Li RS, tang. L’ (1+ tang? L’). (e). 
Thefe formule are not difficult in their vitae and 
do not require a great many logarithms, as they have man 
elements in commen, as will be {een in the ae 
