DEGREE. 
Brought over 9.99903 
RY = 5.31442 
I 
JS= 5-49564 
n U= 2.6786 
0.19828 
4863".9 = 3- poege 
= 1" 21°30 
This method of | Legendre admits of being gen 
when the fecond aD is confiderable, by a more judi- 
cious ufe of the formula; | = e*d.Lco 
For, having found yas “ the ee rales, the fecond 
correGtion i is obtained by A ee by e? cof? L. 
Exanple. To jind the Latitude of Dunkirk, Vid. Example I. 
1.4760768 
03797755 
_ A 
45 Tab. IIf. 
Tang. L 
= I. as = 86.977 
Tab. VIIL. 2 oe ae. 72.92 
9-3 912 s 
L—L'= By", iia 
Oo. ae 
This formula may be verified by the following trigono- 
metrical oo which is perfectly accurate. 
t the latitude of the pot oint A, L the lat. of B, 
P the Geen. of longitude. 
Sin. L — L’ = fia. L fin. ¢ tang. x (1 +e? cof.? L) 
Sin. L — L’ = fin. L’ tang. otang. = (1 + &cof’?L.) 
Example. ibid and Dunkirk. See Example I. 
= 4 = 3-7311373 = 5384%.4 = 1° 29! 44.4 
Tang. ? = 8.4168110 
Cof. L. = 9.9983070 
P=2° 22! 44” = 8.61 1850 
olds == 61° 2°40" = 198907248 
Tang. ? = 8.4108110 
© 11’ 22" = 8.9172539 
2 
r+ ecoi?L a s 
Tang. — 
0.0010194 
din LL — L'= 87".16 = 6. Orehsed 
The method of M. Delambre is quite independent pee 
diftances from the perpendicular and meri 
confidering the diftance of two points, eee on a 
great circle paffing Pate san aa verticals, he takes in pre- 
ference the chords of the 
The obferved angles are elie to thofe of the chords ; 
the bafe muft Kkewife be reduced to a-chord, if the utmof 
precifion is required. 
If B be the length of an arc, its excefs above its cher z 
is thus -_ p being the radius of the earth, 2° = 34 
Zz Q* 
> 
8 method, there is no occafion for a figure, and th 
aitance beiween the parallels may be found without lee: 
lating the diftances to the meridian. There is one inconvee 
nience which, as the author obferves himfelf, is not inconfidere 
able, that the latitudes of the ftations muft be calculated 
previoufly to the diftance between the parallels: but, as the 
laticades of the ftations are generally required, this cannot be 
confidered as a ferious objection, 
Let us fuppofe at firft the earth to be duet rac and the 
triangle P A B to be no longer reétangular. 
Cof, PB=cof. A fin. PA fin. + cof, pact AB; 
and agreeable to. the above a n, 
Sin. L’ = cof. Acof. L. fin 9 + fin. Leof. @: 
Hence, fin. L—fin.L'= fis. na Leof.g— = in geal Acof.L 
==(7 1—co f.%) fin. L—fin. @ cof. iL 
=2hn?igfia.L — fin. dcof. Ac .; oF 
But if A be taken exteriorly, that is, if, inflead of the 
cof. A, we fubfitute its value taken from the equation 
A = 180° — Z — cof. A=co 
then, fin. L.—fin. L’= fin. @ cof. Z. ai L+2 fin. "2? fia. L: 
fin.¢@cof. Zcof. L + 2 fin.? 19 fin, L 
1 _Try . 
or,2fin.§ (L—-L’)= oral) : 
confequently, ' : : 
2i » 
afin tdi fin. @ cof. Z cof. L + 2 fin? 1 o»fin. L 
cof. (L —Z 
__ fin. ¢cof. Z cof. L +2 fin. £9 fin. L 
~ cof, £dL (cof.L+ fin. ae idLy 
Now, very nearl n. Z . aL, and 
cof. § dL =1; hence dividing’ by the cot 1 Tid 
fin. @ col. Z + 2 fin.? Egtang. L 
1 + tang. Ltang. $¢dL 
binomial formula 
fin oar ae eg Z + 2 fin.’ 
tang. $ 
Since ie is sry se the tang. d L may, without fen-. 
fible error, be taken = fin. =i fin.d L: then the pre- 
: this expanded by the 
tang. L) (1— tang. L 
tang. 4 ~L2+ ee hues L)(1 — tang. 
L tang. 3dL-+4+tang.?*Ltang2idh). (1). 
ay i hee and rejecting the ae of the third ri 
£$dL= ae @ cof. Z + fin.” 4 > tang. L 
° yee 7 tan : 
Subititute for tang. 3 1d L its approximate value £ fin, @ 
cof. Z: then, 
tang. dL zi fin. ¢ cof. Z + fin.’ 2 tang. L — iin. © 
ng. 
et fince cof, 2 Z=1—fin.?Z ; 
£dL= $ fin. Qcof. Z un in 
£@tang.L — 
@ a one 1 fine @ cof.? Z'tang. L. 
4 fin? 
Tang. d Z. +5 fin? phat tang L; (2) 
hence, tang. d LordL =fin. gent. Z + 5fin. °@ fin, V2 ta0g. L 
=¢colZ+ z Lofin, ofin.Zta 
d L expreffes the dircace of the parallel ee dna 
the two extremities of the arc @; for ore exact value, - 
may in the ur of equation (1) prefer the terms of 
tang? i. e fhall find in. d7L = cof. Z + 3 in.2@ 
fin.? Z tang. L —i fine dcof, Z — fin? Z tang? L. 
But by the refolution of the triangles, the chord of ? is 
given: fo that 
fin. dL= Kol $0¢ col. Z (1 + tang. i 9 fin. Z tang. Z 
tang. L — 2 fin. § > fin.” Z tang.? L). 
This 18 fe value of fin, di, expreffed i in terms of the fam 
meafure as the ch : forto have d L itfelf, to this ex 
preffion muft be added the excefs of the arc above the fign, or, 
F cof, £ 2 ¢ cof, Z)\ — 
2 finsdL=% 3 
p being the radius of the ai in toifes, fathoms, or metres. 
y| Alfo 
