DEGREE, ° 
- the fun and tower by the = us which will always 
be exa&t enough for this correct 
On the Calculation of the Triangles. 
When the obferved angles are reduced to the horizon by 
the method above fated, the triangles may be either con~ 
fidered as rectilinear, Baan or oo the firft cafe 
fuppofes the sngles reduced to their oo and that the 
fides of the triangles ‘ee e the wine of an irregular poly- 
hedron infcribed within ae oni ad 3; the bafe in 
like manner fhould be reduced to a chord line, by fubdu- 
ing from its meafured length the excefs of the arc above the 
chord. 
But the {pherical computation is rendered very eafy by 
a theorem fit inveftigated by Legendre, by which it ap- 
pears that a {pherical triangle, whofe fides are very f{mall, 
may be calculated without fenfible error by the rules of plane 
trigonometry, es a {mall correction be firft made in 
the obferved angle 
When the Englith {urvey was a began, this theorem 
o was the firft perf 
fectly accurate 5 i t 
the reft inferred, as in calculating diftances from the meridian 
and its perpendicular, the method of Legendre feems to be 
the moft eafy and leaft liable to error. 
General Principles of the Method of tracing and calculating 
a Meridian Line. 
If we imagine a plane ‘pafling through the axis of the 
earth and the zenith of any place; this plane, extended to 
the limits of the celcftial {phere, will there trace a great 
circle, which will be the meridian of that place: and if a line 
refponding terreftrial meridian. From the immenfe length of 
the radius of the heavens, the verticals of all thefe points 
may be confidered as parallel to the 
meridian: fo that the terreftrial neta may be defined a 
curve paffing through and connedting thofe points, in whic 
ut, ont 
lic entirely in it, if it be a {phere, or any regular figure 
revolution. 
If in any given point, we fix a fignal, and by means of 
the optic axis of a telefcope, direéted exa@ly north or fouth 
of that fi goal, we place others in this axis, and by removing 
the telefeope cs continue, in the fame manner, to place other 
direction, we fhould trace a meridian line. 
Let ABC DE, Plate VII. fig. 60, be achain of triangles 
extended in the direCtion of the meridian, and whofe fides may 
be confidered as arcs of the terreftrial {pheroid. Suppofe, 
4 
plane of the celeftial 
by an obfervation of ek azimuth, er a or direétion 
of the fide A C with the firft fide € meridian { be 
known, The poi tM, where cae ai cuts BC, may be 
found by ies onetey. The points A, B, C, being in aie 
fame horizontal plane, A M will likewele be in the fame 
plane; but from the ir pga of the earth the continuation 
of this line, M M’, will be above the furface of i: next tri- 
angle BCD: if ten, hab aby Sprain the angle CM M’, 
the line M M’ be nto the e of the triangle 
» by fup ae it to turn rou oe BC as 
point M’ will pee ea {mall arc of a circle 
confidered as a ftraight line perpeudiculae to the plane 
B 
From this it ea that the ar gata — in bend.» 
ing down this line in the direction ofav ae and in cal- 
ena the dilcncs M M’, to find the - 
y carrying on the a line in ia r throu 
ugh 
the whole feries of triangles, we may by trig paouae al cal- 
culation find the dire ere ane length of this meridian from 
one extremity a the o 
If the earth be ana an irre regular figure, abe line differs a 
little from the terreftrial meridian ; age it always as thie 
edad cal it 13 the fhorteft line that dra 
twee o extremities, over the ice of the ear rth 
the obferver, 
at place 
In the {phere thefe perpendiculars are ie circles which 
cut each other on the equator. But on foid, and 
ftill more if that be ee thefe penpendliculers will be 
curves of double curvatur 
Whatever be the cae of the terreftrial fpheroid, the 
parallels to the equator are curves in which all the points 
have the fame latitude. 
The-fituation of a place is determined when the perpen- 
dicular to che metidian or its parallel is known, and alfo the 
plained, we obtain 
angles, by means of their co-ordinates, ‘or diftances 
the id al and from the meridian of e principal 
ftation 
Su uppo ofe the oS ABC, 
BCD, to make part of a 
chain of other t 
iangles whofe fides are arcs of great circles 
Ww 
its rele 2 ne meridian, ob- 
i rft calculate aie hee excefs ¢in the 
right me triangle A eC, and then refolve it by the two 
ro propor 
. (g0° te: cat ane vee Ac=s 
Sin, (90°—4:) : fin. (z— 48) ‘Cem 7 
The azimuth of AB is = en ZBAX= 
ZCAB—CAX; and by calculating the apc as exe 
cefs of the triangle A ia we have A M’B = go 
—A , 
To determine the fides AM’, BM’, we mult dedu& $ 
of the fpherical excefs, 4 ¢, from each of the angles of the 
triangle A B M’, to ene ig a usiees 
Si — ABg AM’. 
a B: BM’. c 
