45 * 
ASTROI^OM Y. 
find the angle POr, and we (hall have the angle POZ. 
Mow confider PO and Z O as two circles upon the Earth’s 
furface, then the angle PO Z between them is equal to 
the angle POZ of projection, and therefore known: alfo 
the arc PO is the complement of the Sun’s declination; 
and to find the arc ZO, we mu ft confider ZO in the pro¬ 
jection to be the fine of the arc projected ; hence the arc 
ZO is that whofe fine is to radius as O Z to O A, there¬ 
fore we know the fine of the arc ZO, and confequently 
we get Z O itfelf; hence, in the fpherical triangle OPZ, we 
know PO, OZ, and the angle POZ, to find P Z the com¬ 
plement of the latitude of the place where the eclipfe is 
central. Find alfo the angle OPZ; then the time at the 
meridian PB being known, the angle OPB (the Sun’s 
dillance from the meridian) is known ; hence we know 
the angle BPZthe longitude of the point Z from the 
meridian PB ; therefore, the latitude and longitude of Z 
beingknown, the point Z is determined where the eclipfe 
was central at the given time. Make this calculation for 
every quarter or half hour, for all the time the penumbra 
is def'eribing dc, and you will trace out upon the furface 
of the Earth the path of the centre of the penumbra, or 
that trad: where the eclipfe is central. If we bring Z to 
d, we get the place where the Sun riles centrally eclipfed; 
and if Z be brought to e, we (hall find where the Sun fets 
centrally eclipfed. If Z coincide with r, we get the place 
where the Sun is centrally eclipfed upon the meridian. Let 
y be the centre of the penumbra when it firft touches the 
Earth, and .v the centre when it leaves the Earth, and 
draw On) perpendicular to LM. Then knowing Ov and 
the angle Ovzu, we can find O w and vO w\ alfo O y— 
femid. ©-j- femid. penumb. known ; hence in the right- 
angled triangle yOtv, we get the anglejyOrc, and there¬ 
fore we know yOv ; and, POv being already found, we 
know iO P ; hence in the triangle iOP, we know bO 
(=zgo°), PO, and the angle POi ; hence, we find P£ the 
complement of the latitude of Z; find alfo O Pi, and we 
get iPB the longitude of b, from the given meridian PB ; 
thus we get the place b where the eclipfe firft begins at the 
Sun riling. In like manner we get the place a where the 
eclipfe laft ends at fun-fetting. 
Ex. In the folar eclipfe which we have here computed, 
let it be required to find that place upon the Eath’s fur¬ 
face where the Sun is centrally eclipfed at one o’clock, ap¬ 
parent time at Greenwich. In this calc, Orerzqq'sg", 
and the angle OvZ— 5 0 42', and as' the time at v is 41' 
30", and the centre of the penumbra is at Z at one o’clock, 
the time through uZ —19' 30", which gives vZ—tf 3" ; 
hence ZO=35' 59", the angle OZv—%\° 22', and ZOu 
—13°56'. Now the radius, O 0=54'56" ; hence the 
arc OZ upon the furface (correfponding to its projection 
—35' 59") =41° 4' ; alfo PO—84° 30', and POv=23° 
22'; hence POZ-9 0 26'; confequently PZ2345 0 43', 
the complement of which is 44 0 17'the latitude of the 
place; alfq Z P O—S° 51'; but, atone o’clock apparent 
time at Greenwich, its meridian B P makes an angle of 
fifteen degrees with PO, Greenwich being upon that me¬ 
ridian at twelve o’clock ; hence BPZ=223 0 51' the longi¬ 
tude of the place, weft from Greenwich. In like manner 
may any of the other phenomena be calculated. 
To determine the Orbit of a Comet. 
The orbit of a comet may be computed from three ob¬ 
fervations; but, although that data be fufficient, the direiT 
folution of the problem is impracticable. Aftronomers 
therefore have folved this problem by indirect methods, 
firft finding an orbit very near to the truth by mechanical 
and graphical operations, and then, by computation, cor¬ 
recting it, until fuch a parabola was found as would fatisfy 
the obfervations. We ftiall therefore begin, by (hewing 
the methods by which the orbit may be nearly determined; 
and then explain the manner in which it may be corrected 
by calculation. 
Take a firm board perfectly plane, and fix on paper for 
the projection ; let a groove be cut near the edge, and five 
perpendiculars be moveable in it, fo that they may be 
fixed at any diftances. Let S reprefent the Sun, and de- 
feribe any number of circles about it. Compute five ge¬ 
ocentric latitudes and longitudes of the comet, from which 
you will have the five elongations of the comet at the times 
of therefpeCtive ob- 
lervations. Draw 
SA, SB, SC, 
S D, SE, making 
the angles A S B, 
BSC, CSD.DSE, 
equal to the Sun’s 
motion in the inter¬ 
vals of the obferva¬ 
tions ; and on any 
one of the circles, 
make the angles 
Saa, S bj 3 , Scy, 
S dfr, Se = , equal to 
the refpective elon- 
gationsin longitude, 
and fix the five per¬ 
pendiculars, fo that 
«■, ( 3 , 7 , s . From the points a, b, c, d, e, extend threads 
to the refpeCtive perpendiculars, making angles with the 
plane equal to the geocentric latitudes of the comet; then 
fix the focus of the parabola in S, and apply its edge to 
the threads ; and, if it can be made to touch them all, it 
will be the parabola required, correfponding to the mean 
diftance $>a of the Earth, which we here fuppofe to re¬ 
volve in a circle, as it will be fufficiently accurate for our 
purpofe. If the parabola cannot be made to touch all the 
threads, change the points a, b, c, d, e, to fuch of the 
other circles as you may judge, from your prelent trial, 
will be molt likely to iiicceed, and try again ; and by a 
few repetitions you will get fuch a diftance for the Earth, 
that the parabola ftiall touch all the threads, in which po- 
fition, find the inclination, obferve the place of the node, 
and meafure the perihelion diftance, compared with the 
Earth’s diftance, and you will get very nearly the elements 
of the orbit. 
Another method by which we may readily get the orbit 
very nearly, is this. Let S be the Sun, T, t, r, three 
places of the Earth at the times of the three obfervations; 
S 
extend three threads T p, T n, rm, in the directions of the 
comet, as directed above. AITume a point y for the place 
of the comet at the fecond obfervations, and meafure Sjy ; 
then if STz=i, and the velocity of the Earth ben, the 
velocity of the comet at_y will be-- 2 ^ — ; let v be repre- 
y/Sj v 
fented by T t, <r ; and upon a ftraight edge PCbj fet off 
V*XT t 
and ed- 
rr~; then apply the point 
V s y . V s y 
e tojy, and, by turning about the edge, try w hether you 
can make the point c fall in T p, and the point d in tm ; 
if you find this cannot be done, the error will direCt you 
to alfume another diftance ; and by a very few trials you 
will find the point y where the points c and d will fall in 
