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Rev. S. VINCE, A.M. F.R.S. 




printed hy J. Smith, Printer to the University ; 




Chap. I. Definitions 1 

II. On the Doctrine of the Sphere ... 13 
III. To determine the right Ascension, DecU- 
nation, Latitude and Longitude of the 

Heavenly Bodies 41 

IV. On the Equation of Time 47 

V. On the Length of the Year, the Precession 
of the Equinoxes from Observation 

and Obliquity of the Ecliptic ... 58 

VI. On Parallax 65 

VII. On Refraction 78 

VIII. On the System of the World .... 83 
IX. On Kepler's Discoveries 90 

X. On the Motion of a Body in an Ellipse 

about the Focus 94 

XI. On the Opposition and Conjunction of the 

Planets 106 

XII. On the mean Motion of the Planets . . 110 

XIII. On the greatest Equation, Excentricity, 

and Place of the Aphelia of the Orbits 

of the Planets 114 

XIV. On the Nodes and IncUnations of the 

Orbits of the Planets 118 

XV. On the apparent Motions and Phases of 

the Planets 124 

XVI. On the Moon's Motion from Observation, 

and it's Pliaenomena 134 

XVII. On the Rotation of the Sun and Planets 137 

XVIII. On the Satellites iQy 

XIX. On the Ring of Saturn 184 

XX. On the Aberration of Light .... 190 

XXI. On the Eclipses of the Sun and Moon . 208 

XXII. On the Transits of Mercury and Venus . 234 

XXIII. On Comets 244 

XXIV. On the Fixed Stars 257 

XXV. On the Longitude of Places upon the Earth 283 






Art. 1. Astronomy is that branch of Natural 
Philosophy which treats of the heavenly bodies. The 
determination of their magnitudes, distances^ and the 
orbits which they describe, is called plane or pure 
Astronomy ; and the investigatioti of the causes of 
their motions is called physical Astronomy. The 
former discoveries are made from observations on 
their apparent magnitudes and motions ; and the 
latter from analogy, by applying those principles and 
laws of motion by which bodies on and near the 
earth are governed, to the other bodies in the system. 
The principles of plane Astronomy only are what we 
here propose to treat of, and we shall begin with the 
explanation of such terms as are the foundation of the 

(2.) A great circle QRST of a sphere is ouej 
whose plane passes through it's center C; and w 
small circle BDHK is that whose plane does nuv" 
pass through it's center. 



(3.) A diameter PCE of a sphere, perpendicular 
to any great circle QRST, is called the a .vis of that 
great circle ; and the extremities P_, E, of the axis, 
are called it's Poles. 

(4.) Hence, the pole of a great circle is 90° from 
every point of it upon the sphere ; because every 
angle PCR being a right angle, the arc P R is every 
where 90°* And as the axis P E is perpendicular to 
the circle QRST when it is perpendicular to any 
two radii CQ, CR, a point on the surface of the 
sphere 90^ distant from two points of a great circle 
wherever taken, will be the pole. 

(5.) All angular distances on the surface of a 
sphere, to an eye at the center, are measured by the 
arcs of great circles. 

(6.) Hence, all the triangles formed on the surface 
6f a sphere, for the solution of spherical problems, 
must be formed by the arcs o^ great circles. 

(7.) Any two great circles bisect each other ; for 
both passing through the center of the sphere, their 
common section must be a diameter of each, and 
every diameter bisects a circle. 

(8.) Secondaries to a great circle, are great circles 
which pass through it's poles ; thus, P R E is a 
secondary to QRST. 

(9.) Hence, secondaries must be perpendicular to 
their great circles; for if one line be perpendicular 
to a plane, any plane passing through that line will 


also be perpendicular to it ; therefore as the axis 
PE of the great circle QRST is perpendicular to 
it, and is the common diameter of all the secondaries, 
they must all be perpendicular to the great circle. 
Hence also, every secondary, bisecting it's great circle 
(*7)j must bisect every small circle B DH K parallel 
to it ; for the plane of the secondary passes notonly 
through the center C of the great circle, but also the 
center A of the small circle jjarallel to it. 

(10.) Hence, a great circle passing through the 
poles of two great circles, must be perpendicular to 
each ; and vice versa, a great circle perpendicular to 
two other great circles must pass through their poles. 

(11.) If an eye be in the plane of a circle, that 
circle appears a straight line ; hence, in the repre- 
sentation of the surface of a sphere upon a plane, 
those circles whose planes pass through the eye, are 
represented by straight lines. 

(12.) The angle formed by the circumferences of 
two great circles on the surface of a sphere, is equal 
to the angle formed by the planes of those circles; 
and is measured by the arc of a great circle inter- 
cepted between them, and which is a secondary to 

For let C be the center of the sphere, P QE, 
PRE two great circles ; then as the circumferences 
of these circles at Pare perpendicular to the common 
intersection P CE, the angle at P between them is 
equal to the angle between the planes, by Euc. 
B. XI. Def 6. Now draw CQ, CR perpendicular 
to P CE, then as these lines are respectively parallel 
to the directions of the circumferences P Q, PR, at 
the point P, the angle QCR is equal to the angle 
at P formed by the two circles, Euc. B. XI. 
Prop. 10. ; and the angle QCR is measured by the 
arc QR of Si great circle whose pole is P, because 
PQ, PR are each 90^ 

^ The figures in the parentheses refer to the Articles. 
A 2 


(13.) If at the intersection P of two great circles 
as a pole, a great circle QRST be described, and 
also a small circle BDHK parallel to it, the arcs 
QR, B D of the great and small circles intercepted 
between the two o^reat circles, contain the same 
number of degrees. 

For C and A are the centers of the respective 
circles, and QC is parallel to BA, and RC is parallel 
to DA; therefore by Euc. B. XI. Prop. 10. the 
angle BAD is equal to the angle QCR, consequently 
the arcs B D, QR contain the same number of 

Hence, the arc BD oi such a small circle measures 
the angle at the pole between the two great circles. 
Also, QR.BD:. QC: B^ :: radius : cos. BQ. 

(14.) The ajcis of the earth pep'q is that diameter 
pOp' about which it performs it's diurnal rotation ; 
and the extremities p, p', of this diameter, are called 
it's Poles. 

(15.) The terrestrial Equator is a great circle 
erqs of the earth perpendicular to it's axis. Hence, 
the axis and poles of the earth are the axis and poles 
of it*s equator. That half of the earth (suppose epq) 
which lies on the side of the equator which we inhabit, 
is called the iwrthern hemisphere, and the other ep'q 
the southern ; and the poles are respectiv^ely called 
the tiorth and south poles. 

(1 6.) The Latitude of a place on the earth's surface, 
is it's angular distance from the equator, measured 
upon a secondary to it ; thus, the arc e h measures 
the latitude of h. The secondaries to the equator are 
called Meridians. 

(17.) The Longitude of a place on the earth's 
surface, is an arc of the equator intercepted between 
the meridian passing through the place, and another, 
called the Jirst meridian, passing through that place 
from which you begin to measure ; thus, the longitude 
of the place z? on the meridian prp' measured from 
the first meridian pep\ is er. 


(18.) If the plane of the terrestrial equator erqs 
be produced to the sphere of the fixed stars, it marks 

out a circle E RQS called the celestial equator; and 
if the axis of the earth pOp' be produced in like 
manner, the points P, P' , in the Heavens to which 
it is produced, are called Poles, being the poles of the 
celestial equator. The star nearest to each pole is 
called the Pole star. 

(19.) Secondaries, as PxP', to the celestial equa- 
tor are called Circles of Declination ; of these, 24 
which divide the equator into equal parts, each con- 
taining 15°, are called hour circles. 

(20.) Small circles, as dfgh, parallel to the ce- 
lestial equator, are called Parallels of Declination. 

(21.) The sensible horizon is that circle ahc in the 
heavens whose plane touches the earth at the spectator 
h. The rational horizon is a great circle HO R in 
the heavens, passing through the earth's center^ 
parallel to the sensible horizon. 

(22.) Almacanter is a small circle parallel to the 

(23.) If the radius O 6 of the earth at the place h 
where the spectator stands, be produced both ways 
to the heavens, that point Z vertical to him is called 
the Zenith, and the opposite point A^ the Nadir. 
Hence, the zenith and nadir arc the poles of the 



rational horizon (3) ; for the radius produced being 
perpendicular to the sensible, must also be perpen- 
dicular to the rational horizon. 

(24.) Secondaries to the horizon are called veHical 
circles ; and being (9) perpendicular to the horizon, 
the altitude of an heavenly body is measured upon 

(25.) A secondary P E H P' common to the ce- 
lestial equator and the horizon of any place b, and 
which therefore (lo) passes through the poles P, Z, 
of each, is the celestial Meridian of that place. 
Hence, the plane of the celestial meridian of any 
place, conicides with the plane of the terrestrial 
meridian of the same place. 

(26.) The meridian ZP R H cuts the horizon in 
the point R, called the north point, and in the point 
H, called the south point ; P being the north pole. 

(27.) The meridian of any place divides the 
heavens into two hemispheres lying to the east and 
west; that lying to the east is called the eastern 
hemisphere, and the other, lying to the west, is called 
the tvestern hemisphere. 

(28.) The vertical circle which cuts the meridian 
of any place at right angles, is called the prime 
vertical ; and the points where it cuts the horizon 
are called the east and west points. Hence, the 
east and west points are 90^ distant from the north 
and south points. These four are called the cardinal 

(2^.) If a body be referred to the horizon by a 
secondary to it, the distance of that point of the 
horizon from the north or south points, is called it's 
Azimuth. The Amplitude is the distance from the 
east or west point. 

(30.) The Ecliptic is that great circle in the 
heavens which the sun appears to describe in the 
course of a year. 

(31.) The ecliptic and equator being great circles 
Biiist (7) bisect each other, and their inclination is 


called the obliquity of the eliptic ; also, the points 
where they intersect are called the equinoctial points. 
The times when the sun comes to these points are 
called the Equinoxes. 

(32.) The ecliptic is divided into 12 equal parts, 
called Sigfis: Aries t , Taurus b, Gemini n, 
Cancer <b , Leo SI , Virgo iik , Libra =■& , Scorpio ttj. , 
Sagittarius ^ , Capricornus vj , Aquarius ^ , Pisces 
X . The order of these is according to the motion 
of the sun. The first point of Aries coincides with 
one of the equinoctial points, and the first point of 
Libra with the other. The first six signs are called 
northern, lying on the north side of the equator ; 
and the last six are called southern, lying on the 
south side. The signs v^ , ^, X , t? b, n, are 
called ascending, the sun approaching our (or the 
north) pole whilst it passes through them; and 
?s, SI, 11K, ===, TU, f , are called descending, the 
sun receding from our pole as it moves through them. 

(33.) When the motion of the heavenly bodies is 
according to the order of the signs, it is called dii^ect, 
or in consequentia ; and when the motion is in the 
contrary direction, it is called retrograde, or in ante- 
cedentia. The real motion of all the planets fs 
according to the order of the signs, but their apparent 
motion is sometimes in an opposite direction. 

(34.) The Zodiac is a space extending on each 
side of the ecliptic, within which the motions of all 
planets are performed, 

(35.) The right ascension of a body is an arc of 
the equator intercepted between the first point of 
Aries and a declination circle passing through the 
body, measured according to the order of the signs. 

(36.) The oblique ascension is an arc of the equator 
intercepted between the first point of Aries and that 
point of the equator which rises with any body, 
measured according to the order of the signs. 

(37.) The ascensional difference is the difterence 
between the right and oblique ascension. 


(38.) The Declination of a body is it's angular 
distance from the equator, measured upon a secondary 
to the equator drawn through the body. 

(39.) The Longitude of a star is an arc of the 
ecliptic intercepted between the first point of Aries 
and a secondary to the ecliptic passing through the 
star, measured according to the order of the signs. 
If the body be in our system, and seen from the sun, 
it is called the heliocentric longitude; but if seen 
from the earthy it is called the geocentric longitude ; 
the body in each case being referred perpendicularly 
to the ecliptic in a plane passing through the eye. 

(40.) The Latitude of a star is it's angular distance 
from the ecliptic, measured upon a secondary to the 
ecliptic drawn through the star. If the body be in 
our system, it's angular distance from the ecliptic 
seen from the earth is called the geocentric latitude ; 
but if seen from the sun, it is called the heliocentric 

(41.) Thus, if cy, Q be the equator, ^C the 
ecliptic, V the first point of Aries, s a star, and the 

<v^ p 

great circles sr, sp be drawn perpendicularly to f C 
and tQ; then ^p is it's right ascension, sp it's 
declination, s r it's latitude, and <^ r its longitude. 
The circle 5 r is called a Circle of Latitude. 

Hence, if we know the right ascension <^p, and 
declination ps of a body s, we know it's place ; for, 
take T/? = to the given right ascension, draw the 
meridian ps, and take ps = io the given declination, 
and s is the place of the body. In like manner, if 
we know the longitude t ^ of a body s, and latitude 
rs, we know the place of the body. 

(42.) The Tropics aje two parallels of declination 


touching the ecliptic. One, touching it at the be- 
ginning of Cancer, is called the Tropic of Cancer ; and 
the other_, touching it at the beginning of Capricorn, 
is called the Tropic of Capricorn. The two points 
vvliere the tropics touch the ecliptic, are called the 
Solstitial points. 

(43.) The Colures are two secondaries to the ce- 
lestial equator; one, passing through the equinoctial 
points, is called the equinoctial colure ; and the other, 
passing through the solstitial points, is called the 
solstitial colure. The times when the sun comes to 
ihe solstitial points are called the Solstices. 

(44.) The Arctic and Antarctic circles are two 
parallels of declination, the former about the north 
and the latter about the south pole, the distances of 
which from the two poles are equal to the distances 
of the tropics from the equator. These are also 
called polar circles. 

(45.) The two tropics, and two polar circles, when 
referred to the earth, divide it into five parts, called 
Zones; the two parts within the polar circles are 
called the frigid zones ; the two parts between the 
polar circles and the tropics are called the temperate 
zones ; and the part between the tropics, is called the 
torrid zone. Small circles in the heavens are referred 
to the earth, and the contrary, by lines drawn to the 
earth's center. 

(46.) A body is in Conjunction with the sun, when 
it has the same longitude ; in Opposition, when the 
difference of their longitudes is 180°; and in Quad- 
7'ature, when the difference of their longitudes is 
90°. The conjunction is marked thus c5 , the oppo- 
sition thus 8 , and quadrature thus d . 

(47.) Syzygy is either conjunction or opposition. 
(48.) The Elongation of a body from the sun, is 
it's angular distance from the sun when seen from 
the earth. 

(49.) The diurnal parallax is the difference be- 
tween the apparent places of a body in our system 


when referred to the fixed stars, if seen from the 
center and surface of the earth. Tlie annual parallax 
is the difference between the apparent places of a 
body in the heavens, when seen from the opposite 
points of the earth's orbit. 

(50.) The Argument is a term used to denote any 
quantity by which another required quantity may 
be found. For example, the argument of a planet's 
latitude is it's distance from it's node, because upon 
that the latitude depends. 

(51.) The Nodes are the points where the orbits 
of the primary planets cut the ecliptic, and where the 
orbits of the secondaries cut the orbits of their pri- 
maries. That node is called ascending where tlie 
planet passes from the south to the north side of the 
ecliptic ; and the other is called the descending node. 
The ascending node is marked thus Sh , and the de- 
scending node thus y . The line which joins the 
nodes is called the line of the nodes. 

(52.) If a perpendicular be drawn from a planet 
to the ecliptic, the angle at the sun between two 
lines, one drawn from it to that point where the per- 
pendicular falls, and the other to the earth, is called 
the angle of Commutation. 

(53.) The angle of Position is the angle at an 
heavenly body formed by two great circles, one 
passing through the pole of the equator, and the 
other through the pole of the ecliptic. 

(54.) Apparent noon is the time when the sun 
comes to the meridian. 

(55.) True or mean noon is 12 o'clock, by a clock' 
adjusted to go 24 hours in a mean solar day. 

(56.) The Equation of' time at noon, is the interval 
between true and apparent noon. 

(57.) A star is said to rise or set Cosmically, when 
it rises or sets at sun-rising; and when it rises or sets 
at ^un-setting, it is said to rise or set Achronically. 

(58.) A star rises Heliacally, when, after having 
been so near to the sun as not to be visible, it emerges 


out of the sun's rays, and just appears in the morning; 
and it sets Heliacallif, when the sun approaches so 
near to it, that it is about to immerge into the sun's 
rays and to become invisible in the evening- 

(59.) The curtate distance of a planet from the 
sun or earth, is the distance of the sun or earth from 
that point of the ecliptic where a perpendicular to it 
passes through the planet. 

(60) Aphelion is that point in the orbit of a planet 
which is furthest from the sun. 

(61.) Perihelion is that point in the orbit of a 
planet which is nearest the sun. 

(62.) Apogee is that point of the earth's orbit 
which is furthest from the sun, or that point of the 
moon's orbit which is furthest from the earth. 

(63.) Perigee is that point of the earth's orbit 
which is nearest the sun, or that point of the moon's 
orbit which is nearest the earth. 

The terms aphelion and perihelion are also applied 
to the earth's orbit. 

(64.) Apsis of an orbit, is either it's aphelion or 
perihelion, apogee or perigee ; and the line which 
joins the apsides is called the line of tlie apsides. 

(6*5.) Anomaly {true) of a planet, is it's angular 
distance at any time from it*s aphelion, or apogee — 
{mean) is the angular distance from the same point 
at the same time, if it had moved uniformly with it's 
mean angular velocity. 

{QQ.) Equation of the center is the difference 
between the true and meaw anomaly; this is some- 
times called the prosthapheresis. 

{67.) Nonagesimal degree of the ecliptic, is that 
point which is highest above the horizon. 

(68.) The mean place of a body, is the place where 
a body (not moving with an uniformly angular velo- 
city about the central body) would have been, if it 
had moved with it's mean angular velocity. The 
true place of a body, is the place where the body 
actually is at any time. 




(6^.) Equations are corrections which are appHed 
to the mean place of a body, in order to get it's true 

(70.) A Digit is a twelfth part of the diameter of 
the sun or moon. 

(71.) Those bodies which revolve about the sun in 
orbits nearly circular, are called Planets, or pyi- 
mart/ planets for the sake of distinction ; and those 
bodies which revolve about the pr^imary planets are 
called secondary planets, or Satellites. 

(72.) Those bodies which revolve about the sun 
in very elliptic orbits are called Comets. The sun, 
planets, and comets, comprehend all the bodies in 
what is called the Solar System. 

(73.) All the other heavenly bodies are called 
Fixed Stars, or simply Stars. 

(74.) Constellation is a collection of stars contained 
within some assumed figure, as a ram, a dragon, an 
hercules, 8^c. the whole heaven is thus divided into 
constellations. A division of this kind is necessary^ 
in order to direct a person to any part of the heavens 
which we want to point out. 

Characters used for the Sun, Moon and Planets. 




The Sun. 

The Moon. 



The Earth. 









Characters used for the Days of the PFeeh. 

© Sunday. 

D Monday. 

{? Tuesday. 

^ Wednesday. 





Chap. IL 


(75.) A SPECTATOR upon the earth's surface 
conceives himself to be placed in the center of a 
concave sphere in which all the heavenly bodies are 
situated ; and by constantly observing them, he 
perceives that by far the greater number never change 
their relative situations, each rising and setting at the 
same interval of time, and at the same points of the 
horizon, and are therefore csiWed Jiied stars ; but that 
a few others, called planets, together with the sun 
and moon, are constantly changing their situations, 
each continually rising and setting at different points 
of the horizon, and at different intervals of time. 
Now the determination of the times of the rising and 
setting of all the heavenly bodies ; the finding of 
their position at any given time in respect to the 
horizon or meridian, or the time from their position ; 
the causes of the different lengths of days and nights, 
and the changes of seasons; the principles of dialling, 
and the like, constitute the doctrine of the sphere. 
And as the apparent diurnal motion of all the bodies 
has no reference to any particular system, or dispo- 
sition of the planets, but may be solved, either by 
supposing them actually to perform those motions 
every day, or by supposing the earth to revolve about 
an axis, we will suppose this latter to be the case, 
the truth of which will afterwards appear. 

{'jQ.) hetpep'q represent the earth, O it's center, 
b the place of a spectator, HZ R Nthe sphere of the 
fixed stars ; and although the fixed stars do not lie 
in the concave surface of a sphere of which the center 



of the earth is the center, yet, on account of the 
immense distance even of the nearest of them, their 

^ N 

relative situations from the motion of the earth, and 
consequently the place of a body in our system 
referred to them, will not be affected by this suppo- 
sition. The plane ahc touching the earth in the 
place of the spectator, is called (21) the sensible 
horizon, as it divides the visible from the invisible 
part of the heavens ; and a plane HOR parallel to 
abc^ passing through the centre of the earth, is called 
the rational horizon ; but in respect to the sphere 
of the fixed stars, these may be considered as coin- 
ciding, the angle which the arc Ha subtends at the 
earth becoming then insensible, from the immense 
distance of the fixed stars. Now if we suppose the 
earth to revolve daily about an axis, all the heavenly 
bodies must successively rise and set in that time, 
and appear to describe circles whose planes are per- 
pendicular to the earth's axis, and therefore parallel 
to each other, because each body continues at the 
same distance from the equator, during the revolution 
of the earth about it's axis. Thus, all the stars will 
appear to revolve daily about the earth's axis, as if 
they were placed in the concave surface of a sphere 
having the earth in the center. Let therefore pp 
be that diameter of the earth about which it must 


revolve in order to give the apparent diurnal motion 
to the heavenly bodies, then />, />', are called it's 
poles; and if pp' be produced both ways to P, P\ 
in the heavens^, these points are called (18) the poles 
of the heavens, and the star nearest to each of these 
is called the pole star. Now, although the earth, 
from it's motion in it's orbit, continually changes 
it's place, yet as the axis always continues parallel to 
itself, the points P, P', will not, from the immense 
distance of the fixed stars, be sensibly altered ; we 
may therefore suppose these to be fixed points*. 
Produce O h both ways to Z and iV, and Z is the 
zenith, and xYthe nadir (23). Draw the great circle 
PZ HNR, and it will be the celestial meridian (25), 
the plane of which coincides with the terrestrial 
meridian pbp' passing through the place b of the 
spectator. Let erqs represent a great circle of the 
earth perpendicular to it's axis pp', and it will be the 
equator (15); and if the plane of this circle be ex- 
tended to the heavens, it marks out a great circle 
ERQS called the celestial equator (18). Hence, 
for the same reason that we may consider the points 
jP, P', as fixed, we may consider the circle ERQS 
as fixed. Now as the latitude of any place b on the 
earth's surface is measured by the degrees of the arc 
he (l6), it may be measured by the degrees of the 
arc Z E ', hence, as the equator, zenith, and poles in 
the heaven, correspond to the equator, place of the 
spectator, and poles of the earth, we may leave out 
the consideration of the earth in our further enquiries 
upon this subject, and only consider the equator, 
zenith, and poles in the heavens, and H R the 
rational horizon to the spectator. 

{77') Let the annexed figure represent the position 

* This is not accurately true, the earth's axis varying a little 
from its parallelism from the action of the moon. This is called 
the Nutation of the earth's axis, and was discorered by Dr. 



of the Iieavens to Z the zenith of a spectator in north 
latitude, EQ, the equator, P, P, it's poles, HOR 

the rational horizon, PZHP' R the meridian of the 
spectator, and draw the great circle ZON perpen- 
dicular to ZP RH, and it is the prime vertical (28) ; 
R will he the north point of the horizon, and H the 
south (26), and O will be the east or west point, 
(28) according as this figure represents the eastern or 
western hemisphere. Draw also a great circle POP' 
perpendicular to the meridian. We must therefore 
conceive this figure to represent half a globe, and all 
the lines upon it to represent circles ; and if we 
conceive the eye to be vertical to the middle point O 
of the figure, all the circles which pass through that 
point will appear right lines; therefore the right lines 
ZON, POP, EOQ, HOR, must be considered as 
semicircles. Now as each circle HR, EQ, ZN, PP' 
is perpendicular to the meridian, it's pole must be in 
each (8, 9), therefore their common intersection O 
is the pole of the meridian. Draw also the small 
circles wH, nit, ae, Rv, yx\ parallel to the equator; 
and as the great circle POP' bisects EQ. in O, it 
must also bisect the small circles mt, a e, in r and c; 
for as £0 = 90°, tr and ec are each =90° (13); and 
as QO = 90°, m r and a c are each = 90" ; hence, 
ac-=-C€, and m r — r t. 


(78.) As all the heavenly bodies, in their diurnal 
motion, describe either the equator, or small circles 
parallel to the equator, according as the body is in or 
out of the equator ; if we conceive this figure to repre- 
sent the eastern hemisphere, QE, ae, mt, may repre- 
sent their apparent paths from the meridian under the 
horizon to the meridian above, and the points b, 0,s, 
are the points of the horizon where they rise. And as 
ae, QE, mt, are bisected in c, O, r, eb must be greater 
than ba, QO equal to OE, and ts less than s m. 
Hence, a body on the same side of the equator with 
the spectator, will be longer above the horizon than 
below, because eb is greater than ba; a. body in the 
equator will be as long above as below, because QO = 
O E ; and a body on the cow ^rary side will be longer 
below than above, because m s is greater than s t. iVnd 
the further ae, or m t^ are from the equator, the greater 
will be the difference of ab, be, and rns, sf, or of the 
times of continuing above and below the horizon; and 
the further they will rise from O. The bodies de- 
scribing ae, mt, rise at b and s ; and as O is the east 
point of the horizon, and R and /Tare the north and 
south points, a body, on the same side of the equator 
with the spectator, rises between the east and the 
north, and a body on the contran/ side rises between 
the east and the south, the spectator being supposed 
to be in the north latitude; and a body in the equator 
rises in the east at O. When the bodies come to d or 
n, they are in tlie prime vertical, or in the east ; hence, 
a body on the same side of the equator with the 
spectator comes to the east after it is risen, and a body 
on the contrary side, before it rises. The body which 
describes the circle Rv, or any circle nearer to P', 
never sets ; and such circles are called circles of per- 
petual apparition ; and the stars which describe tliem 
are called circumpolar stars. The body which de- 
scribes the circle wH, just becomes visible at H, and 
then it instantly descends below the horizon ; but tiie 
bodies which describe the circles nearer to P' are never 



visible. Such is the apparent diurnal motion of the 
heavenly bodies, when the spectator is situated any 
where between the equator and poles; and this is 
called an oblique sphere, because all the bodies rise 
and set obliquely to the horizon. As this figure may 
also represent the western hemisphere, the same circles 
ea, tm will represent the motions of the heavenly 
bodies as they descend from the meridian above the 
horizon to the meridian under. Hence, a body is at 
the greatest altitude above the horizon, when on the 
meridian, and at equal altitudes when equidistant on 
each side, from it, if the body have not changed it's 

{7^') If the spectator be at the equator, then E 
coincides with Z, and consequently EQ with ZiV, and 


therefore P P' with H R. Hence, as the equator EQ 
is perpendicular to the horizon, the circles ace, mrt, 
parallel to EQ must also be perpendicular to it; and 
as these circles are always bisected by PP\ they must 
now be bisected by H R. Hence, all the heavenly 
bodies are as long above the horizon as below, and rise 
and set at right angles to it, on which account this is 
called a i^ight sphere. 

(80.) If the spectator be at the pole, then P coin- 
cides with Z, and consequently P P' with Z N, and 
therefore EQ with HR. Hence, the circles mt, ae, 
parallel to the equator, are also parallel to the horizon ; 
therefore as a body in it's diurnal motion describes a 
circle parallel to the horizon, those fixed bodies in the 
heavens, which are above the horizon, must always 


continue above, and those which are below must 
always continue below. Hence, none of the bodies, 



by their diurnal motion, can either rise or set. This 
is called a joara//e/ sphere, because the diurnal motion 
of all the bodies is parallel to the horizon. These 
apparent diurnal motions of the fixed stars remain 
constant, that is, each always describes the same 
parallel of declination. 

(81.) The ecliptic, or that great circle in the 
heavens which the sun appears to describe in the 
course of a year, does not coincide with the equator, 
for during that time it is found to be only twice in 
the equator ; let therefore COL represent half the 
ecliptic, or half the sun's apparent annual motion ; 
C the first point of Capricorn, and L the first point 
of Cancer ; and this being a great circle, must cut the 
equator into two equal parts (7). Hence, as the 
apparent motion of the sun is nearly uniform, the sun 
is nearly as long on one side of the equator as on the 
other. (See Fig. in p. l6.) When therefore the sun 
is at q, on the same side of the equator with the 
spectator, describing the parallel of declination ae by 
it's apparent diurnal motion, the days are longer than 
the nights, and it rises at h to the north of the east 
point ; but when it is on the contrary side, at />, de- 
scribing mt, the days are shorter than the nights, and 
it rises at s to the south of the east point, the spectator 
being on the north side of the equator ; but when the 
j*un is in the equator, at O, describing QE, the days 

B 2 



and nights are equal, and it rises in the east, at O *. 
If ae, mt be equidistant from EQy then will he = 7ns, 
and ab = st; hence, when the sun is in these opposite 
parallels, the length of the day in one is equal to the 
length of the night in the other; therefore the mean 
length of a day at every place is 12 hours. Hence, 
at every place, the sun, in the course of a year, is half 
a year above, and half a year below the horizon f. It 
is manifest, also, that the days increase from the time 
the sun leaves C the beginning of Capricorn, till he 
comes to L the beginning of Cancer ; and that they 
decrease from the time the sun leaves the beginning 
of Cancer till he comes to the beginning of Capricorn. 
When the spectator is at the Equator, the sun at p or 
q describing the circles m t, ae, by it's apparent diurnal 


motion, and these being bisected by the horizon, the 
sun will be always as long above as below the horizon. 

* The different degrees of heat in summer and winter, do not 
altogether arise from the different times which the sun is above the 
horizon, but partly from the different altitudes of the sun above 
the horizon ; the higher the sun is above the horizon, the greater 
is the number of rays which fall on any given space, and the 
greater also is the force of the rays. From all these circumstances 
arise the different degrees of heat in summer and winter. The 
increase of heat also as you approach the equator, arises from the 
two latter circumstances. 

t This is not accurately true, because the sun's motion in the 
ecliptic is not quite uniform, on which account it is not exactly as 
long on one side of the equator as on the other. If the major axis 
of the earth's orbit coincided with the line joining the equinoctial 
points, the times would be equal. This happened at the Creatioij. 



and consequently the days and nights will be always 
12 hours long. There will however be some variety 
of seasons, as the sun will recede 23° . 28' on each side 
from the spectator. In this situation of the spectator, 
the sun will be vertical to him at noon when it is in 
the equator. And when the spectator is any where 
between the tropics, the sun will be vertical to him at 
noon, when it's declination is equal to the latitude of 
the place, and of the same kind, that is, when they 
are both north, or both south. When the spectator is 
at the Pole, the sun at j) or q is carried, by it's apparent 




a V 


V.^ "i 



diurnal moWon, in the circles mpt, aqe, parallel to 
the horizon ; hence, it never sets when it is in that 
part OL of the ecliptic which is above the horizon, 
nor rises when in that part OC which is below ; con- 
sequently there is half a year day, and half a year 

night. As the sun illuminates one half of the earth 
or 90° all round about that place to which he is ver- 


tical *, when he is i?i the equator, he will illuminate 
as far as each pole ; when he is on the north side of 
the equator, the north pole will be within the illumi- 
nated part, and the south pole will be in the dark 
part ; and when the sun is on the south side of the 
equator, the south pole will be within the illuminated 
part, and the north pole in the dark part. And when 
the sun is at the tropic, he illuminates 23° . 28' beyond 
one pole, and the other pole is 23^ . 28' within the 
dark part. Hence, the variety of seasons arises from 
the axis of the earth, which coincides with PP\ not 
being perpendicular to the plane of the ecliptic LOC, 
for if it were, the ecliptic and equator would coincide, 
and the sun would then be always in the equator, and 
consequently it would never change it's position in 
respect to the surface of the earth. If QR=.EH=z 
23° . 28', the sun's greatest declination, then on the 
longest day the sun describes the parallel Rv, which 
just touching the horizon at R, shows that the sun 
does not descend on that day below the horizon, and 
therefore that day is 24 hours long. But when the 
sun comes to its greatest declination on the other side 
of EQj it describes ivH, and consequently does not 
ascend above the horizon for 24 hours, and therefore 
that night is 24 hours long. This therefore happens 
when EH, the complement of EZ the latitude (l6), 
is 23*^ . 28', or in latitude 66"^ . 32'. If EH, the com- 
plement of latitude, be less than 23° . 28', the sun will 
be above the horizon in summer, and below in winter, 
for more than 24 hours, and the longer above or 
below, as you approach the pole, where, as was before 
observed, it will be six months above, and as long 
below the horizon. The orbits of all the planets, and 
of the moon, are also inclined to the equator, as ap- 
pears by tracing their motions amongst the fixed 

* This is not accurately true, because, as the sun is greater than 
the earth, he will illuminate beyond 90°, by a quantity which is 
nearly equal, in minutes of a degree, to his apparent semidiameter. 


stars ; therefore, in the time in which each makes one 
revolution in its orbit, the same appearances will take 
place, as in the sun. All these different appearances 
in the motion of the moon, must therefoie happen in 
every month. It is also evident, that these variations 
of rising and setting must be greater or less, as the 
orbits are more or less inclined to the equator, as ap- 
pears by Art. 78. Hence, they must be greater in the 
moon than in the sun*. The apparent annual motion 
of the sun, and the real motion of the moon and 
planets, is from west to east, and therefore contrary to 
their apparent diurnal motion. 

(82.) Hitherto we have considered the motion of 
the heavenly bodies in the eastern hemisphere ; but if 
the figure represent the western hemisphere, all the 
reasoning will equally apply. The bodies will be just 
as long in descending from the meridian to the 
horizon, as in ascending from the horizon to the 
meridian ; the paths described will be similar ; and 
they will set in the same situation in respect to the 
west point of the horizon, as they rise in respect to the 
east ; that is, if a body rise to the north or south of 
the east, it will set at the same distance from the west 
towards the north or south. 

(83.) Having thus explained all the apparent 
diurnal motions of the heavenly bodies, with the cause 
of the variety of seasons, we shall proceed in the next 
place to show the method of determining the positions 
of the different circles, and the situation of the bodies 
in respect to the horizon, meridian, or any other 
circles, at any given time ; and having given their 
situation, to find the time ; for the understanding of 

* On account of the continual change of declination of the sun, 
moon, and planets, their apparent diurnal motions will not be ac- 
curately parallel to the equator ; in those cases therefore, where 
the declination alters sensibly in the course of a day, and where 
great accuracy is required, we must in our computations, take into 
consideration, the change of declination. 



which, a knowledge of" plane and spherical trigonome- 
try is all that is requisite. 

(84.) The altitude PR ofthe }wU above the horizon, 
is equal to the latitude of the place. 

For the arc ZE is (l6) the measure of the latitude ; 
but PE = ZR, each being— 90"; take away ZP which 
is common to both, and EZ=PR*. 

(85.) To find the latitude of a place, observe the 
greatest and least altitude of a circumpolar star, and 
apply the correction for refraction , in order to get 
tlie true altitudes, and half the sum icill he the alti- 
tude of the pole. 

For if yx be the true circle described by the star, 
then, as Px = Py, PR = ^x Ry + Rx. See the last 

The latitude may also be thus found. 

Let eOt be the ecliptic; then when the sun comes 
to e, it's declination is the greatest, and eH is the 


greatest meridian altitude; when the sun comes to the 
ecliptic at t, let ts be the parallel described on that 

* Fro-Ti hence arises the method of measuring the circumference 
ofthe earth ; for if a man travel upon a meridian till the height of 
the pole has altered one degree, he must then have travelled one 
de^rree; hence, by measuring that distance and multiplying it by 
360 we get the circumterence ofthe earth. This was undertaken 
bv our countryman Mr. Norwood, who measured the distance 
between London and York, and observed the different altitudes of 
the pole at those places. Afterwards, the French mathematicians 
^ measured 


dav, and then s H is t!ie least meridian altitude; and 
as Ee = Es, \x He \ Hs=HE the complement of 
the latitude. 

(86.) Half the difference of the suns greatest and 
least meridian altitudes. Is equal to the inclination of 
the ecliptic to the equator. 

For half He- Hs, or half Ae, is eqiial to Ee which 
(12) measures the angle EOe, the inclination of the 
ecliptic to the equator. 

(87.) The angle which the equator makes with the 
horizon, or the altitude of that point of the equator 
which is on the meridian, is equal to the complement 
of the latitude. 

For ZH is 90*^, and therefore EH is the comple- 
ment of EZ; and as OE=OH=Q0\ ^//measures 
(12) the angle EOH*. 

(88.) Let abcdxe be a parallel of declination de- 
scribed by an heavenly body in the eastern hemisphere, 
and draw the circles of declination Ph, Pc, Pd, Px, 
and the circles of altitude Zh, Zc, Zd, Zx. Now, as 
has been already explained, when the body comes to 
Z>, it rises ; at c it is at the middle point between a and 
e ; and at d it is due east ; and let x be it's place at 
any other time. Let us suppose this body to be the 
sun, and not to change it's declination in it's passage 
from a to e, and let us suppose a clock to be adjusted 
to go 24 hours in one apparent diurnal revolution of 
the sun, or from the time it leaves any meridian till it 
returns to it again, then the sun will always approach 
the meridian, or any other circle of declination, at the 

measured a degree. Cassini measured one in France. After 
that, Clairaut, 3Taupertuis, and several other mathemati- 
cians went to Lapland, and measured a degree, the length of 
which appears to be 60,2 English miles in the latitude of 45°; for 
the earth being a spheroid, the degrees in different latitudes are 

* See my Treatise on Plmie and Spherical Trigonometry, Art. 173. 
This is the Trigonometry referred to in the future part of this 


rate of 15*^ in an hour ; also, the angle which the sun 
describes about the pole will vary at the same rate, 
because (13) an arc Joe, which the sun at x has to de- 
scribe before it comes to the meridian, measures the 
angle xPe, called the hour angle. If therefore we 
.suppose the clock to show 12 when the sun is on the 
meridian at a or e, it will be 6 o'clock when he is at c. 
And as the sun describes angles about the pole P at the 
rate of 15" in an hour, the angle between any circle, 
Pt, of declination passing through the sun at x, and 
the meridian PE, converted into time at the rate of 
15° for an hour, will give the time from apparent 
noon, or when the sun comes to the meridian. 

(89.) Given the sun's declination, and latitude of 
the place, tojind the time of' his rising, and azimuth 
at that time. 

The sun rises at h-, and in the triangle ^Z/?, hZ — 
90°, hP = co-dec. P^ = co-lat. Now when one side 
of a triangle = 90°, it may be solved by the circular 
parts, taking the angles adjacent to the side = 90", and 
the complements of the other three parts, for the 
circular parts. Hence, (Trig. Art. 215.) rad. X cos. 
ZPb=^cot. bP X cot. ZP, or, rad. x cos. hour angle= 
tan. dec. X tan. lat. therefore (Trig. Art. 213). 
Log. tan. dcc.-{-log. tan. lat. — 10,=: log. cos. hour ang.fro??iapp,noo?i; 

which converted into time, at the rate of 15° for an 
hour, (see Table I. at the end), and subtracted from 
12 o'clock, gives the apparent time of rising. Also, 
(Trig. Art. 215.) rad. x cos. bP = s'in. ZPx cos. PZb, 
or, rad. x sin. dec. = cos. lat. x cos. azi. therefore 

JOj-^-Iog. sin. dec. —log. cos. lat. = log. cos. azi. from North, 

Ex. Given the latitude of Cambridge 52^ . 12' . 35", 
to find the time of the sun's rising on the longest day, 
and azimuth at that time, assuming the greatest decli- 
nation of the sun 23" . 28'. 


Dec. 23° . 28' . 0". - tan. 9,6376106 
Lat. 32° . 12'. 35. - tan. 10,1104699 

- COS. 9,7480805 

Convert this into time (Tab. I.) and it gives 
Sh. 19'.6", which subtracted from 1 2, gives 3h. 40' . 54'', 
the time when the sun's center is upon the rational 
horizon on the longest day ; Also, 

Dec. 23°. 28'. O". - 10, + sin. 19,6001 181 
Lat. 52. 12. 35. - - cos. 9.7872996 

Azi. 49. 28. 9. - - cos. 9,8128185 

Hence, on the longest day, the sun rises 40°. 3 1'. 5 1" 
from the east towards the north. 

(90.) Tojind the sidHs altitude at six d clock. 

The sun is at c at 6 o'clock, and the angle ZPc is 
a right one ; hence, (Trig. Art. 212.) rad. x cos. Zc = 
cos. ZP X COS. Pc, or rad. x sin. alt. = sin. lat. x sin. 
dec. therefore 

Log. sir}, lat. + log. sin. dec. — 10, = /oo-. sin. alt. 

Ex. Taking the data of the last example, we have, 
Lat. 52«. 12'. 35" - - sin. 9,8977695 

Dec. 23. 28. O - - sin. 9,6001181 

Alt. 18. 20.32. - - sin. 9,4978876 

(91.) To find the time when the sun comes to d the 
prime vertical^ and it^s altitude at that time. 

In this case, the angle dZP — (^0^; hence, (Trig. 
Art. 112.) rad. x cos. dP = cos. ZP x cos. Zd, or, rad. 
X sin. dec. = sin. lat. x sin. alt. therefore 

* This log. 9.7480805 is found in the tables to be the log. cosine 
of 55°. 57'. 13", but as the angle is manifestly greater than 90°, we 
must take its supplement. In the solution of spherical triangles, 
ambiguous cases will frequently arise ; for the determination of 
which, where the case is not evident, the reader is referred to my 
Treatise on Trigo?iometrj/. 


10, -\-log. sin. dec- log. sin. lat. = log. sin. alt. 

Also, (Trig. Art. 212)rad. x cos.ZPd=cot. Pdx tan. 
PZ, or, rad. x cos. hour angle=tan. dec. x cot. lat. 

log. tan. dec. + log. cot. lat. — 1 0, = log. cos. hour angle ; 

which, converted into time (Tab. I.), gives the time 
from apparent noon. 

Ex. Taking the data of the last example, we have, 
Dec. 23^28'. 0" - 10 + ,sin. 19,6*001 181 
Lat.52.12.35 - - sin. 9,8977695 

Alt. 30.15.31 - - sin. 9,7023486 

Dec. 23.28. o - - tan. 9,6376106 
Lat.52.12.35 - - cot. 9,8895301 

Hour z 70. 19. 44. - - COS. 9,5271407 

This angle 'JO'^. I9'. 44", converted into time, gives 
4h. 41'. 19" the time from apparent noon. 

(22.) Given the latitude of the place, the suit's 
decUnation, and altitude, to jind the hour, and his 

Let X be the sun's place ; then, (Trig. Art. 239) sin. 
Px X sin. PZ : rad.^ :: sin, j- X Px j-PZ+Zxx sin. 
I X Px + PZ - Zx : cos. I ZPx^ ; hence, ZPx is 
known, which converted into time (Tab. L) gives the 
time from apparent noon. Also, (Trig. Art. 239) 
sin. Zx X sin. ZP : rad.'* :: sin. | x Zx + ZP+Px x 
sin.^xZx + ZP — Px : cos. § xZP' ; hence, tlie 
azimuth xZP from the north is known. 

Ex. Given the lat. 34°. 55' N, sun's declination 
23^ 22'. 57" N, and true altitude SS"". 59'. 39", to find 
the apparent time. 

Here, ZP = 55°. 5', Zx = 53°. O'. 21", Px = 67°. 
37'. 3" : hence (Trig. Art. 239) 


P.r=67''. 37'. 3" - ar. CO. sin. 0,034019 
ZP = 55. 5. - ar. CO. sin. 0,086193 
Zx=53. O. 21 

Sum 175. 42. 24 

i Sum 87. 51.12 - - - sin. 9,999694 
Z,v = 53. O. 21 

DifF. 34. 50. 51 - - r. sin. 9,756932 


the cosine of 29^ 47' . 44", half the angle ZPjj, .-. ZPjc 
= 59°. 35'. 28", which reduced into time gives 3 h. 
58'. 22", the time from apparent noon. By the very 
same process, the angle JcZP is found. 

(93.) Given the error in altitude, tojind the error 
in time. 

Let m n be parallel to the horizon, and nx represent 
the error in altitude ; then, as the calculation of the 
time is made upon supposition that there is no error 
in the declination, we must suppose the body to be at 
m instead of x, and consequently the angle fnPx, or 
the arc q r, measures the error in time. 

Now nx : xm :: sifi. nmx : rad. (Trig. Art. 125) 

xm '. qr '.: cos.rx : rad. (Art. 13). 

hence, nx : qr :: sin. nmx x cos. rx : rad.^ .-. qr=nx x 


— '■ ; but ZxP =znmx, nxm beintr the 

sm. nmx x cos. rx 

complement of both ; also, (Trig. Art. 221 .) sin. ZxP, 

or nmx, : sin. ZP :: sin. xZP : sin. xP, or cos. rx, 

.'. sin. n7nx x cos. rx = sin. ZP x sin.xZP ; hence, qr = 

rad.' rad.- 

nx X ^-f5 : 7yn - ^^^ ^ TT '■ '■ — • 

sm. ZP X sm. xZP cos. lat. x sm. azmi. 

Hence, the error is least on the prime vertical. All 
altitudes therefore, for the purpose of deducing the 



time, ought to be taken on, or as near to, the prime 
vertical as possible. 

Ex. In lat. 50*^. 12', if the error in alt. at an azim. 

44°. 22' be l', then qr= l' x -^ ^— = 2',334 of 

' ^ ,012 X, 690 

a degree=9",336 in time. 

Hence, the perpendicular ascent of a body is 
quickest when it is on the prime vertical ; for nx 
varies as sin. azim. when qr and the lat. are given, 

(94.) Given the lat. of the place, and the suns 
declination, to find the time when twilight begins. 

Twilight is here supposed to begin when the sun is 
18° below the horizon ; draw therefore the circle hyk 
parallel to the horizon, and 18" below it, and twilight , 
will begin when the sun comes to y, and Zy — 108° ; 
hence, (Trig. Art. 239) sin. Py x sin. PZ : rad.* :: 
sin. I X PZ + Py + 108^ x sin. § x PZ-i-Py— 108° : 
cos. I yPZ'^ ; therefore yPZ is known, which con- 
verted into time (Tab. I.), gives the time from appa- 
rent noon. The operation is the same as that in 
Art. 92. 

(95 .) To find the time when the apparent diurnal 
motion of a fixed star, is perpendicular to the 

Let yx be the parallel described by the star ; draw 
the vertical circle Zh, touching it at o, and when the 

star comes to o, it's motion is perpendicular to the 
horizon ; and as ZoP is a right angle, we have (Trig. 



Art. 212) rad. x cos. ZPo — tan. Po X cot. PZ, or 
rad. X cos, hour angle = cot. dec. x tan. lat. therefore, 
log. cot. dec. + log. tan. lat. — lO, = log. cos. hour angle; 
which converted into time (Tab. I.), gives the time 
from the star's being on the meridian. Hence, the 
time of the star's coming to the meridian being known, 
the time required will be known. 

(96.) To find the time of the shortest tiviUght. 

Let a 6 be the parallel of the sun's declination at the 
time required, draw cd indefinitely near and parallel 
to it, and TW 2i parallel to the horizon 18° below it; 
then vPw., sPt measure the duration of twilight on 
each parallel of declination, and when the twilight is 
shortest, the increment of the duration is=0, and 
these must be equal; hence, vPr = wPz, therefore 
vr =w z ; and as rs = tz, and r and z are right angles, 
rvs=zivt; but Pvr = dO° = Zvs, take Zvr from both, 
and PvZ=-rvs', for the same reason PwZ = zwt ; 
hence, PvZ=PwZ. Take re = wZ = 90°, and join 
Pe; and as Pv=:Piv, ve=ivZ, and Pve — PwZ, we 

have Pe — PZ; and if Pi/ be perpendicular to eZ, 
then will Zy—ye. Now, (Trig. Art. 224) cos. Pv : 
cos. Pe, or PZ, :: cos. vy : cos. ey, that is, sin. dec. ; 
sin. lat. :: sin. ey : cos. ey :: tan. ey = g° : rad. or, rad. 
: sin. lat. :: tan. 9° : sin. of the suns declination at 
the time of the shortest twilight ; and the logarithmic 
operation is, 

log. sin. lat. -^ log. tan. 9^^— 10, = log. sin. dec. 
Because PZ is never greater than 90^^, and Zy = 9", 


therefore Py is never greater than 90\ and it's cosine 
is positive; also, vif is ahvays greater than 90*^, there- 
fore it's cosine is negative; hence, (Trig. Art. 212. 
rad. being unity) cos. Pv (= cos. Pyx cos. vy) is ne- 
gative ; consequently Pv is greater than 90^^ ; there- 
fore the sun's declination is south. 

If, instead of taking RIV = 18^, we take it = the 
sun's diameter (2.v), we shall get the time of the year 
when the body of the sun is the least time in ascend- 
ing above the horizon ; hence, 

log. sin. lat. + log. tan. s — 10, = log. sin. dec. 
Thus we get the declination when the sun is the least 
time in rising ; and as the declination must be always 
very small, this event must happen when the sun is 
very near the equinox. 

(97 ■) Tojindthe duration of the shortest twilight. 

As zi?Psr=i;Pe, therefore ZPe = vPw, which mea- 
sures the shortest time. Now (Trig. Art. 212) rad. 
X sin. Zy=zii\n. PZ x sin. ZPy, or, rad. x sin. 9'^ = cos. 
lat. X sin. ZPy, therefore, 

10, + log. sin. 9° — log. cos. lat. = log. siti. ZPy, 
which doubled gives ZPe, or vPw, which, converted 
into time (Tab. I.), gives the duration of the shortest 

Ex. To find the time of the year at Cambridge, 
when the twilight is shortest ; and the length of that 

Lat. 52°. 12'. 35" - - - - sin. 9,8977695 
9' ----- tan. 9,1997125 

Dec. 7^ 11'. 25" - - - - sin. 9,0974820 

This declination of the sun gives the time about 
March 2, and October 11. 

9°. 0'. O" - - 10,+sin. 19,1943324 
Lat. 52. 12.35 - - - cos. 9,787299^ 

ZPy 14. 47. 27 - - - sin. 9,4070328 



The double of this gives 29°. 34'. 54", which, con- 
verted into time, gives \h. 58'. 20'' for the duration of 
the shortest twilight, it being supposed to end when 
the sun is 18° below the horizon. 

(98.) To Jind the sun's declination, ivhen it is just 
twilight all night. 

Here the sun at a (Fig. p. 25.) must be 18° below 
the horizon ; therefore 1 8*^ + dec. Qa = RQ = EH— 
comp. of lat. of place ; hence, the sun's dec. = comp, 
lat. — 18° ; look therefore into the Nautical Alnianachy 
and see on what days the sun has this declination, 
and you have the time required. The sun's greatest 
declination being 23°. 28' it follows, that if the com- 
plement of latitude be greater than 41°. 28', or if the 
latitude be less than 48°. 32', there can never be twi- 
light all night. If the sun be on the other side of the 
equator, then it's dec.= 18°, -comp. lat. 

{99-) If the sun's declination Ee be greater than 
EZ, then the sun comes to the meridian at e to the 
north of the zenith Z of the spectator ; and if we draw 
the secondary Zqm touching the parallel ae of declina- 
tion described by the sun, then Rm is the greatest 
azimuth from the north which the sun has that day, the 


azimuth increasing till the sun comes to q, and then 
decreasing ; for a circle from Z to any other point of 
€ a will cut RO nearer to R, and it will also cut ea in 
two points which have the same azimuth, they being 
in the same vertical circle; in this case, therefore, the 


sun has the same azimuth twice in the morning. If, 
therefore, we draw the straight Hne Zu perpendicular 
to the horizon, the shadow of this Hne, being always 
opposite to the sun, will, in the morning, first recede 
from the south point //, and then approach it, and 
therefore will go backwards upon the horizon. But 
if we consider PP' as a straight line, or the earth's 
axis produced, the shadow of that line will not go 
backv^ards upon the horizon, because the sun always 
revolves about that line, whereas it does not revolve 
about the perpendicular Zv, it never getting to the 
south of it. Hence it appears, that the shadow of the 
sun upon a dial can never go backwards, because the 
gnomon of a dial is parallel to PP', and therefore the 
sun must always revolve about the gnomon. 

The time when the azimuth is greatest is found from 
the right angled triangle PqZ; for (Trig. Art. 212) 
rad. X COS. ZPq = tan. qP x cot. PZ, or^ rad. x cos. hour 
angle = cot. dec. x tan. lat. ; therefore, 
log. cot. dec. + log. tan. lat. — \0,r:zlog. cos. hour angle 
from apparent noon. 

(100.) It has hitherto been supposed, that it is 12 
o'clock when the sun comes to the meridian ZHN 
(Fig. p. 20) and that the clock goes just 24 hours in 
the interval of the sun's passage from any meridian till 
it returns to it again. But if a clock be thus adjusted 
for one day, it will not continue to show 1 2 o'clock 
every day when the sun comes to the meridian, be- 
cause it is found by observation, that the intervals of 
time from the sun's leaving any meridian till it returns 
to it again, are not always equal ; this difference be- 
tween the sun and the clock is called the Equation of 
Time, as will be explained in Chap. IV. Hence, 
when the clock does not agree with the sun, and the 
sun is at ^, any arc xe is not the measure of the time 
from 12 o'clock, but from the time when the sun comes 
to the meridian, or from apparent noon, as it is called. 
(101.) The method of finding the hour angle for 
the time at which a body rises, has been upon the sup- 


position that the body is upon the rational horizon at 
the instant it appears, or 90° from the zenith ; but all 
bodies in the horizon are elevated by refraction 33' 
above their true places ; this therefore would make 
them appear when they are 33' below the rational 
horizon, or 90*^ + 33' from the zenith; also, all the 
bodies in our system are depressed below their true 
places by parallax, as will be afterwards explained ; 
therefore from this cause they would not appear till 
they were elevated above the rational horizon by a 
quantity equal to their horizontal parallax, or when 
distant from the zenith 90° — hor. par. Hence, from 

both causes together, a body becomes visible when it's 
distance ZFhom the zenith = 90° -}- 33' — hor. parallax, 
f^ being the place of the body when it becomes visible, 
Z the zenith, and P the polej hence, knowing ZF^, 
also ZP the complement of latitude, and Pf^the com- 
plement of declination, we can find the hour angle 
ZPf^. A fixed star has no parallax, therefore in this 
case ZF=90^ 33'. 

Tojindthe Time in which the Sun passes the 3Ieridian, 
or the horizontal Wire of a Telescope. 

(102.) Let moj be the diameter of ^" of the sun, esti- 
mated in seconds of a great circle ; then, as the se- 
conds in mx, considered as a small circle, must be 
increased in proportion as the radius is diminished, 
because (Trig. Art. 75) when the arc is given, the 
angle is inversely as the radius, we have, sin. Px, or 

c 2 



COS. dec. rjc, : rad. :: seconds d' in ma: of a great circle : 
the seconds in w^r ofthe small circle ea, which (13) is 

equal to the seconds in qr, or, in the angle rPq, and 

therefore the angle rPq — d" divided by cos. dec. (rad. 

being unity) = c?" x sec. dec, which measures the time 

in which the sun passes over a space equal to it's 

diameter, and consequently the time the diameter 

will be in passing over the meridian ; hence, 15" in 

space (corresponding to l" in time) : d" x sec. dec. in 

space :: l" in time : the time in seconds of passing the 

... d'y. sec. dec. 

meridian = —-r, ■ • 


(103.) Hence, qr, the sun's diameter in right 

ascension, is equal to d" y. sec. dec. If, therefore, the 

sun's diameter=32'=1920", and it's dec. = 20°, it's 

diameterinrightascension=1920"x l,o64=34'.2,"88. 

The same is true for the moon, if ^" = it's diameter. 

XT, * . ^ rad." 

(104.) By Art. 93. qr=znx x ^ : : — 

^ ^ •' ^ COS. lat. X sin. azim. 

,. . ,., rad.'- 

= (if«a?=r?",thesmi sdiameterja x r— : ; — 

^ ' ' COS. lat. X Sin. azim. 

hence, as before, the time of describing qr, or the 

time in which the sun ascends perpendicularly through 

a space equal to if s diameter, or the time of passing 

, . , . . , d" rad.^ 

an horizontal wire, is equal to ■— 77X ^ . ^ ■ = — 

' ' 15' COS. lat. X sin. azim. 


The same expression must also give the time which 
the body of the sun is in ascending above the horizon. 
If ^"=1980" the horizontal refraction, then d" di- 
vided by 15" = 132"; hence, refraction accelerates the 

rid * 

rising of the sun by 132" x ; ^—^ : — . 

° "^ cos. lat. X sm. azmi. 

On the Principles of DiaUing. 
(105.) As the apparent motion of the sun about the 
axis of the earth, is at the rate of 1 5° in an hour, very 
nearly, let us suppose the axis of the earth to project it's 
shadow into the meridian opposite to that in which the 
sun is, and then this meridian will move at the rate of 1 b"* 
in an hour. Hence, let ^P/?/?^^ represent a meridian 

on the earth's surface, POp the earth's axis, z the place 
of the spectator, HKRf^B. great circle, of which z is the 
pole ; draw the meridians Pip, P2p, &c. making angles 
with PRpoflb^, 30°, &c. respectively, then, sup- 
posing PR to be the meridian into which the shadow 
of OP is projected at 12 o'clock, Pi, P2, &c. are the 
meridians into which it is projected at 1,2, &c. o'clock, 
and the shadow will be projected on the plane HKRF^ 
into the lines OR, 0\, 02, &c. and the angles ROl, 
RO2, &c. will be the angles between the 12 o'clock 
line and the 1,2, &c. o'clock lines. Now in the right 



angled triangle PRl, we have (84) PR the latitude 
of the place, and the angle i?Pl = 15*; hence, (Trig. 
Art. 210) rad. : tan. 15° :: sin. PR : ta?i. Rl ; in the 
same manner we may calculate the arcs H2, RS, &c. 
In this case, we make the earth's axis the gnomon, 
and the shadow is projected upon the plane HKRV. 
Take a plane ahcd at %, parallel to HKRV, and it is 
the sensible horizon (2l), and draw zr parallel to 
JPOp ; then, on account of the great distance of the 
sun, we may conceive it to revolve about zr in the 
same manner as about PO, and consequently the 
shadow will be projected upon the plane ahcd^ in the 
same manner as the shadow of PO is projected upon 
the j^hine HKRV^ and therefore the hour angles are 
calculated by the same proportion. This is an hori- 
zontal dial. 

( 106.) Now let NLzK be a great circle perpendicu- 
lar to PRpH, and consequently perpendicular to the 

horizon at Zy and the side next to H is full south. 
Then, for the same reason as before, if the angles Npl, 
Np2j &c. be 15°, vSO", &c. the shadow of ^O will be 


jDrojected into the lines Ol, 02, &c. at 1, 2, &c. 
o'clock, and the angles A^Ol, N^02, &c. will be mea- 
sured by the arcs A'"!, A 2, &c. Hence, in the right 
angled triangle 7?A^l, p^V=the complement of the 
latitude, and the angle Npl = l5^; therefore (Trig. 
Art. 210) rad. : tan. 15° :: ym. pN : tan. Nl ; in the 
same manner we find A^2, A^3, &c. Hence, for the 
same reason as for the horizontal dial, if zabc be a 
plane coinciding with NLzK, and sthe parallel to Op, 
.st will project it's shadow in the same manner on the 
plane zabc, as Op does on the plane NLzK, and 
therefore the hour angles from the 12 o'clock line are 
computed by the same proportion. This is a vertical 
south dial. In the same manner the shadow may be 
projected upon a plane in any position, and the hour 
angles calculated. 

(107.) In order to fix an horizontal dial, we must 
be able to tell the exact time of the sun's coming to 
the meridian ; for which purpose, find the time (92) 
by the sun's altitude when it is at the solstices, that 
being the best time of the year for the purpose, be- 
cause then the declination does not vary, and set a 
well-regulated watch to that time ; then, when the 
watch shows 12 o'clock, the sun is on the meridian ; 
at that instant, therefore, set the dial, so that the 
shadow of the gnomon may coincide with the 12 
o'clock line, and it stands right. 

(108.) Hence, we may easily draw a meridian line 
upon an horizontal plane. Suspend a plumb line so 
that the shadow of it may fall upon the plane, and 
when the watch shows 12, the shadow of the plumb 
line is the true meridian. The common way is to 
describe several concentric circles upon an horizontal 
plane, and in the center to erect a gnomon perpendi- 
cularly to it, with a small round well defined head, 
like the head of a pin ; make a point upon any one 
©f the circles where the shadow of the head falls upon 


it in the morning, and again where it falls upon the 
same circle in the afternoon ; draw two radii from 
these two points, and bisect the angle between them^ 
and the bisecting line will be a meridian line. This 
should be done when the sun is at the tropic, v/hen it 
does not sensibly change it's declination in the interval 
of the observations ; for if it do, the sun will not be 
equidistant from the meridian at equal altitudes. But 
this method is not capable of very great accuracy ; for 
the shadow not being very accurately defined, it is not 
easy to say at what instant of time the shadow of the 
head of the gnomon is bisected by the circle. If, how- 
ever, several circles be made use of, and the mean of the 
whole number of meridians so taken, be drawn, the 
meridian may be found with sufficient accuracy for all 
common purposes. 

(109.) To find whether a wall be full south for a 
vertical south dial, erect a gnomon perpendicularly to 
it, and hang a plumb line from it ; then when the 
watch, as above adjusted, shows 12, if the shadow of 
the gnomon coincide with the plumb line, the wall is 
full south. 


Chap. III. 


(110.) The foundation of all Astronomy is to deter- 
mine the situation of the fixed stars, in order to find, 
by a reference to such fixed objects, the places of the 
other bodies at any given time, and thence to deduce 
their proper motions. The positions of the fixed stars 
are found from observation, by knowing their right 
ascensions and declinations (4l); and these are found 
by means of the transit telescope and astronomical 
quadrant, as explained in my Treatise on Practical 
Astronomy ; and then, by computation, their latitudes 
and longitudes may be found. 

(ill.) As the earth revolves uniformly about it's 
axis, the apparent motion of all the heavenly bodies, 
arising from this motion of the earth, must be uni- 
form ; and as this motion is parallel to the equator 
{j^^^ the intervals of the times, in which any two 
stars pass over the meridian, must be in proportion to 
the arc of the equator intercepted between the two 
secondaries passing through them, because (13) this 
arc of the equator contains the same number of degrees 
as the arc of any small circle parallel to it, and com- 
prehended between the same secondaries ; and there- 
fore, if one increase uniformly, the other must. 
Hence, the right ascension of stai-s passing the 
meridian at different times, will differ in proportion to 
the difference of the times of their passing, that is, if 
one star pass the meridian 1 hour before another, the 
difference of their right ascensions is 15°. Hence, if 
the clock be supposed to go uniformly, we have the 


following rule : As the interval of the times of the 
succeeding passages of any one jixed star over the 
meridian : the interval of' the passages of any two 
stars :: 360° : their difference of 7'ight ascensio?is'^. 
By the same method we may find the difference of 
right ascensions of the sun or moon, when they pass 
the meridian, and a star, and therefore if that of the 
star be known, that of the sun or moon will j which 
conclusion will be more exact, if we compare them 
with several stars, and take the mean. 

(112.) Now to determine the right ascension of a 
fixed star, Mr. Flamstead proposed a method, by com- 
paring the right ascension of the star with that of the 
sun when near the equinoxes, the sun having the 
same declination each time ; and as this method has 
not been noticed by any writers, we shall give an ex- 
planation. Let AGCKE be the equator, ABCWE 
the ecliptic, -iS" the place of the star, Sm a secondary 
to the equator, and let the sun be at P, near to A, 
when it is on the meridian, and take CT-=PA, and 
diaw PL, TZ, perpendicular to AGC, and ZL is 
parallel to AC, and the sun's declination is the same 
at T as at P. Observe the meridian altitude of the 
sun when at P, and also the time of the passage of it's 

center over the meridian ; observe also at what time 
the star passes over the meridian, and then (111) find 

* A small correction must here be applied for the aberration of 
the star, in order to get the true difference of right ascensions, as 
will be explained ; because there is a small djfierence between the 
true and apparent places. 


the apparent difference Lm of their right ascensions. 
When the sun approaches near to T, observe it's me- 
ridian altitude for several days, so that on one of them, 
at t, it may be greater, and on the next day, at e, it 
may be less than the meridian altitude at jP, so that in 
the intermediate time it must have passed through T; 
and drawing thy es, perpendicular to AGCE, observe, 
on these two days, the difference hrn, sm of the sun's 
right ascension and that of the star ; draw also sv 
parallel to Zo. Then, to find Zb, we may consider 
the variation both of the right ascension and declina- 
tion, to be uniform for a small time, and consequently 
to be proportional to each other; hence, vb (the 
change of meridian altitudes in one day) : oh (the dif- 
ference of the meridian altitudes at t and T, or the 
difference of declination) :: sh (the difference oi sm, 
bm found by observation) : Zh, which added to b m, 
or subtracted from it, according to the situation of m, 
gives Zm, to which add Lm, or take their difference, 
according to circumstances, and we get ZL, which 
subtracted from AGC, or 180", half the remainder 
will be ALj the sun's right ascension at the first ob- 
servation, to which add Z/m, and we get the star's 
right ascension at the same time. Instead of finding 
hZ, we might have found sZ, by taking TZ — es for 
the second term, and thence we should have got Zm. 
Thus we should get the right ascension of a star, upon 
supposition that the position of the equator had re- 
mained the same, and the apparent place of the star 
had not varied in the interval of the observations. But 
the intersection of the equator with the ecliptic has a 
retrograde motion, called the Precession of the 
Equinoxes ; also, the inclination of the equator to the 
ecliptic is subject to a variation, called the Nutation ; 
and from the aberration of the star, it's apparent place 
is continually changing; these must therefore be al- 
lowed for, by considering how much they have varied 
in the interval of the observations ; but these are not 
subjects to be treated of in an elementary treatise. 


Having thus determined the right ascension of one 
star, that of the rest may be found from it (ill). 

(113.) The practical method of finding the right 
ascension of a body from that of a fixed star, by a 
clock adjusted to sidereal time*, is this: Let the 
clock begin it's motion from O^ O'. o" at the instant 
the first point of Aries is on the meridian ; then, when 
any star comes to the meridian, the clock will show 
the apparent right ascension of the star, the right 
ascension being estimated in the time, at the rate of 
15° for an hour, provided the clock is subject to no 
error, because it will then show, at any time, how far 
the first point of Aries is from the meridian. But as 
the clock is necessarily liable to err, we must be able, 
at any time, to ascertain what it's error is, that is, what 
is the difference between the right ascension shown by 
the clock, and the right ascension of that point of the 
equator which is at that time on the meridian. To do 
this, we must, when a star, whose apparent right ascen- 
sion is known, passes the meridian, compare it's appa- 
rent right ascension with the right ascension shown by 
the clock, and the difference will show the error of the 
clock. For instance, let the apparent right ascension 
of Aldeharan be 4h. 23'. 50" at the time when it's 
transit over the meridian is observed by the clock, and 
suppose the time shown by the clock to be Ah. 23'. 52", 
then there is an error of 2" in the clock, it giving the 
right ascension of the star 2" more than it ought. If 
the clock be compared with several stars, and the 
mean error taken, we shall have, more accurately, the 
error at the mean time of all the observations. These 
observations being repeated every day, we shall get the 
rate of the clock's going, that is, how fast it gains or 
loses. The error of the clock, and the rate of it's 

* A clock is said to be adjusted to sidereal time, when it is ad- 
justed to go 24 hours from the time a fixed star leaves the meridian 
till it returns to it, or it is the time of a revolution of the earth 
about it's axis. 


coing, being thus ascertained, if the time of the true 
transit of any body be observed, and the error of the 
clock at the time be appUed, we shall have the right 
ascension of the body. This is the method by which 
the right ascension of the sun, moon, and planets are 
regularly found in observatories. 

(114.) The right ascension of the heavenly bodies 
being thus ascertained, the next thing to be explained 
is, the method of finding their declinations. Take the 
apparent altitude of the body, when it passes the 
meridian, by an astronomical quadrant, as explained 
in mv Treatise on Practical Astronomy ; correct it 
for parallax and refraction (Chap. VI. and VII.) and 
you get the true meridian altitude, Ht, or He, (Fig. 
page 25), the difference between which and the alti- 
tude HE of the equator (which, by Art. 87, is equal 
to the complement of the latitude previously deter- 
mined) is the declination Et, or Ee, required. 

Ex. On April 27, 1774, the zenith distance of the 
moon's lower limb, when it passed the meridian at 
Greenwich, was 68°. 19'. 37",3 ; it's parallax in alti- 
tude was 56'. 19",2, allowing for the spheroidical 
figure of the earth ; the barometer stood at 29, 58, 
and the thermometer at 49 ; to find the declination. 
Observed zenith distance of L.L. 68°. I9'. 37''3 
Refr. cor. for bar. and ther. - + 2. 23 

68. 22.00,3 
Parallax - - 56. 19,2 

True zenith distance of L.L. - 67. 25. 41,1 
Semidiametcr ------ — 16. 35 

Xrue zenith distance of the center Q7 . 9. 6,1 
Latitude -51.28.40 

Declination 50Mf/i - - - - ~ 15. 40. 26,1 

The horizontal parallax and semidiameter may be 
taken from the Nautical Almanack; and the parallax 


in altitude may be found, as will be explained when 
we come to treat of the parallax. 

Given the Right Ascension and Declination of an 
Heavenlif Body, and the OhUquiti/ of the Ecliptic, 
to find the Latitude and Longitude. 

(115.) Let ^> be the body, t C the ecliptic, ^ Q the 
equator, sr, sp perpendicular to r C, t Q- Then 
(Trig. Art. 212) tan. sp : rad. :: sin ^p : cot. s^^p. 
Hence, srp-hQTC=srr. Also, 

COS. s^p : rad. :: tan. p^ ; tan. s^ (Trig. Art. 21 9) 
rad. : cos.^v r :: tan. s t : tan. r^p (Trig. Art. 21 9) 

,*. cos. s^p : COS. 5T r :: tan. p^ : tan. r ^ = 

COS. s rrx tdin. pr ., . . r .^ i -,7 1 
i— the tangent of the lonmtude ; and 

COS. s ^p ° O ' 

the logarithmic operation is, 

ar. CO. log. COS. s Tp-\-^og. cos. s Tr-\-log. (mi. pY—lO,=: log. tan. r T. 

Also, (Trig Art. 210) rad. : sin. rr :: tan. rys : tan. 
sr the tangent oi latitude y and the logarithmic opera- 
tion is, 

log. sin. r T-\-log. tan. rxs— 10, = log. tan. sr. 

In this manner, the right ascensions and declinations 
of the fixed stars being found from observation, their 
latitudes and longitudes may be computed, and their 
places become determined (41) ; hence, a catalogue of 
the fixed stars may be made for any time. 

If the latitude and longitude be given, the right 
ascension and declination may be found in the same 
manner; considering y^C the equator, and y Q thq 

Chap. IV. 


(116.) Having explained, in the last Chapter, 
the practical methods of determining the place of any 
body in the heavens, we come next to the considera- 
tion of another circumstrtiice not less important, that 
is, the irregula rit y of time as mea s ured by the s un. 
The best measure of time which we have, is a clock 
regulated by the vibration of a pendulum. But with 
whatever accuracy a clock may be made, it must be 
subject to go irregularly, partly from the imperfection 
of the workmansliip, and partly from the expansion 
and contraction of the materials by heat and cold, by 
which the length of the pendulum, and consequently 
the time of vibration, will vary. As no clock, there- 
fore, can be depended upon for keeping time accu- 
rately, it is necessary that we should be able to ascer- 
tain, at any time, how much it is too fast or too slow, 
and at what rate it gains or loses. For this purpose, 
it must be compared with some motion whicli is uni- 
form, or of which, if it be not uniform, you can ascer- 
tain the variation. The motions of the heavenly 
bodies have therefore been considered as most proper 
for this purpose. Now the earth revolving uniformly 
about it's axis, the apparent diurnal motion of the 
fixed stars about the axis must be uniform. If a 
clock, therefore, be adjusted to go 24 hours from the 
passage of any fixed star over the meridian till it 
returns to it again, it's rate of going may be at any 
time determined by comparing it with the transit of 
any fixed star, and observing whether the interval con- 
tinues to be 24 hours; if not, the difference shows 


how much it gains or loses in that time. A clock ad- 
justed to go 24 hours in this interval, is said to be 
adjusted to sidereal time. But if we compare a clock 
with the sun, and adjust it to go 24 hours from the 
time the sun leaves the meridian on any day, till he 
returns to it the next day, which is a true solar day, 
the clock will not, even if it go uniformly, continue to 
agree with the sun, that is, it will not show 12 when 
the sun comes to the meridian. 

(117.) For let P be the pole of the earth, vivyz it's 
equator, and suppose the earth to revolve about it's 
axis in the order of the letters v wy z;\et<^ DLE be 
the celestial equator, and ^ CL the ecliptic, in which 



the sun moves according to that direction. Let the 
sun be at a when it is upon the meridian of any place 
on any one day, and m the place when it is on the 
meridian the next day, and draw Pvae, Prmp, se- 
condaries to the equator, and let the spectator be at s 
on the meridian Pv, with the sun at a on his meridian. 
Then when the earth has made one revolution about 
it*s axis, P sv is come again into the same position ; 
but the sun having moved forward, the earth must 
continue to revolve, in order to bring the meridian 
Psv into the position Prni, so that the sun at m may 
be again in the spectator's meridian. Now the angle 
vPr js measured by the arc ep, which is the increase 
of the sun's right ascension in a true solar day, the 
right ascension being measured upon the equator 


T DLE (41) ; hence, the length of a true solar dai/t 
is equal to the time of the earth's rotation about it's 
axis + the time of it's describing an angle equal to 
the increase of the sun's right ascension on a true 
solar day . Now if the sun moved uniforrnlif^ and in 
the equator t DLE, this increase, ep, would be al- 
ways the same in the same time, and therefore the 
solar days would be always equal ; but the sun moves 
in the ecliptic t CL^ and therefore, if it's motion 
were uniforin, equal arcs (a m) upon the ecliptic 
would not give equal arcs {ep) upon the equator *. 
But the motion of the sun is not uniform, and there- 
fore am, described in any given time, is subject to a 
variation, which must, on this account also, necessarily 
make ep variable. Hence, the increase, ep, of the 
sun's right ascension in a day, varies from two causes, 
that is, from the ecliptic not coinciding with the 
equator, and from the unequal motion of the sun in 
the ecliptic ; therefore the length of a true solar day 
is subject to a continual variation ; consequently a 
clock, adjusted to go 24 hours for any one true solar 
day, will not continue to shew 12 when the suncomes 
to the meridian, because the intervals bythe clock will 
continue equal (the clock being supposed neither to 
gain nor lose), but the interval of the sun's passage 
over the meridian will vary. 

(118.) As the sun moves through 36o° of right 
ascension in 365^ days very nearly, therefore 36,5 1 
days : 1 day :: 36o°. : 59'. 2>",2 the increase of right 

* For tjraw int parallel to ep, and suppose ?na to be indefinitely 

small ; then by plain trigonometry, 

ma : mt :: rad. : sin. mat, or Y" «e (Trig. Art. 125.) 

mf : ep :: cos. ae : rad. (Art. 13) 

,'.ma : ep :: cq5. ae : sin. 7" ae :: (because sin. f a e =z 

COS. a Tex rad. _, • a ^ ^^^\ a «<> j- l 

Trig. Art. 212) cos. ae : cos. a Y^ex radius : hence, 

COS. ae 

the ratio of ma to ep is variable ; if therefore the sun's motion ma 
were uniform the corresponding increase ep of right ascension 
would not be uniform. 



ascension in one day, if the increase were uniform, in 
which case the solar days would be equal, and these 
days are called mean solar days. If therefore a clock 
be adjusted to go 24 hours in a meffw solar day, it can- 
not continue to coincide with the sun, that is, to show 
12 when the sun is on the meridian ; but the sun will 
pass the meridian, sometimes before 1 2 and some- 
times after. This difference is called the Equation of 
Time. A clock thus adjusted, is said to be adjusted 
to 7}iean solar time*. The time shown by the clock 
is called tj^ie or mea)i tiriie, and that shown by the 
sun is called apparent time. 

(119.) A clock adjusted to go 24 hours in a 7nean 
solar day, would coincide with an imagi nary star moy- 
ing uniformly in the equator wit h the sun's _rnean 
niotion^ 5,9'. 8'', 2 in rig h t ascens ion, if the star werelo 
set off f rom a ny given meridian when th e clo ck shows 
12 ; that is, the clock would always show 12 when the 
star came to the meridian, because the interval of the 
passages of this star over the meridian would be a mean 
solar day. This star, therefore, if we reckon it's mo- 
tion from the meridian, in time, at the rate of 1 hour 
for 15^, would always coincide with the clock; that 
is, \vheii the clock shows 1 hour, the star's motion 
would be 1 hour in right ascension, reckoned in time 
at the rate of 1 5*^ for an hour ; when the clock shows 
2 hours, the star's motion would be 2 hours ; and so 
on. . Hence, this star may be substituted instead of 
the clock; therefore, when the sun passes the given 
meridian, the difference between it's right ascension 
and that of the star, converted into time, is the differ- 
ence between the time when the sun is on the me- 

* As the earth describes an angle of 360°. 59'. 8",2 about it's 
axis in a mean solar clay of 24 hours, and an angle of 360o in a 
sidereal day, therefore 300°. 59'. 8",2 : 360°. ;: 2ik. : 23k. 56'. 
4,"098 the length of a sidereal day in mean solar time, or the time 
from the passage of a fixed star over the meridian, till it returns to 
it again. 


ridian and 12 o'clock, or the equation of time; 
because the given meridian passes through the star at 
12 o'clock, and it's motion, in respect to the star, is 
at the rate of 16° in an hour (121). 

(120.) Now, to compute the equation of time, let 
APLS be the ecliptic, ALv the equator, A the first 


point of aries, P the sun's apogee, S any place of the 
. sun ; draw Sv perpendicular to the equator, and take 
An = AP. When the sun departs from P, let the 
imaginary star set out from n with the sun's mean 
motion in right ascension, or in longitude, or at the 
rate of 59'. 8", 2 in a day, and when ii passes the 
meridian, let the clock be adjusted to 12, as described 
in the last article : these are the corresponding posi- 
tions of the clock and sun, as assumed by astronomers. 
Take nm=Ps, and when the star comes to m, the 
place of the sun, if it moved uniformly with it's mean 
motion, would be at s, but at that time let S be the 
place of the sun, and let the sun at S, and consequently 
V, be on the meridian ; then as ?n is the place of the 
imaginary star at that instant, 7nv is the equation of 
time. Let a be the mean equinox*, and draw az per- 
pendicular to A L ; then z on the equator would have 
coincided with a, if the equinox had moved uniformly; 
therefore we must reckon the mean right ascension 
from z. Now mv = Av—Am; but Am = Az-{-zm 
= Aax cos. aAz-\-zm= (because cos. aAz (23°. 28') 
=4-^ very nearly) \±-Aa + zm; hencejmv=Av — Z7n 

* The equinox has a retrograde motion, and that motion is not 
uniforni ; we here therefore suppose a to be the point where the 
equinox wouhl have been, if it moved uniformly with it's fitean 

D 2 


— -14-^^ ; but ^i; is the sun's true right ascension, z 7ti 
is the mean right ascension, or mean longitude, and 
i-i-^a jAz ^ is the equation of the eq uinoxes in right 
ascension ;| hence, the equation of time is equal toTIie 
difference of the suri^s true right ascension, and it's 
mean longitude corrected hi/ the equation of the equi- 
noxes in right ascension. When Am is less than Av, 
mean time precedes apparent^ and when greater, 
apparent time precedes mean ; for as the earth turns 
about it's axis in the direction Av, or in the order of 
right ascension, that body whose right ascension is 
least must come upon the meridian first ;fthat is, when 
the sun's true right ascension is ^rea/er than it's mean 
longitude corrected as above, we must add the equa- 
tion of time to the apparent, to get the mean time; 
and when it is less, we must subtract. To convert 
mean time into apparent, we must subtract in the 
former case, and add in the latter. This rule for 
computing the equation of time, was first given by 
Dr. Mashelyne in the Phil. Trans. 1764. 

(121.) As a meridian of the earth, when it leaves 
m, returns to it again in 24 hours, it may be con- 
sidered, when it leaves that point, as approaching 
a point at that time 360° from it, and at which g:i^^^'') 
it arrives in 24 hours. Hence, the relative velocity 
with which a meridian accedes to or recedes from m, 
is at the rate of 15° in an hour. Therefore, when the 
meridian passes through v, the arc v m, reduced into 
time at the rate of 15° in an hour, gives the equation 
of time at that instant. Hence, the equation of 
time is computed for the instant of apparent noon, 
or when the sun is on the meridian. Now the time of 
apparent noon in mean solar time, for which we com- 
pute.can only be known by knowingthe equation of time. 
To compute, therefore, the equation on any day, you 
must assume the equation the same as on that day 
four years before, from which it will differ but very 
little, and it will give the time of apparent noon, suffi- 
ciently accurate for the purpose of computing the 


equation. If you do not know the equation four years 
before, compute the equation for noon mean tirae^ and 
that will give apparent noon accurately enough. 

Ex. To find the equation of time on July 1, 
1792, for the meridian of Greenwich, by Mayer'' s 

The equation on July 1, 1 788, was, by the Nautical 
Almanac, 3'. 28", to be added to apparent noon, to give 
the corresponding mean time; hence, for July 1, 1792, 
at 0//. 3'. 2", compute the true longitude*. 

* The reason of this operation will appear to those who under- 
stand the method of computing the place of the sun from the solar 
tables. The explanation of such matters comes not within the 
plan of this work. See tny Complete System of Astronomy. 



o ' 

o ' 

CO C^ 

r-< CO 




o * 


1— 1 

o * 


-1 CO 
CI r-i 






•0 CO 
tP CO 




•^ CO 

C< F-H 







1^ C< '^ ^ 

tfs «> «\ «^ 


^ c^ 

a^ ^ 

"^ -^ lO J^ vC ^ 

CO rH 


2 1 + 1 + 1 




I— 1 

Epoch for 1792. 
Mean Mot. July 1, 



Mean Longitude 
Equat. of Center 
Equat. ]) I. 
2^ II. 

2 in. 

S3 IV. 
True Longitude 

With this true longitude, and obhquity 23°. 27'. 
48"j4 of the ecliptic, the true right ascension of the 


sun is found to he 3\ 11°. 5'. 4r',25 ; also, the equa- 
tion of the equinoxes in longitude = -0",6; hence, 
The mean longitude - - 3*. 10°. 13'. 25",4 
-|-4-of-0",6 ----- - 0,55 

Mean longitude corrected 3. 10. 13. 24,85 

True right ascension - - 3. 11. 5. 41,25 

Equation - _ - . _ 52. l6,4 

Which converted into time, gives 3'. 29", 1 for the true 
equation of time; which must be added to apparent, 
to give the true time, because the true right ascension 
is greater than the mean longitude. 

(122.) The sun's apogee, P, has a progressive 
motion, and the equinoctial points, A, L, have a 
regressive motion ; the inclination also of the equator 
to the ecliptic is subject to a constant variation. Hence, 
the same Table of the equation of time cannot continue 
to serve for the same degree of the sun's longitude. 
Also, the sun's longitude at noon at the same place 
is different for the same days on different years, and it 
is for apparent noon that the equation is computed. 
For these reasons, the equation of time must be 
computed a-new for every year. 

(123.) The two inequalities are sometimes sepa- 
rately considered, thus: First, that arising from the 
obliquity of the ecliptic. Let the sun and the imagi- 
nary star set off together from L, and let us now assume 
LS= Lm ; and let each move uniformly with the mean 
velocity, and then they will come to S and m together. 
Now when L S h greater than 90*^, the hypothenuse, 
LS, is less than the base, Lv (Trig, Art. 197) ; there- 
fore Lm is less than Lv (the case represented by the 
Figure); the star therefore, being left behind, comes upon 
the meridian first, and consequently true time precedes 
apparent. But when LS is less than 90°, the 
hypothenuse, LS, is greater than the base, Lv\ there- 
fore Lm is greater than Lv, and m lies on the other 
side of V ; therefore the sun comes upon the meridian 


first ; consequently apparent time precedes true. 
Hence, from equinox to tropic, apparent time pre- 
cedes true ; and from tropic to equinox, true time 
precedes apparent. Secondly^ that arising from the 
unequal motion of the earth in it's orbit. Let us 
suppose the sun to move about the earth, instead of the 
earth about the sun, the effect here being just the 
same, and this supposition will render the explanation 
easier. Let the sun depart from the apogee, and let 
the imaginary star set off from thence at the same time, 
with the mean angular velocity of the sun. Now 
when the sun is at it's greatest distance, it's angular 
velocity it less than it's mean angular velocity. (Note 
to Art. 165), and consequently less than the velocity of 
the star; the star therefore getting forwarder than the 
sun, the sun comes upon the meridian first, as shown 
in Art, 120, and therefore apparent time precedes 
true ; and this will continue till the sun comes to it's 
least distance, where, having performed half it's re- 
volution, and the star also having performed half it's 
revolution, the sun and star will coincide, (see Art 
168). Hence, from apogee to perigee, apparent time 
precedes true. Now the sun and star departing together 
from perigee, the sun's velocity is greater than that 
of the star ; the star therefore being left behind, comes 
upon the meridian first, and true time precedes appa- 
rent ; and this will continue till the sun comes to the 
apogee, where they again coincide. Hence, from 
perigee to apogee, true time precedes apparent. 

(124.) Whenever the time is computed from the 
sun's aUitude, that time must be apparent time, be- 
cause we compute it from the time when the sua 
comes to the meridian, which is noon, or 12 o'clock 
apparent time, and will differ from the time shown by 
a well-regulated watch or clock, by the equation of 
time. A clock or watch may therefore be regulated 
by a good dial ; for if you apply the equation, as be- 
fore directed, to the apparent time shown by the dial. 


it will give the mean time, or that which the clock or 
watch ought to show. 

(125.) The equation of time was known to, and 
made use of by, Ptolemy. Tycfio employed only one 
part, that which arises from tlie unequal motion of the 
sun in the ecliptic; but Kepler made use of both 
parts. He further suspected, that there was a third 
cause of the inequality of solar days, arising from the 
unequal motion of the earth about it's axis. But the 
equation of time, as now computed, was not generally 
adopted till 1672, when Flamstead published a dis- 
sertation upon it, at the end of the works of Horrox. 


Chap. V. 


(126.) From comparing the sun's right ascension 
every day with that of the fixed stars lying to the east 
and west, the sun is found constantly to recede from 
those on the west, and approach to those on the east ; 
hence, it's apparent annual motion is found to be from 
west to east; and the interval of time from it's leaving 
any fixed star till it returns to it again, is called a 
sidereal year, being the time in which the sun com- 
pletes it's revolution amongst the fixed stars, or in the 
ecliptic. But the sun, after it leaves either of the 
equinoctial points, returns to it again in a less time 
than it returns to the same fixed star, and this interval 
is called a solar or tropical year, because the time 
from it's leaving one equinox till it returns to it, is the 
same as from one tropic till it comes to the same 
again. This is the year on which the return of the 

On the Sidereal JTear. 

(127.) To find the length of a sidereal year' On 
any day when the sun is at Z on the meridian, (Fig. 
page 42), take the difference, Zm, between the sun's 
right ascension when it passes the meridian, and that 
of a fixed star, S •, and when the sun returns to the 
same part of the heavens the ne^tt year, compare it's 
right ascension with that of the same star for two days, 
one when their difference, b m, of right ascensions is 


Jess, and the other when the difference, sm, is greater 
than the difference, Zm, before observed ; then bs is 
the increase of the sun's right ascension in the time, t ; 
and as the increase of right ascension may be con- 
sidered as uniform for a small time, we have bs : bZ :: 
t : the time, T, in which the right ascension is in- 
creased from b to Z ; this time, T, therefore, added to 
the time of the observed right ascension at b, gives the 
time when the sun is at the same distance, Zm, in 
right ascension from the star, which it was when ob- 
served at ^the year before ; the interval of these times 
is therefore the length of a sidereal year. The best 
time for these observations is about March 25, June 
20, September 17, December 20, the sun's motion in 
right ascension being then uniform. Instead of ob- 
serving the difference of the right ascensions, you may 
observe that of their longitudes. 

If, instead of repeating the second observations the 
year after, there be an interval of several years, and you 
divide the observed interval of time when the difference 
of their right ascensions was found to be equal, by the 
number of years, you will have the length of a sidereal 
year more exactly. 

Ex. On April 1, 1669, at oh. 0.47", mean solar 
time, M, Picard observed the difference between the 
sun's longitude and that of Procyon to be 3\ 8°. 59'. 
36", which is the most ancient observation of this 
kind, the accuracy of which can be depended upon; see 
Hist. Celeste, par 31. le Monnier, page 37. And on 
April 2, 1745, M. dela Caille foum], by taking their 
difference of longitudes on the 2d and 3d, that at 
llh. 10'. 45", mean solar time, the difference of their 
longitudes was the same as at the first observation. 
Now as the sun's revolution was known to be nearly 
365 days, it is manifest that it had made seventy-six 
complete revolutions, in respect to the same fixed star, 
in the space of 76 years 1^/. llh. 6'. 58". Now in 
these 76 years, there were 58 of 365 days and 18 bis- 
sextiles of 366 days ; that interval therefore contains 


27759d. 1 1 A. 6'. 58"; which being divided by 76, the 
quotient is 365c?. 6h. 8'. 47". the length of a siderea! 
year. From the most accurate observations, the length 
of a sidereal year is found to be 3656?. 6h. g\ 11",5. 

On the Tropical Year. 

(128.) Observe the meridian altitude, «, of the sun 
on the day nearest to the equinox ; then the next year 
take it's meridian altitude on two following days, one 
when it's altitude, m, is less than a, and the next when 
it's altitude, ??, is greater than a, then n - m is the in- 
crease of the sun's declination in 24 hours; also, when 
the declination has increased by the quantity a — m, 
from the time when the meridian altitude m, was ob- 
served, the declination will then become a; and as we 
may consider the increase of declination to be uniform 
for a day, we have n~m : a — m v. 24 hours : the in- 
terval from the time when the sun was on the meridian 
on the first of the two days, till the sun has the same 
declination a, as at the observation the year before ; and 
thistinle, added to the time when the sun's altitude m 
was observed, gives the time when the sun's place in 
the ecliptic had the same situation in respect to the 
equinoctial points, which it had at the time of the ob- 
servation the preceding year; and the interval of these 
times is the length of a tropical year. 

If instead of repeating the second observation the 
next year, there be an interval of several years, and vou 
divide the interval between the times when the decli- 
nation was found to be the same, by the number of 
years, you will get the tropical year more exactly. 

Ex. M. Cassini informsus, that on March 20, 1672, 
his father observed the meridian altitude of the sun's 
upper limb at the Royal Obprvatory at Paris, to be 
41°. 43' ; and on March 20, 1716, he himself observed 
the meridian altitude of the upper limb to be 41°. 27'. 
10"; and on the 21st to be 41°. 51': therefore the 
difference of the two latter altitudes was 23'. 50", and 


of the two former 15', 50"; hence, 23'. 50'.: 15'. 50" 
:: 24 hours : I5h. 56'. 39"; therefore, on March 20, 
1716, at Ibh. 56'. 39", the sun's declination was the 
same as on March 20, 1672. Now the interval 
between these two observations was 44 years, of which 
34 consisted of 365 days each, and 10 of 366 ; there- 
fore the interval in days was 16070 ; hence, the whole 
interval between the equal declinations was 16070 
days 15/?-. 56'. 39", which divided by 44, gives 365d. 
5h. 49'. 0". 53'", the length of a tropical year from 
these observations. From the best observations, the 
length of a tropical year is found to be 365 c?. 5h, 
48'. 48". 

To Jind the Precession of the Equinoxes from 

(129.) The sun returning to the equinox every 
year before it returns to the same poin t in th e heaven s, 
shows that the equinoctial points have a reLrograde^ 
motion, and this arises from the moti on of the 
e quator, which is caused by the attraction of the sun 
and moon upon the earth, in consequence of it's 
spheroidical figure. The eiFect of this is, that the 
longitude of the stars must constantly increase ; and 
by comparing the longitude of the same stars at dif- 
ferent times, the motion of the equinoctial points, or 
the precession of the equinoxes, may be found. 

(130.) Hipparchus was the first person who ob- 
served this motion, by comparing his own observations 
with those which Timocharis made 155 years before. 
From this he judged the motion to be one degree in 
about 100 years; but he doubted whether the observa- 
tions of Timocharis were accurate enough to deduce 
any conclusion to be depended upon. In the year 
128 before J. C. he found the longitude of Firgin's 
Spike to be 5*. 24° ; and in the year 1750 its longi- 
tude was found to be 6\ 20**. l', the difference of 


which is 26". 21'. In the same year he found the 
longitude of the Lyons Heart to be 3'. 29°. 50' ; and 
in 1750 it was 4'. 26°. 21', the difference of which is 
26°. 31'. The mean of these two gives 26"^. 26' for 
the increase of longitude in I878 years, or 50". 40'" in 
a year for the precession. By comparing the observa- 
tions of Albategnius, in the year 878, with those 
made in 1738, the precession appears to be 51". 9'". 
From a comparison of 15 observations of Tychoj with 
as many made by M. de la Caille, the precession is 
found to be 50". 20"'. But M. de la Lande, from the 
observations of M. de la Caille, compared with those 
in Flatiistead's Catalogue, determines the secular pre- 
cession to be 1°. 23'. 45", or 50", 25 in a year. 

(131.) The precession being given, and also the 
length of a tropical year, the length of a sidereal year 
may be found by this proportion ; 36o° — 50",25 : 
360°. :: 365d. bh. 48'. 48" : 365c?. 6h. 9'. 11"^ the 
length of the 'sidereal ye^x. 

0n the Anomalistic Year. 


(132.) The year, called the anomalistic year, is 
sometimes used by astronomers, and is the time from 
the sun's leaving its apogee, till it returns to it. Now 
the progressive motion of theapogeein a year is 1 1"5 75, 
and hence, the anomalistic must be longer than the 
sidereal 3^ear, by the time the sun takes in moving over 
ll",75 of longitude at it's apogee; but when the sun 
is in it's apogee, it's motion in longitude is 58'. 13" in 
24 hours; hence, 58'. 13" : ll",75 :: 24 .hours : 4'. 
50"-^-, which added to 365c?. 6h. 9'. 11|", gives liGbd. 
6d. 14'. 2"^ the length of the anomalistic y ear. 
M. de la Lande determined this motion of the apogee, 
from the observations of M. de la Hire and those of 
Dr. Mashelyne. Cassini made it the same. 


On the OhUqidtij of the Ecliptic. 

(133.) The method used by astronomers to deter- 
mine the obliquity of the ecliptic, is that explained in 
Art. 86, by taking half the difference of the greatest 
and least meridian altitudes of the sun. The follow- 
ing is the obliquity, as determined by the different 
astronomers. »* 

Eratosthenes 230 years before «/. C. 23°. 51'. 20" 

Hipparchus 140 years before J. C. 

Ptolemy 140 years after J. C. 

Pappus in the year 390 - - - 

Alhategnius in 880 - _ - - 

Arzachel in 1070 - - . - , 

Prophatius inl3(50 - - - - 

Reg iomojit anus in 1460 - - - 

IValtherus in 1490 - - _ _ 

Copernicus in 1500 - - - - 

Tycho in 1587 ----- 23. 

Cassini (the Father) in 1656 

Cassini (the Son) in 1672 - 

Flamstead in 1690 - - - 

De la Caille in 1/50 . . - 

Dr* Bradley in 1/50 - - 

Mayer in 1750 . _ . . 

Dr. Mashelyne in 1769 - - 

M. de la Landem 1768 - - 

The observations of Alhategnius. an Arabian, are 
here corrected for refraction. Those of Walt her us ^ 
M. de Caille computed. The obliquity by Tycho is 
here put down as correctly computed from his obser- 
vations. Also, the obliquity, as determined by Flam- 
stead, is corrected for the nutation of the earth's axis. 
These corrections M. de la Lande applied. 

(134.) It is manifest, from the above observations, 
that the obliquity of the ecliptic continually decreases ; 



















































and the irregularity which here appears in the diminu- 
tion, we may ascribe to the inaccuracy of the observa- 
tions, as we know that they are subject to greater 
errors than the irregularity of this variation. If we 
compare the first and last observations, they give a 
diminution of 70" in IGO years. If we compare the 
last with that of Tycho, it gives 45". The last, com- 
pared with that of Flamstead, gives 50". If we com- 
pare that of Dr. Mashelyne with Dr. Bradley* s and 
Mayer's, it gives 50". The comparison of Dr. 
Mashelyne's determination, with that of M. de la 
Lande, which he took as the mean of several results, 
gives 50". We may therefore state the secular diminu- 
tion of the obliquity of the ecliptic, at this time, to be 
50", as determined from the most accurate observations. 
This result agre^es very well with that deduced from 


Chap. VI. 


(135.) The center of the earth describes that circle in 
the heavens which is called the ecliptic ; but as the 
same object would appear in different positions in 
respect to this circle, when seen from the center and 
surface, astronomers always reduce their observations 
to what they would have been, if they had been made 
at the center of the earth, in consequence of which, the 
places of the heavenly bodies are computed as seen 
from the ecliptic, and it becomes a fixed plane for that 
purpose^ on whatever part of the earth's surface the 
observations are made. 

(136.) Let C be the center of the earth, ^ the place 
of the spectator on it's surface, S any object, Zjfl the 
sphere of the fixed stars, to which the places of all the 
bodies in our system are referred ; Zthe zenith, i/the 
horizon ; draw CSm, ASn, and m is the place of S 
seen from the center, and ?i from the surface. Now 
the plane S^^^C passing through the centerof the earth, 
must be perpendicular to it's surface, and consequently 
it will pass through the zenith Z ; and the points, m, 
n, lying in the same plane, the arc of parallax, m/z, 
must he in a circle perpendicular to the horizon ; and 
hence, the azimuth is not affected, if the earth be a 
sphere. Now the parallax, ?)in, is measured by the 
angle mSn, ov ASC; and (Trig. Art. 128) CS : CA 
:: sin. SAC, or sin. SAZ, : sin. ^^-S'Cthe parallax=: 

CA X sin. SAZ \ r^ i • + ^ • ^i 

— -; As CA IS constant, supposmg the 

earth to be a sphere, the sine of the parallax varies as 
the sine of the apparent zenith distance directly, and 



the distance of the body from the center of the earth 
inversely. Hence, a body in the zenith has no pa- 
rallax, and at a^ in the horizon it is the greatest. And 

if the object be at an indefinitely great distance, it has 
no parallax ; hence, the apparent places of the fixed 
stars are not altered by it. As ii is the apparent place, 
and m is called the true place, the parallax depresses 
an object in a vertical circle. For diflTerent altitudes 
of the same body, the parallax varies as the sine, .9, of 
the apparent zenith distance; therefore, if/>=:the hori- 
zontal parallax, and radius be unity, we have 1 :s :: p : 
ps, the sine of the parallax. To ascertain, therefore, 
the parallax at all altitudes, we must first find it at 
some given altitude. 

(137.) First method, for the siin. Aristarchus 
proposed to find the sun's parallax, by observing it's 
elongation from the moon at the instant it is dichoto- 
mized^ at which time the angle at the moon is a right 
angle; therefore we may find the angle which the 
distance of the moon subtends at the sun ; which, di- 
minished in the ratio of the moon's distance from the 
earth's center to the radius of the earth, would give 
the sun's horizontal parallax. But a very small error 
in the time when the moon is dichotomized (and it is 
impossible to be very accurate in this), will make so 
very great an error in the sun's parallax, that nothing 


can he deduced from it to be depended upon. Ven- 
(lelinus determined the angle of elongation when 
the moon was dichotomized, to be 89°. 45', from which 
the sun's parallax was found to be 15". But P. Riccioli 
found it to be 28" or 30" from like observations. 

(153.) Second method. Hipparchus proposed to 
find the sun's parallax from a lunar eclipse, by the 
following method. Let S be the sun, E the earth, Ev 
the length of it's shadow, 7nr the path of the moon in 
a central eclipse. Observe the length of this eclipse, 
and then, from knowing the periodic time of the moon, 
the angle mEr, and consequently the half of that 
angle, or nEr, will be known. Now the horizontal 
parallax, ErB, of the moon being known, we have 
the angle Evr = ErB — nEr; hence, we know EAB* 

=AES- Evr=JES- ErB + nEr; that is, the sun's 
horizontal parallax = the apparent semidiameter of the 

* Supply the line AE, 
£ 2 


sun - the horizontal parallax of the moon + the semi- 
diaYneter of the earth's shadow where the moon passes 
through it. The objection to this method, is, the great 
difficulty of determining the angle nEr, with suffi- 
cient accuracy; for any error in that angle will make 
the same error in the sun's parallax, the other quanti- 
ties remaining the same. By this method, Vtohmij 
made the sun's horizontal parallax 2'. 50". Tycho 
made it 3'. 

(139.) Third method, for the moon. Take the 
meridian altitudes of the moon, when it is at it's 
greatest north and south latitudes, and correct them 
for refraction; then the difference of the altitudes, 
thus corrected, would be equal to the sum of the two 
latitudes of the moon, if there were no parallax ; con- 
sequently the difference between the sum of the two 
latitudes and the difference of the altitudes, will be the 
difference between the parallaxes at the two alti- 
tudes. Now, to find from thence the parallax itself, 
let S^ s, be the sines of the greatest and least apparent 
zenith distances, P, p, the sines of the corresponding 
parallaxes; then as, when the distance is given, the 
parallax varies (136) as the sine of the zenith dis tance, 

^ ^ , ^ ^ s X P — p 

S : s :: P : p; hence, S—s : s :: P - p : /»= — ^— — 

the parallax at the greatest altitude. This supposes 
that the moon is at the same distance in both cases ; 
but as this will not necessarily happen, we must cor- 
rect one of the observations, in order to reduce it to 
what it would have been, had the distance been the 
same, the parallax at the same altitude being inversely 
as the distance (136). If the observations be made in 
those places where the moon passes through the zenith 
of one of the observers, the difference between the 
sum of the two latitudes and the zenith distance at 
the other observation, will be the parallax at that 

(140.) Fourth method. Let a body, P, be observed 
from two places, A, B, in the same meridian, then 


the whole angle APB, is the sum of the two parallaxes 
of the two places. The parallax (136) APC=hor. 
par. X sin. PJL, taking A PC for si«. A PC, 


and the parallax BPC = hor. par. x sin PB3I; hence, 
hor. par. x sin PAL -{-sm.PB3I=APB, .*. hor. par. 
= APB divided by the sum of these two sines. If the 
two places be not in the same meridian, it does not 
signify, provided we know how much the altitude 
varies from the change of declination of the body in the 
interval of the passages over the meridians of the two 

Ex. On Oct. 5, 1751, M. de la Ca'ille, at the 
Cape of Good Hope, observed Mars to be l'. 25",8 
below the parallel of X in Aquarius, and at 25° distance 
from the zenith. On the same day, at Stockholm, 
Mars was observed to be T. 57",/ below the parallel 
of \, and at 68°. 14' zenith distance. Hence, the 
angle APB is 3l",9, and the sines of the zenith dis- 
tances beingO,4226and 0,9287, the horizontal parallax 
was 23",6. Hence, if the ratio of the distance of the 
earth from Mars to it's distance from the sun be found, 
we shall have the sun's horizontal parallax, the hori- 
zontal parallaxes being inversely as their distances from 
the earth (136). 

(141.) Fifth method. Let EQ be the equator, 
P it's pole, Z the zenith, i; the true place of the 


hoAy, and r the apparent place, as depressed by paral- 
lax in the vertical circle ZK, and draw the secondaries. 

Pva, Prh ; then ab is the parallax in right ascension, 
and r 5 in declination. Now, 

m' I vs :: 1 (rad,) : sin. vrs^ or ZvP (Trig.Art.125) 

vs : ab :: cos. va : 1 (rad.) Art. 13. 
/. V7' : ab :: cos. va : sin. ZvP ; 

.^ r ■, vrx sin, ZvP , ^ , 

therefore ao =- .; but t;r = ^or. par. X 

COS. va 

sin.vZ (136), and (Trig. Art. 22l)sin.i;Z : sin. ZP 

:: sin. ZPv : sin. ZvP^'^h^ijlJlIhlE^ , there- 

sin. i^Z 

r V 1 i^i. i- I. hor.par. x sin..^P x sin.Z^P?' 
iore,by substitution, ab = L . 

COS. va 

Hence, for the same star, where the hor. par. is given, 

the parallax in right ascension varies as the sine of the 

T 1 A 1 i.1 z. ^b X COS. va 
hour angle. Also, the hor.par. = — — — — _ -.-. 

sm. ZP X sin. ZPv 
For the eastern hemisphere, the apparent place b, lies 
pn the equator to the east of a, the true place, and 
therefore the right ascension is increased by parallax ; 
but in the western hemisphere, b lies to the west of a, 
and therefore therightascension isdiminished. Hence, 
if the right ascension be taken before and after the 
meridian, the whole change of parallax in right ascen- 
sion between the two observations, is the sum (s) of 
the two parts before and after the meridian, and the 


s X COS. vet 
hor, par.='-. — t^jt- — o» where 8= sum of the sines 

of the two hour angles. On the meridian there is 

11 • ■ 1 4. • r 1 vr X sin. ZvP 
no parallax m right ascension^ tor flo= 

cos. va ' 
where the angle ZvP, and consequently it's sine, 

(142.) To apply this rule, observe the right ascen- 
sion of the planet when it passes the meridian, compar- 
ed with that of a fixed star, at which time tliere is no 
])arallax in right ascension ; about 6 hours after, take 
the difference of their right ascensions again, and ob- 
serve how much the difference, dy between the appa- 
rent right ascensions of the planet and fixed star has 
changed in that time. Next, observe the right ascen- 
sion of the planet for 3 or 4 days when it passes the 
meridian, in order to get it's true motion in right 
ascension; then, if it's motion in right ascension in the 
above interval of time, between the taking of the right 
ascensions of the fixed star and planet on and off tlie 
meridian, be equal to d, the planet has no parallax in right 
ascension; but if it be not equal to d, the difference is 
the parallax in right ascension ; and hence, by the last 
article tiie horizontal parallax will be known. Or one 
observation may be made before the planet comes to 
the meridian, and one after, by which a greater differ- 
ence will be obtained. 

Ex. On Aug. 15, 1/19? Mars was very near a star 
of the 5th magnitude in the eastern shoulder of Aqua- 
rius, and SitQh. 18' in the evening. Mars followed the 
star in lO'. 17", and on the iGth, at 4h. 21' in the 
morning, it followed it in 10'. l", therefore, in that 
interval, the apparent right ascension of Mars had 
increased 16" in time. But, according to observations 
made in the meridian for several days after, it appeared 
that Mars approached the star only 14" in that time, 
from it's motion; therefore 2" in time, or 30" in motion, 
is the effect of parallax in the interval of the ob- 
servations. Now the declination of Mars was 15°, 
the co-latitude 41°. 10', and the two hour angles 


49°. 15', and 56°. 3g'; therefore the hor. par.=^ 

30" X COS. 15° 

■ - -^27^" But 

sin. 41°. 10' X sin. 49°. 15' + sm.56°.39' 

at that time, the distance of the earth from Mars was 
to it's distance from the sun, as 37 to 100, and there- 
fore the sun's horizontal parallax comes out 10",17. 

(143.) But, besides the effect of parallax in right 
ascension and declination, it is manifest that the latitude 
and longitude of the moon and planets must also be 
affected by it; and as the determination of this, in respect 
to the moon, is in many cases, particularly in solar 
eclipses, of great importance, we shall proceed to show 
how to compute it, supposing that we have given the 
latitude of the place, the time, and consequently the 
sun's right ascension, the moon's true latitude and lon- 
gitude, with her horizontal parallax. 

(144.) Let HZR be the meridian, t EQ the 
equator, p it's pole; y C the ecliptic, P it's pole j t 

the first point of Aries, JYQi? the horizon, Zthe zenith, 
Z^Z/ a secondary to the horizon passing through the true 
place r, and aparent place t, of the moon ; draw Pt,Pr^ 
which produce to a, drawing the small circle ts, parallel 
to ov ; then rs is the parallax in latitude, and ov the 
parallax in longitude. Draw the great circles, f P, 
PZAB, Ppde, and ZW, perpendicular to pe ; then, 
as X Pz=^o°, and t 7^=90°, Y must (4) be the pole 
oi Ppde, and therefore 6^7" =9^°; consequently d is 
one of the solstitial points,® orv;? ; also, draw Zxper:- 


peiidicular to Pr, and join Z'r ,p<v • Now ^ E, or the 
angle c^ pE, or Zp ^ , is the right ascension of the 
mid-heaven, which is known; PZ~AB (because 
AZ is the complement of both) the altitude of the 
highest point, A^ of the ecliptic above the horizon, 
called the nonagesimal degree, and ^y' Ay or the angle 
<Y' PA, is it's longitude; also, Zp is the co-latitude of 
the place, and the angle Zp W is the difference be- 
tween the right ascension of the mid-heaven ^cpEy 
and ^ e. Now, in the right-angled triangle Zp JVy 
(Trig. Art. 212) rad. x cos. /; = tan. p W y, cot. pZ; 

log. tan. pW=\0,-\-log. COS. p — log. cot. pZ, (Trig. Art. 213); 

hence, P PV—pTV-^pP, where the upper sign takes 
place when the right ascension of the mid-heaven is 
less than 180°, and the lower sign, when greater. 
Also, in the triangles WZp, IVZ P, we have (Trig. 
Art. 231) sin. Wp : sin. JVP :: tan. WP Z : tan. 
IVpz :: (Trig. Art. 82) cot. WpZ : cot. WPZ, or, 
tan. APcv ; therefore (Trig. Art. 49), 

Log.tan. APT = ar. co. lo. sin. Wp-\-log. sin.WP-\-lo. cot. WpZ—lO; 

and as we know ^p o,ov <^ Po, the true longitude of the 
moon, we know APo, or ZPcc. Also, in the triangle 
IVPZ, we have (Trig, Art. 21 9) cos. WP Z, or sin, 
AP^f : I'ad. :: tan. WP : tan. ZP ; therefore, 
Log. tan. ZP = 10, -|- log. tan. WP - log. sin. AP^ • 
Again, in the triangle ZPr, we know ZP, Pr, and 
the angle, P, from which the angle ZrP, or t rs, may 
be thus found. In the right-angled triangle ZPx, we 
know -ZP, and the angle P; hence, (Trig. Art. 212) 
rad. X cos. ZPx — cot. PZ x tan. P.r; therefore, 

Log. tan. Px= 10, -\-cos. ZPx — log. cot. PZ ; 
therefore we know rx-, hence, (Trig. Art. 231) sin. 
rx : sin. Px :: tan. ZPx : tan. Zrx, or trs; there- 
Log. tan. Z)\v=ar. co, log. sin. rx-\-log.si7i. Px-\-log.tan.ZPx— 10; 

also, in the right-angled triangle Zrx, we have (Trig. 


Art. 212) rad. x cos. Z?'x=cot. Zrx tan. rx; there- 

Log. cot. Zr = 10, + log. COS. Z roc — log. tan. ?'.r. 
With this true zenith distance Z?', find (136) the 
parallax, as if it were the apparent zenith distance, 
and it will give you the true parallax nearly ; add this 
therefore to the true zenith distance, and you will get 
nearly the apparent zenith distance, to which com- 
pute again (136) the parallax, and thus you will get 
the true parallax, rt, extremely nearly; then, in the 
right-angled triangle rst, which may be considered as 
plane, we have (Trig. Art. 125) rad. : cos. r ;: rt : rs, 
the parallax in latitude ; therefore, 
log. r.s=log. rt +log. cos. r— 10, = log. par. latitude. 
Also, rad. : sin. r :: rt : ts; therefore, 

log. ts. = log. rt + log. sin. 7' — 1 ; 

hence, (13) cos. tv : rad. :: ts : ov, the parallax in 

longitude ; therefore, 

log. ov = 10, + log. ts—log. COS. tv = log. par. longitude. 

Ex. On January I, 177^5 at 9^. apparent time, in 
lat. 53° N. the moon's true longitude was 3^ 18'^. 27'. 
35", and latitude 4°. 5'. 30" S. and it's horizontal 
parallax 61'. 9"; to find its parallax in latitude and 

The sun's right ascension was 282°. 22'. 2" by the 
Tables, and its distance from the meridian 135^; 
also, the right ascension op E, of the mid-heaven, was 
57°. 22'. 2" ; hence, the whole operation for the solu- 
tion of the triangles will stand thus : 

^ CZpff^ =32°. 37'. 58" - 10, + cos. 19.9253864 
^jZp =37. 0. 0. - - cot. 10.1228856 

^ (p^ =32. 23. 57 - - tan. 9.8025008 

Pp' =23. 28. O 

PfV =55. 51. 67 


^ ^pJV =32°. 23'. 57" - ar. CO. sin. O.2709805 

fe ipfV =55. 51. 57 - - - sin. 9-9178865 

N^<Z;?^=32. 37. 58 - - - cot. 10.1935941 


I^^JPr =^7- 29. 8 - - - tan. 10.3824661 


oPt =108. 27. 35 

oPJ = 40. 58. 27 

N rwp =55. 51. 57 - 10,+tan. 20.1688210 

^ )APr =67. 29. 8 - - sin. 9.9655700 


^ IZP =57. 56. 36 - - - tan. 10.2032510 

(;g (ZPv =40. 58. 27 - 10,+cos. 19,8779500 

S3 )ZP =57. 56. 36 - - - cot. 9.7967445 

'x =50. 19. 33 - - -tan. 10.0812055 

*r =94. 5. 30 

= 43. 45. 47 - ar. co. sin. O.1600743 

= 50. 19. 33 - - - sin. 9.8863144 

U^lZPx =40.58.27 - - -tan. 9.9^87676 


^^Zrx =44. 1. 16 - - -tan. 9.9851563 

'Zrx =44. 1. 16 - 10,-hcos. 19.8567795 
= 43. 45. 57 - - - tan. 9.98 12846 

r =53. 6. 10 - - -cot. 98754949 

Zr =53. 6. 10 - - -sin 9.9029362 
Hor. par. = 6l'. 9" = 3669" - log. 3.5645477 

rt uncorrected = 2934" = 48'.54"log. 3.4674839 

Hot. par. =6l'. 9"=3669" - log. 3.5645477 


^ rPar. rt cor. = 2965" = 49'. 25" log. 3.4720519 
^ J/r5 = 44°. 1'. 16" - - - - COS. 9.8567795 

^ U^^par. in Zaf. = 2132"=35'. 32" log. 3.3288314- 

>^cor. = 2965" log. 3.4720519 

^ ')trs = A4°.l'. iQ" . - - - 6111.9.8419369 

.J^5 = 206l"=34'. 21" - - - log. 3.3139888 
True lat. ?-o = 4°. 5'. 30" 

App. Iat.^i> = 7'0 + r5=:4°. 41'. 2" cos. 9.9985472 
#^=2061" ----- 10, + log. 13.3139888 

o?;««f. = 2067"=34'.27" log. 3.3154416 

The value of /y is 7'o — or + r*, according as the 
moon has N. or S. latitude. 

The order of the signs being from West to East, 
from A towards C is eastward, and from A towards ^ 
is westward ; now the parallax depressing the body 
from r to t, increases the longitude from o to ?; ; but 
if the point had been on the other side of ^, ov would 
have lain the contrary way; hence, when the body is 
to the East of the nonagesimal degree, the parallax 
increases the longitude ; and when it is to the West, 
it diminishes the longitude. 

(145.) According to the Tables of Mayer, the 
greatest parallax of the moon (or when she is in her 
perigee and in opposition) is 61'. 32"; the least 
parallax (or when in her apogee and conjunction) is 
53'. 52", in the latitude of Paris; the arithmetical mean 
of these is 57'. 42"; but this is not the parallax at the 
mean distance, because the parallax varies inversely as 
the distance, and therefore the parallax at the mean 
distance is 57'. 24", an harmonic mean between the 
two. M. de Lamhre re-calculated the parallax from 
the same observations from which Mayer calculated it, 
and found it did not exactly agree with. Mayer s. He 
made the equatorial parallax 57'. H",4. M. de la 
Lande makes it57'. 3" at the equator, 56'. 53",2 at the 


pole, and 57'. l" for the mean radius of the earth, sup- 
posing the difference of the equatorial and polar 

diameters to be of the whole. From the formula 


of Mm/er, the equatorial parallax is 57'. 11", 4. 

(146.) To find the mean distance, Cs,oi the moon, 

we have AC, the mean radius (?•) of the earth, : Cs, 

the mean distance (Z)) of the moon from the earth, :: 
sin. 57'. l"=JsC {145) : radius :: 1 : 6o,3 ; conse- 
quently Z>=6o,3r; but r = 3964 miles; hence, Z) = 
239029 miles. 

(147.) According to M. de la Lande, the horizontal 
semidiameter of the moon : it's horizontal parallax for 
the mean radius (r) of the earth -.: 15' : 54'. 57",4, or 
very nearly as 3 : 11; hence, the semidiameter of the 
moon is -iV ^ = tV X 3964=1081 miles ; and as the 
magnitude of spherical bodies are as the cubes of their 
radii, we have the magnitudes of the moon and earth 
as 3^ : 1 1^, or as 1 : 49 nearly. 


Chap. VI. 


(148.) VVhen a ray of light passes out of a vacuum 
into any medium, or out of any medium into one of 
greater density, it is found to deviate from it's rectilinear 
course towards a perpendicular to the surface of the 
medium into which it enters. Hence, light passing 
out of a vacuum into the atmosphere will, where it 
enters, be bent towards a radius drawn to the earth's 
center, the top of the atmosphere being supposed to be 
spherical and concentric with the center of the earth ; 
and as, in approaching the earth's surface, the density 
of the atmosphere continually increases, the rays of 
light, as they descend, are constantly entering into a 
denser medium, and therefore the course of the rays 
will continually deviate from a right line, and describe 
a curve ; hence, at the surface of the earth, the rays of 
light enter the eye of the spectator in a different direc- 
tion from that in which they would have entered, if 
there had been no atmosphere; consequently the ap- 
parent place of the body from which the light comes, 
must be different from the true place. Also, the re- 
fracted ray must move in a plane perpendicular to the 
surface of the earth ; for, conceiving a ray to come in 
that plane before it is refracted, then the refraction 
being always in that plane, the ray must continue to 
move in that plane. Hence, the refraction is always 
in a vertical circle. The ancients were not unac- 
quainted with this effect. Ptolemy mentions a dif- 
ference in the rising and setting of the stars in different 
states of the atmosphere; he makes, however, no 
allowance for it in his computations from his observa- 


tlonsj this correction^ therefore, must be appUed, 
where great accuracy is required. Archimedes ob- 
served the same in water, and thought the quantity of 
refraction was in proportion to the angle of incidence. 
Alhazen, an Arabian optician, in the eleventh century, 
by observing the distance of a circumpolar star from 
the pole, both above and below, found them to be 
different, and such as ought to arise from refraction. 
Snellkis, who first observed the relation between the 
angles of incidence and refraction, says, that PFalthei^us, 
in his computation, allowed for refraction; but Tyclm 
was the first person who constructed a table for that 
purpose, which, however, was very incorrect, as he 
supposed the refraction at 45° to be nothing. About 
the year l()6o, Cassini published a new table of refrac- 
tions, much more correct than that of Tycho ; and, 
since his time, astronomers have employed much at- 
tention in constructing more correct tables, the niceties 
of modern astronomy requiring their utmost accuracy. 

To find the quantity of refraction. 

(149.) First method. Take the altitude of the sun, 
or a star whose right ascension and declination are 
known, and note the time by the clock ; observe also 
the times of their transits over the meridian, and that 
interval gives the hour angle. Now, in the triangle 


PZx, we know PZ and Px, the complements of lati- 
tude and declination, and the angle xPZ, to find the 


side Zx (92), the complement of whicli is the true 
altitude, the difference between which and the oh- 
served altitude, is the refraction at that altitude. 

Ex. On May 1, 1738. at bh. 20' in the morning, 
Cassini observed the altitude of the sun's center at 
Paris to be 5°. O'. 14'', and the sun passed the meridian 
at I2h. 0'. O", to find the refraction, the latitude bein^y 
48°. 50'. 10", and the declination was 15°. 0'. 25". 
The sun's distance from the meridian was 6h. 40', 
which gives 100° for the hour angle xPZ ; also, PZ 
= 41°. 9'. 50", and Px=74\ 59'. 35"; hence, (Trig. 
Art. 233) Zx=85°. lO'. 8", consequently the true alti- 
tude was 4°. 49'. 52". Now to 5°. O'. 14", the appa- 
rent altitude, add 9" for the parallax, and we have 
5°. 0'. 23" the apparent altitude corrected for parallax ; 
hence, 5°. O'. 23" -4°. 49'. 52"=10'.3l", the refrac- 
tion at the apparent altitude 5°. O'. 14". 

(150.) Second method. Take the greatest and least 
altitudes of a circumpolar star which passes through, 
or very near, the zenith, when it passes the meridian 
above the pole ; then the refraction being nothing in 
the zenith, we shall have the true distance of the star 
from the pole at that observation, the altitude of the 
pole above the horizon, being previously determined ; 
but when the star passes the meridian under the pole, 
we shall have it's distance affected by refraction, and 
the difference of the two observed distances, above and 
below the pole, gives the refraction at the apparent 
altitude below the pole. 

Ex. M. de la Caille, at Paris, observed a star to 
pass the meridian within & of the zenith, and conse- 
quently, at the distance of 41°. 4' from the pole ; 
hence, it must pass the meridian under the pole at the 
same distance, or at the altitude 7°- 46'; but the ob- 
served altitude at that time was 7°- 52'. 25" ; hence, 
the refraction was 6'. 25" at that apparent altitude. 

(151.) Let CAn be the angle of incidence, CAm the 
angle of refraction, and consequently 7nAn the quan- 
tity of refraction ; let TC be the tangent of C?n, mv 


its sine, nw the sine of Cn, and draw rm parallel to 
vw ; then, as the refraction of air is very small, we 

W V 

may consider m rn as a rectilinear triangle ; and 

hence, by si-milar triangles, Av : Am :: rn : mn = 

Am X rn , ^ . , , • r 
-J — ; but Am is constant, and as the ratio ot mv 

to nw is constant by the laws of refraction, their dif- 
ference, rn, must vary as nw ; hence, m?z varies as 

'"^ , ^rr^ AmXniv , . , . mv ^ 

■j^; but CT=: — j^ , which varies as ^, be- 
cause Am is constant; hence, the refraction, mn, varies 
as CT, the tangent of the apparent zenith distance of 
the star, because the angle of refraction, CAm, is the 
angle between the refracted ray and the perpendicular 
to the surface of the medium, which perpendicular is 
directed to the zenith. Whilst, therefore, the refrac- 
tion is very small, so that rm« may be considered as 
a rectilinear triangle, this rule will be sufficiently 
accurate *. 

(152.) The twilight in the morning and evening, 
arises both from the refraction and reflection of the 
sun's rays by the atmosphere. 

It is probable that the reflection arises principally 
from the exhalations of various kinds with which the 
lower parts of the atmosphere are charged ; for the 
twilight lasts till the sun is further below the horizoa 
in the evening, than it is in tlie morning when it 

* For further information on this subject, see ray Complete 
Si/stem of Astronomj/. 



begins ; and it is longer in summer than in winter. 
Now, in the former case, the heat of the day has 
raised the vapours and exhalations ; and in the latter, 
they will be more elevated from the heat of the 
season ; therefore, upon supposition that the reflection 
is made by them, the twilight ought to be longer in 
the evening than in the morning, and longer in 
summer than in winter. 

(153.) Another effect of refraction is, that of giving 
the sun and moon an oval appearance, by the refrac- 
tion of the lower limb being greater than that of the 
upper, whereby the vertical diameter is diminished. 
For suppose the diameter of the sun to be 32', and the 
lower limb to touch the horizon, then the mean refrac- 
tion at that limb is 33", but the altitude of the upper 
limb being then 32', it's refraction is only 28'. 6", the 
difference of which is 4', 54", the quantity by which 
the vertical diameter appears shorter than that parallel 
to the horizon. When the body is not very near the 
horizon, the refraction diminishing nearly uniformly, 
the figure of the body is very nearly that of an ellipse. 


Chap. VIII. 


(154.) When any effect or phaenomenon is discovered 
by experiment or observation, it is tbe business of 
Philosophy to investigate it's cause. But there are 
very few, if any, enquries of this kind, where we can 
be led from the effect to the cause by a train of mathe- 
matical reasoning, so as to pronounce with certainty 
upon the cause. Sir /. Newton, therefore, in his 
Principia, before he treats on the System of the 
World, has laid down the following Rules to direct us 
in our researches into the constitution of the universe. 

Rule I. No more causes are to be admitted, than 
what are sufficient to explain the phaenomenon. 

Rule II. Of effects of the same kind, the same 
causes are to be assigned, as far as it can be done. 

Rule III. Those qualities which are found in all 
bodies upon which experiments can be made, and 
which can neither be increased nor diminished, may 
be looked upon as belonging to all bodies. 

Rule IV. In Experimental Philosophy, proposi- 
tions collected from phaenomena by induction, are to 
be admitted as accurately, or nearly true, until some 
reason appears to the contrary. 

The principles, upon which the application of these 
Rules is admitted, are, the supposition that the opera- 
tions of nature are performed in the most simple 
manner, and regulated by general laws. And al- 
though their application may, in many cases, be very 
unsatisfactory, yet, in the instances to which we shall 
here want to apply them, their force is little inferior to 

F 2 


that of direct demonstration, and the mind rests equally 
satisfied as if the matter were strictly proved. 

(155.) The diurnal motion of the heavenly bodies 
may be accounted for, either by supposing the earth 
to be at rest, and all the bodies daily to perform their 
revolutions in circles parallel to each other ; or by sup- 
posing the earth to revolve about one of it's diameters 
as an axis, and the bodies themselves to be fixed, in 
which case their apparent diurnal motions would be 
the same. If we suppose the earth to be at rest, all 
the fixed stars must make a complete revolution, in 
parallel circles, every day. Now the nearest of the 
fixed stars cannot be less than 400000 times further 
from us than the sun is, and the sun's distance from 
the earth is not less than 93 millions of miles. Also, 
from the discoveries which are every day making by 
the improvement of telescopes, it appears that the 
heavens are filled with almost an infinity of stars, to 
which the number visible to the naked eye bears no 
proportion; and whose distancesare, probably, incom- 
parably greater than what we have stated above. But 
that an almost infinite number of bodies, most of them 
invisible, except by the best telescopes, at almost infi- 
nite distances from us and from each other, should 
have their motions so exactly adjusted, as to revolve in 
the same time, and in parallel circles, and all this 
without their having any central body, which is a 
physical impossibility, is an hypothesis, which, by the 
Rules we have here laid down, is not to be admitted, 
when we consider, that all the phaenomena may be 
solved, simply by the rotation of the earth about one 
of it's diameters. If therefore we had no other reason, 
we might rest satisfied that the apparent diurnal mo- 
tions of the heavenly bodies are produced bj^ the earth's 
rotation. But we have other reasons for this supposi- 
tion. Experiments prove that all the parts of the 
earth have a gravitation towards each other. Such a 
body, therefore, the greatest part of whose surface is 
a fluid, if it remain at rest, must, from the equal gra^ 


vitatlon of it's parts, form itself into a perfect sphere. 
But if we suppose the earth to revolve, the parts most 
distant from the axis must, from their greater velocity, 
have a greater tendency to fly off, and therefore that 
diameter which is perpendicular to the axis must be 
increased. That this must be the consequence appears 
from taking an iron hoop, and making it revolve 
swiftly about one of it's diameters, and that diameter 
will be diminished, and the diameter perpendicular to 
it will be increased. Now it appears from mensura- 
tion, that the earth is not a perfect sphere, but a 
spheroid, having the equatorial longer than it's polar 
diameter. That diameter therefore, about which the 
earth must revolve, in order to solve all the phaenomena 
of the apparent revolution of the heavenly bodies, is 
the shortest ; and as it necessarily must be the shortest, 
if the earth be supposed to revolve, this agreement 
affords a very satisfactory proof of the earth's rotation. 
Another reason for the earth's rotation is from analogy. 
The planets are opaque and spherical bodies, like to 
our earth ; now all the planets, on which sufficient ob- 
servations have been made to determine the matter, 
are found to revolve about an axis, and the equatorial 
diameters of some of them are visibly greater than their 
polar. When these reasons, all upon different prin- 
ciples, are considered, they amount to a proof of the 
earth's rotation about it's axis, which is as satisfactory 
to the mind, as the most direct demonstration could 
be. These, however, are not all the arguments which 
might be offered ; the situations and motions of the 
bodies in our system necessarily require this motion of 
the earth. 

(156.) Besides this apparent diurnal motion, the 
sun, moon, and planets, have another motion ; for 
they are observed to make a complete revolution 
amongst the fixed stars, in different periods; and 
whilst they are performing these motions in respect to 
the fixed stars, they do not always appear to move in 
the same direction, or in that direction in which their 


complete revolutions are made, but sometimes appear 
stationary, and sometimes to move in a contrary direc- 
tion. We will here briefly describe and consider the 
different systems which have been invented, in order to 
solve these appearances. Ptolemy supposed the earth 
to be perfectly at rest, and all the other bodies, that is, 
the sun, moon, planets, comets and fixed stars, to re- 
volve about it every day; but that, besides this diurnal 
motion, the sun, moon, planets and comets had a mo- 
tion in respect to the fixed stars, and were situated, in 
respect to the earth, in the following order ; the Moon, 
Mercury, Venus, the Sun, Mars, Jupiter, Saturn. 
These revolutions he first supposed to be made in 
circles about the earth, placed a little out of the center, 
in order to account for some irregularities of their 
motions; but as their retrograde motions and stationary 
appearances could not thus be solved, he supposed 
them to revolve in epicycloids, in the following manner. 
Let ABC be a circle, S the center, E the earth, ah c d 
another circle, whose center v is in the circumference 
of the circle ABC. Conceive the circumference of the 
circle ABC to be carried round the earth every 
twenty-four hours, according to the order of the letters, 


and at the same time let the center v of the circle ahcd 
have a slow motion in the opposite direction, and let 


a body revolve in this circle in the direction abed; 
then it is manifest, that by the motion of the body in 
this circle, and the motion of the circle itself, the body 
will describe such a curve as is represented bv Mrnhnop-, 
and if we draw tlie tangents El, Em, the body woald 
appear stationary at tlie points / and m, and it's mo- 
tion would be retrograde through /m, and then direct 
again. Now to make Venus and Mercurv always 
accompany the sun, the center v of the circle «^c</ was 
supposed to be always very nearly in a right line be- 
tween the earth and the sun, but more nearly so for 
Venus than for Mercury, in order to give each it's 
proper elongation. This system, althougli it will ac- 
count for all the motions of the bodies, yet it will not 
solve the phases of V^enus and Mercury ; for in this 
case in both conjunctions with the sun, they ought to 
appear dark bodies, and to lose their lights both ways 
from their greatest elongations ; wdiereas it appears 
from observation, that in one of their conjunctions 
they shine with full faces. This system therefore 
cannot be true. 

(157-) The system received by the Egyptians was 
this: That the earth is immoveable in the center, 
about which revolve in order, the Moon, Sun, Mars, 
Jupiter, and Saturn ; and about the Sun revolve Mer- 
cury and Venus. This disposition will account for the 
phases of Mercury and Venus, but not for the apparent 
motions of Mars, Jupiter, and Saturn. 

(158.) The next system which we shall mention, 
though posterior in time to the true, or Copernican 
System, as it is usually called, is that of Tycho Brake, 
a Polish nobleman. He was pleased with the Coper- 
nican System, as solving all the appearances in the most 
simple manner; but conceiving, from takiugtheliteral 
meaning of somepassages in scripture, that it was ne- 
cessary to suppose the earth to be absolutely at rest, he 
altered the system, but kept as near to it as possible. 
And he further objected to the earth's motion, because 
it did not, as he conceived^ affect the motions of 


comets observed in opposition, as it ought ; whereas, 
if he had made observations on some of them, he would 
have found that their motions could not otherwise 
have been accounted for. In his system the earth is 
supposed to be immoveable in the center of the orbits 
of the sun and moon, without any rotation about an 
axis ; but he made the sun the center of the orbits of 
the other planets, which therefore revolved with the 
sun about the earth. By this system, the different 
motions and phases of the planets may be solved, the 
latter of which could not be by the Ptolemaic System ; 
and he was not obliged to retain the epicyclods, in 
order to account for their retrograde motions and 
stationary appearances. One obvious objection to this 
system is, the want of that simplicity by which all the 
apparent motions maybe solved, and the necessity that 
all the heavenly bodies should revolve about the earth 
every day; also, it is physically impossible that a large 
body, as the sun. should revolve in a circle about a 
small body, as the earth, at rest in its center ; if one 
body be much larger than another, the center about 
which they revolve must be very near the large body; 
an argument which holds also against the Ptolemaic 
System. It appears also from observation, that the 
plane in which the sun must, upon this supposition, 
diurnally move, passes through the earth only twicein 
a year. It cannot therefore be any force in the earth 
which can retain the sun in it's orbit; for it would 
move in a spiral, continually changing it's plane. In 
short, the complex manner in which all the motions 
are accounted for, and the physical impossibility of 
such motions being performed, is a sufficient reason 
for rejecting this system ; especially when we consider, 
in how simple a manner all these motions may be ac- 
counted for, and demonstrated from the common prin- 
ciples of motion. Some of Tycho's followers, seeing 
the absurdity of supposing all the heavenly bodies 
daily to revolve about the earth, allowed a rotatory 
motion to the earth, in order to account for their 


diurnal motion ; and this was called the Semi- 
Tycho7iic System ; but the objections to this system 
are in other respects, the same. 

(159.) The system which is now universally re- 
ceived is called the Copcrnican. It was formerly 
taught by Pythagoras, who lived about 500 years 
before J. C. and Pkilolaus, his disciple, maintained 
the same ; but it was afterwards rejected, till revived 
by Copernicus. Here the Sun is placed in the center 
of the system, about which the other bodies revolve in 
the following order ; Mercury, Venus, the Earth, 
Mars, Jupiter, Saturn, and the Georgian Planet, which 
was lately discovered by Dr. Herschel ; beyond which, 
at immense distances, are placed the fixed stars ; the 
moon revolves about the earth, and the earth revolves 
about an axis. This disposition of the planets solves 
all the phaenomena, and in the most simple manner. 
For from inferior to superior conjunction, Venus and 
Mercury appear, first horned, then dichotomized, and 
next gibbous; and the contrary, from superior to infe- 
rior conjunction ; they are always retrograde in the 
inferior, and direct in their superior conjunction. 
Mars and Jupiter appear gibbous about their quadra- 
tures ', but in Saturn and the Georgian this is not 
sensible, on account of their great distances. The 
motions of the superior planets are observed to be 
direct in their conjunction, and retrograde in their 
opposition. All these circumstances are such as ought 
to take place in the Copernican System. The motions 
also of the planets are such as should take place upon 
physical principles. We may also further observe, 
that the supposition of the earth's motion is necessary, 
in order to account for a small apparent motion which 
every fixed star is found to have, and which cannot 
otherwise be accounted for. I'he harmony of the 
whole is as satisfactory a proof of the truth of this 
system, as the most direct demonstration could be ; 
we shall therefore assume this system. 


Chap. IX. 

ON Kepler's discoveries. 

(l6o.) Kepler was the first who discovered the 
figures of the orbits of the planets to be elhpses, having 
the sun in one of the foci; this he determined in the 
following manner. 

(l6l.) Let S be the sun, 71/ Mars, I>, E, two places 
of the earth when Mars is in the same point M of it's 
orbit. When the earth was at D, he observed the 

difference between the longitudes of the sun and 
Mars, or the angle MDS; in like manner, he ob- 
served the angle MES. Now the places D, E, of the 
earth in it's orbit being known, the distances DS, ES, 
and the angle DSE, will be known ; hence, in the 
triangle DSE, we know DS, SE, and the angle DSE, 
to find DE, and the angles SDE, SED ; hence, we 
know the angles MDE, MED, and DE, to find MD; 
and lastly, in the triangle MDS, we know MD, DS, 
and the angle MDS, to find MS, the distance of 
Mars from the sun. He also found the angle 3ISD, 
the difference of the heliocentric longitudes of Mars 
and the earth. By this method, Kepler, from observa- 
tions made on Mars when in aphelion and perihelion 


(for he had determined the position of the hne of the 
apsides, by a method which we sj^all afterwards_ex- 
plain^ independent of the form of the orbit, deter- 
mined the former distance from the sun to be 166/80, 
and the latter 138500, the mean distance of the earth 
from the sun being 100000; hence, the mean distance 
of Mars was 15 264 0_, and the excentricity of it's orbit 
14140. He then determined, in hke manner, three 
other distances, and found them to be 14/7^0, 163100, 
166255. Fie next calculated the same three distances, 
upon supposition that the orbit was a circle, and found 
them to be 14853.9, 16*3883, l66605 ; the errors 
therefore of the circular hypothesis were '^8Q, 7^^i 
350. But he had too good an opinion of Ti/cIio*s 
observations (upon which he founded all these calcu- 
lations) to suppose that these differences might arise 
from their inaccuracy; and as the distance between 
the aphelion and perihelion was too great, upon suppo- 
sition that the orbit was a circle, he knew that the 
form of the orbit must be an oval ; Itaque plane hoc 
est : Orhita planet ce non est cir cuius, sed ingrediens 
ad latera utraque paulatim, iterumque ad circuli 
amplitudinem in perigceo ejciens, cujusmodi Jiguram 
itinens ovalem nppellitant, pag. 213*. And as, of 
all ovals, the elhpse appeared to be the most simple, 
he first supposed the orbit to be an ellipse, and placed 
the sun in one of the foci ; and upon calculating the 
above observed distances, he found they agreed toge- 
ther. He did the same for other points of the orbit, 
and found that they all agreed ; and thus he pro- 
nounced the orbit of Mars to be an ellipse, having the 
sun in one of it's foci. Having determined this for 
the orbit of Mars, he conjectured the same to be true 
for all the other planets, and upon trial he found it 
to be so. Hence, he concluded, That the six primary 
planets revolve about the sun in ellipses, having the 
sun in one of the foci. 

* See his Work, De Motihus StellcB Martis. 

92 Kepler's discoveries. 

The relative mean distances of the planets from the 
sun are as follows: Mercury, 38^10; Venus, 72333 ; 
Earth, 100000; Mars, 15236,9; Jupiter, 320279; 
Saturn, 954072; Georgian, 1918352. 

(162.) Having thus discovered the relative mean 
distances of the planets from the sun, and knowing 
their periodic times, he next endeavoured to find if 
there was any relation between them, having had a 
strong passion for finding analogies in nature. On 
March 8. 16I8, he began to compare the powers of 
these quantities, and at that time he took the squares 
of the periodic times, and compared them with the 
cubes of the mean distances, but, from some error in 
the calculation, they did not agree. But on May 15, 
having made the last calculations ag^ain, he discovered 
his error, and found an exact agreement between 
them. Thus he discovered that famous law. That the 
squares of the periodic times of all the planets are as 
the cubes of their mean distances J roni the sun. Sir 
/. Newton afterwards proved that this is a necessary 
consequence of the motion of a body in an ellipse, re- 
volving about the focus. Prin. Phil. Lib. I. Sec. \. 
Pr. 15. 

(163.) Kepler also discovered, from observation, 
that the velocities of the planets, when in their apsides, 
are inversely as their distances from the sun ; whence 
it followed, that they describe, in these points, equal 
areas about the sun in equal times. And although he 
could not prove, from observation, that the same was 
true in every point of the orbit, yet he had no doubt 
but that it was so. He therefore applied this principle 
to find the equation of the orbit (as will be explained 
in the next Chapter), and finding that his calculations 
agreed v»'ith observations, he concluded that it was 
true in general, That the playlets describe about the 
sun. equal areas in equal times. This discoveiy was, 
perhaps, the foundation of the Priticipia, as it might 
probably suggest to Sir /. Newton the idea, that tlofi 


proposition was true in general, which he afterwards 
proved it to be. These important discoveries are the 
foundation of all Astronomy. 

(164.) Kepler also speaks of Gravity as a power 
which is mutual between all bodies ; and tells us, that 
the earth and moon would move towards each other, 
and meet at a point as much nearer to the earth 
than the moon, as the earth is greater than the moon, 
if their motions did not hinder it. He further adds, 
that the tides arise from the gravity of the waters 
towards the moon. 


Chap. X. 



(165.) As the orbits which are described by the 
primary planets revolving about the sun, are ellipses 
having the sun in one of the foci, and each describes 
about the sun equal areas in equal times, we next pro- 
ceed to deduce, from these principles, such conse- 
quences as will be found necessary in our enquiries 
respecting their motions. From the equal description 
of areas about the sun in equal times, it appears * that 

* For if APSi be an ellipse described by a planet about the sun 
at S in the focus, the indefinitely small area, PSpy described in a 

given time, will be constant; draw P?- perpendicular to Sp ; and 
as the area SPp is constant for the same time, Pr varies as tt: 

Pr . 1 

but the angle pSP varies as -— >, and therefore it varies as-^-^^; that 

is, in the same orbit, the angular velocity of a planet varies in- 
versely as the square of it's distance from the sun. For different 
planets, the areas described in the same time are not equal, and 

^ • area SPp ^1 ^, , r>^ 

therefore Pr varies as — ^ — i., consequently the angle p.5P vanes 



as a; 1 > ^h^t is, the angular velocities of different planets, are 

as the areas described in the same time directly, and the squares of 
their distances from the sua inversely. 



the planets rnave with unequal angular velocities about 
the sun. The proposition therefore, which we here 
propose to solve, is, given the periodic time of a 
planet, the time of it's motion from it's aphelion, and 
the excentricity of it's orbit, to find it's angular dis- 
tance from the aphelion, or it's true anomaly, and it's 
distance from the sun. This was first proposed by 
Kepler, and therefore goes by the name of Kepler's 
Problem. He knew no direct method of solving it, 
and therefore did it by very long and tedious tentative 

(li6.) Let AGQ be the ellipse described by the 
body about the sun at S in one of it's foci, AQ the 

major, GC the semi-minor axis, ^ the aphelion, Q 
the perihelion, P the place of the body, AP^QE a 
circle, C it's center ; draw NPI perpendicular to AQ^ 
join PS, NS, and NC, on which produced Jet fall the 
perpendicular ST. Let a body move uniformly in the 
circle from A to D with the me«w angniar velocity 
of the body in the ellipse, whilst the body moves in the 
ellipse from ^ to P; then the angle ACD is the 
mean, and the angle ASP tlie true anomaly .: and the 
difference of these two angles is called the equation of 


the planefs center, or prosthapheresis. Let p s= the 
periodic time in the ellipse or circle (the periodic 
times being equal by supposition), and ^=:the time of 
describing AP or AD ; then, as the bodies in the 
ellipse and circle describe equal areas in equal times 
about *S' and C respectively, we have 

area ADC : area of the circle :: t : p, 

area of the ellipse : area ASP :: p : t ; also, 

area of circ. : area of ellip. :: area A S ]V:sir en ASP;* 

therefore, area ADC : area ASP :: area ASN: area 
ASP ; hence, ADC = ASN ; take away the area 
ACN, which is common to both, and the area DCN 
= SNC, but DCN=l DNx CN, and SNC=i ST 
X CN ; therefore ST= DN. Now if t be given, the 
arc AD will be given ; for as the body in the circle 
moves uniformly, we have p \ t w 36o° : AD. Thus 
we may find the mean anomaly at any given time, 
knowing the time when the body was in the aphelion ; 
hence, if we can find ST, or ND, we shall know the 
angle NCA, called the excentric anomaly, from 
whence, by one proportion (167) we shall be able to 
find the angle ASP the true anomaly. The problem 
is therefore reduced to this ; to find a triangle CST, 
such, that the angle C+ the degrees of an arc equal to 
ST^ may be equal to the given angle ACD. This 
may be expeditiously done by trial in the following 
manner, given by M. de la Caille in his Astronomy. 
Find what arc of the circumference of the circle 
ADQE is equal to C^, by saying, 355 : 113 :: 180°. 
: 57°. 17'-44",8, the number of degrees of an arc equal 
in length to the radius CA ; hence, CA : CS :: 57°. 
17'- 44",8 : the degrees of an arc equal to CS. Assume 
therefore the angle SCT, multiply it's sine into the 

* See my Treatise on the Conic Sections, second edit. prop. 7. of 
the Ellipse, cor. 3 and 4. And this is the Treatise referred to in 
the future part of this Work. 


degrees in CS, and add it to the angle SCT, and if it 
be equal to the given angle ACD, the supposition was 
right ; if not, add or subtract the difference to or from 
the first supposition, according as the result is less or 
greater than ACD, and repeat the operation, and in a 
very few trials you will get the accurate value of the 
angle SCT. The degrees in ST" may be most readily 
obtained, by adding the logarithm of CS to the loga- 
rithm of the sine of the angle SCT, and subtracting 
10 from the index, and the remainder will be the lo- 
garithm of the degrees in ST. Having found the 
value of AN, or the angle ACN, we proceed next to 
find the angle ASP. 

(167.) Let V be the other focus, and put AC=l ; 
then by Eucl. B. II. P. 12. SP ' - Pv' = vS'-{-2vSx 
vl=.vS+2vIxvS=2Cv-{-2vIx2SC = 2CTx2SC; 
hence, SP-^ Pv:2C I:: 2SC : SP - Pv, or 2 : 2CI :: 
2SC: SP-2-SP,OYl : CI .: SC : SP-l;hence, 
SP = 1 + C.V X CI= 1 + CSx COS. z ACN. But 

. I —COS. ASP — =-„ 

(Trig. Art. 94), ^ _^^^^^^^p = tan. ^ ASP-, also, 

(Trig. Art. 1 25) SP, or 1 + C^x cos..^ CiV ; rad. = 1 
:: SI, or CS + CI, or CS + cos. ANC, : cos. ASP = 

CS + cos.ACN 

, ■ ' r'L' i^,\- . Hence, tan. i ASP^ ; ( = 

1 + Cc) X cos.^CiV ' ^ ' ^ 

1- COS. ASP \ _\ +CSx COS. ACN- CS- cos. ACN 

I 4- COS. A SP^ ~ 1 + CS X cos.^6'iV-|- C6'+ cos.^C'iV 

_ 1 - CS+cos.ACNx C8^ _ 8a-cos.^CiYx8Q 
~ 1 + CS-^cos.ACN-\-CS+l~ SA + COS. ACNxSA 

1 - COS.ACN SQ ^rj. . A • , X 

= l+cos..^CW ^ SA= (Trigonometry, Article 95) 

tan. I ACN'^ X tti ; therefore sJ SA : yj SO, :: tan. 

lACNi tan. | ASP, consequently we get ASP the 

true anomaly. 


Ex. Required the true place of Mercury on Aug. 
526, 174O5 at noon, the equation of the center, and it's 
distance from the sun. 

By M. de la Caille's Astronomy, Mercury was in 
it's aphehon on Aug. 9, at 6h. Sf. Hence, on Aug. 
26, it had passed it's apheHon l6d. l^h. 23' ; there- 
fore 82 d. 23h. 15'. 23" (the time of one revolution) : 
l6d. I7h. 23' :: 36o° : 68^ 26'. 28". the arc JD, or 
mean anomaly. Now (according to this author) CA : 
CS :: 1011276 : 211l65 (166) :: 57°. 7l'.44",8 : 1 1°. 
57'. 50"= 43070", the value of CS reduced totlie arc 
of a circle, the log. of which is 4,6341749. Also, 68**, 
26'. 28" = 246388". Assume the angle ^C7' to be 
60°. =216000", and the operation (166) to find the- 
angle ACN, will stand thus. 



9,9375306 log. of. 2i6'ooo=a 

4,5717055 37300 



6912 = 6 


9,9287987 2O9088=a- 6 = 58^4'. 48" — C 

4,5629736 36557 




9,9297694 ...... 209831 =c + f/ = 58°.l7'.l I " = r 

4,5639443 36639 




9,9296626 209749=^e-/ = 58^1o'.49'' = o- 

4,5638375 36630 


9 = //; hence, as the dif- 
ference between the value deduced from the assump- 
tion and the true value, is now diminished about nine 
times every operation, the next difference would be l" ; 

G 2 


if therefore we add h to g, and then subtract 1", we 
get 58°. 15'. 57", for the true value of the angle ACN, 
the excentric anomaly. Hence, (167) find the true 
anomaly ASP, from the proportion there given, by 
logarithms, thus. 

Log. tang. 29°. 7'. 58"! . . 9,7461246 
I Log. *S'Q=800lll . . . 2,9515751 

J Log. 5^=1222441 . . . 3,0436141 

Log. tang. 24M6M5" . . . 9,6540856 

Hence, the true anomaly is 48°. 32'. 30". Now the 
aphelion A was in 8\ 13°. 54'. 30" ; therefore the true 
place of Mercury was \0\ 2°. 2/'. Hence, (166), 68°. 
26'. 28" -48°. 32'. 30"= 19°. 53'. 58", the equation of 
the center. Also, 8P=1 + CS y. cos. z ACN= 
1310983 the distance of Mercury from the sun, the 
radius of the circle, or the mean distance of the 
planet, being unity. Thus we are able to compute,at 
any time, the place of a planet in it's orbit, and it's 
distance from the sun ; and this method of computing 
the excentric anomaly appears to be the most simple 
and easy of application of all others, and capable of 
any degree of accuracy. 

(168.) As the bodies D and P departed from A at 
the same time, and will coincide again at Q, ADQ^ 
^PQ being described in half the time of a revolution ; 
and as at A the planet moves with it's least angular 
velocity (by the Note to Art. l65), therefore from A to 
Q, or '\x\ the first ^va signs of anomaly, the angle ^CZ> 
will be greater than ASP, or the mea?i will be greater 
than the true anomaly ; but from Q to A, or in the 
last six signs, as the planet at Q moves with it's 
greatest angular velocity, the true will be greater than 
the 7nean anomaly. When the equation is greatest in 
going from A to Q, the mean place is before the true 
place, by the equation, and from Q to A, the true 



place is before the mean place, by the equation; 
therefore, from the time the equation is greatest till it 
becomes greatest again, the difference between the 
true and mean motions, is twice the equation. From 
apogee to perigee, the true and mean motions are the 

(169.) The method ascribed by some writers to 
Seth IVard, Professor of Astronomy at Oxford, and 
published in l654, although, as M. de la Lande ob- 
serves, it is given both by JVard and Mercator to 
BuUialdiis, is less accurate than these we have already 
given; yet as it may, in many cases, serve as a useful 
approximation, it deserves to be mentioned. He 

assumed the angular velocity about the other focus v to 
be uniform*, and therefore made it represent the mean 

* That this is not true may be thus shown. With the center 5* and 
radius SlVzzV'^^x C£ describe the circle zH'^, then the area of this 


circle=area of the ellipse [Conic Sections, Ellipse, Prop. 7. Cor. 5): 



anomaly. Produce v P io r, and take Pr=PS; 
then in the triangle Svr, 7'v-\-vS -. rv-vS :: tan. 
i . A vS r-\-vrS: tan. J. z vSr — vrS (Trig. Ar t. 135); 
now ^ . rv+vS=^ AQ + l vS-JS, and j. ri ; - t?^ 
= I ^Q - I Z78= SQ ; also, tan. | . z t?«Sr + vrS - tan. 
J. z AvPi and tan. |. z vSr-vrS=: (as Pr = PS) tan. 
|. zy-Sr — P8r = tan. I . a ASP-, hence, ^Ae aphelion 
distance : perihelion distance :: fi/7?. o/' | the mean 
anomaly : ^«». ^ ^rwe anomaly. This is called the 
simple elliptic hypotlvesis. In the orbit of the earth, 
the error is never greater than I7"; in the orbit of the 
moon, it may be l'. 35". By this hypothesis, for 90° 
from aphelion and perihelion, the computed place is 
hackivarder than the true ; and for the other part it is 
foi uoarder. 

(170.) The greatest equation of the center may be 
easily found from the Note to Article 169, giving the 
dimensions of the orbit. For as long as the angular 

Jet a body, moving uniformly in it, make one revolution in the 
same time liie body does in the ellipse ; and let the bodies set otF 
at the same time from A and z, and describe AP, ?p, in tlie same 
time ; then the angle zSv is the mean, and ASP the true anomaly. 
Draw pS indefinitely near to PS, and Pr, po, perpendicular to 

Sp, FP ; then Pr—po. Now the angle PFp A^nes as -rrp^ 

— — r; ^ut, in a given time, the area PSp is given, therefore Pr 

varies as -r-rr ; hence, the angle PF/), described in a given time, 
\, which is not a constant quantity. Also, Z PFp 

: Z PSp :: PS : PF :: ^^^p^ • 7^- And by the Note to 

Art. 165, as equal areas are described in equal times in the circle 
and ellipse about S, the angular velocity about S in the circle, be- 
comes ;:• Hence, the angular velocity about S is greater or 

less than the mean angular velocity, according as PFx PS is less 
or greater than Sl'V-, or than ACk CE. Also, the angular velocity 
about .S is the same in similar points of the ellipse ia respect to the 
center, or at equal distances from the center. 



velocity of the body in the circle is greater than that in 
the ellipse about S, the equation will increase, the 


bodies setting out from A and z j and when they be- 
come equal, the equation must be the greatest ; this 

therefore happens when-^, ='~sW' " AC x CE ' '''' 
when ^Cx C£ = SP^ hence, SP is known. Let 
SfV represent this value of SP ; then as we know 


Sfr, FIV{-2AC-SIV) will be known, and as 
SF is known, we can find the angle FSIV the 
true anomaly. Hence (167) yj SQ, : sj SA :' 
tan. ^ true anom. : tan ^ excen. anom. ACN, or tan. 
J SCT ; hence, as we know SC^ we can find SJ\ or 


ND ; and to convert that into degrees, say, Rad. = 1 
: ND :: 57°. 17'. 44",8 : the degrees in ND\ which 
added to or subtracted from the angle ACN, gives 
ACD the mean anomaly, the difference between which 
and the true anomaly is the «Tea^e*f equation- Thus 
we may find the equation at any other time, having 
given SP. 

(171.) The excentricity, and consequently the di- 
mensions of the orbit, may be known from knowing 
the greatest equation. For (170) the equation is 
greatest when the distance is a mean between the semi- 
axes major and minor, and therefore in orbits nearly 
circular, the body must be nearly at the extremity of 
the minor axis, and consequently the angle NCA, or 
SCT (Fig. p. 103) will be nearly a right angle, there- 
fore ST\s nearlv equal to SC ; also N'SJ will be very 
nearly equal to' PSA. Now the angle NCA-NSA 
(PS'A)=SNC, and DCA- NCA=nCN, add these 
together, and DCA - PSA = DCN+ SNC,wh\ch (as 
iVC is nearly parallel to DS) is nearly equal to 2 DCN\ 
that is, the difference between the true and mean 
anomaly, or the equation of the center, is nearly equal 
to twice the arc DN, or twice ST, or very nearly 
twice SC. Hence, 57°. if. 44", 8 : half the greatest 
equation :: rad. = l : SC the excentricity. But if the 
orbit be considerably excentric, compute the greatest 
equation to this excentricity ; and then, as the equa- 
tion varies very nearly as SC, say, as the computed 
equation : excentricity found :: given greatest equa- 
tion : true excentricity. 

Ex. If we suppose, with la Caille, that Mer- 
cury's greatest equation is 24*^. 3'. 5"; then 57°. 17'. 
44",8 : 12°. r. 32",5 :: 1 : ,209888 the excentricity 
very nearly. Now the greatest equation, computed 
from this excentricity, is 23°. 54'. 28",5 ; hence, 23°. 
54'. 28",5 : 24°. 3'. 5" :: ,209888 : ,211165 the true 
excentricity. M. de la Lande makes the greatest 
equation 23°. 40', and the excentricity ,207745. 

(172.) The converse of this problem^that is, given 


the excentricity and true anomaly, to find the mean, 
may be very readily and directly solved. The excen- 
tricity being given, the ratio of the major to the minor 
axis is known*, vvhich is the ratio of AY to PI\ hence, 
the angle ASP being given, we have PI : NI :.- tan. 
ASP : tan. ASN; therefore, in the triangle NCS, 
we know NC, CS, and the angle CSN, to find the 
angle SCN' (Trig.Art.128), the supplement of which 
is the angle ACN, or SCT; hence, in the right- 
angled triangle STC, we know SC and the angle SCT, 
to find 87' (Trig Art 128), which is equal to ND, 
the arc measuring the equation, which may be found 
by saying, radius : 87' :: 57°- I?'- 44",8 : the degrees 
in ND, which added to ACN, gives ACD the mean 

To Jind the hourly Motion of a Planet in ifs Orbit, 
having give?! the mean hourly Motion. 

(173.) The hourly motion of a planet in it's orbit is 
found immediately from the Note to Art. 169; for it 
appears from thence, that the angles PSp, ff'Sw, de- 
scribed by the body at P in the ellipse, and the body 
W'\n the circle in the same time, are as SIV^'^ : SP^^ 
or as (see Fig. p. 103) ACx CE : SP- ; hence, SPp=: 

/4C V CP 

Pf^Sw X — rrjji — the hourly motion of a planet in 

it's orbitf the angle flTSw being the mean motion of 
the planet in an hour. For greater accuracy, SP must 
be taken at the middle of the hour. Thus we may 
easily compute a table of the hourly motions of the 
planets in their orbits. 

* For as ^C, CS ar e known, we have GC=\/(jfS~— ^C'=: 
^AC'—SC''=s/AC-\-SCx AL—SC, for thecoraputation of vvhich 
by logarithms, see Trig. Art. 52. 


Chap. XI. 


(174.) The place and time of the opposition of a su- 
perior, or conjunction of an inferior planet, are the 
most important observations for determining tlie ele- 
ments of the orbit, because at that time the observed 
is the same as the true longitude, or that seen from the 
sun ; whereas, if observations be made at any other 
time, we must reduce the observed to the true longi- 
tude, which requires the knowledge of their relative 
distances, and which, at that time, are supposed not 
to be known. They also furnish the best means of 
examining and correcting the tables of the planets' 
motions, by comparing the computed with the ob- 
served places. 

(175.) To determine the time of opposition, observe, 
when the planet comes very near to that situation, the 
time at which it passed the meridian, and also it's right 
ascension (ill or 113) ; take also it's meridian alti- 
tude ; do the same for the sun, and repeat the obser- 
vations for several days. From the observed meridian 
altitudes find the declinations (114), and from the 
right ascensions and dechnations compute (115) the 
latitudes and longitudes of the planet, and the longi- 
tudes of the sun. Then take a day when the differ- 
ence of their longitudes is nearly 180% and on that 
day reduce the sun's longitude, found from observation 
when it passed the meridian, to the longitude found at 
the time (t) the planet passed, by finding from obser- 
vation, or computation, at what rate the longitude 
then increases. Now in opposition the planet is 
retrograde, and therefore the difference between the 


longitudes of the planet and sun increases by the sum 
of their motions. Hence, the following rule : As the 
sum (S) of their daily motions in longitude : the 
difference (D) between 180° and the difference of 
their longitudes reduced to the same time [t)* (sub- 
tracting the sun's longitude from that of the planet to 
get the difference reckoned from the sun according to 
the order of the signs) :: 2Ah. : interval between that 
time {t) and the time of opposition. This interval 
added to or subtracted from the time it), according as 
the difference of their longitudes at that time was 
greater or less than 180°^ gives the time of opposition. 
\i this be repeated for several days, and the mean of 
the whole taken, the timew^ill be had more accurately. 
And if the time of opposition found from observation, 
be compared with the time by computation from the 
Tables, the difference will be the error of the Tables, 
which may serve as means of correcting them. 

Ex. On October 24, 17b'3, M. de la Lawf/e observed 
the difference between the right ascensions of ^ Aries, 
and Saturn, which passed the meridian at 12/t. 1/'. 17" 
apparent time, to be 8^. 5'. 7", the star passing first. 
Now the apparent right ascension of the star at that 
time was 25°. 24'. 33",6; hence, the apparent right 
ascension of Saturn was 1'. 3*^. 29'. 40",6' at 12 /z. 17'. 
17" apparent time, or \2h. T. 37" mean time. On 
the same day he found, from observation of the me- 
ridian altitude of Saturn, that it's declination was 10°. 
35'. 20" N. Hence, from the right ascension and de- 
clination of Saturn, it's longitude is found to be V. 4°. 
50'. 56", and latitude 2^ 43'. 25" S. At the same 
time the sun's longitude was found by calculation to be 
T. l^ 19'. 22", which subtracted from P. 4°. 50'. 56", 
gives 6'. 3 \ 31'. 34"; hence, Saturn was 3°. 31'. 34'' 

* For this diflirencc shows how far the planet is from opposi- 
tion ; and the proposition is founded on this principle, that the sun 
approaches the star by spaces in proportion to the times; the spaces 
S and D must therefore be as the time 24//, and the time {t) to 


beyond opposition, but being retrograde will after- 
wards come into opposition. Now, from the observa- 
tions made on several days at that time, Saturn's longi- 
tude was found to decrease 4'. 50" in 24 hours, and 
by computation, the sun's longitude increased 59'. 59" 
in the same time, the sum of which is 64'. 49"; hence, 
64\ 49" : 3". 31'. 34" :: 24h. : 78/?. 20'. 20", which 
added to October 24, I2h. l'. 37", gives 27c?. I8h. 21'. 
57" for the time of opposition. Hence, we may find 
the longitude of Saturn at the time of opposition, by 
saying, 24A. : 78^. 20'. 20" :: 4'. 50" : 15'. 47" the re- 
trograde motion of Saturn in 78A. 20'. 20", which sub- 
tracted from 1*. 4°. 50'. 56", leaves 1*. 4°. 35'. 9" the 
longitude of Saturn at the time of opposition. In like 
manner we may find the sun's longitude at the same 
time, in order to prove the opposition; for 24/z. : 78^. 
20'. 20" :: 59'. 59" : 3°. 15'. 47", which added to 7». 
1°. 19'. 22", the sun's longitude at the time of observa- 
tion, gives 7*. 4°. 35^ 9" for the sun's longitude at the 
time of opposition, which is exactly opposite to that 
of Saturn. Hence, we may also find the latitude of 
Saturn at the same time, by observing, in like manner, 
the daily variation, or by computation from the Tables 
after the elements of it's motions are known, and the 
Tables constructed ; by which it appears, that in the 
interval between the times of observation and opposi- 
tion, the latitude had increased 6", and consequently 
the latitude was 2**. 43'. 31". Thus we find the times 
of opposition of all the superior planets. 

(176.) The place and time of conjunction of an in- 
ferior planet may be found in like manner, when the 
elongation of the planet from the sun, near the time 
of conjunction, is sufficient to render it visible ; the 
most favourable time therefore must necessarily be 
when the geocentric latitude of the planet at the time 
of conjunction is the greatest. In the year 1689, 
Venus was in it's inferior conjunction on June 25, and 
it was observed on 21, 22, and 28 ; from which observa- 
tions it's conjunction was found to be at 1 3 A. 46' appa- 


rent time at Paris, in longitude as 4°. 53'. 40", and 
latitude 3°. l'. 40"N. The state of the air must be 
very favourable, that the time and place of the superior 
conjunction may be thus observed; for as Venus is 
then about six times as far from the earth as at it's in- 
ferior conjunction^ it's apparent diameter and the 
quantity of light which we receive from it, are so small, 
as to render it difficult to be perceived. But the most 
accurate method of observing the time of an inferior 
conjunction both of Venus and Mercury, is from ob- 
servations made upon them in their transits over the 
sun's disc. 


Chap. XII. 


(177.) The determination of the mean motions of the 
planets, from their conjunctions and oppositions, 
would very readily follow, if we knew the place of the 
aphelia and excentricities of their orbits ; for then we 
could (166) find the equation of the orbit, and reduce 
the true to the mean place ; and the mean places being 
determined at two points of time, give the mean motion 
corresponding to the interval between the times. But 
the place of the aphelion is best found from the mean 
motion. To determine therefore the mean motion, 
independent of the place of the aphelion, we must 
seek for such oppositions or conjunctions, as fall very 
nearly in the same point of the heavens; for then the 
planet being very nearly in the same point of it's orbit, 
the equation will be very nearly the same at each ob- 
servation, and therefore the comparison between the 
true places will be nearly a comparison of their mean 
places. If the equation should differ much in the two 
observations, it must be considered. Now, by compar- 
ing the modern observations, we shall be able to get 
nearly the time of a revolution ; and then, by com- 
paring the modern with the ancient observations, the 
mean motion may be very accurately determined ; for 
any error, by dividing it amongst a great number of 
revolutions, will become very small in respect to one 
revolution. As this will be best explained by an ex- 
ample, we shall give one from M. Cassini {Elem. 
(iAstron. p. 362), with the proper explanations as we 


Ex. On September l6, 1701, Saturn w&s in oppo- 
sition at 2h. when the place of the sun was nj^ 23°. 21'. 
16", and consequently Saturn in K 23°. 21'. 16", with 
2°. 27'. 45" south latitude. On September 10, 1730, 
the opposition was at 12^. 27', and Saturn in k 17°« 
53'. 57", with 2°. 19'. 6" south latitude. On Sep- 
tember 23, 1731, the opposition was at I5h. 51' in 
T 0°.30'. 50", with 2°. 36'. 55" south latitude. Now 
the interval of the two first observations was 29 years 
(of which seven were bissextiles) wanting 5<^. l3/i.33'; 
and the interval of the two last was I3/. 136?. 3h. 24'. 
Also, the difference of the places of Saturn in the two 
first observations was 5°. 27'. I9", and in the two last it 
was 12°. 36'. 53". Hence, in ly. 13d. 3h. 24', Saturn 
had moved over 12°. 06'. 53"; therefore 12°. 36'. 53" : 
5°. 27'. 19" :: ly. 136?. 3h, 24' : l63d. l2/i. 41', the 
time of moving over 5°. 27'- 19" very nearly, because 
Saturn, being nearly in the same part of it's orbit, will 
move nearly with the same velocity ; this therefore, 
added to the interval between the two first observations 
(because at the second observation Saturn wanted 5°. 
27'. 19" of being up to the place at the first observa- 
tion), gives 29 common years l64c?. 23/i. 8', for the 
time of one revolution. Hence say 29?/. l64c?. 23h. 
8' : 365^. :: 36o°. : 12°. 13'. 23". 50"' the mea?z an- 
nual motion of Saturn in a common year of 365 days, 
that is, the motion in a year if it had moved uni- 
formly. If we divide this by 365, we shall get 2'. O". 
28'" for the mean daily motion of Saturn. The mean 
motion thus determined will be sufficiently accurate to 
determine the number of revolutions which the planet 
must have made, when we compare the modern with 
the ancient observations, in order to determine the 
mean motion more accurately. 

The most ancient observation which we have of the 
opposition of Saturn was on March 2, in the year 228, 
before J. C. at one o'clock in the afternoon, in thie 
meridian of Paris, Saturn bejng then in m^ S°. 23', 


with 2°. 50' north lat. On February 26, 1714, at 8h. 
15', Saturn was found in opposition in ijp 7'^. 56'. 46", 
with 2°. 3' north lat. From this time we must sub- 
tract 1 1 days, in order to reduce it to the same style 
as at the first observation, and consequently this oppo- 
sition happened on February 15, at 8^. 15'. Hence, 
the difference between these two places was only 26'. 
14". Also, the opposition in 1/15 was on March 11, 
at l6^. 55', Saturn being then in nj 21°. 3'. 14", with 
2°. 25' north lat. Now between the two first opposi- 
tions there were 1 942 years (of which 485 were bissex- 
tiles) wanting 14^. l6/i. 45', that is, 1943 common 
years, and I05d. 7/1. 15' over. Also, the interval be- 
tween the times of the two last oppositions was S^Sd 
8h. 40', during which time Saturn had moved over 13°. 
6'. 28"; hence,i3°. 6'.28" : 26'. 14" :: 378d. 8h. 40': 
13d. \Ah. which added to the time of the opposition 
in 1714, gives the time when the planet had the same 
longitude as at the opposition in 228 before J. C. 
This quantity added to 1943 common years 105^. 7/?. 
15', gives 1943?/. 118c?. 2\h. 15', in which interval of 
time Saturn must have made a certain complete num- 
ber of revolutions. Now having found, from the 
modern observations, that the time of one revolution 
must be nearly 29 common years l6Ad. 23h. 8', it fol- 
lows that the number of revolutions in the above 
interval was QQ\ dividing therefore that interval by 
66, we get 292/. 1626?. Ah. 27' for the time of one re- 
volution. From comparing the oppositions in the 
years 1714 and 1715, the true movement of Saturn 
appears to be very nearly equal to the mean move- 
ment, which shows that the oppositions have been ob- 
served very near the mean distance ; consequently the 
motion of the aphelion cannot have caused any con- 
siderable error in the determination of the mean 
motion. Hence the mean annual motion is 12°. 13'. 
45". 14'", and the mean daily motion 2'. O". 35'". Dr. 
Halley makes the annual motion to be 12°. 13'. 21". 
M. de Place makes it 12°. 13'. 36",8. As the revolu- 


tion here determined is in that respect to the longitude 
of the planet, it must be a tropical revohition. Hence, 
to get tlie sidereal revolution, we must say, 2'. 0". 35'" 
: 24'. 42". 20'" (the precession in the time of a tropical 
revolution, Art. 130) :: 1 day : 12fi?. "jh. l'.57", which 
added to l^y. l62d. All. 2f, gives 29;/. 174^. \\h. 
28'. 57" the length of a sidereal year of Saturn. Thus 
we find the periodic times of all the superior planets. 
The periodic times of the inferior are found from their 

The periodic times of the planets are as follows ; 
Mercury, 87^/. 23/i. 15'. 43",6 ; Venus, 224^. l67i. 
49'. 10",6; Mars, \y. 32ld. 23h. 30'. 35",6 ; Jupiter, 
lly.3l5d. I4h. 27'. 10",8 ; Saturn, 29,y. 174^. 1/l 
31'. ir',2 ; the Georgian, 83/y. 150^. 18//. 


Chap. XIII. 


(178.) Having determined the mean motions of the 
planets, we proceed next to show the method of find- 
ing the greatest equation of their orbits, the excen- 
tricity, and place of their aphelia. For although, in 
order to determine the mean motions very accurately, 
these tilings were supposed to be known, yet, without 
them, the mean motions may be so nearly ascertained, 
that these elements may from thence be very accu- 
rately settled. By Art. 161, we may find the distance 
of a planet from the sun in any point of it's orbit. The 
problem therefore is, having given in length and posi- 
tion, three lines drawn from the focus of an ellipse, to 
determine the ellipse. 

(179.) Let SB, SC, SD, be the three lines ; pro- 
duce CB, CD, and take SB : SC :: EB : EC, and 


SC : SD :: CF : DF, then SC- SB , SC :. BC : 
EC=^^y^, and SC'-SD : SC v. DC: CF= 


—p^ rvT^. Join FE, and draw DK, CI, BH, per- 

pendicularto it. Now, by similar triangles, IC : IIB 
:: EC: EB :: (by const.) SC : SB; also, IC : KD 
:: CF: DF:: SC : SD. Hence, the proportion of 
/C, HB, KD, is the same as SC, SB, SD, conse- 
quently EF is the directrix of the ellipse passing 
through B, C, D, (Con. Sect. p. 31). Through .S' 
draw y^.^QG perpendicular to FE ; take GA : AS :: 
CI: CS, and GQ ; SQ :: CI: CS ; then Cl+CS : 

CS :: GS : -SQ = ~jr^ — ^rij '■> in like manner we find 

Kyi -j- Oo 

AS=j=;y — y^^ and y/, Q, will be the vertices of the 

conic section. 

(180.) Calculation. In the triangles *S5C, 5(71), 
we know two sides and the included angles, they be- 
ing the distances of the observed })laces in the orbit; 
hence, (Trig. Art. 135) we can find BC, CD, and the 
angles BCS, SCD, and consequently BCD. Hence 
(179) we know CE and CF, and the angle ECF be- 
ing also known, the angle CEF can be found. There- 
fore in the right-angled triangle CIE, CE and the 
angle E are given; hence, (Trig. Art. 128) C7 is 
known. Join SI; then in the triangle SIC we know 
CI, CSand the angle SCI [=BCS - ECI), hence, 
we know SI, and the angles CIS, CSI, and hence 
the angle SIG is known ; therefore, in the right-angled 
triangle SIG, we know SI and the angle SIG, from 
whence SG is found. Hence, (179) we know SA, 
SQ, half the difference of which is the excentricitv, 
and their sum = ^Q. Lastly, in the triangle BSO 
(O being the other focus) we know all the sides, to 
find the angle BSA (Trig. Art. 133), the distance of 
the aphelion from the observed place B. 

In the year 1/40, on July 17, August 26, Septem- 
ber 6, M. de la Caille found three distances of Mer- 
cury (the mean distance being lOOOO) as follows .- SB 


= 10351,5, lC==ll325,5,SD = 9672,l66,the angle 
BSC = 3\27°.0'.35", and CSD = 4r. 40'. 4". Hence, 
BSC=29\ 55'. 5", BC= 18941, 5CZ)=56°. 49', Ci> 

= 8124,5, i?CF=86^ 44'. 5", C£ = 215004, C^ = 
55647, C£F=14°.41'.44", C/=54543, C^/--=124\ 
47'. 45", C7^=9*'- 49'. 4", 6T=47281, -S/G=:8b<'. 
10'. 56", ^'G= 465 89, ^P = 8010,5, 5^=12209, SO 
= 4198,5 ; hence, the excentricity = 2099,75, BSA 
= 7V. 37'. 23, or 2^ 1 r. 37'. 23", which ifdded to 6^ 
2^ 13'.'51", the position of SB, gives 8\ 13^ 51', 14" 
for the place of the aphelion. Hence, the greatest 
equation is 24". 3'. 5". 

(181.) Or from the same data, the place of the 
aphelion and excentricity may be thus found. Put 
the semi-axis major = 1, SB = a, SD = b, SC=c, the 
angle BSD = v. BSC=u, BSQ=^x, OS=e, half the 
parameter = r. Then, by a well-known property of the 

ellipse (Conic Sect. EUipse, Prop. 16), (^ = YTec^x' 

b^ ^^ , c= ===; hence, r=« 

1 -{-e.cos. v+.r I -\-e. COS. u + jc 

-]-ae.cos.x = h + be.cos.v7^=:c-\- ce.cos.u-h x ; there- 

b-a c-a 

fore, — T= = e = — , 

a.cos.x — b.cos.v + x a.cos.x- c .cos.u + x 

now for cos-iJ + o^, and cos. u + x, substitute cos. ?;.cos. 
X-- sin. t;sin..r, and cos. u. cos. cT — sin. m. sin. x (Trig. 
Art. 102,) and we shall have 
b — a 
a . cos. x — b . COS. v . cos. x -\- b . sin. v sm. x 
c — a 

a . cos. x — c . COS. u . cos, x-\-c. sin. z<. sin, x ' 
each denominator by cos. x, and we have 
it — a c — a 


a — b.cos.v-\-b.sin:i\tain.x a— c. cos. u-\ ' 

, b.c — a. COS. V — c. b- a. cos. u - a.c — b 

hence, tan. x = == — ; , 

b . c — t7.sin.t; — c .b — a . sin. u 

which gives the place of the perihelion. Hence^ vvc 


c—a , . . 

know e = — ===theexcentncity; con- 

a . COS. X — c. COS. II + X 

sequently I - e and 1 +e, the perihelion and aphehon 
distances, are known. The species of the elhpsc being 
determined, it's major axis may be thus found : Com- 
pute the mean anomaly corresponding to the angle 
CSBy then say, as that mean anomaly : 36o° :: the 
time of describing the angle CSB : the periodic time. 
The periodic time being known, the major axis is 
found (162) by Kepler* s Rule. For other practical 
methods, see my Complete System of Astronomy. 

(182.) All the epochs in our Astronomical Tables 
are reckoned from noon on December 31, in the com- 
mon years, and from January 1, in the bissextiles. 

The places of the aphelia for che beginning of 1/50, 
are. Mercury, S\ 13°. 33'. 58"; Venus, 10^ 7°. 46'. 
42"; the Earth, 3^ 8^ 37'. 16"; Mars^ 5^ r. 28'. 14"; 
Jupiter, 6M0°. 21'. 4"; Saturn, 8\ 28°. 9'. 7", the 
Georgian, 11\ l6°. I9'. 30". 

Tlie excentricities of the orbits, the mean distance 
of the earth from the sun being 1 00000, are, M-^rcury, 
7955,4; Venus, 498; the Earth, l681,395; Mars, 
14183,7 ; Jupiter, 25013,3 ; Saturn, 53640,42 ; the 
Georgian, 908O4. 

The greatest equations are. Mercury, 23°. 40'. o" ; 
Venus, 0°. 47'. 20"; the Earth, V. 55'.36",5 ; Mars, 
10°. 40'. 40" ; Jupiter, 5\ 30'. 38 ',3 ; Saturn, 6°. 26'. 
42"; the Georgian, 5°. 27'. 16". 

The aphelia of the orbits of the planets have a mo- 
tion, which may be found, from finding tlie places of 
the aphelia of each at two different times. Those 
motions in longitude in 100 years are. Mercury, 1°. 
33'. 45"; Venus, 1°. 21'. O"; the Earth, 1°. 43'."35"; 
Mars, 1°. 51'. 40"; Jupiter, 1°. 34'. 33"; Saturn, 1°. 
50'. 7". 

According to the calculation of M. de la Grange, 
the aphelion of the Georgian Planet is progressive 
t5",17 in the year, from the action of Jupiter and 
Saturn; consequendy it's motion in longitude is 


Chap. XIV. 


(183.) From observing the course of the planets for 
one revolution, their orbits are found to be incHned to 
the ecliptic, for they appear only twice in a revolution 
to be in the ecliptic ; and as it is frequently requisite 
to reduce their places in the ecliptic, ascertained from 
observation, to the corresponding places in their orbits, 
it is necessary to know the inclinations of their orbits 
to the ecliptic, and the points of the ecliptic where 
their orbits intersect it, called the Nodes. But, pre- 
vious to this, we must show the method of reducing 
the places of the planets seen from the earth to the 
places seen from the sun, and how to compute the 
heliocentric latitudes. 

(184.) Let E be the place of the earth, P the 
planet, S the sun, v the first point of aries ; draw Pv 

perpendicular to the ecliptic, and produce Es to a. 
Compute, at the time of observation, the longitude of 
the sun seen at a (115), and you have the longitude 
of the earth at E, or the angle r 8E ; compute also . 
the longitude of the planet, or the angle y* Sv (115), 


and the difFerence of these two angles is the angle ESv 
of convmitation. Observe the place of the planet in 
the ecliptic, and the place of the sun being known, 
we have the angle vES of elongation in respect to 
longitude; hence, we know the angle SvE, which 
measures the difference of the places of the planets 
seen from the earth and the sun ; therefore, the place 
of the planet seen from the earth being known, the 
place seen from the sun will be known. Also, 

tan. PEv : rad. :: vP : Ev (Trig. Art. 123) 

rad. : tan. PSv :: vS : vP 

.-. tan. PEv : tan. PSv :: vS: Ev :: sin. SEv : sin. 
ESu; that is, the sine of elongation in longitude : sin. 
of the difference of the longitudes of the earth and 
planet :: tan. of the geocentric latitude ; tan. of the 
heliocentric latitude. When the latitude is small, Sv 
: Ev is very nearly as PS : PE, which, in opposition, 
is very nearly as PS : PS— SE. Or we may com- 
pute (167) the values of PS and SE, which we can 
do with more accuracy than we can compute the 
angfles SEv and ESv. The curtate distance Sv of 
the planet from the sun may be found, by saying, 
rad. : cos. PSv :: PS : Sv. 

(185.) Now to determine the place of the node, 
find the planet's heliocentric latitudes just before and 
after it has passed the node, and let a and b be the 
places in the orbit, 7n and n the places reduced to the 
ecliptic; then the triangles arnN, bnN (which we 

may 'consider as rectilinear) being similar, we have 
am : hn :: Nm : N^n; therefore, am-\-bn : am w Nm 
+ Nn (mn) : Nm, or «m-F bn : mn :: am : Nm, that 
is, the sum of the two latitudes : the difference of the 
longitudes :: either latitude: the distance of the node 
from the longitude corresponding to that latitude. Or 



if we take the two latitudes from the earth, it will be 
very nearly as accurate when the observations are 
made in opposition. If the distance of the observa- 
tions should exceed a degree, this rule will not be suffi- 
ciently accurate, in which case we must 'make our 
computations for spherical triangles thus. Put m n = 
a, am = (i, bn=zh, Nm = x\ then (Trig. Art. 212) 

sin. a- X , ,., sin. cr ,. , . 

^- ; — = cot. A = -, radms being unity; but 

tan. b tan. /^ o j ' 

(Trig. Art. lOl) sin. «— a: = sin. ax cos. <i' — sin..rx 

, sin. a X cos. x - sin. x x cos. a sin. x 
cos. a\ hence, 


tan. b 
sin. a X tan /3 

sin. X 

= tan.a?. 

tan. b + COS. a x tan./3 cos. x 
Ex. Mr. Bugge observed the right ascension and 
declination of Saturn, and thence deduced (1 14,184) 
the following heliocentric longitudes and latitudes. 

1784. Apparent Time. Heliocentric Lon. 

Heliocentric Lat\ 

Julyl2,atl2\ 3'. 1" 

9s. 20°. 3 7'. 29'' 


3'. 13" N. 

20,-11.29. 9 

9. 20. 51. 53 


2. 41 

Aug. 1,-10.38.25 

9- 21. 13. 17 


1. 34 

8,-10. 9. 

9. 21. 26. 2 


0. 56. 

21,- 9. 14.59 

g. 21 49. 27 


0. 2 

27,- 8. 50. 19 

9. 22. 0. 12 


0. 27 S. 

31,- 8. 33.47 

9. 22. 7. 32 


0. 50 

Sept. 5,- 8. 13. 45 

9. 22. 16. 28 


1. 21 

15,- 7. 33. 45 

9. 22. 34. 32 


1. 59 

Oct. 8- 6. 4. 23 

— — ' 

9. 23. 16. 15 


3. 35 

From the observations on August 21 and 27, by 
considering the triangles as plane, cr = 44",5; from 
those on 21 and 31, :r=44",5; and from those on 
August 21, and September 5, ^ = 40"; the mean of 
these is x = 42" ; Mr. Bugge makes ^ = 41", probably 
by taking the mean of a greater number, or computing 
from considering them as spherical triangles; hence, 
the heliocentric place of the descending node was 9*. 


21°. 50'. 8",5. Now on August 21, at gh. 12'. 26" 
true time, Safurns heliocentric longitude was 9'. 21°. 
49'. 27", and on 27, at 8/^ 49'. 23" true time, it was 9. 
22°. 0'. 12"; therefore, in 5d, 23h. 36'. 57" Saturn 
moved lO'. 45" in longitude; hence, lO'. 45" : 41" :: 
5d. 23h. 36'. 57" : 9/?. 7'. 44" the time of describing 
41" in longitude, which added to August 21, 9^. 12'. 
26", gives August 21, 18/^ 20'. 10", the time when 
Saturn was in it's node. 

The longitudes of the nodes of the planets for the 
beginning of 17^0^ are, Mercury, V. 15°. 20'. 43" 
Venus, 2^ 14°. 26'. 18"; Mars, 1^ 17°. 38'. 38" 
Jupiter, 3^ 7°. 55'. 32"j Saturn, 3^. 21°. 32'. 22" 
Georgian, 2^ 12°. 47'. 

(186.) To determine the inclination of the orbit, we 
have am the latitude of the planet, and m N it's dis- 
tance upon the ecliptic from the node ; hence, Trig. 
Art. 210) sin. j}iN : tan. am :: rad. : tan. of the ano-Ie 
iV. But the observations which are near the node 
must not be used to determine the inclination, as a 
very small error in the latitude will make a consider- 
able error in the angle. If we take the observation on 
July 20, it gives the angle 2^. 38'. 15": if we take that 
on October 8, it gives the angle 2°. 22'. 13"; the mean 
of these is 2°. 30'. 14", the inclination of the orbit to 
the ecliptic, from these observations. Or the inclina- 
tion may be found thus, 

(I87.) Find the angle PSv (184), then the place 
of the planet and that of it's node being given^ we 

know vN; hence (Trig. Art. 210), sin. v N : tan Pv 
:: rad. : tan. PNv the inclination of the orbit. 


On March 27, 1694, at 7/?. 4'. 40", at Greenwich, 
Mr. Flamstead determined the right ascension of Mars 
to be 115°. 48'. 55'', and it's decHnation 24\ lo'. 50" 
north; hence, (184) the geocentric longitude was 
2B 23°. 26'. 1 2", and latitude 2°. 46'. 38". Let 8 be the 
sun, E the Earth, P Mars, v it's place reduced to the 
ecliptic. Now the true place of Mars (by calculation) 
seen from the sun was Si 28°. 44'. 14", and the place 
of the sun was t 7^- 34. 25"; hence, subtracting the 
place of the sun from the place of Mars seen from the 
earth, we have the angle i'>ES between the sun and 
Mars 105°. 51'. 4'/"; and the place of the earth being 
==b 7°. 34'. 25", take from it the place of Mars, and we 
have the angle ^^V = 38°. 50'. 11"; hence, (I87) sin. 
105°. 51'. 47" : sin. 38°. 50'. 11" :: tan. PEv = 2''.46'. 
38" : tan. PSv- 1°. 48'. 36". Now the place of the 
node was in « 17". 15', which subtracted from SI 28°. 
44'. 14", gives 101°. 29'. 14" for the distance vN of 
Mars from it's node; hence, sin. vN=: 101°. 29'. 14" : 
tan. Pv=l^. 48'. 36" :: rad. : tan. PNv=zl^. 50'. 50", 
the inclination of the orbit. Mr. Bugge makes the 
inclination to be 1°.50'. 56",56, for Mart 1 788. M. de 
la Lande makes it 1°. 51' for 178O. 

The inclination of the orbits of the planets are, 
Mercury, 7°. O'. o"; Venus, 3°. 23'. 35"; Mars, 1°. 
51'. 0";' Jupiter, 1°. 18'. 56"; Saturn, 2^ 29'. 50"; 
Georgian, 46'. 20". 

(188.) The motion of the nodes is found, by com- 
paring their places at two different times ; from 
whence, that of Mercury in 100 years is found to be 
1°. 12'. 10"; Venus, 0°. 51'* 40"; Mars, 0°. 46'. 40"; 
Jupiter, 0°. !^^' . 30"; Saturn, 0°. 55'. 30". This mo- 
tion is in respect to the equinox. 

The Georgian planet has not been discovered long 
enough to determine the motion of it's nodes from 
observation. M. de la Grange has found the annual 
motion to be 1 2",5 by theory. But if we take the 
density of Venus according to M. de la Lande, it will 
be 20". 40"', which he uses in his tables. 


Thus we have determined all the elements necessary 
for computing the place of a planet in it's orbit at any 
time ; but to facilitale the operatioD, which would be 
extremely tedious if we had only the elements thus 
irivcn, astronomers have constructed tables of their 
motions, by which their places at any tune may be 
very readily computed. 

Since the discovery of the Georgium Sldus, four 
other primary planets have been discovered : The 
first called Ceres, was discovered by 31. Piazzi at 
Palermo, Jan. 1, 1801 ; the second, called Pallas, 
was discovered by Dr. Others at Bretnen, March 28, 
1802 ; the third, called Juno, was discovered by 
M. Harding at LUienthal, September 1, 1804; and 
the fourth, called /"e.vfa, was discovered by Dr. Olbers, 
March 29, I807. The following Table contains 
the elements of the orbits of the three first ; the orbit 
of the fourth is not yet computed. 


Epocfi of Mean Long. 1 803 

for Merid. of Seeberg 
Long, of Aphelion - - 
Long, of ascending Node 
inclination of the Orbit - 
Excentricity - _ - - 
Log. of Mean Distance - 
iVIean Din in. Trop. Mot. 


30». 1 2'. 1",1 

326. 2S. 4,4 
81. 0.41 
10. 37. 86 
0,07 84699 


18° 13' I",] 

301. 3.24,3 
J72. 30.47 
31-. 37.43 
771 ",6802 



233. 11.40 
171. 4.15 
13. 3.38 

Dr. Herschel makes the diameter of Pallas 147 
miles: and that of Ceres l6l,6 miles. 


Chap. XV. 


(I89.) As all the planets describe orbits about the 
sun as their center, it is manifest, that to a spectator at 
the sun they would appear to move in the direction in 
which they really do move, and shine with full faces. 
But to a spectator on the earth, which is in motion, 
tliey will sometimes appear to move in a direction con- 
trary to their real motion, and sometimes appear sta- 
tionary ; and as the same face which is turned towards 
the earth, is not turned towards the sun, except in con- 
junction and opposition, some part of the disc which 
is towards the earth will not be illuminated. These, 
with some other appearances and circumstances which 
are observed to take place among the planets, we shall 
next proceed to explain ; and as they are matters in 
which great accuracy i.s never requisite, being of no 
great practical use, but rather subjects of curiosity, we 
shall consider the motions of all the planets as per- 
formed in circles about the sun in the center, and 
lying in the place, of the ecliptic. 

(190.) To find the position of a planet when sta- 
tionary. Let S be the sun, E the earth, P the co- 
temporary position of the planet, Xy the sphere of the 
fixed stars, to which we refer the motions of the 
planets ; let EF, PQ, be two indefinitely small arcs 
described in the same time, and let EPy FQ pro- 
duced, meet at L ; then it is manifest, that whilst the 
earth moves from E to F, the planet appears stationary 
at L ; and on account of the immense distance of the 
fixed stars, EPL, FQL may be considered as parallel. 
Draw SE, SFw, SvP, and SQ ; then, as EP and 


Fa are parallel, the angle QFS--PES=PwS^ 
PES = ESF, and SPw - SQF= SvF- SaF= PSQ ; 
that is, the cotemporary variations of theangles E and 
P are as ESF : PSQ, the cotemporary variations of 
the angular velocities of the earth and planet, or (be- 
cause the angular velocities are inversely as the periodic 
times, or inversely in the sesquiplicate ratio of the 

distances) as SP^ : SE^, or, (if SP : .S'^ :: « 1) as 
ffli : 1^. But sin. SEP : sin. SPE being as SP :SE, 

or, a : 1, the cotemporary variations of these angles 
will be as their tangents*. Hence, if x and i/ be the 
sines of the angles SEP and SPE. we have x -. y ■.-. a 

: 1, and 


Jl-x'' s/^ 


av : 1, whence 0?^ = 



and a:r=: 

. — the sine of the 

a'-fa + r Va^ + u-\-\ 

planet's elongation from tlie sun, when stationary. 

Ex. If P be the earth, and E Venus ; and we take 
the distances of the earth and Venus to be 100000 and 
72333, we find :r = 0,4826*4 the sine of 28". 51'. 5", 

See the Optics, Art. 421. 


the elongation of Venus when stationary, upon the 
supposition of circular orbits. 

For excentric orbits, the points will depend upon 
the position of the apsides and places of the bodies at 
the time. We ma}-, however, get a very near approxi- 
mation thus. Find the time when the planet would 
be stationary if the orbits were circular, and compute 
for several days, about that time, the geocentric place 
of the planet, so that you get two days, on one of 
which the planet was direct, and on the other retro- 
grade, in which interval it must have been stationary, 
and the point of time when this happened may be de- 
termined by inter])olation. 

(191.) To find the time when a planet is stationary, 
we must know the time of it's opposition, or inferior 
conjunction. Let m and n be the daily angular ve- 
locities of the earth and planet about the sun, and v 
the angle PSE when the planet is stationary ; then 
m — ti, or n — m, is the daily variation of the angle at 
the sun between the earth and planet, according as it 
is ^ superior or inferior planet ; hence, m — ?i, or 

'V V 

n—rn, -. v '.: \ day : , or , the time from 

•^ 711 — 71 n — 771 

opposition or conjunction to the stationary points both 
before and after. Hence, the planet must be stationary 
twice every synodic* revolution. 

Ex. Let P be the earth, E Venus ; then by the 
Example to Art. I90, the angle SPE = 28°. 51',5 ; 
therefore, PSE =13° ; also, n — T}i = 3f; hence, 3/' : 
13^ :: 1 day : 21 days the time between the inferior 
conj miction and stationary positions. 

(192.) If the elongation be observed when sta- 
tionary, we may find the distance of the planet from 
the sun, compared with the earth's distance, supposed 

* A Synodic revolution is the time betweea two conjunctions of 
the same sort, or two oppositions of a planet. 



to be unity. For (190) a.' = 

hencOj «- + 



X a= — 

— = (if ^ = the tangent of the angle 
whose sine is x) a* - 1' a=t' ; consequently a = ~t^ + t 

\/ 1 H — J upon the supposition of circular orbits. 

(193.) A superior planet is retrograde in opposition, 
and an inferior planet is retrograde in it's inferior 
conjunction ; for let E be the earth, P a superior 
planet in opposition ; then, as the velocities are as the 
inverse square roots of the radii of the orbits, the su- 
perior planet moves slowest ; hence, if EF, PQ, be 
two indefinitely small cotemporary arcs, PQ is less 
than E F, and on account of the immense distance of 
the sphere yZ of the fixed stars, FQ must cut EP in 
some point x between P and ?«, consequently, the 
planet appears retrograde from m to ??, If P be the 

earth, and E an inferior planet in inferior conjunction, 
it will appear retrograde from v to w. These retro- 
grade motions must necessarily continue till the' 



planets become stationary. Hence, from this and the 
last Artide, a superior planet appears retrograde from 
it's stationary point before opposition to it's stationary 
point after ; and an inferior planet, from it's stationary 
point before inferior conjunction to it's stationary point 

(194.) If S be the sun, E the earth, F Venus or 
Mercury, and EFsl tangent to the orbit of the planet. 

then will the angle SEf^ be the greatest elongation 
of the planet from the sun ; which angle, if the orbits 
were circles having the sun in their center, would be 
found by saying, ES : *S'^ :: rad, : sin. SEV. But 
the orbits are not circular, inconsequence of which the 
angle E V S will not be a right angle, unless the 
greatest elongation happens vrhen the planet is at one 
of it's apsides. The angle SEP^ is also subject to an 
alteration from the variation oi S E and S f^. The 
greatest angle SJ5/^ happens, when the planet is in it's 
aphelion and the earth in it's perigee ; and the least 
angle SEV, when the planet is in it's perihelion and 
the earth in it's apogee. M. de la Lande has calcu- 
lated these greatest elongations, and finds them 47°. 
48'. and 44°. 57' for Venm, and 28^. 20' and 17°. 3& 
for Mercury. If we take the mean of the greatest 
elongations of Venus, which is 46°. 22',5, it gives the 
angle FSE=43°. 37',5 : and as the difi'erence of the 
daily mean motions of Venus and the earth about the 
sun is 37', we have 37' : 43°. 37',5 :: 1 day : 70,7 


i^ays, the time that would elapse between the greatest 
elongations and the inferior conjunction, if the mo- 
tions had been uniform, which will not vary much 
from the true time. 

(195.) To delineate the appearance of a planet at 
any time. Let S be the sun, JE the earth, /"^ Venus, 

for example; aPl) the plane of illumination perpen- 
dicular to SFy cFd, the plane of vision perpendicular 
to E Vy and draw av perpendicular to cf/; then ca is 
the breadth of the visible illuminated part, which is 
projected by the eye into cv, the versed sine of CVa, 
or SVZ, for S Vc is the complement of each. Now 
the circle terminating the illuminated part of the 
planet, being seen obliquely, appears to be an ellipse 
(Con. Sect. p. 37); therefore, if cmdn represent the 
projected hemisphere of Venus next to the earth, mw, 
cd, two diameters perpendicular to each other, and we 
take ci;=the versed sine of SVZ, and describe the 
ellipse mvn, then cv is the axis minor, and mcnvm will 
represent the visible enlightened part, as it appears at 
the earth ; and from the property of the ellipse (Con. 
Sect. Ell. Prop. 7. Cor. 6.), this area varies as cv. 
Hence, the visible enlightened part : the whole disc ;: 
the versed sine of SFZ : diameter. 

Hence, Mercury and Fenus will have the same 
phases from their inferior to their superior conjunc- 
tion, as the moon has from the new to the full : and 



the same from the superior to the inferior conjunction, 
as the moon has from the full to the new. Mars will 
appear gibbous in quadratures, as the angle SFZ will 
then differ considerably from two right angles, and 
consequently the versed sine will sensibly differ from 
the diameter. For Jiipiter\ Saturn, and the Georgian, 
the angle SVZ never differs enough from two right 
angles to make those planets appear gibbous, so that 
they always appear full-orbed. 

(196.) Let /^ he the moon; then as EV'\s\evy 
small compared with P^ S, ES, these lines will be very 
nearly parallel, and the angle SVZ very nearly equal 
to SEV\ hence, the visible enlightened part of the 
moon varies very nearly as the versed sine of ifs 

(197-) Dr. Halley proposed the following problem : 
To find the position of Venus when brightest, suppos- 
ing it's orbit, and that of the earth, to be circles, having 
the sun in their center. Draw Sr perpendicular to 
EVZ, and put a - SE, b=SV, x = EV, y - Vr ; then 
h—y is the versed sine of the angle SVZ, which 
versed sine varies as the illuminated part; and as the 
intensity of light varies inversely as the square of it's 
distance, the quantity of light received at the earth 

varies as ~^= A- *^; but by Euclid, B. II. P. 12. 

X X' X- '' 

rt* _ fc2 _ J^.1 
a'^=l/--\-x^ + 2xy ; hence, y= — ^ ; substitute 

Ji X 

this for y, and we get the quantity of light to be as 

b a^-b^-x^ 2bx-a' + b--\-x^ 

*-T - • 5 = ^— " = a maximum ; put 

X 2x^ 2x^ ^ 

the fluxion=0, and we get x = ^3a^ -{- //' —2b. Now, 
if «=1, Z>=,72333, as in Dr. ^ff/Ze/s Tables, then 
.r = , 43036; hence, the angle ES V=22\ 21', but the 
angle ESV, at the time of the planet's greatest elonga- 
tion, is 43°. 40'; hence, Venus is brightest between it's 
inferior conjunction and it's greatest elongation ; also, 
the angle SEV=3()°. 44' y th§ elongation of Venus 


from tbt3 sun at the same time, and z SVZ=- VSE'\- 
VES=62°. 5', the versed sine of which is 0,53, radius 
being unity ; hence (195), the visible enhghtened 
part : whole disc. :: 0,53 : 2 ; Venus therefore appears 
a little more than one fourth illuminated, and answers 
to the appearance of the moon when five days old. 
Her diameter here is about 39 ", and therefore the en- 
lightened part is about 10",25. At this time, Venus 
is bright enough to cast a shadow at night. This 
situation happens about 36 days before and after it's 
inferior conjunction ; for, supposing Venus to be in 
conjunction with the sun, and when seen from the 
sun to depart from the earth at the rate of 3/' in 1 day, 
we have 3f : 22°. 21' :: 1 day : 36 days nearly, the 
lime from conjugation till Venus is brightest. 

(198.) If we apply this to Mercury, b=,3l7li and 
jc= 1,00058 ; hence, the angle ESF=78°. 55'l; but 
the same angle, at the time of the planet's greatest 
elongation, is 67°. 13'|. Hence, Mercury is brightest 
between it's greatest elongation and superior conjunc- 
tion. Also, the angle SEF=22°. 18'4, the elonga- 
tion of Mercury at that time. 

(199.) When Venus is brightest, and at the same 
time is at it's greatest north latitude, it can then be 
seen with the naked eye at any time of the day, when 
it is above the horizon; for when it's north latitude is 
the greatest, it rises highest above the horizon, and 
therefore is more easily seen, the rays of light having 
to come through a less part of the atmosphere, the 
higher the body is. This happens once in about eight 
years, Venus and the earth returning to the same 
parts of their orbits after that interval of time. 

(200.) Venus is a morning, star from inferior to su- 
perior conjunction, and an evening star from superior 
to inferior conjunction. For let S be the sun, E the 
earth, ACBD the orbit of Venus, arm, csn, two tan- 
gents to the earth, representing the horizon at </and c. 
Then the earth, revolving about it's axis according to 
the order abcj when a spectator is at a, the part rCm 

I 2 


of the orbit of Venus is above the horizon, but the sun 
is not yet risen ; therefore Venus, in going from r 
through C to m, appears in the morning before sun- 
rise. When the spectator is carried by the earth's ro- 
tation to c, the sun is then set, but the part nDs of 
Venus' orbit is still above the horizon ; therefore. 

Venus, in going from n through D to s, appears in 
the evening after sun-set. 

(201.) If two planets revolve in circular orbits, to 
find the time from conjunction to conjunction. Let 
P:=the periodic time of a superior planet, /? = that of 
an inferior, ^=the time required. Then P : 1 day :; 

360° : — p-the angle described by the superior planet 

in 1 day ; for the same reason, is the angle de- 

scribed by the inferior planet in 1 day; hence, 

— p- is the daily angular velocity of the inferior planet 



from the superior. Now if they set out from con- 
junction, they will return into conjunction again after 

the inferior planet has gained 3oO° ; hence, — 


: 360" 

1 day : t- jr^. 
J P—p 

This will also give 

the time between two oppositions, or between any two 
similar situations. 


Chap. XVI. 


(202.) The moon being the nearest, and after the 
sun, the most remarkable body in our system, and 
also useful for the division of time, it is no wonder 
that the ancient astronomers were attentive to discover 
it's motions ; and it is a very fortunate circumstance 
that their observations have come down to us, as from 
thence it's mean motion can be more accurately 
settled, than it could have been by modern observations 
only ; and it moreover gav-e occasion to Dr. Halley, 
from the observations of- some ancient eclipses, to dis- 
cover an acceleration in it's mean motion. The proper 
motion of the moon, in it's orbit about the earth, is 
from west to east ; and from comparing it's place with 
the fixed stars in one revolution, it is found to describe 
an orbit inclined to the ecliptic ; it's motion also ap- 
pears not to be uniform ; and the position of the orbit, 
and the line of it's apsides are observed to be subject 
to a continual change. These circumstances, as they 
are established by observation, we come now to explain. 

To determine the Place of the Moon^s Nodes. 

(203.) The place of the moon's nodes may be de- 
termined as in Art. 185, or by the following method. 

In a central eclipse of the moon, the moon's place at 
the middle of the eclipse is directly opposite to the sun, 
and the moon must also then be in the node; calcu- 
late therefore the true place of the sun, or, which is 
more exact, find it's place by observation, and the 


opposite point will be the true place of the moon, and 
consequently the place of it's node. 

Ex. M. Cassini, in his Astronomy, p. 281, informs 
us, that on April l6, 1707, a central eclipse was ob- 
served at Paris, the middle of which was determined 
to be at I3h. 48' apparent time. Now the true place 
of the sun, calculated for that time, was 0^ 26**. 19' 
17" ; hence, the place of the moon's node was 6^. 26". 
19'. 17". The moon passed from north to south 
latitude, and therefore this was the descending node. 

(204.) To determine the mean motion of the nodes, 
find (203) the place of the nodes at different times, 
and it will give their motion in the interval ; and the 
greater the interval, the more accurately you will get 
the mean motion. Mayer makes the mean annual mo- 
tion of the nodes to be 12^ 19'. 43", 1. 

On the Inclination of' the Orbit of the Moon to the 

(205.) To determine the inclination of the orbit, 
observe the moon's right ascension and declination 
when it is 90° from it's nodes, and thence compute it's 
latitude (114), which will be the inclination at that 
time. Repeat the observation for every distance of 
the sun from the earth, and for every position of the 
sun in respect to the moon's nodes, and you will get 
the inclination at those times. From these observa- 
tions it appears, that the inclination of the orbit to the 
ecliptic is variable, and that the least inclination is 
about b", which is found to happen when the nodes 
are in quadratures; and the greatest is about S*". 18', 
which is observed to happen when the nodes are in 
syzygies. The inclination is also found to depend upon 
the sun*s distance from the earth. 

On the mean Motion of the Moon. 

(206.) The mean motion of the moon is found 
from observing it's place at two different times, and 



you get the mean motion in that interval, suppositTg 
the moon to have had the same situation in respect to 
it's apsides at each observation ; and if not, if there be 
a very great interval of the times, it will be sufficiently 
exact. To determine this, we must compare together 
the moon's places, first at a small interval of time from 
each other, in order to get nearly the mean time of a 
revolution; and then at a greater interval, in order to it more accurately. The moon's place may be 
determined directly from observation, or deduced from 
an eclipse, 

(207.) M. Cassini, in his Astronomy, p. 394, ob- 
serves, that on September 9, 17 18, the moon was 
eclipsed, the middle of which eclipse happened at 8/^. 
4', when the sun's true place was 5^. 16". 40'. This 
he compared with another eclipse, the middle of which 
was observed at 8 A. 32'. on August 29, 17^9, when the 
sun's place was 5*. 5°. 47'- In this interval of 35 id^ 
28' the moon made 12 revolutions and 349". 7' over; 
divide therefore 354^. 28' by 12 revolutions + 349°. 
7'. part of a revolution, and it gives 27^i 'jh. 6' for the 
time of one revolution. From two eclipses in 1699, 
1717, the time was found to be 22d. *jh. 43'. 6". 

(208.) The moon was observed at Paris to be 
eclipsed on Sept. 20, 1717, the middle of which eclipse 
was at 6A. 2'. Now Ptolcmif mentions, that a total 
eclipse of the moon was observed at Babylon on 
March 19, 720 years before J. C. the middle of which 
happened at ^h. 30', at that place, which gives 6/l 48' 
at Paris. The interval of these times was 2437 years 
(of which 609 were bissextiles) ^]r4f days wanting 46'; 
divide this by 27^. lh. 43'. 6", and it gives 32585 re- 
volutions and a little above |. Now the difference of 
the two places of the sun, and consequently of the 
moon, at the times of observations, was 6^ 6°. 12'^. 
Therefore, in the interval of 24373/. 17 4d. wanting 46', 
the moon had made 32585 revolutions 6\6°. 12', 
which gives 27c?. 7h. 43'. 5" for the mean time of a 
revolution. This determination is very exact, as the 


moon was at each time very nearly at the same dis- 
tance from it's apside. Hence, the mean diurnal mo- 
tion is 13°. 10'. 35", and the mean hourly motion 32'. 
56". 27'"|. M. de la Lande makes the mean diurtial 
motion 13". lO'. 35",02784394. This is the mean 
time of a revolution in respect to the equinoxes. The 
place of the moon at the middle of the eclipse has here 
been taken the same as that of the sun, whicli is not 
accurate, except for a central eclipse ; it is sufficiently 
accurate, however, for this long interval. 

(209.) As the precession of the equinoxes is 50",25 
in a year, or about 4" in a month, the mean revolution 
of the moon in respect to the fixed stars must be 
greater than that in respect to the equinox, by the 
time the moon is describing 4" with it's mean motion, 
which is about 7". Hence, the time of a sidereal re- 
volution of the moon is 27<^. "^h. 43'. 12". 

(210.) Observe accurately the place of the moon 
for a whole revolution as often as it can be done, and 
by comparing the true and mean motions, the greatest 
difference will be double the equation. If two ob- 
servations be found, where the difference of the true 
and mean motions is nothing, the moon must then 
have been in it's apogee and perigee (168). Mayer 
makes the mean excentricity 0,05503568, and the 
corresponding greatest equation 6°. 18'. 3l"6. It is 
6°. 18'. 32" in his last Tables, published by Mr. 
Mason, under the direction of Dr. Maskelyne. 

(211.) To determine the place of the apogee, from 
M. Cass'ml's observations, we have the greatest equa- 
tion = 5\ 1'. 44"5; therefore (171), 57°. 17'. 48"8 : 
2°. 30'. 52"25 :: ^C= 100000 : CS=4'688 for the 
moon's excentricity at that time*. Now (Fig. p. 101.) 
let V be the focus in which the earth is situated ; then 

* The excentricity of the moon's orbit is subject to a variation, 
it being greatest wh«n the apsides lie in syzygies, and least when 
in quadratures. 

138 moon's motion 

(169) supposing QSP to be the mean anomaly, as 
QvP is the true anomaly, their difference SPv is the 
equation of the orbit, which equation is here 37'. 
50",5 ; and as PS=Pr, the angle t^r 6'= 18'. 55",25 ; 
hence, (Trigonometry, Art. 128) vS=z817^ • ^^ = 
200000 :: sin. vrS= 18'. 5 5", 25 : sin. vSr, or QSr,=: 
7°. 12'. 20", from which take t;r^=18'. 55",25, and 
we have QvP = 6°. 53'. 25" the distance of the moon 
from it's apogee; add this to 2^ 19". 40', the true 
place of the moon, and it gives 2^ 26'\ 33'. 25" for the 
place of the apogee on December 10, l685, at loh. 
38'. 10" mean time at Paris. This therefore may be 
considered as an epoch of the place of the apogee. 

To determine the mean Motion of the Apogee. 

(212.) Find it's place at different times, and com- 
pare the difference of the places with the interval of 
the time between. To do this, we must first compare 
observations at a small distance from each other, lest 
we should be deceived in a whole revolution; and then 
we can compare those at a greater distance. The 
mean annual motion of the apogee in a year of 365 
days is thus found to be 40°. 39'. 50", according to 
Mayer. Horrox, from observing the diameter of the 
moon, found the apogee subject to an annual equation 
of 12°,5. 

(213.) The motion of the moon having been ex- 
amined for one month, it was immediately discovered 
that it was subject to an irregularity, which sometimes 
amounted to 5° or 6°, but that this irregularity disap- 
peared about every 14 days. And by continuing the 
observations for different months, it also appeared, that 
the points where the inequalities were the greatest, 
were not fixed, but that they moved forwards in the 
heavens about 3° in a month, so that the motion of the 

moon, in respect to it's apogee, was about less than 

it's absolute motion ; thus it appeared that the apogee 


had a progressive motion. Ptolemy determined this 
first inequality, or equation of the orbit, from three 
lunar eclipses observed in the years 7^9 ^"d 720, 
before J. C. at Babylon by the Chaldeans; from 
which he found it amounted to 5°. l', when at it's 
greatest. But he soon discovered that this inequality 
would not account for all the irregularities of the 
moon. The distance of the moon from the sun, ob- 
served both by Hipparchus and himself, sometimes 
agreed with this inequality, and sometimes it did not. 
He found that when the apsides of the moon's orbit 
were in quadratures, this^r*^ inequality would give 
the moon's place very well ; but that when the apsides 
w^ere in syzygies, he discovered that there was a further 
inequality of about 2°~, which made the whole ine- 
quality about 7°-T. This second inequality is called 
the Evection, and arises from a change of excentricity 
of the moon's orbit. The inequality of the moon was 
therefore found, by Ptolemi/, to vary from about 5° to 
7°!^, and hence the mean quantity was 6". 20'. Mayer 
makes it 6°. 18'. 3l",6. It is very extraordinary, that 
Ptolemy should have determined this to so great a 
degree of accuracy. We cannot here enter any further 
into the inequalities of the motion of the moon. They 
who wish to see more on this subject, may consult my 
Complete System of Astronomy. 



(214.) Times of the Revolutions of the Moon, ofifs 
Apogee and Nodes, as determined hy M. de la 

27''.7\43'. 4",6r95 
27. 7. 43. 11,5259 
44. 2,8283 
18. 33,9499 


27. 5. 5. 


8". 311. 8. 34. 57,6177 
8. 312. 11. 11. 39,4089 
18. 228. 4. 52. 52,0296 
18. 223. 7. 13. 17,744 

- - 13°. 10'.35"02784394 

Tropical revolution 
Sidereal revolution 
Synodic revolution 
Anomalistic revolution 
Revolution in respect 

to the node - - 
Tropical revolution 

of the apogee 
Sidereal revolution of 

the apogee - - 
Tropical revolution 

of the node - - 
Sidereal revolution of 

the node - - • 
Diurnal motion of- 

the moon in respect | 

to the equinox - 
Diurnal motion of the apogee 0. 6.41,069815195 
Diurnal motion of the node - O. 3. 10,638603696 

The years here taken are the common years of 365 

On the Diameter of the Moon. 

(215.) The diameter of the moon may be measured, 
at the time of it's full, by a micrometer; or it may be 
measured by the time of it's passing over the vertical 
wire of a transit telescope ; but this must be when the 
moon passes within an hour or two of the time of the 
full, before the visible disc is sensibly changed from a 
circle. To find the diameter by the time of it's 
passage over the meridian, let <f' = the horizontal di- 
ameter of the moon, c=sec. of it's declination, and m 
=the length of a lunar day, or the time from the 

moon's diameter. 


passage of the moon over the meridian on the day we 
calculate, to the passage over the meridian the next 
day. Then (102) cd" is the moon's diameter in right 
ascension ; hence, 360'' : cd' :: m : the time {t) of 

passing the meridian ; therefore c?" = 360** x • If we 

'^ ° cm 

observe when the limb of the moon comes to the 

meridian, we can find the time when the center comes 

to it, by adding to, or subtracting from, the time when. 

the first or second limb comes to the meridian, half 

the time of the passage of the moon over the meridian. 

The time in which the semidiameter of the moon 

passes the meridian, may be found by two Tables, in 

the Tables of the moon's motion. 

(21 6.) Alhategnius made the diameter of the moon 
to vary from 29'. 30" to 35'. 20", and hence the mean 
is 32'. 25". Copernicus found it from 2f. 34". to 35'. 
38", and therefore the mean 31'. 3&'. Kepler made 
the mean diameter 31'. 22". M. de la Hire made it 
31'. 30". M. Ca^^im made the diameter from 29'. 
30" to 33'. 38". M. de la Lande, from his own ob- 
servations, found the mean diameter to be 31'. 26"; 
the extremes from 29'. 22" when the moon is in 
apogee and conjunction, and 33'. 31" when in perigee 
and opposition. The mean diameter here taken, is 
the arithmetic mean between the greatest and least 
diameters ; the diameter at the mean distance is 
31'. 7". 

(217.) When the moon is at different altitudes 

above the horizon, it is at different distances from the 

J42 moon's phases. 

spectator^ and therefore there is a change of the appa- 
rent diameter. Let C be the center of the earth, A 
the place of a spectator on it's surface, Z his zenith, M 
the moon; then (Trig. Art. 128) sin. CAM, or ZAM : 

«in. ZCM'.CM : AM^^^L^L^^:^^ ; but the 

sni. ZAM 

apparent diameter is inversely as it's distance ; hence, 

the apparent diameter varies as ". ' ^^,,^ , CM being 
* * sm. ZCM *' 

supposed constant. Now, in the horizon, -— -v^Wrv 

sm. ZCM 

may be considered as equal to unity ; hence, 1 : 

—. — TyTTnfi or sm. ZLM : sm. ZAM, or cos. true alt. 
sm. ZCM ' 

(a) : cos. apparent alt. (A) :: the horizontal diameter : 

the diameter at the apparent altitude (A). Hence, 

the horizontal diameter : it's increase :: cos.« : cos. A 

—cos. a = (Trig. Art. Ill) 2 sin. ^a + i A x sin. 

I a-j;A; therefore the increase of the semidiameter 

, • 1 •, sin ^ « -{- ^A X sm.^a-^A 

= nor. semidiameter x = = 1 £ — • 

cos, a 

from this we may easily construct a table of the increase 

of the semidiameter for any horizontal semidiameter, 

and then for any other horizontal semidiameter, the 

increase will vary in the same proportion. 

On the Phases of the Moon. 

(218.) By Art. I96, the greatest breadth of the 
visible illuminated part of the moon's surface, varies as 
the vei*sed sine of the moon's elongation from the sun, 
very nearly; and the circle terminating the light and 
dark part, being seen obliquely, appears an elhpse ; 
hence, the following delineation of the phases. Let E 
be the earth, 8 the sun, Mthe moon ; describe the 
circle a 6crf, representing the hemisphere of the moon 
uhich is towards the earth, projected upon the plane 


of vision; ac, db, two diameters perpendicular to each 
ether ; take dv=zthe versed sine of elongation SEM, 

and describe the ellipse avc, and (195) adcva will re- 
present the visible enhghtened part; which will be 
horned between conjunction and quadratures ; a semi- 
circle at quadratures ; and gibbous between quadra- 
tures and opposition ; the versed sine being less than 
radius in the first case, equal to it in the second, and 
greater in the third. The visible enlightened part 
varying as dv, vie have, the visible enlightened pai^t ' 
whole : : versed sine of' elongation ; diameter. 

On the Libration of the Moon. 

(219,) Many Astronomers have given maps of the 
face of the moon ; but the most celebrated are those 
of Hevelius in his Selenographia, in which he has re- 
presented the appearance of the moon in it's different 
states from the new to the full, and from the full to the 
new; these figures Mayer prefers. Langrenus and 
Ricciolus denoted the spots upon the surface by the 
names of Philosophers, Mathematicians, and other 
celebrated men, giving the names of the most cele- 
brated characters to the largest spots; Hevelius marked 
them with the geographical names of places upon the 
earth. The former distinction is now generally fol- 

The spots upon the moon are caused by the moun- 
tains and vallies upon it's surface; for certain parts are 
found to project shadows opposite to the sun ; and 


when the sun becomes vertical to any of them, they 
are observed to have no shadows ; these therefore are 
mountains; other parts are always dark on that side 
next to the sun, and illuminated on the opposite side; 
these therefore are cavities. Hence, the appearance 
of the face of the moon continually varies, from it's 
altering it's situation in respect to the sun. The tops 
of the mountains, on the dark part of the moon, are 
frequently seen enlightened at a distance from the 
cosines of the illuminated part. The dark parts have, 
by some, been thought to be seas, and by others, to be 
only a great number of caverns and pits, the dark sides 
of which, next to the sun, would cause those places to 
appear darker than others. The great irregularity of 
the line bounding the light and dark part, on every 
part of the surface, proves that there can be no very 
large tracts of water, as such a regular surface would 
necessarily produce a line, terminating the bright part, 
perfectly free from all irregularity. If there was much 
water upon it's surface, and an atmosphere, as con- 
jectured by some Astronomers, the clouds and vapours 
might easily be discovered by the telescopes which we 
have now in use ; but no such phsenomena have ever 
been observed. 

(220.) Very nearly the same face of the moon is 
always turned tovvards the earth, it being subject only 
to a small change within certain limits, those spots 
which lie near to the edge appearing and disappearing 
by turns ; this is called it's Libratlon, and arises from 
four causes. 1. Galileo, who first observed the spots 
of the moon after the invention of telescopes, discovered 
this circumstance ; he perceived a small daily variation '^ 
arising from the motion of the spectator about the 
center of the earth, which, from the rising to the setting 
of the moon, would cause a little of the western limb 
of the moon te disappear, and bring into view a little 
of the eastern limb. 2. He observed likewise, that 
the north and south poles of the moon appeared and 
disappeared by turns; this arises from the axis of the 


itioon not being ; perpendicular to the plane of it*s 
orbit, and is ^called a libration in latitude. 3. From 
the unequal angular motion of the moon about the 
earth, and the uniform motion of the moon about it's 
axis, a little of the eastern and western parts must 
gradually appear and disappear by turns, the period of 
which is a month, and this is called a libration in 
longitude; the cause of this libration was first assigned 
by Ricciolus, but he afterwards gave it up, as he made 
many observations which this supposition would not 
satisfy. Hevelius, however, found that it would solve 
all the phaenomena of this libration. 4. Another 
cause of libration arises from the attraction of the earth 
upon the moon, in consequence of it's spheroidical 

(221.) If the angular velocity of the moon about it*s 
axis were equal to it's angular motion about the earth, 
the libration in longitude would not take place. For 
if E be the earth, abed the moon at v and w, and avc 

be perpendicular to Ebvd ; then aba is that hemi- 
sphere of the moon at v next to the earth. When the 
moon comes to w, if it did not revolve about it's axis 
bwd would be parallel to bvd, and the same face would 
not lie towards the earth. But if the moon, by re- 
volving about it's axis in the direction abed, had 
brought b into the line Ew, the same face would have 
been turned towards the earth ; and the moon would 
have revolved about it's axis through the angle bwE, 



which is equal to the alternate angle wEv, the angle 
\vhich the moon has described about the earth. 

(222.) When the moon returns to the same point 
of it's orbit, the same face is observed t6 he towards 
the earth, and therefore (221) the time of the revolu- 
tion in it's orbit is equal to the time about it's axis. 
But in the intermediate points it varies, sometimes a 
little more to the east, and sometimes to the west, be- 
comes visible ; and this arises from it's unequal angular 
motion wEv about the earth, whilst the angular mo- 
tion about it's axis is equal, in consequence of which 
these two Angles cannot continue equal, and therefore, 
by the last article, the same face cannot continue to- 
wards the earth. Hence, the greatest libration in 
longitude is nearly equal to the equation of the orbit, 
or about 7^2 at it's maximum, and would be accu^ 
rately so, if the axis of the moon were perpendicular 
to it's orbit ; for the difference of the moon's mean 
motion and true motion, or the equation of the orbit, 
is the same as the difference of the moon's motion about 
it's axis and it's true motion, which is the libration. 
The same face will be towards the earth in apogee and 
perigee, for at those points there is no equation of the 
orbit. If E be the earth, M the moon, pq it's axis, 

J" P 

M^r-7 E TV" yM 

not perpendicular to the plane of the orbit ah ; then at 
a the pole/? will be visible to the earth, and at b the 
pole q will be visible ; as the moon therefore revolves 
about the earth, the poles must appear and disappear 
by turns, causing the libration in latitude. This is 
exactly similar to the cause of the variety of our sea- 
sons, from the earth's axis not being perpendicular to 
the plane of it*s orbit. Hence, nearly one half of the 
moon is never visible at the earth. Also, the time of 
it's rotation about it's axis being a month, the length 
of the lunar days and nights will be about a fortnight 


each, they being subject but to a very small change, 
on account of the axis of the moon being nearly per- 
pendicular to the ecliptic. 

(223.) Hevelms (Selenographia, p. 245.) observed, 
that when the moon was at it's greatest north latitude, 
the libration in latitude was the greatest, the spots 
which are situated near the northern limb being then 
nearest to it; and as the moon departed from thence, 
the spots receded from that limb, and when the moon 
came to it's greatest south latitude, the spots situated 
near the southern limb were then nearest to it. This 
variation he found to be about l'. 45", the diameter of 
the moon being 30'. Hence it follows, that when the 
moon is at it's greatest latitude, a plane drawn through 
the earth and moon perpendicular to the plane of the 
moon's orbit, passes through the axis of the moon ; 
consequently the equator of the moon must intersect 
the ecliptic in a line parallel to the line of the nodes of 
the moon's orbit, and therefore, in the heavens, the 
nodes of the moon's orbit and of it's equator coincide. 

(224.) It is a very extraordinary circumstance, that 
the time of the moon's revolution about it's axis should 
be equal to that in it's orbit. Sir /. Newton, from the 
altitude of the tides on the earth, has computed that 
the altitude of the tides on the moon's surface must be 
23 feet, and therefore the diameter of the moon per- 
pendicular to a line drawn from the earth to the moon, 
ought to be less than the diameter directed to the earth, 
by 186 feet ; hence, says he, the same face must al- 
ways be towards the earth, except a small oscillation; 
for if the longest diameter should get a little out of 
that direction, it would be brought into it again by 
the attraction of the earth. The supposition of D. de 
Mairan is, that that hemisphere of the moon next the 
earth is more dense than the opposite one, and hence, 
the same face would be kept towards the earth, upon 
the same principle as above. 

(225.) When the moon is about three days from 
the new, the dark part is very visible, by the light re- 



fleeted from the earth, which is moon-light to the 
Lunarians, considering our earth as a moon to them; 
and in the most favourable state, some of the principal 
spots may then be seen. But when the moon gets 
into quadratures, it's great light prevents the dark part 
from being visible. According to Dr. Smith, the 
strength of moon-light, at the full moon, is ninety 
thousand times less than the light of the sun; but, 
from some experiments of M. Bouguer, he concluded 
it to be three hundred thousand times less. The light 
of the moon, condensed by the best mirrors, produces 
no sensible effect upon the thermometer. Our earth, 
in the course of a month, shows the same phases to the 
Lunarians, as the moon does to us ; the earth is at the 
full at the time of the new moon, and at the new at the 
time of the full moon. The surface of the earth being 
about 13 times greater than that of the moon, it affords 
13 times more light to the moon than the moon does 
to the earth. 

On the Altitude of the Luna?' Mountains. 

(226.) The method used by Hevelius, and others 
since his time, to determine the height of a lunar 
mountain is this. Let SLM be a ray of light from 

L M 

tlie sun, passing the moon at L, and touching the top 
of the mountain at M ; then the space between L and 
M appears dark. With a micrometer, measure LM^ 
and compare it with LC \ then, knowing LC, we 
kn ow LM, a nd by Eucl. B. L p. 4/. CM = 
^ CL- + L3I~ is known ; from which subtract Cpy 



and we get the height jo 71/ of the mountain. But as 
Dr. Herschel ohserves, in the Phil. Trans. 178I, this 
method is only applicable when the moon is in quadra- 
tures ; he has therefore given the following general 
method. Let E be the earth ; draw EMn and Lo 
perpendicular to the moon's radius RC, and Lr 
parallel to on, also ME' perpendicular to SM. Now 
ML would measure it's full length when seen from the 
earth in quadratures at E\ but seen from E, it only 
measures the length of Lr. As the plane passing 
through SM, EM, is perpendicular to a line joining 
the cusps, the circle RLp may be conceived to be a 
section of the moon perpendicular to that line. Now 
it is manifest, that the angle SLo or LCR, is very 
nearly equal to the elongation of the moon from the 
sun ; and the triangles LrM, LCo, being similar, Lo 

: LC :: Lr : Z.M= ""r,^' = __, ^,„,,^ , ra- 

Lo sine of elongation' 

dius being unity. Hence, we find Mp as before. 

Ex. On June, 178O, at seven o'clock, Dr. Herschel 
found the angle under which LM, or Lr appeared, to 
be 40", 625, for a mountain in the south-east quadrant; 
and the sun's distance from the moon was 125°. 8', 
whose sine is ,8104 ; hence, 40",625 divided by ,8104, 
gives 50", 13, the angle under which LM would ap- 
pear, if seen directly. Now the semidiameter of the 
moon was 16'. 2'\6, and taking its length to be 10^ 


miles, wehavel6'.2",6 : 50",13 :: IO9O: LiI/=56,73 
miles ; hence, 3Ip~l,4'^ miles. 

(227.) Dr. Herschel found the height of a great 
many more mountains, and thmks he has good reason 
to believe, that their altitudes are greatly over-rated ; 
and that, a few excepted, they generally do not exceed 
half a mile. He observes, that it should be examined 
whether the mountain stands upon level ground, 
which is necessary, that the measurement may be 
exact. A low tract of ground between the mountain 
and the sun will give it higher, and elevated places 
between will make it lower, than it's true height above 
the Common surface of the moon. 

(228.) On April 19, 1787, Dr. ^(?7'*c/ie/ discovered 
three volcanos in the dark part of the moon; two of^ 
them seemed to be almost extinct^ but the third 
showed an actual eruption of fire, or luminous matter, 
resembling a small piece of burning charcoal covered 
by a very thin coat of white ashes ; it had a degree of 
brightness about as strong as that with which such a 
coal would be seen to glow in faint day-hght. The 
adjacent parts of the volcanic mountain seemed faintly 
illuminated by the eruption. A similar eruption ap- 
peared on May 4, 1783. Phil. Trans. 1787. On 
March 7j 1794, a few minutes before eight o'clock in 
the evening, Mr. Wilk'ms^ of Norwich, an eminent 
architect, observed, with the naked eye, a very bright 
spot upon the dark part of the moon ; it was there 
when he first looked at the moon ; the whole time he 
saw it, it was a fixed, steady light, except the moment 
before it disappeared, when it's brightness increased ; 
he conjectures that he saw it about five minutes. The 
same phaenomenon was observed by Mr. T. Stretton, 
in St. John's-square, Clerkenwell, London. Phil. 
Trans. 1794. On April 13, 1793, and on February 
5, 1794, Mr. Piazzi, Astronomer Royal at Palermo, 
observed a bright spot on the dark part of the moon, 
near Aristarchus. Several other Astronomers have 


observed the same phaenomenon. See the Memoirs 
de Berlm, for 1788. 

(229.) It has been a doubt amongst Astronomers, 
whether the moon has any atmosphere ; some suspect- 
ing that at an occultation of a fixed star by the moon, 
the star did not vanish instantly, but lost it's light 
gradually ; whilst- others could never observe any such 
appearance. M. Schroetar of Lilianthan, in the duchy 
of Bremen, has endeavoured to establish the existence 
of an atmosphere, from the following observations. 
1. He observed the moon when two days and an 
half old, in the evening soon after sun-set, before the 
dark part was visible, and continued to observe it till it 
became visible. The two cusps appeared tapering in 
a very sharp, faint prolongation, each exhibiting it's 
farthest extremity faintly illuminated by the solar rays, 
before any part of the dark hemisphere was visible. 
Soon after, the whole dark limb appeared illuminated. 
This prolongation of the cusps beyond the simicircle, 
he thinks, must arise from the refraction of the sun's 
rays by the moon's atmosphere. He computes also 
the height of the atmosphere, which refracts light 
enough into it's dark hemisphere to produce a twilight, 
more luminous than the light reflected from the earth 
when the moon is about 32° from the new, to be 1356 
Paris feet ; and that the greatest height capable of 
refracting the solar rays is 53/6 feet, 2. At an occul- 
tation of Jupiter's satellites, the third disappeared, 
after having been about l" or 2" of time indistinct; the 
fourth became indiscernible near the limb ; this was 
not observed of the other two. Phil. Trans. 1792. 
If there be no atmosphere of the moon, the heavens, to 
a Lunarian, must always appear dark like night, and 
the stars be constantly visible ; for it is owing to the 
reflection and refraction of the sun's light by tlie at- 
mosphere, that the heavens^ in every part, appear 
bright in the day. 



On the Phcenomenon of the Harvest Moojt. 

(230.) The full moon which happens at, or nearest 
to, the autumnal equinox, is called the Harvest moon ; 
and at that time there is a less difference between the 
times of it's rising on two successive nights, than at 
any other full moon in the year; and what we here 
propose, is to account for this phaenomenon. 

(231.) Let P be the north pole of the equator 
Q^U, HAO the horizon, EAC the ecliptic, A the 


first point of Aries ; then, in north latitudes, A is the 
ascending node of the ecliptic upon the equator, AC 
being the order of the sines, and AQ that of the appa- 
rent diurnal motion of the heavenly bodies. When 
Aries rises in north latitudes, the ecliptic makes the 
least angle with the horizon ; and as the moon's orbit 
makes but a small angle with the ecliptic, let us first 
suppose EAC to represent the moon's orbit. Let A 
be the place of the moon at it's rising on one night; 
now, in mean solar time, the earth makes one revolu- 
tion in 23A. 56'. 4", and brings the same point ^ of the 
equator to the horizon again ; but, in that time, let 
the moon have moved in it's orbit from A to c, and 
draw the parallel of declination tcm, then it is mani- 
fest that 3'. 56" before the same hour the next night, 
the moon, in it's diurnal motion^ has to describe c n 


before it rises. Now en is manifestly the least possible, 
when the angle CAn is the least, ^£^ being given. 
Hence, it rises more nearly at the same hour, wHen 
it's orbit makes the least ano;le with the horizon. Now 
at the autumnal equinox, when the sun is in the first 
point of Libra, the moon, at that time at it's full, will 
be at the first point of Aries, and therefore it rises 
with the least diiference of times, on two succeslsive 
nights; and it being at the time of it's full, it is more 
taken notice of; for the same thing happens every 
month when the moon comes to Aries. 

(232.) Hitherto we have supposed the ecliptic to 
represent the moon's orbit, but as the orbit is inclined 
to it at an angle of 5°. 9' at a mean, let xAsz represent 
the moon's orbit when the ascending node is at A, and 
As the arc described in a day; then the moon's orbit 
making the least possible angle with the horizon in 
that position of the nodes, thearcs'w, and consequently 
the difference of the times of rising, will be the least 
possible. As the moon's nodes make a revolution in 
about 19 years, the least possible diflference can only 
happen once in that time. In the latitude of London 
the least difi^erence is about 17'. 

(233.) The ecliptic makes the greatest angle with 
the horizon when the first point of Libra rises, conse- 
quently, when the moon is in that part of it's orbit, 
the diflference of the times of it's rising will be the 
greatest; and if the descending node of it's orbit be 
there at the same time, it will make the diflTerence the 
greatest possible ; and this diflference is about \h. 17' 
in the latitude of London. This is the case with the 
vernal full moons. Those signs which make the least 
angle with the horizon when they rise make the 
greatest angle when they set, and vice versa; hence, 
when the diflference of the times of rising is the least, 
the diflference of the times of setting is the greatest, 
and the contrary. 

(234.) By increasing the latitude, the angle zAuj 
and consequently sn is diminished; and when the 


time of describing sn, by the diurnal motion, is 3'. 56"' 
the moon will then rise at the same solar hour. Let 
us suppose the latitude to be increased until the angle 
sAn vanishes, then the moon's orbit becomes coinci- 
dent with the horizon every day, for a moment of time, 
and consequently the moon rises at the same sidereal 
hour, or 3'. .56" sooner, by solar time. Now take a 
globe, and elevate the north pole to this latitude, and, 
marking the moon's orbit in this position upon it, turn 
the globe about, and it will appear, that at the instant 
after the above coincidence, one half of the moon's 
orbit, corresponding to Capricorn, Aquarius, Pisces, 
Aries, Taurus, Gemini, will rise; hence, when the 
moon is going through that part of it's orbit, or for 13 
or 14 days, it rises at the same sidereal hour. Nowj 
taking tlie angle ocA E^b". 9', and the angle EAQ= 
23". 28', the angle QAjo, or QA H, when the moon's 
orbit coincides with the horizon, is 28°. 37' ; hence, 
(87) the latitude is 6v. 23' where these circumstances 
take place. If the descending node be at A, then x 
lying above JE;, QAt, or QAH=lS^. I9', and the 
latitude is 71°- 41'. In any other situation of the 
orbit, the latitude will be between these limits. When 
the angle QA,v is greater than the complement of 
latitude, the moon will rise sooner the next day. As 
there is a complete revolution of the nodes in about 18 
years 8 months, all the varieties of the intervals of the 
rising and setting of the moon will happen within that 

On the Horizontal Moon. 

(235.) The phaenomenon of the horizontal moon is 
this, that it appears larger in the horizon than in the 
meridian; whereas, from it's being nearer to us in 
the latter than in the former case, it subtends a greater 
angle. Gassendus thought that, as the moon was less 
bright in the horizon, we looked at it there with a 
greater pupil of the eye, and therefore it appeared 


larger. But this is contrary to the prhiciples of 
Optics, since the magnitude of the image upon the 
retina does not depend upon the pupil. This opinion 
was supported by a French Ahhe, who supposed that 
the opening of the pupil made the chrystalline humour 
flatter, and the eye longer, and thereby increased the 
image. But there is no connection between the 
muscles of the iris and the other parts of the eye, to 
produce these effects. Des Cartes thought that the 
moon appeared largest in the horizon, because, when 
comparing it's distance with the intermediate objects, 
it appeared then furthest off; and as we judge it's 
distance greatest in that situation, we of course think 
it larger, supposing that it subtends the same angle. 
This opinion was supported hy Tiw Wallis, in the 
Phil. Trans. N°. I87. Dr. Berkley accounts for it 
thus. Faintness suggests the idea of greater distance; 
the moon appearing most faint in the horizon, sug- 
gests the idea of greater distance, arid, supposing the 
visual angle the same, that must suggest the idea of a 
greater tangible object. He does not suppose the 
visible extension to be greater^ but that the idea of a 
greater tangible extension is suggested, by the altera^ 
tion of the appearance of the visible extenision. He 
says, l.That which suggests the idea of greater mag- 
nitude, must be something perceived ; for what is not 
perceived can produce no? visible effect. 2. It must be 
something which is variable, because the nioon does 
not always appear, of the same magnitude in the 
horizon. 3. It cannot lie in the intermediate objects, 
they remaining the same ; also, when these objects are 
excluded from sight, it makes no alteration. 4. It 
cannot be the visible magnitude, because that is least 
in the horizon ; the cause, therefore, must lie in the 
visible appearance, which proceeds from the greater 
paucity of rays coming to the eye, producing ^am^- 
ness. Mr. Rouoning supposes that the moon appears 
furthest from us in the horizon, because the portion of 
the sky which we see, appears not an entire hemisphere, 


but only a portion of one; and in consequence of this, 
we judge the moon to be furthest from us in the 
horizon, and therefore to be then largest. Dr. Smithy 
in his Optics^ gives the same reason. He makes the 
apparent distance in the horizon to be to that in the 
zenith as 10 to 3, and therefore the apparent diameters 
in that ratio. The methods by which he estimated 
the apparent distances, may be seen in Vol. I. page 
65. The same circumstance also takes place in the 
sun, which appears much larger in the horizon than in 
the zenith. Also, if we take two stars near each 
other in the horizon, and two other stars near the 
zenith at the same angular distance from each other, 
the two former will appear at a much greater distance 
from each other, than the two latter. Upon this ac- 
count, people are, in general, very much deceived in 
estimating the altitudes of the heavenly bodies above 
the horizon, judging them to be much greater than 
they are. Dr. Smith found, that when a body was 
about 23° above the horizon, it appeared to be half 
way between the zenith and horizon, and therefore at 
that real altitude it would be estimated to be 45° high. 
The lower part of a rainbow also appears broader than 
the upper part. And this may be considered as an ar- 
gument that the phaenomenon cannot depend entirely 
upon the greater degree of faintness in the object when 
in the horizon, because the lower part of the bow fre- 
quently appears brighter than the upper part, at the 
same time that it appears broader. Also, this cause 
could have no effect upon the distance of the stars ; 
and as the difference of the apparent distance of the 
two stars, whose angular distance is the same, in the 
horizon and zenith, seems to be fully sufficient to ac- 
count for the apparent variation of the moon's diameter 
in these situations, it may be doubtful, whether the 
faintness of the object enters into any part of the 


Chap. XVII. 


(236.) The times of rotation of the sun, and planets, 
and the position of their axes, are determined from the 
spots which are observed upon their surfaces. The 
position of the same spot, observed at three different 
times, will give the position of the axis ; for tliree 
points of any small circle will determine it's situation, 
and hence we know the axis of the sphere which is 
perpendicular to it. The time of rotation may be 
found, either from observing the arc of the small 
circle described by a spot in any time, or by observing 
the return of a spot to the same position in respect to 
the earth. 

On the Rotation of the Sun. 

(237.) It is doubtful by whom the spots on the 
sun were first discovered. Scheiner, Professor of 
Mathematics in Ingolstadt, observed them in May, 
1611, and published an account of them in 1612, in 
a work entitled, Rosa Urs'ina. Galileo^ in the Pre- 
face to a work entitled, Tsforia, D'lmostrazioni^ 
intorno alle Macchte Solari, Roma, l6'l3, says, that 
being at Rome in 161I, he then showed the spots of 
the sun to several persons, and that he had spf)ken of 
them, some months before, to his friends at Florence. 
He imagmed them to adhere to the sun. Kepler, m 
his Ephemeris, says, that they were observed by the 
son of David Fabricius, who published an account of 
them in 1611. In the papers of Harriot, not yet 
printed, it is said, that spots upon the sun were ob- 
served on December 8, 1610. As telescopes were in 


use at that time, it is probable that each might make 
the discovery. Admitting these spots to adhere to the 
sun's body, the reasons for which we shall afterwards 
give, we proceed to show how the time of it's rotation 
may be found. 

(238.) M. CassiJii determined the time of rotation, 
from observing the time in which a spot returns to the 
same situation upon the disc, or to the circle of lati- 
tude passing through the earth. Let t be that interval 
of time, and let m be equal to the true motion of the 
earth in that time, and n equal to it's mean motion ; 
then 360° +m : 36o°+n :: t : the time of return if 
the motion had been uniform, and this, from a great 
number of observations, he determines to be 2'jd. I2h, 
20' ; now the mean motion of the earth in that time is 
27°. ;'. 8" ; hence, 360° + 27''. -7'. 8'' : 36o° :: 27d. I2h 
29' : 25d. 14h. 8', the time of rotation. Elem. 
d'Astron. p. 104. 

(239.) When the earth is in the nodes of the sun's 
equator, and consequently in it's plane, the spots ap- 
pear to describe straight lines : this happens about the 
beginning of June and December. As the earth re- 
cedes from the nodes, the path of a spot grows more 
and more elliptical, till the earth gets 90° from the 
nodes, which happens about the beginning of Sep- 
tember and March, at which time the ellipse has it's 
minor axis the greatest, and is then to the major axis, 
as the sine of the inclination of the ?olar equator to 

(240.) There has been a great difference of opinions 
respecting the nature of the solar spots. Scheiner sup- 
posed them to be solid bodies revolving about the sun, 
very near to it ; but as they are as long visible as they 
are invisible, this cannot be the case. Moreover, we 
have a physical argument against this hypothesis, 
which is, that most of them do not revolve about tlie 
sun in a plane passing through it's center, which they 
necessarily must, if they revolved, like the planets, 
about the sun. Galileo confuted Schelne7''s opinion, 


by observing that the spots were not permanent ; that 
they varied their figure; that they increased, and 
sometimes disappeared. He compared them to smoak 
and clouds. Hevelius appears to have been of the 
same opinion ; for in his Cometographla, p. 36o, 
speaking of the solar spots, he says, Hoec materia 
nunc ea ipsa est evaporatio et exhalatio [quia aliunde 
minime oriri potest) quae ex ipso corpore solis, ut 
supra ostensum est, expirafur et exhalatur. But the 
permanency of most of the spots is an argument 
against this hypothesis. M. de la Hire supposed them 
to be solid, opaque bodies, which swim upon the 
hquid matter of the sun, and which are sometimes en- 
tirely immersed. M. de la Lande supposes that the 
sun is an opaque body^ covered with a liquid fire, and 
that the spots arise from the opaque parts, like rocks, 
which, by the alternate flux and reflux of the liquid 
igneous matter of the sun, are sometimes raised above 
the surface. The spots are frequently dark in the 
middle, with an umbra about them ; and M. de la 
Lande supposes that the part of the rock which stands 
above the surface, forms the dark part in the center, 
and those parts which are but just covered by the ig- 
neous matter, form the umbra. Dr. f^il.son, Professor 
of Astronomy at Glasgow, opposes this hypothesis of 
M. c?e la Lande, by this argument. Generally speak- 
ing, the umbra immediately contiguous to the dark 
central part, or nucleous, instead of" being very dark, 
as it ought to be, from our seeing the immersed parts 
of the opaque rock through a thin stratum of the 
igneous matter, is, on the contrary, very nearly of the 
same splendour as the external surface, and the umbra 
grows darker the further it recedes from the nucleus; 
this, it must be acknowledged, is a strong argument 
against the hypothesis of M. de la Lande. Dr. fVilson 
further observes, that M. de la Z/a/7rtf<? produces no op- 
tical arguments in support of the rock standing above 
the surface of the sun. The opinion of Dr. If^ilson is, 
that the spots are excavations in the luminous matter 

i6g rotation of the sun. 

of the sun, the bottom of which forms the umbra. 
They who wish to see the arguments by which this is 
supported, must consult the Phil, Trans. 1/74 and 
1783. Dr. Halley conjectured that the spots are 
formed in the atmosphere of the sun. Dr. Herschel 
supposes the sun to be an opaque body, and that it 
has an atmosphere; and if some of the fluids which 
enter into it's composition should be of a shining 
brilhancy, whilst others are merely transparent, any 
temporary cause which may remove the lucid fluid will 
permit us to see the body of the sun through the 
transparent ones. Seethe Phil. Trans. 1795. Dr. 
Herschel, on April 19, 1779? saw a spot which mea- 
sured 1'. 8",o6in diameter, which is equal in length to 
more than 31000 miles ; this was visible to the naked 
eye. Besides the dark spots upon the sun, there are 
also parts of the sun, called Faculce, Lucili, &c. which 
are brighter than the general surface; these always 
abound most in the neighbourhood of the spots them- 
selves, or where spots recently have been. Most of 
the spots appear within the compass of a zone lying 
30° on each side of the equator; but on July 5, 178O, 
M. de la Lande observed a spot 40° from the equator. 
Spots which have disappeared, have been observed to 
break out again. The spots appear so frequently, that 
Astronomers very seldom examine the sun with their 
telescopes, but they see some ; Scheiner saw fifty at 
once. The following phaenomena of the spots are 
described by Scheiner and Hevelius. 

I. Every spot which has a nucleus, has also an 
umbra surrounding it. 

II. The boundary between the nucleus and umbra 
is always well defined. 

III. The increase of a spot is gradual, the breadth 
of the nucleus and umbra dilating at the same time. 

IV. The decrease of a spot is gradual, the breadth 
of the nucleus and umbra contracting at the same 


V. The exterior boundary of the umbra never con- 
sists of sharp angles, but is always curvilinear, how- 
ever irregular the outline of the nucleus may be. 

VI. The nucleus, when on the decrease, in many 
cases changes it's figure, by the umbra encroaching 
irregularly upon it. 

VII. It often happens, by these encroachments, 
that the nucleus is divided into two or more nuclei. 

VIII. The nucleus vanishes sooner than the umbra. 

IX. Small umbrae are often seen without nuclei. 

X. An umbra of any considerable size is seldom 
seen without a nucleus. 

XI. When a spot, consisting of a nucleus and 
umbra, is about to disappear, if it be not succeeded 
by a facula, or more fulgid appearance, the place it 
occupied is, soon after, not distinguishable from any 
other part of the sun's surface. 

On the Rotation of the Planets. 

(24 1 .) The Georgian is at so great a distance, that 
Astronomers, with their best telescopes, have not been 
able to discover whether it has any revolution about 
it's axis. 

(242.) Saturn was suspected by Cassini and Fato, 
in 16*83, to have a revolution about it's axis; for they 
one day saw a bright streak, which disappeared the 
next, when another came into view near the edge of 
it's disc; these streaks are cdWed Belts. In 1719» 
when the ring disappeared, Cassini saw it's shadow 
upon the body of the planet, and a belt on each side 
parallel to the shadow. When the ring was visible, 
he perceived the curvature of the belts was such as 
agreed with the elevation of the eye above the plane 
of the ring. He considered them as similar to our 
clouds floating in the air ; and having a curvature 
similar to the exterior circumference of the ring, he 
concluded that they ought to be nearly at the same 



distance from the planet, and that consequently the 
atmos}3here of Saturn extended to the ring. Dr. 
Hcrschel found that the arrangement of the belts 
aluays followed the direction of the ring ; thus, as the 
ring opened, the belts began to show an incurvature 
answering to it. And during his observations on 
June 19, 20, and 21, 1780, he saw the same spot in 
three different situations. He conjectured, therefore, 
that Saturn revolved about an axis perpendicular to the 
plane of it's ring. Another argument in support of 
this, is, that the planet is an oblate spheriod, having 
the diameter in the dn-ection of the ring to the dia- 
meter perpendicular to it, as about 11 : 10, according 
to Dr. Herschel; the measures were taken vvith a wire 
micrometer prefixed to his 20 feet reflector. The 
truth of his conjecture he has now verified, having 
determined thai Saturn revolves about it's axis in \0h. 
16', 0",4. Phil. Trans. 1794. The rotation is accord- 
ing to the order of the signs. 

(243.) Jupiter is observed to have belts, and also 
spots, by wftich the time of it's rotation can be very 
accurately ascertained. M. Cassim found the time of 
rotation to be 9^'- 5^', from a remarkable spot which he 
observed m l6(j-5. In October 1 69 1, he observed two 
bright spots almost as broad as the belts ; and at the 
end of the month he saw tvvo more, and found them 
to revolve in 9//. 51'; he also observed some other 
spots near Jupiter's equator, which revolved in ^h. 
50'; and, in general, he found that the nearer the 
spots were to the equator, the quicker they revolved. 
It is probable, therefore, that t'le spots are not upon 
Jupiter's surface, but in it's atmosphere; and for this 
reasoi) also, that several spots whicli apj)eared round at 
first, grew obl(^n<^ by degrees in a direction parallel to 
the belts, and divided themselves into two or tliree 
spots. M. Moraldi, from a great many observations 
of the spot observed by Cassini in 1665, found tlie 
time of rotation to be 9/i, 56'; and concluded that the 
spots had a dependence upon the contiguous belt, as 


the spot had never appeared without tlie belt, though 
the belt had without the spot. It continued to appear 
and disappear till l6^4, and was not seen any more 
till 17O8; hence, he concluded, that the spot was 
some effusion from the belt upon a fixed place of 
Jupiter's body, for it always appeared in the same 
place. Dr. Herschel found the time of rotation of 
different spots to vary ; and that the time of rotation 
of the same spot diminished ; for the spot observed in 
1788 revolved as follows. From February 25 to 
March 2, in ^h. 55'. 20" j from March 2 to the 14th, 
in ^h. 54'. 58" ; from April 7 to the 12th, in ^h. 51'. 
35". Also, from a spot observed in 1 799, it's rotation 
was, from April 14 to the 19th, in 9/^. 51'. 45"; from 
April 19 to the 23d, in Qh. 50'. 48". This, he ob- 
serves, is agreeable to the theory of equinoctial winds, 
as it may be some time before the spot can acquire the 
velocity of the wind ; and if Jupiter's spots should be 
observed in different parts of it's revolution to be ac- 
celerated and retarded, it would amount almost to a 
demonstration of it's monsoons, and their periodical 
changes. M. Schroeter makes the time of rotation 
gh. 55'. 36",6 ; he observed the same variations as 
Dr. Herschel. The rotation is according to the order 
of the signs. This planet is observed to be flat at it's 
poles. Dr. Pound measured the polar and equatorial 
diameters, and found them as 12 : 13. Mr. Short 
made them as 13 : 14. Dr. Bradley made them as 
12,5 ; 13,5. Sir /. Newton makes the ratio ^^ : 10^ 
by theory. The belts of Jupiter are generally parallel 
to it's equator, which is very nearly parallel to the 
ecliptic; they are subject to very great variations, 
both in respect to their number and figure ; some- 
times eight have been seen at once, and at other times 
only one ; sometimes they continue for three months 
without any variation, and sometimes a new belt has 
been formed in an hour or two. From their being 
subject to such changes, it is very probable that they 

L 2 


do not adhere to the body of Jupiter, but exist in it's 

(244.) Galileo discovered the phases of Mars; 
after which, some Itahans in l636, had an imperfect 
view of a spot. But in 1666, Dr. Hook and M. Cassini 
discovered some well-defined spots; and the latter 
determined the time of the rotation to be 24h. 40'. 
Soon after, M. Maruldi observed some spots, and 
determined the time of rotation to be 2Ah 39'. He 
also observed a very bright part near the southern 
pole, appearing like a polar zone; this, he says, has 
been observed for 60 years-, it is not of equal bright- 
ness, more than half of it being brighter than the rest; 
and that part which is least bright, is subject to great 
changes, and sometimes disappears. Something like 
this has been seen about the north pole. The rotation 
is according to the order of the signs. Dr. Herschel 
makes the time of a sidereal rotation to be 24//. 39'. 
2l"67, without the probability of a greater error than 
2"34. He proposes to find the time of a sidereal ro- 
tation, in order to discover, by future observations, 
whether there is any alteration in the time of the revo- 
lution of the earth, or of the p!anets, about tiieiraxes; 
for a cha> ge of either would thus be discovered. He 
chose Mars, because it's spots are permanent. See the 
Phil. T'rans. 178I. From further observations upon 
Mars, which he published in Phil. Trans. 1/84, he 
makes it's axis to be inclined to the ecliptic 59°. 42', 
and 61". 18' to it's orbit; and the north pole to be 
directed to 17°. 47' of Pisces upon the ei liptic, and 
19°. 28' on it's orbit. He makes the ratio of the di- 
ameters of Mars to be as 16 : 15. Dr. Maskelyne has 
carefully observed Mars at the timeof opposi ion, but 
could not perceive any difference in it's diameters. 
Dr. Hei^schel observes, that Mars has a considerable 

(245.) Galileo first discovered the phases of Verms 
in 161 1, and sent the discovery to William de" Medici ^ 


to communicate it to Kepler. He sent it in this 
cypher, //cec immaturce a me frustra leguntur, o, 1/ ; 
which put in order, is, Cyntklce figuras cemulatur 
jiiater amorian, that is, Venus emulates the phases of 
the moon. He afterwards wrote a letter to him, "giv- 
ing an account of the discovery, and explaining the 
cypher. In 1666, M. Casshii, at a time when Venus 
was dichotomized, discovered a bright spot upon it at 
the straight edge, like some of the bright spots upon 
the moon's surface ; and by observing it's motion, 
which was upon the edge, he found the sidereal time 
of rotation to be 23/^. 16'. In the year 1726, Bianchini 
made some observations upon the spots of Venus, and 
asserted the time of rotation to be 24^ days; that the 
north pole answered to the 10^^ degree of Aquarius, 
and was elevated from 15° to 20° above it's orbit; and 
that the axis continued parallel to itself. The small 
angle which the axis of Venus makes with it's orbit is 
a singular circumstance, and must cause a very great 
variety in the seasons. M. Cassini, the Son, has vin- 
dicated his Father, and shown, from Blanchims ob- 
servations being interrupted, that he might easily 
mistake different spots, for the same: and he con- 
cludes, that if we suppose the periodic time to be 23/?. 
20', it agrees equally with their observations; but if 
we take it 24^ days, it will not at all agree with his 
Father's observations. M. Schroeter has endeavoured 
to show that Venus has an atmosphere, from observing 
that the illuminated limb, when horned, exceeds a 
semicircle ; this he supposes to arise from the refrac- 
tion of the sun's rays through the atmosphere of Venus 
at the cusps, by which they appear prolonged. The 
cusps appeared sometimes to run 15°. 19' into the dark 
hemisphere; from which he computes, that the height 
of the atmosphere, to refract such a quantity of light, 
must be 15156 Paris ieei. But this must depend on 
the nature and density of the atmosphere, of which we 
are ignorant. Phil. Trans. 1792. He makes the time 
of rotation to be 23A, 21', and concludes, from his ob. 


servationSj tliat there are considerable mountains upon 
this planet, Phil. Trans. 1795- Dr. Herschel agrees 
with M. Schroeter, that Venus has a considerable 
atmosphere; but he has not made any observations, by 
which he can determine, either the time of rotation, or 
the position of the axis. Phil. Trans. 1793. 

(246.) The phases of Meixury are easily distin- 
guished to be like those of Venus ; but no spots have 
yet been discovered, by which we can ascertain whe- 
ther it has any rotation. 

(247.) The fifth satellite of Saturn was observed 
by M. Cassini for several years, as it went through the 
eastern part of it's orbit, to appear less and less, till it 
became invisible ; and in the western part to increase 
again. These phaenomena can hardly be accounted 
for, but by supposing some parts of the surface to be 
incapable of reflecting light, and therefore, when such 
parts are turned towards the earth, they appear to 
grow less, or to disappear. As the same appearances 
returned again when the satellite came to the same 
part of it's orbit, it aiFords an argument that the time 
of the rotation about it's axis is equal to the time of 
it's revolution about it's primary, a circumstance 
similar to the case of the moon and earth. See Dr. 
Herschets account of this in the Phil. Trans. 1792. 
The appearance of this satellite of Saturn is not always 
the same, and therefore it is probable that the dark 
parts are not permanent. Dr. Herschel has discovered 
that all the satellites of Jupiter have a rotatory motion 
about their axes, of the same duration with their 
respective periodic times about their primaries. Phil. 
Tram. 1797. 


Chap. XVIII. 


(248.) On January 8, 1610, Galileo discovered the 
four satellites of Jupiter, and called them Medicea 
Sidera, or Medicean Stars, in honour of the family of 
the Medici, his patrons. This was a discovery, very 
important in it's consequences, as it furnished a ready 
method of finding the longitudes of places, by means 
of their eclipses; the eclipses led M. Roemer to the 
discovery of the progressive motion of light; and 
hence Dr. Bradley was enabled to solve an apparent 
motion in the fixed stars, which could not otherwise 
have been accounted for, 

(249.) The satellites of Jupiter, in going from the 
west to the east, are eclipsed by the shadow of Jupiter, 
and as they go from east to west, they are observed to 
pass over it's disc; hence, they revolve about Jupiter, 
and in the same direction as Jupiter revolves about the 
sun. The three first satellites are always eclipsed, 
when they are in opposition to the sun, and the 
lengths of the eclipses are found to be different at dif- 
ferent times : but sometimes the fourth satellite passes 
through opposition without being eclipsed. Hence it 
appears, that the planes of the orbits do not coincide 
with the plane of Jupiter's orbit, for, in that case they 
would always pass through the center of Jupiter's 
shadow, and there would always be an eclipse, and of 
the same, or very nearly the same, duration, at every 
opposition to the sun. As the planes of the orbits 
which they describe sometimes pass through the eye, 
they will then appear to describe straight lines passing 


through the center of Jupiter ; but at all other times 
they will appear to describe ellipses, of which Jupiter 
is the center. 

On the Periodic Times, and distances of Jupiter's 

(250.) To get the mean times of their synodic re- 
volutions, or of their revolutions in respect to the sun, 
observe, when Jupiter is in opposition, the passage of 
a satellite over the body of Jupiter, and note the time 
when it appears to be exactly in conjunction with the 
center of Jupiter, and that will be the time of con- 
jmiction with the sun. After a considerable interval 
of time, repeat the same observation, Jupiter being in 
opposition, and divide the interval of time by the 
number of conjunctions with the sun in that interval, 
and you get the time of a synodic revolution of the 
satellite. This is the revolution which we have occa- 
sion principally to consider, it being that on which the 
eclipses depend. But, owing to the equation of Jupi- 
ter's orbit, this will not give the mean time of a synodic 
revolution, unless Jupiter was at the same point of it's 
orbit at both observations ; otherwise, we must pro- 
ceed thus. 

(251.) Let J IP R be the orbit of Jupiter, aS* the 
sun in one focus, and F the other focus ; and as the 
excentricity of the orbit is small, the motion about F 
may be considered (169) as uniform. Let Jupiter be 
in it's aphelion at A in opposition to the earth at T; 
and L a satellite in conjunction ; and let / be the 
place of Jupiter at it's next opposition with the earth 
at D, and the satellite in conjunction at G. Then, 
if the satellite had been at O, it would have been in 
conjunction with F, or in mean conjunction ; therefore 
it must describe the angle FIS before it comes to the 
mean conjunction, which angle is (169) the equa- 
tion of the orbit, according to the simple elliptic 



hypothesis, which may be here used, as the excentricity 
of the orbit is but small ; the angle FIS therefore mea- 
sures the difference between the mean synodic revolu- 
tions in respect to F, and the synodic revolutions in 

respect to the sun S. If, therefore, n be the number 
of revolutions which the satellite has made in respect 
to the sun, n x 36o° — *S7/^=the revolutions in respect 
to F; hence, n x 36o°—SIF: 36o° :: the time between 
the two oppositions : the tmie of a mean synodic revo- 
lution about the sun. 

(252.) As the satelHte is at O at the mean conjunc- 
tion, and at G when in conjunction with the sun, it is 
manifest, that if the angle FJS continued the same, 
the time of a revolution in respect to S would be equal 
to the time in respect to F, or to the time of a mean 
synodic revolution ; hence, the difference between the 
times of any two successive revolutions in respect to S 
and irrespectively, is as the variation of the angle FIS^ 
or variation of the equation of the orbit. When 
Jupiter is at j4, the equation vanishes, and the times 
of the two conjunctions at F and 8 coincide. When 
Jupiter comes to /, the mean conjunction at O hap]}eij 3 


after the true conjunction at G, by the time of de- 
scribing the angle SIF, the equation of Jupiter's orbit. 
This is what Astronomers call the^r^Y mequalily ; 
and by this inequality of the intervals of the times of 
the true conjunctions, the times of the eclipses of the 
satellites are affected. 

(253.) But as a conjunction of the satellite may 
not often happen exactly at the time when Jupiter is 
in opposition, the time of a mean revolution may be 
found, when he is out of opposition, thus. Let H be 
the earth when the satellite is at Z in conjunction with 
Jupiter at R; and let ^' be another position of the 
earth when the satellite is at C in conjunction with 
Jupiter at /; and produce RH, IF, to meet in M; 
then the motion of Jupiter about the earth, in this 
interval, is the same as if the earth had been fixed at 
M. Now the difference between the true and mean 
motions of Jupiter is RFI- RMI= FIM+FRM, 
which shows how much the number of mean revolu- 
tions, in respect to jP, exceeds the same number of 
apparent revolution*; in respect to the earth j hence, 
nX36o° - FIM- FRM : 3b"0° :: the time between 
tiie observations : the time of a mean synodic revolu- 
tion of thv" satellite. If C and Z lie on the other side 
of O and J', the angles FJM, FRM, must be added to 
77 y 360° ; and if one lie on one side, and the other on 
tlie Dther, one must be added and the other subtracted, 
according to the circumstances. 

(234.) As it is difficult, from the great brightness 
of .fupi'er, to determine accurately the time when the 
saUlhle is in conjunction with the center of Jupiter as 
iv passes over it's di;^c, the time of conjunction is deter- 
mined by obs'-rving it's enlrance uptm the disc, and 
it's going off: but as tl is cannot be detertnined with 
so.u'uch accuracy as. the tunes of i?imicrsion into the 
shadow of Jupiter, and emersion from it, the time of 
r ' -M-'tion Can be most accurati ly determined from 
..; jL,^t / be the center of Jupiter's shadow FG, 


Nmt the orbit of a satellite, N the node of the satel- 
lite's orbit upon the orbit of Jupiter; draw Iv perpen- 

dicular to 77V, and Ic to Nt ; and when the satellite 
comes to i;, it is in conjunction* with the sun. Now 
both the immersion at m and emersion at t of the 
second, third, and fourth satellites may sometimes be 
observed, the middle point of time between which, 
gives the time of the middle of the eclipse at c ; and 
by calculating cv, from knowing the angle A^and NI, 
we get the time of conjunction at v. If both the im- 
mersion and emersion cannot be observed, take the 
time of either, and after a very long interval of time, 
when an eclipse happens as nearly as possible in the 
same situation in respect to the node, take the time of 
the same phaenomenon, and from the interval of these 
times you will get the time of a revolution. By these 
different methods, M. Cassini found the times of the 
mean synodic revolutions of the four satellites to be as 
follows : 


Second. Third. Fourth. 




(256.) Hence it appears, that 247 revolutions of the 
first satellite are performed in 437^. 3^. 44' ; 123 re- 

* A satellite is said to be in conjunction, both when it is bet'veen 
the Sun and Jupiter, and when it is opposite to the Sun ; the latter 
may be called superior, and the former inferior conjunction. 



volutions of the second, in 437^. Sh. 41'; 6l revolu- 
tions of the third, in 43^d. 3h. 35', and 26 revolutions 
of the fourth, in 435^. 14^. 13'. Therefore, after an 
interval of 437 ^^y^) the three first satellites return to 
their relative situations within nine minutes. 

(257.) In the return of the satellites to their mean 
conjunction, they describe a revolution in their orbits, 
together with the mean angle a° described by Jupiter 
in that time; therefore, to get the periodic time of 
each, we must say, 360° 4- «° : 36o° :: time of a 
synodic revolution : the time of a periodic revolution; 
hence, the periodic times of each are ; 





1*^.1 8\27'.33" 




(258.) The distances of the satellites from the 
center of Jupiter may be found at the time of their 
greatest elongations, by measuring with a micrometer, 
at that time, their distances from the center of Jupiter, 
and also the diameter of Jupiter, by which you get 
their distances in terms of the diameter. Or it may 
be done thus. When a satellite passes over the middle 
of the disc of Jupiter, observe the whole time of it's 
passage, and then, the time of a revolution : the time 
of it's passage over the disc :: 36o° : the arc of it's 
orbit corresponding to the time of it's passage over the 
disc; hence, the sine of half that arc : radius :: the 
semidiameter of Jupiter : the distance of the satellite. 
Thus M. Cassini determined their distances in terms 
of the semidiameter of Jupiter to be, of thejirsty 5,67 ; 
of the *ecowc?, 9 ; of the third, 14,38; and of the 
fourth, 25,3. 

(259.) Or, having determined the periodic times, 
and the distance of one satellite, the distances of the 
others may be found from the proportion of the squares 
of the periodic times being as the cubes of their dis- 
tances. Mr. Pound, with a telescope 15 feet long, 


found at the mean distance of Jupiter from the earth, 
the greatest distance of the fourth satelHte to be 8'. 
16"; and by a telescope 123 feet long;, he found the 
greatest distance of the third to be 4'. 42" ; hence, the 
greatest distance of the second appears to be 2'. 56'". 
47'", and of the first, l'. 5 1". 6'". Now the diameter 
of Jupiter, at it's mean distance, was determined, by- 
Sir /. NeivtoUy to be 3fl ; hence the distances of 
the sateUites, in terms of the semidiameter of Jupiter, 
come out 6,966; 9,494; 15,141, and 26,63 re- 
spectively. Prln. Math. Lib. ter. Phcen. 

(260.) Hence, by knowing the greatest elongations 
of the satellites in minutes and seconds, we get their 
distances from the center of Jupiter, compared with 
the mean distance of Jupiter from the earth, by say- 
ing, the sine of the greatest elongation of the satelhte : 
radius :: the distance of the satellite from Jupiter • the 
mean distance of Jupiter from the earth. 

On the Eclipses of Jupiter's Satellites. 

(261.) Let 8 be the sun, ^Fthe orbit of the earth, 
/Jupiter, ate the orbit of one of it's satellites. When 
the earth is at E before the opposition of Jupiter, the 
spectator will see the immersion at a\ but if it be the 
first satellite, upon account of it's nearness to Jupiter, 
the immersion is never visible, the satellite being then 
always behind the body of Jupiter; the other three 
satellites may have both tiieir immersions and emer- 
sions visible ; but this will depend upon the position 
of the earth. When the earth comes to i^ after oppo- 
sition, we shall then see the emersion of the first, but 
can tiien never see the immersion ; but we may see 
both the emersion and immersion of the other three. 
Draw EIr ; then at, the distance of the center of the 
shadow from the center of Jupiter, referred to the 
orbit of the satellite, is measured at Jupiter by sr, or 
the angle sir, or the angle ELS. The satellite may 
be hidden behind the body at r without being eclipsed. 


which is called an Occultation. When the earth is 
at E, the conjunction of the satellite happens later at 

the earth than at the sun ; but when the earth is at F, 
it happens sooner. 

(262.) The diameter of the shadow of Jupiter, at 
the distance of any of the satellites, is best found by 
observing the time of an eclipse when it happens at 
the node, at which time the satelhte passes through 
the center of the shadow ; for tli^ time of a synodic 
revolution : the time the satellite is passing through 
the center of the shadow :: 36o° : the diameter of the 
shadow in degrees. But when the first and second 
^atelhtes are in the nodes, the immersion and emersion 
cannot both be seen. Astronomers, therefore, compare 
the immersions some days before the opposition of 
Jupiter with the emersions some days after, and then, 
knowing how many synodic revolutions have been 
made, they get the time of the transit through the 
shadow, and thence the corresponding degrees. But 
on account of the excentricity of some of the orbits, 


the times of the cen'^ral transit must vary ; forexample, 
the second satelHte is sometimes found to be ih. 50' 
in passing through the center of tlie shadow, and, 
sometimes 2/?. 54'; this indicates an excentricity. 

(263.) The duration of the echpses being very une- 
qual, shows tliat the orbits are inclined to the orbit of 
Jupiter ; sometimes the fourth satellite passes through 
opposition without suffering an eclipse. The duration 
of the eclipses must depend upon the situation of the 
nodes in respect to the sun, just the same as in a lunar 
eclipse ; when the line of the nodes passes through 
the sun, the satellite will pass through the center of 
the shadow ; but as Jupiter revolves about the sun, 
the line of the nodes will be carried out of conjunction 
with the sun, and the time of the eclipse will be 
shortened, as the satellite will then describe only a 
chord of a section of the shadow instead of the 

On the Rotation of the Satellites of Jupiter, 

(264.) M. Cassini suspected that the satellites had 
a rotation about their axes, as sometimes in their pas- 
sage over Jupiter's disc tliey were visible, and at other 
times not ; he conjectm'ed, therefore, that they had 
spots upon one side and not on the other, and that 
they were rendered visible in their passage when the 
spots were next to the earth. At different times also 
they appear of different magnitudes and of different 
brightness. The fourth appears generally the smallest, 
but sometimes the greatest ; and the diameter of it's 
shadow on Jupiter appears sometimes greater than the 
satellite. The third also appears of a variable magni- 
tude, and the like happens to the other two. Mr. 
Pound also observed, that they appear more luminous 
at one time than another, and therefore he concluded 
that they revolve about their axes. Dr. Herschel has 
discovered that they all revolve about their axes, in the 

17^ Saturn's satellites. 

times in which they respectively revolve about 

On the Satellites of Saturn. 

(265.) In the year \6bb, Huygens discovered the 
fourth satelHte of Saturn ; and pubHshed a table of 
it's mean motion in 1659. In 1671, M. Cassini dis- 
covered the fifth, and the third in 1672 ; and in l684, 
the first and second ; and afterwards pubhshed Tables 
of their motions. He called them Sidera Lodoicea, 
in honour of Louis le Grand, in whose reign, and ob- 
servatory, they were discovered. Dr. Halley found, 
by his own observations in l682, that Huygens's 
Tables had considerably run out, they being about 15" 
in 20 years too forward, and therefore he composed 
new Tables from more correct elements. He also re- 
formed M. CassinVs Tables of the mean motions ; and 
about the year 1720, published them a second time, 
corrected from Mr. Pound's observations. He observes, 
that the four innermost satellites describe orbits very 
nearly in the plane of the ring, which, he says, is, as 
to the sense, parallel to the equator ; and that the 
orbit of the fifth is a little inclined to them. The 
following Table contains the periodic times of the 
five satellites, and their distances in semidiameters of 
the ring, as determined by Mr. Pound, with a micro- 
meter fitted to the telescope given by Huygens to 
the Royal Society. Mr. Pound first measured the 
distance of the fourth, and then deduced the rest from 
the proportion between the squares of the periodic 
times and cubes of their distances, and these are found 
to agree with observations. 




by Pound. 

Dist. in 

semid. oj 

Ring, by 


Dist. in 

semid. of 

Saturn by 


Dist. in 

semid. of 

Ring by 


Dist. at 

the mean 
dist. of 





1 ' * 



2. 17. 41. 22 




0. 56 


4. 12. 25. 12 




1. 18 







79. 7.49. 




8. 42,5 

The last column is from Cassini; but Dr. Het^schel 
makes the distance of the fifth to be 8'. 3l",97, which 
is probably more exact. In this and the two next 
Tables, the satelhtes are numbered from Saturn as 
they were before the discovery of the other two. 

On June 9, 1749, at loh. Mr. Pound found the 
distance of the fourth satellite to be 3'. 7" with a 
telescope of 123 feet, and an excellent micrometer 
fixed to it ; and the satellite was at that time very near 
it's greatest eastern digression. Hence, at the mean 
distance of the earth from Saturn, that distance be- 
comes 2'. 58",21 ; Sir /. Newtoti makes it 3'. 4". 

(266.) The periodic times are found as for the 
satellites of Jupiter (251). To determine these, M. 
Cassini chose the time when the semi-minor axes of 
the eclipses which they describe, were the greatest, as 
Saturn was then 90° from their node, because the 
place of the satellite in it's orbit is then the same as 
upon the orbit of Saturn ; whereas in every other case, 
it would be necessary to apply the reduction in order 
to get the place in it's orbit. 

(267,) As it is difficult to see Saturn and tiie satel- 
lites at the same time in the field of view of a telescope, 
their distances have sometimes been measured, by ob- 
serving the time of the passage of the body of Saturn 



over a wire adjusted as an hour circle in the field of the 
telescope, and the interval between the times when 
Saturn and the satellite passed. From comparing the 
periodic times and distances, M. Cassini observed that 
Kepler^ Rule (162) agreed very well with observa- 

(26s.) By comparing the places of the satellites 
with the ring in difierent points of their orbits, and 
the greatest minor axes of the ellipses which they 
appear to describe, compared with the major axes, the 
planes of the orbits of the first four are found to be 
very nearly in the plane of the ring, and therefore are 
inclined to the orbit of Saturn about 30" ; but the 
orbit of the fifth, according to M. Cassini the Son, 
makes an angle with the ring of about 1 5 degrees. 

(269.) M. Cassini places the node of the ring, and 
consequently the nodes of the four first satellites, in 5^ 
22'^ upon the orbit of Saturn, and 5^ 21° upon the 
ecliptic. M. Hay gens had determined it to be in 5^ 
20\ 30'. M. MirakU, in 17 16, determined the longi- 
tude of the node of the ring upon the orbit of Saturn 
to be b\ 19°. 48'. 30"; and upon the ecliptic to be 5^ 
16°. 20'. The node of the fifth satellite is placed by 
M. Cassini in 5^ 5" upon the orbit of Saturn. M. 
de la Lande makes it 5*. O". 2/'. From the observa- 
tion of M. Bernard, at Marseilles, in 1787, it appears 
that the node of this satellite is retrograde. 

(270.) Dr. Halley discovered that the orbit of the 
fourth satellite was excentric ; for, having found it's 
mean motion, he discovered that it's place by observa- 
tion was at one time 3" forwarder than by his calcula- 
tions, and at other observations it was 2°. 30' behind ; 
this indicated an excentricity ; and he placed the line 
of the apsides in lO^. 22% Phil. Trans. N°. 145. 

Saturn's satellites. 


Tables of their Revolutions and Mean Motions, 
according to M. de la Lande. 


Diurnal Motion. 

Motion in 365 Days. 


6^ lo^4l^53" 

4\ 4°. 44'. 42" 


4. 11. 32. 6 

4. 10. 15. 19 


2. 19. 41. 25 

9- 16. 57. 5 


0. 22. 34. 38 

10. 20. 39. 37 


0. 4. 32. 17 

7. 6. 23. 37- 


Periodic Revolution. 

Sj/7iodic Revolution. 





2. 17. 44. 51,177 

2. 17. 45. 51,013 


4. 12. 25. 11,100 

4. 12. 27. 55,239 


15. 22. 41. 16,022 

15. 23. 15. 23,153 


79. 7. 53. 42,772 

79. 22. 3. 12,883 

(271.) M. Cassini observed, that the fifth sateUite 
disappeared regularly for about half it's revolution, 
when it was to the east of Saturn ; from which he 
concluded, that it revolved about it's axis ; he after- 
wards, however, doubted of this. But Sir /. Newton^ 
in his Pri?icipia, Lib. III. Prop. 17, concludes from 
hence, that it revolves about it's axis, and in the same 
time that it revolves about Saturn ; and that the varia- 
ble appearance arises from some parts of the satellite 
not reflecting so much light as others. Dr. Herschel 
has confirmed this, by tracing regularly the periodical 
ciiange of light througli more than ten revolutions, 

M 2 

180 Saturn's satellites. 

which he found, in all appearances, to be cotemporary 
with the return of the satelHte to the same situation in 
it's orbit. This is further confirmed by some observa- 
tions of M. Bernard, at Marseilles, in 1787 ; and is a 
remarkable instance of analogy among the secondary 

(272.) These are all the satellites which were 
known to revolve about Saturn, till the year 1789. 
when Dr. //er^cAe/, ina Paper in the PJiil. Trans, for 
that year, announced the discovery of a sixth satellite, 
interior to all the others^ and promised a further ac- 
count in another Paper. But in the intermediate time 
he discovered a seventh satellite, interior to the sixth ; 
and in a Paper upon Saturn audit's ring, in the PhiL 
Trans. 1790? he has given an account of the discovery; 
with some of the elements of their motions. He 
afterwards added Tables of their motions. 

(273.) After his observations upon the ring, he 
says, he cannot quit the subject without mentioning 
his own surmises, and that of several other Astrono- 
mers, of a supposed roughness of the ring, or inequality 
in the planes and inclinations of it's flat sides. This 
supposition arose from seeing luminous points on it's 
boundaries^ projecting like the moon's mountains ; or 
from seeing one arm brighter or longer than the other ; 
or even from seeing one arm when the other was 
invisible. Dr. Herschel was of this opinion, till he 
^Tiw one of these points move off the edge of the ring 
in the form of a satellite. With his 20 feet telescope 
he suspected that he saw a sixth satellite ; and on 
August 19, 1787; marked it down as probably being 
one ; and having finished his telescope of 40 feet focal 
length, he saw six of it's satellites the moment he di- 
rected his telescope to the planet. This happened on 
August 28, 1789. The retrograde motion of Saturn 
was then nearly 4'. 30" in a day, which made it very 
easy to ascertain whether the stars he took to be satel- 
lites, were really so ; and in about two hours and an 
half after, he found that the planet had visibly carried 


them all away from their places. He continued his 
observations, and on September 17, he discovered the 
seventh satellite. These two satellites lie within the 
orbits of the other five. Their distances from the 
center of Saturn are 36",7889, and 28",6689 ; and 
their periodic times are id. 8h. 33'. 8",9, and22/i. 37'. 
22",9. The planes of the orbits of these satellites lie 
so near to the plane of the ring, that the diiFerence 
cannot be perceived. 

On the Satellites of the Georgian, 

(274.) On January 11, I787, as Dr. Herschel was 
observing the Georgian, he perceived, near it's disc, 
some very small stars, whose places he noted. The 
next evening, upon examining them, he found that 
two of them were missing. Suspecting, therefore, that 
they might be satellites which had disappeared in con- 
sequence of having changed their situation, he con- 
tinued his observations, and in the course of a month 
discovered them to be satellites, as he had first con- 
jectured. Of this discovery he gave an account in the 
Phil. Trans. 1787. 

(275.) In the Phil. Trans. 1788, he published a 
further account of this discovery, containing their pe- 
riodic times, distances, and positions of their orbits, so 
far as he was then able to ascertain them. The most 
convenient method of determining the periodic time of 
a satellite, is, either from it's eclipses, or from taking 
it's position in several successive oppositions of the 
planet ; but no eclipses have yet happened since the 
discovery of these satellites, and it would be waiting a 
long time to put in practice the other method. Dr. 
Herschel, therefore, took their situations whenever he 
could ascertain them with some degree of precision, 
and then reduced them, by computation, to such situa- 
tions as were necessary for his purpose. In comput- 
ing the periodic times, he has taken the synodic 
revolutions, as the positions of their orbits, at the 


times when their situations were taken, were not suffi- 
ciently known to get very accurate sidereal revokitions. 
The mean of several results gave the synodic revolu- 
tion of the first satellite 8^/. l^jh- i'. 19 ",3, and of the 
second \3d. llh. b'. l",5. The results, he observes, 
of which these are a mean, do not much differ among 
themselves, and therefore the mean is probably toler- 
abl}'' accurate. The epochs from which their situations 
may, at any time^ be computed are, for the^r*^, Oct. 
19j "^l^l, at 19/i. 1 1'. 28"; and for the second, at l/A. 
22'. 40"; at which times they were 'jQ''. 43' north, 
following the planet. 

(276.) The next thing to be determined, in the 
elements of the satellites, was their distances from the 
planet; to obtain which, he found one distance by 
observation, and then the other from the periodic times 
(Article 162). Now, in attempting to discover the 
distance of the second, the orbit was seemingly ellip- 
tical. On March 18, 1787, at 8A. 2'. 50", he found 
the elongation to be 46",46, this being the greatest of 
all the measures he had taken. Hence, at the mean 
distance of the Georgian from the earth, this elonga- 
tion will be 44",23. Admitting, therefore, for the 
present, says Dr. Herschel, that the satellites move in 
circular orbits, we may take 44",23 for the true 
distance, without much error; hence, as the squares 
of the periodic times are as the cubes of the distances, 
the distance of the first satellite comes out 33",09, 
The synodic revolutions were here used instead of the 
sidereal, which will make but a small error. 

(277') The last thing to be done, was to determine 
the inclinations of the orbits, and places of their nodes. 
And here a difficulty presented itself which could not 
be got over at the time of his first observation ; for it 
could not then be determined which part of the orbit 
was inclined to the earth, and which /rom it On the 
two different suppositions, therefore, Dr. Herschel has 
computed the inclinations of the orbits, and the places 
of their nodes, and found them as follows. The orbit 


of the second satellite is inclined to the ecliptic 99''. 
43'. 53",3 or 81°. 6'. 44",4 ; it's ascending node upon 
the ecliptic is in 5^ 18% or 8^. 6° ; and when the planet 
comes to the ascending node of this satellite, which 
will happen about the year 1818, the northern 
half of the orbit will be turned towards the east or 
west, at the time of it's meridian passage. M. de la 
Lambre makes the ascending node in 5^ 21°, or 8*. 9°, 
from Dr. Herschel's observations. The situation of 
the orbit of the first satellite does not materially differ 
from that of the second. The light of the satellites is 
extremely faint ; the second is the brightest, but the 
difference is small. The satellites are probably not 
less than those of Jupiter. There will be eclipses 
of these sateUites about the year 1818, when 
they will appear to ascend through the shadow of the 
planet, in a direction almost perpendicular to the 

Since these discoveries were made, Dr. Herschel has 
discovered four more satellites of the Georgian, and 
found that their motions are all retrograde. Phil. 
Trans. 1798. 


Chap. XIX. 


(278.) Galileo was the first person who observed 
any thing extraordhiary in Saturn. The planet ap- 
peared to him like a large globe between two small 
ones. In the year 1610 he announced this discovery. 
He continued his observations till l6l2, when he was 
surprised to find only the middle globe ; but some 
time after he again discovered the globes on each 
side, which, in process of time, appeared to change 
their form ; sometimes appearing round, sometimes 
oblong like an acorn, sometimes semicircular, then 
with horns towards the globe in the middle, and grow- 
ing, by degrees, so long and wide as to encompass it, 
as it were, with an oval ring. Upon this, Huygens 
set about improving the art of grinding object glasses ; 
and made telescopes which magnified two or three 
times more than any which had been before made, 
with which he discovered very clearly the ring of 
Saturn ; and having observed it for some time, he 
published the discovery in 1656. He made the space 
between the globe and the ring equal to, or rather 
bigger than the breadth of the ring ; and the greater 
diameter of the ring to that of the globe as 9 to 4. 
But Mr. Pound, with a micrometer applied to 
Huygen''?, telescope of 123 feet long, determined the 
ratio to be as 7 to 3. Mr. JVIiiston in his Memoirs of 
the Life of Dr. Clark, relates, that the Doctor's Father 
once saw a fixed star between the ring and the body 
of Saturn. In the year 1675, M. Cassini saw the 
ring, and observed upon it a dark elliptical line, divid- 
ing it, as it were, into two rings, the inner of which 


appeared brighter than the outer. He also observed 
a dark belt upon the planet, parallel to the major axis 
of the ring. Mr. Hadley observed that the outer part 
of the ring seemed narrower than the inner part, and 
that the dark line was fainter towards it's upper edge ; 
he also saw two belts, and observed the shadow of the 
ring upon Saturn. In October, 1714, when the plane 
of the ring very nearly passed through the earth, and 
was approaching it, M. Maraldi observed, that while 
the arms were decreasing both in length and breadth, 
the eastern arm appeared a little larger than the other 
for three or four nights, and yet it vanished first, for, 
after two nights interruption by clouds, he saw the 
western arm alone. This inequality of the ring made 
him suspect that it was not bounded by exactly 
parallel planes, and that it turned about it's axis. But 
the best description of this singular phaenomenon is 
that given by Dr. Herschel, in the Phil. Trans. 1790> 
who, by his extraordinary telescopes, has discovered 
many circumstances which had escaped all other ob- 
servers. We shall here give the substance of his 

(279-) The black disc, or belt upon the ring of 
Saturn, is not in the middle of it's breadth ; nor is the 
ring subdivided by many such lines, as has been re- 
presented by some Astronomers ; but there is one ^ 
single, dark, considerable broad line, belt, or zone, 
which he has constantly found on the north side of 
the ring. As this dark belt is subject to no change 
whatever, it is probably owing to some permanent 
construction of the surface of the ring. This con- 
struction cannot be owing to the shadow of a chain of 
mountains, since it is visible all round on the ring; 

* In a Paper in the Phil. Trans. 1792, Dr. Hcrschel observes, 
that, " since the year 1774 to the present time, I can find only- 
four observations where any other black division of" the ring is 
njientioned, than the one which 1 have constantly observed ; these 
were all in June, 1780.'' 


for at the ends of the ring there could be no shade ; 
and the same argument will hold against any supposed 
caverns. It is moreover pretty evident, that this dark 
zone is contained between two concentric circles, as 
all the phaenomena answer to the projection of such a 
zone. The matter of the ring is undoubtedly no less 
solid than the planet itself; and it is observed to cast 
a strong shadow upon the planet. The light of the 
ring is also generally brighter than that of the planet; 
for the ring appears sufficiently bright, when the 
telescope affords scarcely light enough for Saturn. Dr. 
Herschel next takes notice of the extreme thinness of 
the ring. He frequently saw the first, second, third, 
fourth, and fifth satellites pass before and behind the 
ring, in such a manner that they served as an excel- 
lent micrometer to measure it's thickness by. It may 
be proper to mention a few instances, as they serve 
also to solve some phaenomena observed by other 
Astronomers, without having been accounted for in 
any manner that could be admitted consistently with 
other known facts. July 18, 1789, at \6h. 41'. 9" 
sidereal time, the third satellite seemed to hang upon 
the following arm, declining a little towards the north, 
and was seen gradually to advance upon it towards the 
body of Saturn ; but the ring was not so thick as the 
lucid point. July 23, at \^h. 41'. 8", the fourth 
satellite was a very little preceding the ring, but the 
ring appeared to be less than half the thickness of the 
satellite. July 27, at 20h. 15'. 12", the fourth satellite 
was about the middle, upon the following arm of the 
ring, and towards the south ; and the second at the 
farther end, towards the north ; but the arm was 
thinner than either. August 29, at 22/z. 12'. 25", the 
fifth satellite was upon the ring, near the end of the 
preceding arm, and the thickness of the arm seemed 
to be about ^ or J of the diameter of the satellite, 
which, in the situation it then was, he took to be less 
than one-second in diameter. At the same time the 
first appeared at a little distance following the fifth, 


in the shape of a bead upon a thread, projecting on 
both sides of the same arm ; hence, the arm is thinner 
than the first, which is considerably smaller than the 
second, and a little less than the third. October l6, 
he followed the first and second satellites up to the 
very disc of the planet ; and the ring, which was ex- 
tremely faint, did not obstruct his seeing them gra- 
dually approach the disc. These observations are 
sufficient to show the extreme thinness of the ring. 
But Dr. Herschel further observes, that there may be 
a refraction through an atmosphere of the ring, by 
which the satellites may be lifted up and depressed, so 
as to become visible on both sides of the ring, even 
though the ring should be equal in thickness to the 
smallest satellite, which may amount to 1000 miles. 
From a series of observations upon luminous points of 
the ring, he has discovered that it has a rotation about 
it's axis, the time of which is \0h. 32'. 13l'4. 

(280.) The ring is invisible* when it's plane passes 
through the sun, or the earth, or between them ; in 
the first case, the sun shines only upon it's edge, 
which is too thin to reflect sufficient light to render it 
visible ; in the second case, the edge only being op- 
posed to us, it is not visible, for the same reason ; in 
the third case, the dark side of the ring is exposed to 
us, and therefore the edge being the only luminous 
part which is towards the earth, it is invisible on the 
same account as before. Observers have difl^ered 10 
or 12 days in the time of it's becoming invisible, owing 
to the dift'erence of the telescopes, and of the state of 
the atmosphere. Dr. Herschel observes that the ring 
was seen in his telescope, when we were turned to- 
wards the unenlightened side; so that he either saw 
the light reflected from the edge, or else the reflection 

* The disappearance of the ring is only with the telescopes in 
common use among Astronomers; for Dr. Herschel, with his large 
telescopes, has been able to see it in every situation. He thinks 
the edge of the ring is not flat, but spherical, or spheroidical. 


of the light of Saturn upon the dark side of the ring, 
as we sometimes see the dark part of the moon. He 
cannot, however, say which of the two might be the 
case; especially as there are very strong reasons to 
think, that the edge of the ring is of such a nature as 
not to reflect much light. M. de la Lande thinks that 
the ring is just visible with the best telescopes in com- 
mon use, when the sun is elevated 3' above it's plane, 
or three days before it's plane passes through the sun; 
and when the earth is elevated 2'. 20" above the plane, 
or one day from the earth's passing it. 

(281.) In a paper in the Phil. Trans. 1790, Dr. 
//er^c^^/ ventured to hint at a suspicion that the ring 
was divided; this conjecture was strengthened by fu- 
ture observations, after he had an opportunity of seeing 
both sides of the ring. His reasons are these : l . The 
black division, upon the southern side of the ring, is in 
the same place, of the same breadth, and at the same 
distance from the outer edge, that it always appeared 
upon the northern side. 2. With his seven feet re- 
flector and an excellent speculum, he saw the division 
on the ring, and the open space between the ring and 
the body, equally dark, and of the same colour with 
the heavens about the planet. 3. The black division 
is equally broad on each side of the ring. From these 
observations, Dr. Herschel thinks himself authorised 
to say, that Saturn has two concentric rings, situated 
in one plane, which is probably not much inclined to 
the equator of the planet. The dimensions of the 
rings are in the following proportions, as nearly as they 
could be ascertained. 


Inside diameter of the smaller ring - 5900 

Outside diameter ------ 7510 

Inside diameter of the larger ring - 7740 

Outside diameter ------ 8300 

Breadth of the inner ring - - . - §05 

Breadth of the outer ring - - - - 280 

Breadth of the space between the rings 115 


In the Mem. de VAcad. at Paris, 1787:. M. de la 
Place supposes that the ring may have many divi- 
sions ; but Dr. Herschel remarks, that no observations 
will justify this supposition. 

(282). From the mean of a great many measures of 
the diameter of the larger ring. Dr. Herschel m?ikes it 
46'",677 at the mean distance of Saturn. Hence, it's 
diameter : the diameter of the earth :: 25,8914 : 1. 
From the above proportion, therefore, the diameter of 
the ring must be 204883 miles ; and the distance of 
the two rings 2839 miles. 

(283.) The ring being a circle, appears elliptical 
from it's oblique position ; and it appears most open 
when Saturn is 90° from the nodes of the ring upon 
the orbit of Saturn, or when Saturn's longitude is 
about 2'. 17°, and 8'. 17°. In such a situation, the 
minor axis is extremely nearly equal to half the 
major, when the observations are reduced to the sun ; 
consequently the plane of the ring makes an angle of 
about 30° with the orbit of Saturn. 


Chap. XX. 


(284.) In the year 1725, Mr. Molyneux^ assisted by 
l)r. Bradley, fitted up a zenith sector at Kew, in order 
to discover whether the fixed stars had any sensible 
parallax *, that is, whether a star would appear to 
have changed it's place whilst the earth moved from 
one extremity of the diameter of it's orbit to the 
other; or, which is the same, to determine whether 
the diameter of the earth's orbit subtends any sensible 
angle at the star. The very important discovery 
arising from their observations is so clearly and fully 
related by Dr. Bradley, in a letter to Dr. Halley, that 
I cannot do better than give it to the reader in his 
own words. Phil. Trans. N°. 406. 

(285.) "^ Mr. Molyneuxs apparatus was completed 
and fitted for observing, about the end of November, 
1725, and on the third day of December following, 
the bright star in the head of Draco, marked t by 
Bayer, was for the first time observed as it passed 
near the zenith, and it's situation carefully taken with 
the instrument. The like observations were made on 
the 5th, nth, and 12th of the same month; and 

* Dr. Hook was the first inventor of this mfthod, and in the year 
1669 he put it in practice at Gresham College, with a telescope 36 
feet long. His first observation was July 6, at which time he 
found the bright star in the head of Draco, marked T by Bayer, 
passed about 2'. 12" northward from the zenith ; on July 9, it 
passed at the same distance ; on August 6, it passed 2'. 6" north- 
ward from the zenith; on October 2!, it passed 1'. 48" or 50" 
north from the zenith, according to his observations. See his 
Cutlcrian Lectures. 


there appearing no material difference in the place of 
the star, a farther repetition of them at this season 
seemed needless, it being a part of the year wherein 
no sensible alteration of parallax in this star could soon 
be ex]3ected. It was chiefly, therefore, curiosity that 
tempted me (being then at Kew, where the instru- 
ment was fixed) to prepare for observing the star on 
December 17, when, having adjusted the instrument 
as usual, 1 perceived that it passed a little more south- 
wardly this day than when it was observed before. 
Not suspecting any other cause of this appearance, we 
first concluded that it was owing to the uncertainty of 
the observations, and that either this or the foregoing 
were not so exact as we had supposed ; for which rea- 
son we purposed to repeat the observation again, in 
order to determine from whence this difference pro- 
ceeded; and upon doing it on December 20, I found 
that the star passed still more southwardly than in the 
former observations. This sensible alteration the more 
surprised us, in that it was the contrary way from 
what it would have been, had it proceeded from an 
annual parallax of the star : but being now pretty 
well satisfied that it could not be entirely owing to the 
want of exactness in the observations, and having no 
notion of any thing else that could cause such an ap- 
parent motion as this in the star, we began to think 
that some chansre of the materials, &c. of the instru- 
ment itself might have occasioned it. Under these 
apprehensions we remained some tmie ; but bemg at 
length fully convinced, by several trials, of the great 
exactness of the instrument, and finding, by the gra- 
dual increase of the star's distance from the pole, that 
there must be some regular cause that produced it, we 
took care to examine nicely, at the time of each ob- 
servation, how much it was ; and about the beginning 
of March, 1726, the star was found to be 20" more 
southwardly than at the time of the first observation. 
It now indeed, seemed to have arrived at it's utmost 
jijnit southward, because in several trials made about 


this time, no sensible difference was observed in it's si- 
tuation. By the middle of April it appeared to be re- 
turning back again towards the north ; and about the 
beginning of June it passed at the same distance from 
the zenith as it had done in December when it was 
first observed. 

From the quick alteration of the star's declination 
about this time (it increasing a second in three days) 
it was concluded that it would now proceed northward, 
as it before had gone southward of its present situa- 
tion ; and it happened as was conjectured, for the star 
continued to move northward till September following, 
when it again became stationary, being then near 20" 
more northwardly than in June, and no less than 39 
more northwardly than it was in March. From Sep- 
tember the star returned towards the south, till it 
arrived in December to the same situation it was in at 
that time twelve months, allowing for the difference of 
declination on account of the precession of the equinox. 

This was a sufficient proof that the instrument had 
not been the cause of this apparent motion of the star, 
and to find one adequate to such an eft'ect, seemed a 
difficulty. A nutation of the earth's axis was one of 
the first things that offered itself upon this occasion, 
but it was soon found to be insufficient ; for though it 
might have accounted for the change of declination in 
7 Draconis, yet it would not at the same time agree 
with the phaenomena in other stars, particularly in a 
small one almost opposite in right ascension to 7 
Draconis, at about the same distance from the north 
pole of the equator J for though this star seemed to 
move the same way as a nutation of the earth's axis 
would have made it, yet it changing it's declination 
but about half as much as 7 Draconis in the same 
time, (as appeared upon comparing the observations of 
both made upon the same days at diflferent seasons of 
the year,) this plainly proved that the apparent motion 
of the stars was not occasioned by a real nutation, since, 
if that had been the cause, the alteration in both stars 
would have been nearly equal. 


The great regularity of the observations left no room 
to doubt but that there was some regular cause that 
pioduced this unexpected motion, which did not de- 
pend on the uncertainty or variety of the seasons of the 
year. Upon comparing the observations with each 
other, it was discovered, that in both the fore-men- 
tioned stars, the apparent difference of declination 
from the maxima was always nearly proportional to 
the versed sine of the sun's distance from the equi- 
noctial points. This was an inducement to think that 
the cause, whatever it was^ had some relation to the 
sun's situation with respect- to those points. But not 
being able to frame any hypothesis at that time, suffi- 
cient to solve all the phaenomena, and being very 
desirous to search a little farther into this matter, I 
began to think of erecting an instrument for myself 
at Wansted ; that, having it always at hand, I might 
with the more ease and certainty enquire into the laws 
of this new motion. The consideration, likewise, of 
being able, by another instrument, to conlirm the 
truth of the observations hitherto made with Mr. 
Moll/Helix's, was no small inducement to me ; but the 
chief of all was, the opportunity 1 should thereby 
have of trying in what manner other stars were affected 
by the same cause, whatever it was. For Mr. Moly- 
tieuxs instrument being originally designed for ob- 
serving 7 Draconis (in order, as I said before, to try 
whether it had any sensible parallax) was so contrived 
as to be capable of but little alteration in it's direction, 
jiot above seven or eight minutes of a degree; and 
there being few stars within half that distance from 
the zenith of Kew, bright enough to be well observed, 
he could not with his instrument thoroughly examine 
how this cause affected stars differently situated with 
respect to the equinoctial and solstitial points of the 

These considerations determined me; and by the 
contrivance and direction of the very ingenious person 
Mr. Graham, my instrument was fixed up August 



1.9; 1727' As I had no convenient place where I 
conld make use of so long a telescope as Mr. Moly- 
neux's, I contented myself with one of but little more 
than half the length of his (viz. of about 12^ feet, his 
being 24 1) judging, from the experience which I had 
already had, that this radius would be long enough to 
adjust the instrument to a sufficient degree of exact- 
ness, and I have had no reason since to change my 
opinion ; for, from all the trials I have yet made, 1 am 
well satisfied, that when it is carefully rectified, it's 
situation may be securely depended upon to half a se- 
cond. As the place where my instrument was to be 
hung, in some measure determined it's radius, so did 
it also the length of the arch or limb on which the 
divisions were made to adjust it ; for the arch could 
not conveniently be extended farther than to reach to 
about 6°. 15' on each side my zenith. This indeed 
was sufficient, since it gave an opportunity of making 
choice of several stars very different both in magnitude 
and situation, there being more than two hundred in- 
serted in the British Catalogue, that may be observed 
with it. I needed not to have extended the limb so 
far, but that I was willing to take in Capella, the only 
star of the first magnitude that comes so near to my 

My instrument being fixed, I immediately began to 
observe such stars as I judged most proper to give me 
light into the cause of the motion already mentioned. 
There was variety enough of small ones, and not less 
than twelve that I could observe through all the sea- 
sons of the year, they being bright enough to be seen 
in the day-time, when nearest the sun. 1 had not 
been long observing, before I perceived that the notion 
we had before entertained, of the stars being farthest 
north and south when the sun was about the equinoxes, 
was only true of those that were near the solstitial 
colure ; and after I had continued my observations a 
few months, I discovered, what I then apprehended 
to be a general law, observed by all the stars^ viz. that 


each of them became stationary, or was farthest north 
or soutli when they passed over my zenith at six o'clock 
either in the morning or evening. I perceived hke- 
wise, that whatever situation the stars were in, with 
respect to the cardinal points of the ecliptic, the appa- 
rent motion of every one tended the same way when 
they passed my instrument about the same hour of 
the day or night; for they all moved southward while 
they passed in the day, and northward in the night ; 
so that each was farthest north when it came about 
six o'clock in the evening, and farthest south when it 
came about six in the morning. 

Though 1 have since discovered that the maxima in 
most of these stars do not happen exactly when they 
come to my instrument at those hours, yet, not being 
able at that time to prove the contrary, and supposing 
that they did, I endeavoured to find out what propor- 
tion the greatest alterations of declination in different 
stars bore to each other ; it being very evident that 
they did not all change their declinations equally. I 
have before taken notice that it appeared from Mr. 
Molyneux's observations that 7 IJraconis altered it's 
declination about twice as much as the fore-mentioned 
small star almost opposite to it; but examining the 
matter more particularly, I found that the greatest 
alteration of declination in these stars was as the sine 
of the latitude of each respectively. This made me 
suspect, that there might he the like proportion be- 
tween the maxima of other stars; but finding that the 
observations of some of them would not perfectly cor- 
respond with such an hypothesis, and not knowing 
whether the small difterence 1 met with might not be 
owing to the uncertainty and error of the observations, 
I deferred the farther examination into the truth of 
this hypothesis, till I should be furnished with a series 
of observations made in all parts of the year; which 
might enable me, not only to determine what errors 
the observations are liable to, or Iiow far they may 

N 2 

\q6 aberration of light. 

safely be depended upon ; but to judge whether there 
had been any sensible change in the parts of the in- 
strument itself. 

Upon these considerations I laid aside all thoughts 
at that time about the cause of the fore-mentioned 
phaenomena, hoping that I should the easier discover 
it, when I was better provided with proper means to 
determine more precisely what they were. 

When the yetir was completed, 1 began to examine 
and compare my observations, and having pretty well 
satisfied myself as to the general laws of the phse- 
nomena, I then endeavoured to find out the cause of 
them. I was already convinced, that the apparent 
motion of the stars was not owing to a nutation of the 
earth's axis. The next thing that oflfered itself, was 
an alteration in the direction of the plumb-line with 
which the instrument was constantly rectified; but 
this, upon trial, proved insufficient. Then I con- 
sidered what refraction might do; but here also 
nothing satisfactory occurred. At last I conjectured, 
that all the phaenomena hitherto mentioned, proceeded 
from the progressive motion of light and the earth's 
annual motion in it's orbit. For I perceived, if light 
was propagated in time, the apparent place of a fixed 
object would not be the same when the eye is at rest, 
as when it is moving in any other direction than that 
of the line passing through the eye and object; and 
that when the eye is moving in different directions, the 
apparent place of the object would be diflferent." 

This is Dr. Bradley's account of this very important 
discovery ; we shall therefore proceed to show that 
his principle will solve all the phaenomena. 

(28G.) The situation of any object in the heavens 
is determined by the position of the axis of the tele- 
scope annexed to the instrument with which we mea- 
sure ; for such a position is given to the telescope, that 
the rays of light from the object may descend down 


the axis, and in that situation the index shows the 
angular distance required. Now if hght be progres- 
sive, when a ray from any object descends down the 
axis, the position of the telescope must be different 
from what it would have been, if light had been instan- 
taneous, and therefore the place to which the telescope 
is directed, will be different from the true place of the 
object. For let S' be a fixed star, FF the direction of 
the earth's motion, S'F the direction of a particle of 
light, entering the axis ac of a telescope at a, and 
moving through aF while the earth moves from c to 
F; then, if the telescope continue parallel to itself, the 
light will descend in the axis. For let the axis, wm, 
Fw, continue parallel to « c ; then, considering each 
motion * as uniform, the spaces described in the same 

time will continue in the same proportion ; but cF: 
aF:: en : av, and by supposition cF, aF, are described 

* The motion of the spectator arising from the rotation of the 
earth about it's axis is not here taken into consideration, it being 
so small as not to produce any sensible effect. 


in the same time, therefore en, av, are described in 
the same time; hence, when the telescope comes into 
the situation «m, the particle of light will be in the 
axis at v ; and this being true for every instant, in this 
position of the telescope the ray will descend down the 
axis, and consequently the place of the star, deter- 
mined by the telescope at F, is *', and the angle S' Fs 
is the abcrratioiiy or tfie difference between the true 
place of the star and the place determined by the in- 
strument. Hence, if we take any line FS : Ft :: ve- 
locity of light : the velocity of the earth, and join St, 
and complete the parallelogram FtSs, the aberration 
will be equal to the angle FSt. Also S represents 
the true place of the star, and s the place determined 
by the instrument. 

(287.) As the place measured by the instrument 
differs from the true place, let us next consider how 
the progressive motion of light may effect the place of 
the star seen by the naked eye. If a ray of light fall 
upon the eye in motion, it's relative motion, in respect 
to the eye, will be the same as if equal motions were 
impressed in the same direction upon each, at the in- 
stant of contact ; for equal motions in the same direc- 
tion, impressed upon two bodies, will not affect their 
relative motions, and therefore the effect of one upon 
the other will not be altered. Let l^F be a tangent 
to the earth's orbit at F, which will represent the di- 
rection of the earth's motion at F, S' the star, join S'F, 
and produce it to G, and take FG : Fn :: the velocity 
of light : the velocity of the earth's in it's orbit, and 
complete the parallelogram nFGH, and draw the 
diagonal FH. Now as FG, nF, represent the mo- 
tions of light and of the earth in it's orbit, conceive a 
motion F?i equal, and opposite to JiFto be impressed 
upon the eye at F, and upon the ray of light, then the 
eye will be at rest, and the ray of light, by the two 
motions FG, F?i, will describe the diagonal FH -, this, 
therefore, is the relative motion of the ray of light in 
respect to the eye itself. Hence, the object appears 


in the direction HF, and consequently it's apparent 
])lace differs from it's true place by the angle GFH= 
FSt. It appears, therefore, that the apparent place 
of the object to the naked eye, is the same as the place 
determined by the instrument. We may therefore 
call the place, measured by the instrument, the appa- 
rent place. Many writers have given the explanation 
in this article, as the proof of the aberration, not con- 
sidering that the aberration is the difference between 
the true place and that determined by the instrument, 
or the instrumental error; indeed, in this case, the 
apparent place to the naked eye, coincides with the 
place determined by the instrument, and therefore no 
error has been produced by considering it in that point 
of view; but it introduces a vvrong idea of the subject ; 
the correction which we apply, or the aberration, is 
the correction of the place determined by the instru- 
ment, and therefore the investigation ought to proceed 
upon this principle; how much does the place, deter- 
mined by the instrument, differ fiom the true place.'' 

(288.) By Trigonometry, Art. 128, sin. FSt : sin. 
FtS :: Ft : FS :: velocity of the earth : velocity 
of light; hence, sine of aberration = sin. FtS k 

— r — r 1 • ■ ; therefore, if we consider the velocity of 
vel. of light ^ 

the earth and of light to be constant, the sine of abcr- 

ration, or the aberration itself, as it never exceeds 20", 

varies as sin. FtS^ and therefore is greatest when that 

angle is a right angle ; if, therefore, s be put for the 

sine of FtS, we have 1 (rad.) : s :: 20" : s x 20" the 

aberration. Hence, when Ft coincides with FS, or 

the earth is moving directly to or from a star^ there is 

no aberration. 

(289.) As (by observation) the angle FSt = 20", 
when FtS~()0°, we have, the velocity of the earth ; 
velocity of light :; sin. 20" : radius :: 1 : 10314. 

(290.) The aberration S's' lies, from the true place 
of the star, in a direction parallel to the direction of the 
earth's motion, and towards the same part. 



(291.) Whilst the earth makes one revolution in 
it's orbit, the curve, parallel to the echptic, described 
by the apparent place of a fixed star, is a circle. For 
let AFBQ, be the earth's orbit, K the focus in which 
the sun is, H the other focus ; on the major axis AB 

describe a circle in the same plane ; draw a tangent yFZ 
to the point F, and Ky,HZ, perpendicular to it; then 
(Conic Sect. Ellipse, prop. 5), the points y and Z will 
be always in the circumference of the circle. Let S' 
be the true place of the star, any where out of the 
plane of the ecliptic, which therefore must be conceived 
as elevated above the plane AFBQ, and take tF : FS 
as the velocity of the earth to the velocity of light, and 
complete the parallelogram FtSs, and s will (286) be 
the apparent place of the star. Draw FL perpendi- 
cular to AB, and let WsVx be the curve described by 
the point s, and JVSV be parallel to Fh. Now (from 
physical principles) the velocity of the earth varies as 

rrr-, or as HZ (Con. Sec. El. p. 6) ; but tF, or Ss 

represents the velocity of the earth ; hence, Ss varies 
as HZ, Also, as Ss^ SF, are parallel to Fy, FL, the 


angle sSF'=the angle z/FL, which is = theangle Z/fL, 
because the angieLFZ added to each makes two right 
angles, for in the quadrilateral figure LFZH, the 
angles L and Z are right ones. Hence, as Ss varies 
as HZ, and the angle iiSP=ZHA, the figures de- 
scribed by the points s and Z must be similar; but Z 
describes a circle in the time of one revolution of the 
earth in it's orbit ; hence, a" must describe a circle 
about S in the same time. And as Ss is always 
parallel to tF which lies in the plane of the ecliptic, 
the circle lVsf^x'\s parallel to the ecliptic. Also, as S 
and H are two points similarly situated in WP^ and 
AB, it appears that the true place of the star divides 
that diameter, which, although in a different plane, we 
may consider as perpendicular to the major axis of the 
earth's orbit, in the same ratio as the focus divides the 
major axis. But as the earth's orbit is very nearly a 
circle, we may consider S in the centre of the circle, 
without any sensible error. 

(292.) As we may, for the purposes which we shall 
here want to consider, conceive the earth's orbit AFBQ. 
to be a circle, and therefore to coincide with AyZB, 
if from the center C we draw Cs parallel to Ss, or yF, 
s will be the point in that circle corresponding to s iu 
the circle WsVx-, and as F/=90°, the apparent place 
of the star in the circle of aberration is always 90*^ 
before the place of the earth in it's orbit, and conse- 
quently the apparent angular velocity of the star and 
earth about their respective centers are always equa). 
It is further supposed, that the star S' is at an indefi- 
nitely great distance ; for the true place of the star is 
supposed not to be altered from the motion of the 
earth, and considering FH as always parallel to itself, it 
will always be directed to S' as a fixed point in the 
heavens. Hence also, as t!;e apparent place of the 
sun is opposite to that of the earth, the apparent place 
of the star, in the circle of aberration, is 90" hehiml 
that of the sun. 

(293). As a small part of the heavens may be con- 



ceivcd as a plane perpendicular to a line joining the 
star and eye, it follows, from the principles of ortho- 
graphic projection, that the circle amhn parallel to the 
ecliptic described by the apparent place of the star, pro- 
jected upon that plane, will be an ellipse ; the apparent 
path of the star in the heavens will therefore be an 
ellipse, and the major axis will be to the minor, as 
radius to the sine of the star's latitude. For let CE be 
the plane of the ecliptic, P it's pole, PE a secondary 
to it, PC perpendicular to EC, C the place of the eye, 
and let ab be parallel to CE, then it will be that 
diameter of the circle anhm of aberration which is 
seen most obliquely, and consequently that diameter 
which is projected into t!ie minor axis of the ellipse; 
let Dui be perpendicular to ah, and it will be seen di- 
rectly, being perpendicular to a line drawn from it to 

the eye, and therefore will be the m*.ijor axis ; draw Ca, 
Cbd, and ad is the projection oi ab ; and as ad may 
be considered as a straight line, we have (Trig. Art. 
128) mn or ah, the major axis, : ad the minor :: rad. 
: sin. ahd, or ECd the star's latitude. As the angle 
had is the complement of ahd, or of the star's latitude, 
the circle is projected upon a plane making an angle 
with it equal to the complement of the star's latitude. 
(294.) As the minor axis da coincides with a se- 
condary to the ecliptic, it must be perpendicular to it, 
and the major axis myi is parallel to it, it's position not 
being altered by projection. Hence, in the pole of 


the ecliptic^, the sine of the star's latitude being radius, 
the clHpse becomes a circle; and in the plane of the 
ecliptic, the sine of the star's latitude being = 0, the 
minor axis vanishes, and the ellipse becomes a straight 
line, or rather a very small part of a circular arc. 

(295.) To find the aberration in latitude and lotigi- 
tude. Let ABCD be the earth's orbit, supposed to 
be a circle with the sun in the center at x, and con- 
ceive P to be in a line drawn from x perpendicular to 
ABCD, and to represent the pole of the ecliptic ; let 
8 be the true place of the star, and conceive ape q to 
be the circle of aberration ])arallel to the ecliptic, and 
abed the ellipse into which it is projected ; let t 2^ be 
an arc of the ecliptic, and draw the secondary PSG to 
it, and (293) it will coincide with the minor axis bd 
into which the diameter j»</ is projected ; draw GCxA^ 
and it is parallel to pq, and Bx D perpendicular to 
AG must be parallel to the major axis a e ; then, when 
the earth is at A, the star is in conjunction, and in 
opposition when the earth is at C. Now, as the place 
of the star in the circle of aberration (292) is always 
90" before the earth in it's orbit, when the earth is at 
A, B, C, D, the apparent places of the star in the 
circle will be at r/,7;, c, q, and in the ellipse at a, b, c, 
d; and to find the place of the star in the circle, when 
the earth is at any point E, take the angle pSs = ExBj 
and s will be the corresponding place of the star in the 
circle; and to find the projected place in the ellipse, 
draw 51; perpendicular to Se, and vt perjjendicular to 
Sc in the plane of the ellipse, and t will be the appa- 
rent place of the star in the ellipse; join st, and it 
will be perpendicular to v t, because the projection of 
the circle into the ellipse is in lines perpendicular to 
the ellipse; draw the secondary PvtK, which will, as 
to sense, coincide with vt, unless the star be very near 
to the pole of the ecliptic 3 therefore the rules here 
given will be sufficiently accurate, except in that case. 
Now as cvS is parallel to the ecliptic, S and v must 
have the same latitude ; hence, vt is the abenation in 



latitude ; and as G is the true, and K the apparent 
place of the star in the echptic, GK is the aberration 








in longitude. To find these quantities, put m and n 
for the sine and cosine of the angle sSc, or CxE. the 
earth's distance from syzygies, radius being unity ; and 
as (293) the angle 5?;^= the complement of the star's 
latitude, the angle vst = t\ie star's latitude, for the sine 
and cosine of which put v and w, and put r = Sa, or 
S s ; then in the right-angled triangle Ssv (Trig. Art. 
128) 1 : m :: r : sv = rm; hence, in the triangle 
i^ts, \ : V :: rm : tv = rvm the aberration in latitude. 
Also, in the triangle Ssv, 1 : n :: r : vS=:rn; hence, 

w (13) ; I :: 7m : GK= — the aberration in lonsitude 


When the earth is in syzygies, 7«=0, therefore there 
is no aberration in latitude ; and, as n is then greatest, 
tliere is the greatest aberration in longitude ; if the 
earth be at A, or the star in conjunction, the apparent 
place of the star is at a, and reduced to the ecliptic at 
//; therefore GH'is the aberration, which diminishes 
the longitude of the star, the order of the signs being 
<~c GT', but when the earth is at C, or the star in 
opposition, the apparent place c reduced to the ecliptic 
is at F, and the aberration GFincreases the longitude; 
hence, the longitude is the greatest when the star is in 
opposition, and least when in conjunction. When the 
earth is in quadratures at Dor B, thenw = 0, and m 
is greatest ; therefore there is no aberration in longi- 
tude, but the greatest in latitude ; when the earth is 
at D, the apparent place of the star is at d, and the 
latitude is there increased; but when the earth is at 
B, the apparent place of the star is at h, and the lati- 
tude is diminished ; hence, the latitude is least in 
.quadrature before opposition, and greatest in quadra- 
ture after. From the mean of a great number of ob- 
servations. Dr. Bradley determined the value of /• to 
be 20". 

Ex. 1. What is the greatest aberration in latitude 
and longitude of 7 Ursce minoris, whose latitude is 
73^ 13' ? First, 7«= 1, 'v =,9669 the sine of 75°. 13'; 
hence, 20" x 9669=19",34 the greatest aberration in 
latitude. For the greatest aberration in longitude, 

n — \, iv = ,2bb\ ; hence, — — — = 7^"i4 the greatest 


aberration in longitude. 

Ex. 2. What is the aberration in latitude and longi- 
tude of the same star, when the earth is 30" from 
syzygies ? Here m = ,5 ; hence, 19",34 x ,5 =9",G7 
the aberration in latitude. If the earth be 30" beyond 
conjunction or before opposition, the latitude is di- 
minished; but if it be 30° after opposition or before 
conjunction, the latitude is increased. Also, w- = ,866 ; 
hence, 78",4 x ,866'=:67",89 the aberration in longi- 



tude. If the earth be 30° from conjunction, the 
longitude is diminished; but if it be 30° from opposi- 
tion, it is increased. 

Ex. 3. For the Sun, ni=0 and ?i = 1, w=l ; hence, 
it has no aberration in latitude, and the aberration in 
longitude =r = 20" constantly. This quantity 20" of 
aberration of the sun, answers to it's mean motion in 
8'. 7". 30'", which is therefore the time in which the 
light moves from the sun to the earth at it's mean dis- 
tance. Hence, the sun always appears 20" backwarder 
than it's true place. 

(296.) To find the aberration in j-ight ascension 
and declination. Let JEL be the equator, p it's pole ; 

^4CL the ecliptic, P it's pole ; S the true place of the 
star, s the apjmrent place in the ellipse; draw the great 
circles, Psa, Psb, pSw, pSv, and Sv, st perpendicular 
to P h, p V. Now s V = rv m (2^5 ) ; also, Sv = rn\ 
hence, (Trig. Art. 123) rym {vs) : rn [Sxj) :: rad. : 

tan. .S^y = — = 

cotan. earth's dist. syzv. 

—. , — j = cosec. star's 

vm sni. star s lat. 

lat. X cotan. earth's distance from syzygies. Thus we 
immediately compute the angle Ssv\ compute also the 
angle of position Psp from the three sides of that tri- 
angle being given (Trig. Art. 239), ^"^ "^ get the 


anf>le Ssp, it being the sum or difference of Ssv and 
Psp. Put a and b for the sine and (josinc of Ssv, c and 
d for the sine and cosine of Ssp, ;2;=: cosine of the star's 
dechnation ; then (as sv, st, are the cosines of Ssv, Sst, 

to radius sS) b : d :: sv ( =:rvm) : st = rvm x ^= 20" x 

vm X r the aberration in declination ; and (as Sv, St, 

are the sines of Ssv, Sst, to radius 5.9) a :c :: Sv { =rn) 

.8jf = ''— ; hence {I3),vw(= —^^) = 20" X ^ 
a V ' \ COS. doc. ^ «.!£ 

the aberration in right ascension. 


Chap. XXI. 


(297.) An eclipse of the Moon is caused by it's en- 
tering into the earth's shadow, and consequently it 
must happen when the moon is in opposition to the 
sun, or at the full moon. An eclipse of the Sun is 
caused by the interposition of the moon between the 
earth and sun, and therefore it must happen when the 
moon is in conjunction with the sun, or at the new 
moon. If the plane of the moon's orbit coincided with 
the plane of the ecliptic, there would be an eclipse at 
every opposition and conjunction ; but the plane of 
the moon's orbit being inclined to the ecliptic, there 
can be no eclipse at opposition or conjunction, unless 
at that time the moon be at, or near to the node. For 
suppose MMinm' be the orbit of the moon, and let 
the other circle represent the plane of the earth's 
orbit, or that plane in which the sun S, appears as 
seen from the earth E, and let these two planes be 
inclined to each other, so that we may conceive the 
part MM'm to lie above, and the part mm'Mhe\ow the 
plane of the earth's orbit; and M, m, are the nodes. 
Now if the moon be at M, in conjunction, the three 
bodies are then in the same plane, and therefore the 
moon is interposed between the earth and sun, and 
causes an eclipse of the sun. But if the moon be at 
M when the sun comes into conjunction at S\ M is 
iiow elevated above the line joining E and S', and M 
may be so far from My that the elevation of M' above 
the line ES' may be so much, that the moon may not 
be interposed between E and S', in which case there 



will be no eclipse of the sun. Whether, therefore, 
there will be an eclipse of the sun at the conjunction, 
or not, depends upon the distance of the moon from 
the node at that time. If the moon be at m at the 
time of opposition, then the three bodies being in the 
same right line, the shadow EV oi the earth must fall 
upon the moon, and the moon must suffer an eclipse. 
But if the moon be at tri at the time of opposition, rd 
may be so far below the shadow Ev of the earth, that 
the moon may not pass through it, in which case 
there will be no eclipse. Whether, therefore, there 

will be a lunar eclipse at the time of opposition, or 
not, depends upon the distance of the moon from the 
node at that time. If the two planes coincided, there 
would evidently be a central interposition every con- 
junction and opposition, and consequently a total 


eclipse. Meton, who lived about 430 years before 
Christ, observed, that after 19 years, the new and full 
moons returned again on the same day of the month. 
The ancient Astronomers also observed, that at the 
end of 18 years 10 days, a period of 223 lunations, 
there was a return of the same eclipses ; and hence, 
they were enabled to foretel when they would happen. 
This is mentioned by Pliny the Naturalist, Lib. II. 
Ch. 13. and by Ptolemy, Lib. IV. Ch. 2. This 
restitution of eclipses depends upon the return of the 
following elements to the same state. — 1. The sun's 
place. 2. The moon's place. 3. The place of the 
moon's apogee. 4. The place of the ascending node 
of the moon. The exact recurrence of these can 
never take place ; but it so nearly happens in the 
above time, as to produce eclipses remarkably corre- 
sponding. In this manner Dr. Halley predicted and 
published a return of eclipses from 1700 to 17I8, 
many of them corrected from observations ; together 
with the following elements. — 1 . The apparent time 
of the middle. 2. The sun's anomaly. 3. The an- 
nual argument. 4. The moon's latitude. He says, 
that in this period of 223 lunations, there are 18 years 
10 or 11 days (according as there are five or four 
leap-years) 'jh. 43'| ; that if we add this time to the 
middle of any eclipse observed, we shall have the re- 
turn of a corresponding one, certainly within ih. 30'; 
and that, by the help of a few equations, we may find 
the like series for several periods. 

To explain the Principles of the Calculation of an 
Eclipse of the Moon. 

(298.) The first thing to be done, is to find the 
time of the ?wea«* opposition. To get which, from 

* The time of the mean opposition is the time when the opposi- 
tion would have taken place, if the motions of the bodies had been 


the Tables of Epacts*, amongst the Tables of the 
moon's motion, take out the epact for the year and 
month, and subtract the sum from 29c?. \2h. 44'. 3", 
one synodic revolution of the moon, or two if neces- 
sary, so that the remainder may be less than a revolu- 
tion, and that remainder gives the time of the mean 
conjunction. If to this we add 14^:?. 18A. 22'. \'\b, 
half a revolution, it gives the time of the next mean 
opposition; or if we subtract, it gives the time of the 
preceding mean opposition. If it be leap-year, in 
January and February subtract a day from the sum of 
the epacts, before you make the subtraction. When 
the day of the mean conjunction is O, it denotes the 
last day of the preceding month. 

Ex. To find the times of the mean new and full 
moons in February, 1795' 

Epact 1795 - - 9^ 11^ 6'. 17" 
February - --1. 11. 15. 57 

10. 22. 22. 24 
29. 12. 44. 3 

Mean new moon - 18. 14. 21. 49 
14. 18. 22. 1,5 

Mean full moon - 3. 19. 59.-47,5 

(299«) To determine whether an eclipse may happen 
at opposition, find the mean longitude of the earth at 
the time of mean opposition, and also the longitude of 
the moon's nodej then, according to M. Cassini, if 

* The epact for any year is the ag€ of the moon at the beginning 
of the year from the last mean conjunction, that is, from the time 
when the mean longitudes of the sun and moon were last equal. 
The epact for any month is the age which the moon would have 
had at the beginning of the month, if it's age had been nothing at 
the beginning of the year ; therefore, if to the epact for the year, 
the epact for the month be added, the sum taken from 29 d. \2h_. 
44-'. 3". or from twice that quantity if the sum exceed it, must give 
the time oimcan conjunction. 

O 2 


the difference between the mean longitudes of the 
earth and the moon's node be less than 7°- 30', there 
must be an eclipse; if it be greater than 14*^. 30', 
there cannot be an eclipse; but between 7°« 30', and 
14°. 30', there may, or may not, be an eclipse. M. de 
Lambre makes these limits 7°. 47', and 13°. 21'*. 

Ex, To find whether there will be an eclipse at the 
full moon on Februarys, 1795- 
Sun's mean long. at3^ 19'. 59'. 47",6.10M3°.27'.'20"8 

Mean long, of the earth - - -,8 
Long, of the moon's node - - - 4. 8. 1. 48,5 

Difference - - 0. 5.25.32,3 

Hence, there must be an eclipse. 

Examine thus all the new and full moons for a 
month before and a month after the time at which the 
sun comes to the place of the nodes of the lunar orbit, 
and you will be sure not to miss any eclipses. Or, 
having the eclipses for the last 18 years, if you add to 
the times of the middle of these eclipses, 18?/. lO^Z. 7A. 
43' I, or I83/. lid. 7h. 43' |, (297) it will give the 
times when you may expect the eclipses will return. 

(300.) To the time of mean opposition, compute 
the true longitudes of the sun and moon, and the 
moon's true latitude; and find, from the Tables of 
their motions, the horary motions of the sun and 
moon in longitude, and the difference (d) of their 
horary motions is the relative horary motion of the 
moon in respect to the sun, or the motion with which 
the moon approaches to, or recedes from, the sun ; 
find also the moon's horary motion in latitude ; and 
suppose, at the time (t) of mean opposition, the moon 
is at the distance [m] from opposition ; then, as we 

* This may be found from Art. 306. by finding the true hmit, 
and then applying the greatest diflercnce of the true and mean 


may suppose the moon to approach the sun, or recede 
from it, uniformly, d : m :: 1 hour : the time {w) be- 
tween t and the opposition, which added to, or sub- 
tracted from, the time f, according as the moon is not 
yet got into opposition, or is beyond it, gives the time 
of the echptic opposition. 

(301.) To find the place of the moon in opposition, 
let 71 be the moon's horary motion in longitude; then, 
1 hour '. w :: n : the increase of the moon's longitude 
in the time w, which applied to the moon's longitude 
at the time of the mean opposition, gives the true 
longitude of the moon at the time of the ecliptic 
opposition. The opposite point to that must be the 
true longitude of the sun. Find also the moon's true 
latitude at the time of opposition, by saying, 1 hour : 
IV :: the horary motion in latitude : the motion in lati- 
tude in the time iVy which applied to the moon's lati- 
tude at the time of the mean opposition, gives the true 
latitude at the time of the true opposition*. In like 
manner you may compute the true time of the ecliptic 
conjunction, and the places of the sun and moon for 
that time, when you calculate a solar eclipse. 

(302.) With the sun's horary motion in longitude, 
and the moon's in longitude and latitude, find the in- 
clination of the relative orbit, and the horary motion 
upon it. To do this, let L3Ibe the horary motion of 
the moon in longitude, S Al that of the sun ; draw 
Ma perpendicular to LM, and equal to the moon's 
horary motion in latitude ; take Sb = Ma, and parallel 

* For greater certainty, you may compute again, from the Tables^ 
the places of the sun and moon, and if they be not exactly in op- 
position, which probably may not be the case, as the moon's longi- 
tude does not increase uniformly, repeat the operation. This 
accuracy, however, in eclipses is generally unnecessary ; for the 
best lunar Tables cannot be depended upon to give the moon's 
longitude nearer than 10"; therefore the probable error from the 
Tables is vastly greater than that which arises from the motion in 
longitude not being uniform. Unless, therefore, very great accu- 
racy be required, this operation is unnecessary. 



to it, and join La, Lb ; then La is the moon's true 
orbit, and Lb it's relative orbit in respect to the sun. 

Hence, LS (the difference of the horary motions in 
longitude) : Sb the moon's horary motion in latitude 
:: radius : tan. bLS, the inch nation of the relative 
orbit ; and cos. bLS : radius :: LS : Lb, the horary 
motion in the relative orbit. 

(303.) At the time of opposition, find, from the 
Tables, the moon's horizontal parallax, it's semi- 
diameter, and the semidiameter of the sun, the hori- 
zontal parallax of which we may here take = 9". 

(304.) To find the semidiameter of the earth's 
shadow at the moon, seen from the earth. Let AB 
be the diameter of the sun, TR the diameter of the 

earth, O and C their centers ; produce AT, BR, to 
meet at /, and draw OCT; let FGH be the diameter 
of the earth's shadow at the distance of the moon, and 
join OT, CF. Now the angle FCG= CFA- CIA, 
but CIA=OTA^TOC, therefore FCG = CFA- 
OTA + TOC, that is, the angle under which the se- 
midiameter of the earth's shadow, at the moon, ap- 
jyears, is equal to the sum of the horizontal parallaxes 
of the sun and moon diminished by the apparent semi- 
diameter of the sun. In eclipses of the moon, the 
shadow is found to be a little greater than this Rule 


gives it, owing to the atmosphere of the earth. This 
augmentation of the semidiameter is, according to M, 
Cassini, 20" \ according to M, Mowmer, 30" ; and 
according to M. de la Htre^ 6o". Mayer thinks the 

correction is about 77- of the semidiameter of the sha- 

dow, or that you may add as many seconds as the 
semidiameter contains minutes. Some computers 
always add 50" ; but this must be subject to some un- 

(305.) As the angle CIT ( = OTA-TOC) is 
known, we have sin. TIC-, cos. TIC .: TC . C/the 
length of the earth's shadow. If we take the angle 
ATO=z\&. 3" the mean semidiameter of the sun, 
TOC=i^" the horizontal parallax of the sun, we have 
CIT=15\ 54"; hence, sin. 15'. 54" : cos. 15'. 54", or 
1 : 216,2 :: TC: Cr=2l6,2 TC. 

(306.) The different eclipses which may happen of 
the moon, may be thus explained. Let CL represent 
the plane of the ecliptic, OR the moon's orbit, cutting 


the ecliptic in the node A^; ^nd let SH represent a 
section of the earth's shadow at the distance of the 
moon from the earth, and M the moon at the time 
when she passes nearest to the center of the earth's 
shadow. Hence, if the opposition happen as in posi- 
tion I, it is manifest that the moon will just pass by 
the shadow of the earth without entering it, and there 
will be no eclipse. In position II, part of the moon 
will pass through the earth's shadow, and there will be 
a ^ar^k^ eclipse. In position III, the whole of the. 



moon passes through the earth's shadow, and there is 
a total echpse. In position IV, the center of the 
moon passes through the center of the earth's shadow, 
and there is a total and central ecHpse. It is plain, 
therefore, that whether there will, or will not, be an 
eclipse at the time of opposition, depends upon the 
distance of the moon from the node at that time ; or 
the distance of the earth's shadow, or of the earth, 
from the node. Now in lunar eclipses we may take 
the angle at N=b°. 17', and in position I, the value 
of Ev is about l**. 3'. 30"; hence (Trig. Art. 221), sin. 
5". 17' : rad. :: sin. 1°. 3'. 30" : sin. EN:=^\\\ 34'-, 
when, therefore, EN is greater than that quantity at 
the time of opposition, there can be no eclipse. This 
quantity 11**. 34' is called the ecliptic limit. 

(307.) Let Arh be that half of the earth's shadow 
which the moon passes through, NL the relative orbit 

of the moon ; draw Cmr perpendicular to NL, and 
let z be the center of the moon at the beginning of 
the eclipse, m at the middle, x at the end ; also, let 
AB be the ecliptic, and Cn perpendicular to it. Now 
in the right-angled triangle Cnm, we know Cn the 
latitude of the moon at the time of the ecliptic con- 
junction, and (302) the angle Cnm* the complement 
of the angle which the relative orbit of the moon makes 
with the ecliptic; hence (Trig. Art. 125) rad. : cos. 

* If the moon at n have north or south latitude increasing, the 
angle Oim is to be set off to the right j otherwise, to the left of Cn, 


Cnm :: Cn : nni, which is called the Reduction ; and 
rad. : sin. Cnm :: C?i : Cm. The horary motion [h) 
of the moon upon it's relative orbit being known, we 
know the time of describing mti, by saying, h : mn :: 
1 hour : the time of describing mn. Hence, knowing 
the time of the ecliptic conjunction at n, we know the 
time of the middle of the eclipse at m. Next, in the 
right-angled triangle Cmz, we know Cm, and Cz the 
sum of the semidiameters of the earth's shadow and 
the moon, to find mz, which is done thus by loga- 
rithms; 2ismz=sJ Cz' - Cm^ = \/ Cz + Cmx Cz - Cm, 

the log. of mz-=ix log. Cz-{- Cm + log. Cz - Cm, 
(Trig. Art. 52). Hence, the horary motion of the 
moon being known, we know the time of describing 
zm, which subtracted from the time at m gives the 
time of the beginning, and added, gives the time of 
the end. The magnitude of the eclipse at the middle 
is represented by tr, which is the greatest distance of 
the moon within the earth's shadow, and this is mea- 
sured in terms of the diameter of the moon, conceived 
to be divided into 12 equal parts, called Digits, or 
Parts deficient ; to find which, we know Cm, the 
difference between which and Cr gives mr, which 
added to mt, or if m fall out of the shadow, take the 
difference between mr and 7nt, and we get tr ; hence, 
to find the number of digits eclipsed, say, mt : tr :: 6 
digits, or 36o', (it being usual to divide a digit into 
60 equal parts, and call them minutes,) : the digits 
eclipsed. If the latitude of the moon be north, we 
use the upper semicircle ; if south, we take the lower. 
(308.) If the earth had no atmosphere, when the 
moon was totally eclipsed, it would be invisible ; but, 
by the refraction of the atmosphere, some rays will be 
brought to fall on the moon's surface, upon which ac- 
count the moon will be visible at that time, and appear 
of a dusky red colour. M. Maraldi [Mem. de I Acad. 
1723) has observed, that, in general, the earth's umbra, 
at a certain distance, is divided by a kind of penumbra. 


from the refraction of the atmosphere. This will ac- 
count for the circumstance of the moon being more 
visible- in some total eclipses than in others. It is 
said, that the moon, in the total eclipses in 1601, 1620, 
and 1642, entirely disappeared. 

(309.) An eclipse of the moon, arising from it's 
real deprivation of light, must appear to begin at the 
same instant of time to every place on that hemisphere 
of the earth Vk^hich is turned towards the moon. Hence, 
it affords a very ready method of finding the difference 
of longitudes of places upon the earth, as will be after- 
wards explained. The moon enters the penumbra of 
the earth before it comes to the umbra, and therefore 
it gradually loses it's light; and the penumbra is so 
dark just at the umbra, that it is difficult to ascertain 
the exact time when the moon's limb touches the 
umbra, or when the eclipse begins. When the moon 
has entered into the umbra, the shadow upon it's disc 
is tolerably well defined, and you may determine, to a 
considerable degree of accuracy, the time when any 
spot enters into the umbra. Hence, the beginning 
and end of a lunar eclipse are not so proper to deter- 
mine the longitude from, as the times at which the 
umbra touches any of the spots. 

On Eclipses of the Sun. 

(310.) An eclipse of the sun is caused by the inter- 
position of the moon between the sun and spectator, 
or by the shadow of the moon falling on the earth at 
the place of the observer. The different kinds of 
eclipses will be best explained by a figure. Let S be 
the sun, M the moon, JB or Jt'B' the surface of the 
earth; draw tangents pxvs, qzvr, from the sun to the 
same side of the moon, and ocvz will be the moon's 
umbra, in which no part of the sun can be seen ; if 
tangents ptbd, qwac, be drawn from the sun to the 
opposite sides of the moon, the space comprehended 
between the umbra and waCy tbd, is called the 


penumbra, in which part of the sun only is seen. 
Now it is manifest, that if AB be the surface of the 

earth, the space mn, where the umbra falls, will suffer 
a total eclipse; the part am, bn, between the bounda- 
ries of the umbra and penumbra, will suffer a partial 
eclipse ; but to all the other parts of the earth there 
will be no eclipse. Now let A' B' be the surface of 
the earth, the earth being, at different times, at dif- 
ferent distances from the moon; then the space within 
rs will suffer an annular eclipse; for if tangents be 
drawn from any point o within rs to the moon, they 
must evidently fall within the sun, therefore the sun 
will appear all round about the moon in the form of a 
ring; the parts cr, sd, will suffer a partial eclipse; 
and the other parts of the earth will suffer no eclipse. 
In this case, there can be no total eclipse any where, 
as the moon's umbra does not reach the earth. Ac- 


cording to M. du Sejour, an eclipse can never be an- 
nular longer than 12'. 24", nor total longer than 
7'. 58". 

(311.) The umbra xvz is a cone, and the penumbra 
wcdf the frustrum of a cone whose vertex is f^. 
Hence, if these be both cut through their common 
axis perpendicular to it, the section of each will be a 
circle, having a common center in the line joining the 
centers of the sun aud moon, and the penumbra in- 
cludes the umbra. 

(312.) The moon's mean motion about the center 
of the earth is at the rate of about 33' in an hour ; 
but 33' of the moon's orbit is about 2280 miles, which, 
therefore, we may consider as the velocity with which 
the moon's shadow passes over the earth ; but this is 
the velocity upon the surface of the earth where the 
shadow falls perpendicularly upon it, it being the 
velocity perpendicular to Mv; in every other place, 
the velocity over the surface will be increased in the 
proportion of the sine of the angle which Mv makes 
with the surface, in the direction of it's motion, to 
radius. But the earth having a rotation about it's 
axis, the relative velocity of the moon's shadow over 
any given point of the surface will be different from 
this ; if the point be moving in the direction of the 
shadow, the velocity of the shadow, in respect to that 
point, will be diminished, and consequently the time 
in which the shadow passes over it will be increased ; 
but if the point be moving in a direction contrary to 
that of the shadow, as is the case when the shadow 
falls on the other side of the pole, the time will be 
dimini^ed. The length of a solar eclipse is therefore 
affected by the earth's rotation about it's axis. 

(313.) The different eclipses of the sun may Jbe 
thus explained. Let CL represent the orbit of the 
earth, OR the line described by the centers of the 
moon's umbra and penumbra at the earth ; N the 
moon's node; SF the earth, E it's center; pn the 



moon*s penumbra, u the umbra. Then, in position I, 
the penumbra p n just passes by the earth, without 

falHng upon it, and therefore there will be no echpse. 
In position II, the penumbra pn falls upon the earth, 
but the umbra u does not; therefore there will be a 
partial eclipse where the penumbra falls, but no total 
eclipse. In position III, both the penumbra /?/?, and 
umbra u fall upon the earth; therefore, where the 
penumbra falls, there will be a partial eclipse, and 
where the umbra falls there will be a total eclipsp ; 
and to the other parts of the earth there will be no 
eclipse. Now the ecliptic litnit, may be thus found. 
The angle A^ may be taken 5°. 17'j and in position I, 
the value of Eu {u being the center of the umbra) is 
about 1°. 34'. 27"; hence (Trig. Art. 221) sin. 5°. if 
: rad. :: sin. V. 34'. 27" : sin. EN=17\2l\ 2f' the 
ecliptic limit; if therefore, at the time of conjunction, 
the earth be within this distance of the node, there 
will be an eclipse. 

(314.) An eclipse of the sun, or rather of the earth, 
without respect to any particular place, may be calcu- 
lated exactly in the same manner as an eclipse of the 
moon, that is, the times when the moon's umbra or 
penumbra first touches and leaves the earth ; hut to 
find the times of the beginning, middle, and end, at 
any particular place, the apparent place of the moon, 
as seen from thence, must be determined, and conse- 


quently it's parallax in latitude and longitude must be 
computedj which renders the calculation of a solar 
echpse extremely long and tedious. 

To explain the Principles of the Calculation of an 
Eclipse of the Sun for any particular Place. 

(315.) Having determined {X\S) that there will be 
an eclipse somewhere upon the earth, compute, by the 
Astronomical Tables, the true longitudes of the sun 
and moon, and the moon's true latitude, at the time 
of mean conjunction (301) ; find also the horary mo- 
tions of the sun and moon in longitude, and the 
moon's horary motion in latitude ; and compute the 
time of the ecliptic conjunction of the sun and moon, 
in the same manner (300) as the time of the ecliptic 
opposition was computed. At the time of the ecliptic 
conjunction, compute (301) the sun's and moon's 
longitude, and the moon's latitude ; find also the hori- 
zontal parallax of the moon from the Tables of the 
moon's motion, from which subtract the sun's hori- 
zontal parallax, and you get the horizontal parallax of 
the moon from the sun. 

(31 6.) To the latitude of the given place, and the 
horizontal parallax of the moon from the sun (which 
we here use instead of the horizontal parallax of the 
moon, as we want to find what effect the parallax has 
in altering their apparent relative situations,) at_tjie_ 
time of the ecli ptic conjunct ion^ compute (144) the 
moon's parallax in latitude and longitude from the 
sun ; the parallax in latitude applied to the true lati- 
tude gives the apparent latitude [L) of the moon from 
the sun ; and the parallax in longitude shows the 
appare nt difference {D) of the longitudes of the sun 
and moon. 

(317.) Let S be the sun, CE the ecliptic, according 



to the order of the signs; take SM=D, draw MIV 
perpendicular to MSy and take it = L, then N is the 

apparent place of the moon, and SN= ^ D' + U is 
the apparent distance of the moon from the sun. 

(318.) If the moon be to the ea^"^ of the nonagesimal 
degree, tlie parallax increases the longitude ; if to the 
tvest, it diminishes it (Art. 144) ; hence, if the trme 
longitudes of the sun and moon be equal, in the 
former case the apparent place will be from aS" towards 
E, and in the latter, towards C. To some time, as an 
hour, after the true conjunction, if the moon be to 
the west of the nonagesimal degree ; or before the 
true conjunction, if the moon be to the east of the 
nonagesimal degree, find the sun's and the moon's true 
longitude, and the moon's true latitude, from their 
horary motions ; and to the same time compute the 
moon's parallax in latitude and longitude from the 
sun ; apply the parallax in latitude to the true lati- 
tude, and it gives the apparent latitude (/) of the 
moon from the sun; take the difference of the sun's 
and moon's true longitude, and apply the parallax in 
longitude, and it gives the apparent distance {d) of the 
moon from the sun in longitude. From S set off SP 
= d, and on EC erect the perpendicular PQ equal to 
I, and Q is the apparent place of the moon at one hour 
from the true conjunction ; and SQ ( = ^ d^ -\- 1*) is 
the apparent distance of the moon from the sun ; draw 
the straight line NQ, and it will represent the relative 
apparent path of the moon, considered as a straight 
line, in general it being very nearly so j it's value also 


represents the relative horary motion of the moon in 
the apparent orbit, the relative horary motion in 
longitude being MP. 

(319,) The difference between the moon's apparent 
distance in longitude from the sun at the time of the 
true ecliptic conjunction, and at the interval of an 
hour, gives the apparent horary motion (r) in longi- 
tude of the moon from the sun ; the difference (D) 
between the true longitude at the ecliptic conjunction, 
and the moon's apparent longitude, is the apparent 
distance of the moon from the sun in longitude 
at the true time of the ecliptic conjunction; hence, 
r : D :: 1 hour : the time from the true to the 
apparent conjunction, consequently we know the time 
of the apparent conjunction. To find whether this 
time is accurate, we may compute (from the horary 
motions of the sun and moon) their true longitudes, 
and the moon's parallax in longitude from the sun, 
and apply it to the true longitude, and it gives the 
apparent longitude, and if this be the same as the sun's 
longitude, the time of the apparent conjunction is truly 
found; if they be no t the same, find jrom thence the 
true time, as b efore. iTo the true time of the apparent 
conjunctTonTHfind the moon's true latitude from it's 
horary motion, and compute the parallax in latitude, 
and you get the apparent latitude at the time of the 
apparent conjunction. Draw SA perpendicular to 
CE, and equal to this apparent latitude ; then the 
point A will not probably fall in NQ ; but suppose it 
to fall in QiV, to which draw SB perpendicular, and 
NR parallel to PM. Then knowing NR ( = P3I), 
and QR { = QP^MN) we have 

NR : RQ :: rad. : tan. QNR, or ASB {Tng. Art. 123) 
Sin. QNR : rad. :: QR : QiV (Trig. Art. 128) 

The time of describing NQ in the apparent orbit being 
equal to the time from M to P in longitude, NQ is 
the horary motion in the apparent orbit. 


Rad. : sin. ASB :: AS : AB (Trig. Art. 125) 
Rad. : COS. ASB :: AS : SB. 

(320.) At the apparent conjunction the moon apr- 
pears at A, which time (319) is known; when the 
inoon appears at B, it is at it's nearest distance from 
the sun", and consequently the time is that of the 
greatest obscuration, (usually called the time of the 
middle,) provided there is an eclipse, which will 
always be the case, when SB is less than the sum of the 
apparent semidiatneters of the sun and moon. If, 
therefore, it appear that there will be an eclipse, we 
proceed thus to find it's quantity, and the beginning 
and end. As we may suppose the motion to be uni- 
form, QiV: AB :: the time of describing NQ : the 
time of describing AB, which added to or subtracted 
from the time at A, (according as the apparent latitude 
is decreasing or increasing), gives the time of the 
greatest obscuration. 

(321.) From the sum of the apparent semidiameters 
of the sun and moon, subtract BS, and the remainder 
shows how much of the sun is covered by the moon, 
or the parts deficient; hence, semid. o : parts defi- 
cient :: 6 digits : the digits eclipsed. If SB be less 
than the difference of the semidiameters of the sun 
and moon, and the moon's semidiameter be Xhe greater, 
the eclipse will be total; but if it be the less, the 
eclipse will be annular, the sun appearing all round 
the moon ; if B and S coincide, the eclipse will bQ 

(322.) Produce, if necessary, QN, and take SV,SIV, 
equal to the sum of the apparent semidiameters of the 
sun and moon, at the beg inning and end respectively ; 
then BF=JSF'^SB\ax^d BIF=.JSI^F'-.SB^-, 
and to find the times of describing these, say, as the 
hourly motion of the moon in the apparent orbit, or 
NQ, : BF :: 1 hour : the time of describing FB ^ 
and NQ : BfF.. 1 hour : the time of describing J5^A^ 



which times, respectively subtracted from and added to 
the time of the greatest obscuration, give nearly the 
times of the beginning and end. But if accuracy be 
required, a different method must be adopted ; for we 
suppose yfV to be a straight line, which supposition 
will, in general, cause errors, too considerable to be 
neglected. It may, however, always serve as a rule to 
assume the time of the beginning and end. Hence it 
follows, that the time of the greatest obscuration at B, 
is not necessarily equidistant from the beginning and 

(323.) If the eclipse be total, take ST, SIV, equal 
to the difference of the semidiameters of the sun and 
moon, and then BF= Bir= J'SW^ - SB\ from 
whence we may find the times of describing J5^, 
BfV, as before, which we may consider as equal, and 
which applied to the time of the greatest obscuration 
at B, give the time of the beginning and end of the 
total darkness. 

(324.) To find more accurately the time of the 
beginning and end of the eclipse, we must proceed 
thus. At the estimated time of the beginning, find, 
from the horary motions, and the computed parallaxes, 
the apparent latitude f^D of the moon, and it's appa- 
rent longitude DS from the sun, and we have S f^= 
aJ SW^-i-Df'^y and if this be equal to the apparent 
semid. D 4- semid. © (which sum call S), the estimated 
time is the time of the beginning; but if Sl^he not 
equal to S, assume (as the error directs) another time 
at a small interval from it, before, if SFhe less than 
S, but after, if it be greater], to that time compute 
again the moon's apparent latitude mv, and apparent 
longitude Sm from the sun, and find Sv=^ Sm'^ + mv^; 
and if this be not equal to S, proceed thus ; as the 
difference of Sv and SF : the difference of Sv and SL 
{=^S) :: the above-assumed interval of time, or time 
of the motion through Vv, • the time through vLy 


which added to or subtracted from the time at v, ac- 
cording as Sv is greater or less than SL, gives the time 
of the beginning. The reason of this operation is, 
that as F^v, vL, are very small;, they will be very 
nearly proportional to the differences of SF, Sv, and 
Sv, SL. But as the variation of the apparent distance 
of the sun from the moon is not exactly in proportion 
to the variation of the differences of the apparent 
longitudes and latitudes, in cases where the utmost 
accuracy is required, the time of the beginning thus 
found (if it appear to be not correct) maybe corrected_, 
by assuming it for a third time, and proceeding as 
before. This correction, however, will never be ne- 
cessary, except where extreme accuracy is required in 
order to deduce some consequences from it. But the 
time thus found is to be considered as accurate, only 
so far as the Tables of the sun and moon can be de- 
pended upon for their accuracy; and the best lunar 
Tables are subject to an error of 10" in latitude. 
Hence, accurate observations of an eclipse, compared 
with the computed time, furnish the means of correct- 
ing the lunar Tables. In the same manner, the end 
of the eclipse may be computed. 

(325.) As there are not many persons who have an 
opportunity of seeing a total eclipse of the sun, we shall 
here give the phaenomena which attended that on 
April 22, 1715. Capt. Stannyayi, at Bern in Switzer- 
land, says, *' the sun was totally dark for four minutes 
and a half; that a fixed star and planet appeared very 
bright; and that it's getting out of the eclipse was 
preceded by a blood-red streak of light, from it's left 
limb, which continued not longer than six or seven 
seconds of time ; then part of the sun's disc appeared, 
all on a sudden, as bright as Venus was ever seen in 
the night ; nay, brighter, and in that very instant gave 
a light and shadow to things, as strong as moon-light 
used to do.'* The inference drawn from these phae- 
nomena is, that the moon has an atmosphere. 

J. C. FaciSf at Geneva, says, *' ther^ was seen, 
P 2 


during the whole time of the total immersion, a white- 
ness, which seemed to break out from behind the 
moon, and to encompass it on all sides equally ; it's 
breadth was not the twelfth part of the moon's dia- 
meter. Venus, Saturn, and Mercury were seen by 
many ; and if the sky had been clear, many more stars 
might have been seen, and with them Jupiter and 
Mars. Some gentlewomen in the country saw more 
than 16 stars ; and many people on the mountains 
saw the sky starry, in some places where it was not 
overcast, as during the night at the time of the full 
moon. The duration of the total darkness was three 

Dr. J. J. Scheuchzer, at Zurich, says, *' that both 
planets and fixed stars were seen ; the birds went to 
roost; the bats came out of their holes ; and the fishes 
swam about j we experienced a manifest sense of cold; 
and the dew fell upon the grass. The total darkness 
lasted four minutes." 

(326.) Dr. //a//ei/*, who observed this eclipse at Lon- 
don, has thus given the phaenomena attending it. " It 
was unwersally observed, thatwhen the last part of the 
sun remained on it's east side, it grew very faint, and 
was easily supportable to the naked eye, even through 
the telescope, for above a minute of time before the 
total darkness ; wjiereas, on the contrary, my eye 
could not endure the splendour of the emerging beams 
in the telescope from the first moment. To this, per- 

* The Doctor begins his account thus. " Though it be certain, 
■from the principles of Astronomy, that there happens necessarily 
a central eclipse of the sun, in some part or other of the terra- 
queous globe, about twenty»eight times in each period of eighteea 
years; and that of these, no less than eight do pass over the 
parallel of London, three of which eight are total with continuance ; 
yet, from the great variety of the elements, w^hereof the calculus 
of eclipses consists, it has so happened, that since March 20, 1140, 
1 cannot find that there has been a total eclipse of the sun seen at 
London, though in the mean time the shade of the moon has often 
passed over other parts of Great Britain." 


haps, two causes concurred ; the one that the pupil 
of the eye did necessarily dilate itself during the dark- 
ness, which before had been much contracted by look- 
ing on the sun. The other, that the eastern parts of 
the moon, having been heated with a day near as long 
as thitty of our's, must of necessity have that part of 
it's atmosphere replete with vapours, raised by the 
long-continued action of the sun; and, by conse- 
quence, it was more dense near the moon's surface^ 
and more capable of obstructing the lustre of the sun's 
beams. Whereas at the same time the western edge 
of the moon had suffered as long a night, during which 
time there might fall in dews, all the vapours that 
were raised in the preceding long day ; and for this 
reason, that part of it's atmosphere might be seen 
much more pure and transparent. 

About two minutes before the total immersion, the 
remaining part of the sun was reduced to a very fine 
horn, whose extremities seemed to lose their accuteness, 
and to become round like stars. And for the space of 
about a quarter of a minute, a small piece of the 
southern horn of the eclipse seemed to be cut off from 
the rest by a good interval, and appeared like an 
oblong star round at both ends; which appearance 
could proceed from no other cause, but the inequali- 
ties of the moon's surface, there being some elevated 
parts thereof near the moon's southern pole, by which 
interposition, part of that exceedingly fine filament of 
light was intercepted. 

A few seconds before the sun was totally hid, there 
discovered itself round the moon a luminous ring, 
about a digit, or perhaps a tenth part, of the moon's 
diameter in breadth. It was of a pale whiteness, or 
rather pearl colour, seeming to me a little tinged with 
the colours of the iris, and to be concentric with the 
moon ; whence 1 concluded it was the moon's atmo- 
sphere. But the great height of it, far exceeding that 
of our earth's atmosphere ; and the observations of 
some one who found the breadth of the ring to increase 


on the west side of the moon, as the emersion ap- 
proached ; together with the contrary sentiments of 
those, whose judgement I shall always revere, make 
me less confident, especially as in a matter whereto I 
gave not all the attention requisite. 

Whatever it was, this ring appeared much brighter 
and whiter near the body of the moon, than at a dis- 
tance from it ; and it's outward circumference, w^hich 
was ill defined, seemed terminated only by the extreme 
rarity of the matter it was composed of; and in all 
respects resembled the appearance of an enlightened 
atmosphere viewed from far : but whether it belonged 
to the sun or the moon, I shall not at present under- 
take to decide. 

During the whole time of the total eclipse, I kept 
my telescope constantly fixed on the moon, in order to 
observe what might occur in this uncommon appear- 
ance, and I saw perpetual flashes or coruscations of 
light, which seemed for a moment to dart out from 
behind the moon, now here, now there, on all sides, 
but more especially on the western side, a little before 
the emersion ; and about two or three seconds before 
it, on the same western side, where the sun was just 
coming out, a long and very narrow streak of dusky, 
but strong red light, seemed to colour the dark edge 
of the moon, though nothing like it had been seen 
immediately after the immersion. But this instantly 
vanished upon the first appearance of the sun, as did 
also the aforesaid luminous ring;. 

As to the degree of darkness, it was such, that one 
might have ex[3€cted to have seen more stars than were 
seen in London ; the planets Jupiter, Mercury, and 
J'enuSy were all that were seen by the gentlemen of 
the Society from the top of their house, where they 
had a free horizon ; and 1 do not hear that any one in 
town saw more than Capella and Aldeharan of the 
fixed stars. Nor was the light of the ring round the 
moon capable of eflfacing the luster of the stars, for it 
was vastly inferior to that of the full moon, and so 


weak, that I did not observe it cast a shade. But the 
under-parts of the hemispherej particularly in the 
south-east under the sun, had a crepuscular bright- 
ness ; and all round us, so much of the segment of 
our atmosphere as was above the horizon, and was 
without the cone of the moon's shadow, was more or 
less enlightened by the sun's beams; and it's reflection 
gave a diffused light, which made the air seem hazy, 
and hindered the appearance of the stars. And that 
this was the real cause thereof, is manifest by the 
darkness being more perfect in those places near which 
the center of the shade passed, where many more stars 
were seen, and in some, not less than twenty, though 
the light of the ring was to all alike. 

I forbear to mention the chill and damp, with which 
the darkness of this eclipse was attended, of which 
most spectators were sensible, and equally judges; or 
the concern that appeared in all sorts of animals, birds, 
beasts, and fishes, upon the extinction of the sun, since 
ourselves could not behold it without some sense of 

(327.) If a conjunction of the sun and moon happen 
at, or very near, the node, there will be a great solar 
eclipse; but, in this case, at the preceding opposition, 
the earth was not got into the lunar ecliptic limits, 
and at the next opposition it will be got beyond it ; 
hence, at each node there may happen only one solar 
eclipse, and therefore in a year there may happen only 
two solar eclipses. 

There must be one conjunction in the time in which 
the earth passes through the solar ecliptic limits, and 
consequently there must be one solar eclipse at each 
node ; hence, there must be two solar eclipses at least 
in a year. 

If an opposition happen ju«t before the earth gets 
into the lunar ecliptic limit, the next opposition may 
not happen till the earth is got beyond the limit on 
the other side of the node ; consequently there may 
not be a lunar eclipse at the node ; hence, there may 


not be an eclipse of the moon in the course of a year. 
When, therefore, there are only two eclipses in a year, 
they must be both of the sun. 

If there be an eclipse of the moon as soon as the sua 
gets within the lunar ecliptic limit, it will be got out of 
the limit before the next opposition ; consequently 
there can be only one lunar eclipse at the same node. 
But as the nodes of the moon's orbit move backwards 
about 19° iu a year, the earth may come within the 
lunar ecliptic limits, at the same node, a second time 
in the course of a year, and therefore there may be 
three lunar eclipses in a year; and there can be no 

If an eclipse of the moon happen at, or very near to, 
the node, a conjunction may happen before and after, 
whilst the earth is within the solar ecliptic limits; 
hence there may, at each node, happen two eclipses of 
the sun and one of the moon ; and in this case, the 
eclipses of the sun will be small, and that of the moon 
large. When, therefore, the eclipses do not happen a 
second time at either node, there may be six eclipses 
in a year, four of which will be of the sun, and two 
of the moon. But if, as in the last case, an eclipse 
should happen at the return of the earth within the 
lunar ecliptic limits at the same node a second time in 
the year, there may be six echpses, three of the sun 
and three of the moon. 

There may be seven eclipses in a year. For twelve 
lunations are performed in 354 days, or in 1 1 days less 
than a common year. If, therefore, an eclipse of the 
sun should happen before Jan uary 11, and the re be at; 
that, and at the next node, two solar and one lunar 
eclipse at each ; then the twelfth lunation from the 
first eclipse will give a new moon within the y. ar, and 
(on account of the retrograde motion of the moon's 
nodes) the earth may be got within the solar ecliptic 
limits, and there may be another solar eclipse. Hence, 
when here are seven eclipses in a year, five will be 
of the sun and two of the moon. This is upon sup- 


position that the first eclipse is of the sun ; but if the 
first eclipse should be of the moon, there may be three 
of the sun and four of the moon. 

As there are seven eclipses in the year but seldom, 
the mean number will be about four. 

The nodes of the moon move backwards about 19^^ 
in a year, which arc the eartli describes in about I9 
days, consequently the middle of the seasons of the 
ecHpses happens every year about I9 days sooner than 
in the preceding year. 

The ecliptic limits of the sun (313) are greater than 
those of the moon (306), and hence, there will be 
more solar than lunar eclipses, in about the same pro- 
portion as the limit is greater, that is, as 3 : 2 nearly. 
But more lunar than solar eclipses are seen at any 
given place, because a lunar eclipse is visible to a whole 
hemisphere at once ; whereas a solar eclipse is visible 
only to a part, and therefore there is a greater proba- 
bility of seeing a lunar than a solar eclipse. Since the 
moon is as long above the horizon as below, every 
spectator may expect to see half the number of lunar 
eclipses which happen. 

For the calculation of eclipses, and all the circum- 
stances respecting them, see my Complete System of 


Chap. XXII. 


(328.) When Dr. Halley was at St. Helena, whither 
he went for the purpose of making a catalogue of the 
stars in the southern hemisphere, he ohserved a transit 
of Mercury over the sun's disc ; and, by means of a 
good telescope, it appeared to him that he could deter- 
mine the time of the ingress and egress, without it's 
being subject to an error of l"*; upon which he im- 
mediately concluded, that the sun's parallax might be 
determined by such observations, from the difference 
of the times of the transit over the sun, at different 
places upon the earth's surface. But this difference is 
so small in Mercury, that it would render the conclu- 
sion subject to a great degree of inaccuracy; in Venus, 
however, whose parallax is nearly four times as great 
as that of the sun, there will be a very considerable 
difference between the times of the transits seen from 
different parts of the earth, by which the accuracy of 
the conclusion will be proportionably increased. The 
Doctor, therefore, proposed to determine the sun's 
parallax from the transit of Venus over the sun's disc, 
observed at different places on the earth ; and as it was 
not probable that he himself should live to observe the 

* Hence, Dr. Ilallej/ concluded, that by a transit of Venus, the 
sun's distance might be determined with certainty to the 600th 
part of the whole ; but the observations upon the transits which 
happened in !761 and 1769, showed that the time of contact of the 
liaibs of the Sun and Venus could not be determined to that degree 
of certainty. 


next transits, which happened in 1761 and 1769, he 
very earnestly recommended the attention of thein to 
the Astronomers who should be alive at that time. 
Astronomers were therefore sent from England and 
France to the most proper parts of the earth to observe 
both those transits, from the result of which, the 
parallax has been determined to a veiy great degree of 

(329.) Kepler was the first person who predicted 
the transits of Venus and Mercury over the sun's disc; 
he foretold the transit of Mercury in l631, and the 
transits of Venus in l631 and 176'!. The first time 
Venus was ever seen upon the sun, was in the year 
1639, on November 24, at Hoole, near Liverpool, by 
our countryman Mr. Horrox, who was educated at 
Emanuel College in this University. He was em- 
ployed in calculating an Ephemerisfrom the Lausherg 
Tables, which gave, at the conjunction of Venus with 
the sun on that dav, it's apparent latitude less than 
the semidiameter of the sun. But as these Tables had 
so often deceived him, he consulted the Tables con- 
structed by Kepler, according to which, the conjunc- 
tion would be at 8//. l' a. m. at Manchester, and the 
planet's latitude 14'. 10" south; but, from his own 
corrections, he expected it to happen at 3h, 57' p.m. 
with 10' south latitude. He accordingly gave this in- 
formation to his friend Mr. Crahtree, at Manchester, 
desiring him to observe it; and he himself also pre- 
pared to make observations upon it, by transmitting 
the sun's image through a telescope into a dark 
chamber. He described a circle of about six inches 
diameter, and divided the circumference into 360°," 
and the diameter into 120 equal parts, and caused the 
sun's image to fill up the circle. He began to observe 
on the 23d, and repeated his observations on the 24th 
till one o'clock, when he was unfortunately called away 
by business; but, returning at 15' after three o'clock, 
he had the satisfaction of seeing Venus upon the sun's- 
disc, just wholly entered on the left side, so that the 


limbs perfectly coincided. At 35' after three, he found 
the distance of Venus from the sun's center to be 13', 
30"; and at 45' after three, he found it to be 13'; and 
the sun setting at 50' after three o'clock, put an end to 
his observations. From these observations, Mr. Horrox 
endeavoured to correct some of the elements of the 
orbit of Venus. He found Venus had entered upon 
the disc at about 62°. 30' from the vertex towards the 
right on the image, which, by the telescope, was in- 
verted. He measured the diameter of Venus, and 
found it to be to that of the sun, as 1,12 : 30, as near 
as he could measure. Mr. Crabtree, on account of 
the clouds, got only one sight of Venus, which was at 
2>h. Ab'. Mr. Horrox * wrote a Treatise, entitled 
Venus in Sole visa, but did not live to publish it ; it 
was, however, afterwards published by Hevelius. 
Gassendus observed the transit of Mercury which 
happened on November f, l631, and this was the first 
which had ever been observed; he made his observa- 
tions in the same manner that Horrox did after him. 
Since his time, several transits of Mercury have been 
observed, as they frequently happen ; whereas only 
two transits of Venus have happened since the time of 
Horrox. If we know the time of the transit at one 
node, we can determine, in the following manner, 
when they will probably happen again at the same 

(330.) The mean time from conjunction to conjunc- 
tion of Venus or Mercury being known (Art. 201), 
and the time of one mean conjunction, we shall know 
the time of ail the future mean conjunctions ; observe, 
therefore, those which happen near to the node, and 
compute the geocentric latitude of the planet at the 
time of conjunction, and if it be less than the appa- 

* The diflBculties which this very extraordinary person had to 
encounter with in his astronomical pursuits, he himself has de- 
scribed, in the Prolegomena prefixed to his Opera Postfmna^ 
published by Dr. fVallis, 


rent semidiameter of the sun, there will be a transit of 
the planet over the sun's disc ; and we may determine 
the periods when such conjunctions happen, in the 
following manner. Let P = the periodic time of the 
earth, p that of Venus or Mercury. Now that a 
transit may happen again at the same node, the earth 
must perform a certain number of complete revolu- 
tions in the same time that the planet performs a cer- 
tain number, for then they must come into conjunction 
again at the same point of the eai'th's orbit, pr nearly 
in the same position in respect to the node. Let the 
earth jjerform x revolutions whilst the planet performs 

y revolutions; then will Px—j)y^ therefore -=■£. 

Now P = 365,256, and for Mercury, jO = 87,968; 

. P X p 87,968 ., , . -^ . , . , 

tnerelore- = ~ = -n -. = (by resolving; it into its 

y P 365,25b ^ ^ ^ 

continued fractions) -, — , — , — ,— ^, ■, &c. That 

^ 4' 25' 29' 54'137'191' 

is, 1, 6, 7, 13, 33, 46, &c. revolutions of the earth are 
nearly equal to 4, 25, 29, 54, 13 7, I9I, &c. revolu- 
tions of Mercury, approaching nearer to a state of 
equality, the further you go. The first period, or 
that of one year, is not sufficiently exact ; the period 
of six years will sometimes bring on a return of the 
transit at the same node ; that of seven years more fre- 
quently ; that of 13 years still more frequently, and so 
on. Now there was a transit of Mercury at its de- 
scending node, in May, 17S6; hence, by continually 
adding 6, 7, 13, 33, 46, &c. to it you get all the years 
when a transit may be expected to happen at that node. 
In 1789, there was a transit at the ascending node, 
and therefore, by adding the same numbers to that 
year, you will get the years in which the transits may 
be expected to happen at that node. The next tran- 
sits at the descending node will happen in 1832, 
1845, 1878, I891 ; and at the ascending node, in 
1815, 1822, 1835, 1848, I861, 1868, 1881, I894. 


T^ T^ ^ y X p 224,7 

¥ov Jenus, p = 224,7', hence, ^ = -p=^^^-^ 

8 235 713 o r.., ^ , 

=^"-r7 "TTTT-j -, ifcc. Inereiore the periods are 8, 

13' 382' 1159 

235, 713, &c. years. The transits at the same node 
will therefore, sometimes, return in eight years, but 
oftener in 235, and still oftener in 713, &c. Now, in 
1769, a transit happened at the descending node in 
June, and the next transits at the same node will be 
in 2004, 2012, 2247, 2255, 249O, 2498, 2733, 2741, 
and 2984. In 1639, a transit happened at the ascend- 
ing node in November ; and the next transits at the 
same node will be in 18/4, 1882, 21 17, 2125, 236o, 
2368, 2603, 2611, 2846, and 2854. These transits 
are found to happen, by continually adding the periods, 
and finding the years when they may be expected, and 
then computing, for each time, the shortest geocentric 
distance of Venus from the sun's center at the time of 
conjunction, and if it be less than the semidiameter of 
the sun, there will be a transit. 

A new Method of computing the Effect of Parallar, 
in accelerating or t^etarding the Time of the Be- 
ginning or End of a Transit of Venus or Mercury 
over the Sun's Disc. By Nevil Maskelyne, D. D. 
F. R. S. and Astronomer Royal. 

(331.) The scheme which is here given, relates 
particularly to the transit of J'^enu-s over the sun 
which happened in 1769. Let C represent the center 
of the sun LQ, P the celestial north pole of the equa- 
tor, PC a meridian passing through the sun, Z the 
zenith of the place, ADB ?S the relative path of Venus, 
^ being the relative place of the descending node ; A 
the geocentric place of Venus at the ingress, B at the 
egress, and D at the nearest approach to the sun's 
center, as seen from the earth's center, and o the appa- 
rent place of Venus at the egress to an observer whose 


zenith is Z\ draw ouZ^ and u is the true place of 
Venus when the apparent place is at o, and>«^is the 

parallax in altitude of Venus from the suii; and the 
time of contact will be diminished by the time which 
Venus takes to describe u B ; draw 7i d honE parallel 
to A B, meeting Z B produced in E, and Bti, A ill, 
tangents to the circle, and let ChD be perpendicular 
to AB. Now the trapezium aoEB, on account of 
the small ness of it's sides, may be considered as rec- 
tilinear, and from the magnitude oi'ZB compared with 
Bu, BE may be considered as parallel to no, conse- 
quently no E B may be considered as a parallelogram, 
and therefore Eo may be taken equal to Bu. Now 
Eo = Enisino, according as E falls without or within 



the circle LQ of the sun's disc; and (Trig. Art. 128) 
En : EB :: sin. EBn = cos. CBZ : sm. BnE —sin, 

T>/-rrk ^I>T^ U r EBx COS.CBZ , 

BLD — Q.OS. CBD; hence, En = ^^m^ — ; ^"d 

' cos. CBD 

(by Euclid) no = — , = -jj^ very nearly ; but Bn : BE 

:: sin. B En = s\n. Z B D : sin. BnE:^ cos. CBD; 

, ^ „ ^ BE'xsm.ZBD' , 

theretore Bn = ^. „ , ,> ; hence, 720 = 


—TTi ' ^,r> i^n - Put h = horizontal parallax of 

AB X cos. CBD"^ ^ 

Venus from the sun ; then (136) BE = hx sin. -Zo = 

h X sin. Z B ', hence, 11 B = E = E ?i :h no = 

h X s'ln.Z B X COS. CBZ .j^ . „„j s'm. ZBD^ 

^^^ (.jSjj Xfi xsm.Z/i X jji^^^^cBD 

= (Trig. Art. 80) // x sin. ZB x cos.CBZ x sec. CBD 

. /?^ xsin. Zl^^ X sin. ZBD^ x sec. CBD' ^, 

X ' rn • A he pa- 

A iJ 

rallax, therefore, consists of two parts ; one part varies 

as h, and the other as /i% the other quantities being 

the same. Put #" = the time which Venus takes, by 

it's geocentric relative motion, to describe the space // ; 

to find which, let ?« be the relative horary motion of 

,r , 7 1 ^ „ .„ hxSGoo" 

Venus; then ?/i : h :: 1 hour = 3600 : t = . 


Hence, to find the time of describing uB, we have, 

h : h X sin. ZB X cos. CBZ x sec. CB D ± 

h' X sin. ZB' X sin. ZBD' x sec. CBD' ^ ^ . 

— . r-n — ■: t : t x sm. 


ZB X cos. CBZx sec. CJ5/J + 

t xh X sm.ZB' X sin. ZBD' x sec. CBD' 

describing uB, or tlic effect of parallax in accelerating 
or retarding the time of contact; the upper sign is to 
be used when CBZ is acute, and the lower sign when 
it is obtuse. If CBZ be very nearly a right angle, 
but obtuse, it may happen that nE may be less than 

the time of 


no, in which case, nE is to be taken from no, according 
to the rule. The principal part 7iE of the effect of 
parallax will increase or diminish the planet's distance 
from the sun's center, according as the angle ZBC is 
acute or obtuse ; but the small part tio of parallax will 
always increase the planet's distance from the center; 
take, therefore^ the sum or difference of the effects, 
with the sign of the greater, as to increasing or de- 
creasing the planet's distance from the center of the 
sun. The second part of the correction will not ex- 
ceed 9" or lo" of time in the transits of Venus in 
1761 and 1769, where the nearest apjiroach of Venus 
to the sun's center was about 10'. In the transit of 
3Iercurij, the first part alone will be sufficient, except 
the nearest distance be much greater. 

If we suppose the 7nean horizontal parallax of the 
sun to be 8", 8 J, then, by calculation from * the above 
expression, it appears that the total durationat Wardhus 
was lengthened by parallax 1 1'. l6",88,and diminished 
at Otaheite by 12'. 10".07; hence, the computed dif- 
ference of the times is 23'. 26",95 3 but the observed 
difference was 23'. 10". 

(332.) Hence, the correct parallax may be accu- 
rately found as follows. Because the observed differ- 
ence of the total durations at Wardhus and Otaheite 
is 23'. lo", and the computed difference, from the 
assumed mean horizontal parallax of the sun 8",83, is 
23'. 26",95, the true parallax of the sun is less than 
that assumed. Let the true parallax be to that as- 
sumed as 1 — e to 1, and (331) the first parts of the 
computed parallax will be lessened in the ratio of 1 — e 
: 1 ; and the second parts, in the ratio of 1 - ef to 1, 
or of 1 - 2e to 1 nearly. All the first parts, viz. 
406",05 ; 287",05 ; 34l",48;382",47,inall = l417",05, 
combine the same way to make the total duration 
longer at Wardhus than at Otaheite. As to the 

* See my Co?>iplete Sj/stem of Astronomy, Chap. 25. 


second parts, the effects at Wardhus are — 7">3l and 
-8",91, and at Otaheite are+l",63 and_-f4",4g, in 
all =--10", 10. Therefore 1417",05 xl—e- 10",10 
X 1 - 2e= 1390" the excess of the observed total dura- 
tion at Wardhus abo ve that at Qtaheite ; or 1417",05 
— 10",10 — 1390" = 1417",05-20"j20xe; and e = 

^.„ =0,0121. Hence, the mean horizontal 

parallax of the sun^8",83x 1 -0,0121 =8",723l6; 
we assume, therefore, the mean horizontal parallax of 
the sun = 8" J. ♦ 

Hence, the radius of the earth : the distance of the 
sun from the earth ;: sin. 8"! : rad. :: 1 : 23575. 

(333.) The effect of the parallax being determined, 
the transit affords a very ready method of finding the 
difference of the longitudes of two places where the 
same observations were made. For, compute the 
effect of parallax in time, and reduce the observations 
at each place to the time, if seen from the center of 
the earth, and the difference of the times is the differ- 
ence of the longitudes. For example, the times at 
Wardhus and Otaheite, at which the first internal con- 
tact would take place at the earth's center^ are Qh. 40'. 
44",6, and \2h. 38'. 25",07, the difference of which is 
12h. 2'. 19",53 = 180°. 34'. 53", the difference of the 
meridians. From the mean of 63 results from the 
transits of Mercury, Mr. Short found the difference of 
the meridians of Greenwich and Paris to be 9'. 15"; 
and from the transit of Venus in 1761, to be 9'. lO" 
in time. 

(334.) The transit of Venus affords a very accurate 
method of finding the place of the node. For by the 
observations made by Mr. Rittenhouse^ at Norriton in 
the United States of America, the least distance CD 
was observed to be 10'. 10"; hence, drawing C^per- 
pendicular to C?5, cos.DCF=S°. 28'. 54" : rad. :: 
CD= 10'. 10" : CF= 10'. 17", the geocentric latitude 


of Venus at the time of conjunction; and * 0,72626 : 
0,28895 :: 10'. 17" : 4'. 5", the hehocentric latitude CV 
of Venus; hence, considering C?s/^as a right Uned 
triangle, tan. ^^ C=3°. 23'. 35" : rad. :: the helio- 
centric latitude CF=4\ b" : C^ =1°. 8'. 52", which 
added to 2'. 13°. 26'. 34", the place of the sun, gives 
'2\ 14°. 35'. 26" for the place of the ascending node of 
the orbit of Venus. 

(335.) The time of the ecliptic conjunction may be 
thus found. Find, at any time (t), the difference (d) 
of longitudes of Venus and the sun's center; find also 
the apparent geocentric horary motion (jn) of Venus 
from the sun in longitude, and then say, m : d :: 1 
hour : the interval between the time (t) and the con- 
junction, which interval is to be added to or subtracted 
from f, according as the observation was made before 
or after the conjunction. In the transit in 1761, at 
6/1. 31'. 46", apparent time at Paris, M. de la Lands 
found d = 2'. o4:\4: and m = 3'. 57",4; hence, 3'. 57",4 
: 2'. 34",4 :: 1 hour : 39'. l", which subtracted from 
6h. 31'. 46", because at that time the conjunction was 
past, gives 5/?. 52'. 45" for the time of conjunction from 
this observation. We may also thus find the latitude 
at conjunction. The horary motion of Venus in lati- 
tude was 35",4; hence, 60' : 39',1 :: 35",4 : 23", the 
motion in latitude in 39',1, which subtracted from 10'. 
1",2, the latitude observed at 6//. 31'. 46", gives 9'. 
38",2 for the latitude at the time of conjunction. 

* 0,72626 is the distance of Venus from the sun, her distance 
from the earth being 0,28895 ; and the angle subtended by CF is 
inversely as the distance from CV. 

& 2 


Chap. XXIII. 


(336.) Comets are solid bodies, revolving in very ex- 
centric ellipses about the sun in one of the foci, and 
are therefore subject to the same laws as the planets, 
but differ in appearance from them ; for as they ap- 
proach the sun, a tail of light, in some of ibem, begins 
to appear, which increases till the comet comes to it's 
perihelion, and then it decreases again, and vanishes ; 
others have a light encompassing the nucleus, or body 
of the comet, without any tail. The most ancient 
philosophers supposed comets to be like planets, per- 
forming their revolutions in stated times. Aristotle^ 
in his first book of Meteors, speaking of comets, says, 
" But some of the Italians, called Pythagoreans, say, 
that a Comet is one of the Planets, but that they do 
not appear unless after a long time, and are seen but a 
small time, which happens also \.o Mercury T Seneca 
also, in Nat. Quest. Lib. vii. says, " Apollonius af- 
firmed, that the Comets were, by the Chaldeans, 
reckoned among the Planets, and had their periods 
like them." Seneca himself also, having considered 
the phaenomena of two remarkable comets, believed 
them to be stars of equal duration with the world, 
though he was ignorant of the laws that governed 
them ; and foretold, that after-ages would unfold all 
these mysteries. He recommended it to Astronomers 
to keep a catalogue of the comets, in order to be able 
to determine whether they returned at certain periods. 
Notwithstanding this, most Astronomers, from his time 
till Tycho Brake, considered them only as meteors, 
existing in our atmosphere. But that Astronomer, 


finding, from his own observations on a comet, that it 
had no diurnal parallax, placed them above the moon. 
Afterwards Kepler had an opportunity of observing 
two comets, one of which was very remarkable ; and 
from his observations, which aiforded sufficient indica- 
tions of an annual parallax, he concluded, " that 
comets moved freely through the planetary orbs, with 
a motion not much different from a rectilinear one; 
but of what kind he could not precisely determine." 
Hevelius embraced the hypothesis of^a rectilinear 
motion ; but, finding his calculations did not perfectly 
agree with his observations, he concluded, " that the 
path of a comet was bent in a curve line, concave to- 
wards the sun." He supposed a comet to be generated 
in the atmosphere of a planet, and to be discharged 
from it, partly by the rotation of the planet, and then 
to revolve about the sun in a parabola by the force of 
projection and it's tendency to the sun, in the same 
manner as a projectile upon the earth's surface describes 
a parabola. At length came the famous comet in 
168O, which descending nearly in a right line towards 
the sun, arose again from it in like manner, which 
proved it's motion in a curve about the sun. G. S. 
Doerfell^ Minister at Plaven in Upper Saxony, made 
observations upon this comet, and found that it's mo- 
tion might be very well represented by a parabola, 
having the sun in it's focus. He was ignorant, how- 
ever, of all the laws by which the motion of a body in 
a parabola is regulated, and erred considerably in his 
parabola, making the perihelion distance about twelve 
times greater than it was. This was published five 
years before the Principia,\n which work Sir /. Newton 
having proved that Kepler's law, by which the motions 
of the planets are regulated, was a necessary conse- 
quence of his theory of gravity, it immediately foU 
lowed, that comets were governed by the same law ; 
and the observations upon them agreed so accurately 
with his theory, as to leave no doubt of it's truth. 
That comets describe ellipses^, and not parabolas oy 


hyperbolas, Dr. Halley (see his Synopsis of the 
Astronomy of Comets) advances the following reasons. 
" Hitherto I have considered the orbits as exactly 
parabolic, upon which supposition it would follow, 
that comets, being impelled towards the sun by a cen- 
tripetal force, would descend as from spaces infinitely 
distant, and, by their so falhng, acquire such a ve- 
locity, as that they may again fly off into the remotest 
parts of the universe, moving upwards with a perpetual 
tendency, so as never to return again to the sun. But 
since they appear frequently enough, and since none 
of them can be found to move with an hyperbolic mo- 
tion, or a motion swifter than what a comet might 
acquire by it's gravity to the sun, it is highly probable 
they rather move in very excentric elliptic orbits, and 
make their returns after long periods of time ; for so 
their number will be determinate, and, perhaps, not so 
very great. Besides, the space between the sun and 
the fixed stars is so immense, that there is room 
enough for a comet to revolve, though the period of 
it's revolution be vastly long. Now, the latus rectum 
of an ellipsis is to the lotus rectum of a parabola, 
which has the same distance in it's perihelion, as the 
distance in the aphelion, in the eli psis, is to the whole 
axis of the ellipsis. And the velocities are in a sub- 
duplicate ratio of the same; wherefore, in very excen- 
tric orbits, the ratio comes very near to a ratio of 
equality ; and the very small difference which happens, 
on account of the greater velocity in the parabola, is 
easily compensated in determining the situation of the 
orbit. The principle use, therefore, of the Table of 
the elements of their motions, and that which indeed 
induced me to construct it, is, that whenever a new 
comet shall appear, we may be able to know, by com- 
paring together the elements, whether it be any of 
those which have appeared before, and consequently 
to determine it's period, and the axis of it's orbit, and 
to foretel it's return. And, indeed, there are many 
things which make m.e believe that the cornet^ which 


Apian discovered in the year 1531, was the same with 
that which Kepler and Longomontcmus more accu- 
rately described in the year 1607 ; and which I myself 
have seen return, and observed in the year l682. All 
the elements agree, and nothing seems to contradict 
this my opinion, besides the inequality of the periodic 
revolutions; which inequality is not so great neither, 
as that it may not be owing to physical causes. For 
the motion of Saturn is so disturbed by the rest of the 
planets, especially Jupiter, that the periodic time of 
that planet is uncertain for some whole days together. 
How much more, therefore, will a comet be subject to 
such like errors, which rises almost four times higher 
than Saturn, and whose velocity, though increased but 
a very little, would be sutficient to change it's orbit, 
from an elliptic to a parabolical one. And I am the 
more confirmed in my opinion of it's being the same ; 
for in the year 1456,. in the summer-time, a comet was 
seen passing retrograde between the earth and the sun, 
much after the same manner ; which, though nobody 
made observations upon it, yet, from it's period, and 
the manner of it's transit, 1 cannot think difFei-ent 
from those I have just now mentioned. And since 
looking over the histories of comets, I find, at an equal 
interval of time, a comet to have been seen about 
Easter in the year 1305, which is another double period 
of 151 years before the former. Hence, I think, I 
may venture to foretel that it will return again in the 
year 1758." 

(337.) Dr. Halley computed the effect of Jupiter 
upon this comet in l682, and found that it would in- 
crease it's periodic time above a year, in consequence 
of which he predicted it's return at the end of the year 
1758, or the beginning of 1759. He did not make 
his computations with the utmost accuracy, but, as he 
himself informs us, kvi calamo. M. Clairauf com- 
puted the eflfects both of Saturn and Jupiter, and found 
that the former would retard it's return in the last pe- 
riod 100 days, and the latter 511 days; and he deter- 


mined the time when the comet would come to it's 
perihelion to be on April 15, 1/^9^ observing that he 
might err a month, from neglecting small quantities in 
the computation. It passed the perihelion on March 
13, within 33 days of the time computed. Now if we 
suppose the time stated by Dr. Hallei/ to mean the 
time of it's passing the perihelion, then if we add to 
that 100 days, arising from the action of Saturn, 
which he did not consider, it will bring it very near to 
the time in which it did pass the perihehon, and prove 
his computation of the effect of Jupiter to have been 
very accurate. If he meant the time when it would 
first appear, his prediction was very accurate, tor it was 
first seen on December 14, 17^8, and his computation 
of the effects of Jupiter will then be more accurate than 
could have been expected, considering that he made 
his calculations only by an indirect method, and in a 
manner confessedly not very accurate. Dr. Halley, 
therefore, had the glory, first to foretel the return ot a 
comet, and the event answered remarkably to his pre- 
diction. He further observed, that the action of 
Jupiter, in the descent of the comet towards it's 
perihelion in l682, would tend to increase the inclina- 
tion of it's orbit; and accordingly the inclinati(m in 
l682 was found to be 22' greater than in ib'O/. A 
learned Professor (Dr. Long's Astronomy, p. 562) in 
Italy to an English gentleman, writes thus : — 
*' Though M. de la Lande, and some other French 
gentlemen, have taken occasion to find fault with the 
inaccuracies of Halleys calculation, because he him- 
self had said, he only touched upon it slightly; never- 
theless they can never rob him of the honour, — First, 
of finding out that it was one and the same comet 
which appeared in l682, 1607, 1531, 1456, and 1305. 
— Secondly, of having observed, that the planet Jupiter 
would cause the inclination of the orbit of the comet 
to be greater, and the period longer. — Thirdly, of hav- 
ing foretold that the return thereof might be retarded 
till the end of 1/58, or the beginning of 1759." 


From the observations of M, Messier upon a comet 
in 1 770, Mr. Edric Prosperin, Member of the Royal 
Academies of Stockhohn and Upsal, showed, that a 
parabolic orbit would not answer to it's motions, and 
he recommended it to Astronomers to seek for the 
eUiptic orbit. This laborious task M. Lexell under- 
took, and has shown that an ellipse, in which the 
periodic time is about five years and seven months, 
as^rees very well with the observations. See the Phil. 
Trans. 17 79. As the ellipses which the comets de- 
scribe are very excentric, Astronomers, for the ease of 
calculation, suppose them to move in parabolic orbits, 
for that part which lies within the reach of observa- 
tion, by which they can very accurately find the place 
of the perihelion ; it's distance from the sun; the in- 
chnation of the plane of it's orbit to the ecliptic, and 
the place of the node. But it fails not within the 
plan of this work to enter into an investigation of these 
matters. For this, I refer the reader to my Complete 
System of Astronomy. 

(338.) It is extremely difficuk to determine, from 
computation, the elliptic orbit of a comet, to any 
degree of accuracy ; for when the orbit is very excen- 
tric, a very small error in the observation will change 
the computed orbit into a parabola, or hyperbola. 
Now from the thickness and inequality of the atmo- 
sphere with which the comet is surrounded, it is im- 
possible to determine with any great precision, when 
either the limb or center of the comet passes the wire 
at the time of observation. And this uncertainty in 
the observations will subject the computed orbit to a 
great t-rror. Hence, it happened, that M. Bouguer 
determined the orbit of the comet to be an liyperbola. 
M. Elder first determined the same for the comet in 
17/4; but, having received more accurate observa- 
tions, he found it to be an ellipse. The period of the 
comet in l680 appears, from observation, to be 575 
years, which Mr. Eider, by his computalion, deter- 
mined to be l66'| years. The only safe way to get 


the periods of comets, is to compare the elements of 
all those which have been computed, and where you 
find they agree very well, you may conclude that they 
are elements of the same comet, it being so extremely 
improbable that the orbits of two different comets 
should have the same inclination, the same perihelion 
distance, and the places of the perihelion and node 
the same. Thus, knowing the periodic time, we get 
the major axis of the ellipse ; and the perihelion 
distance being known, the minor axis will be known. 
When the elements of the orbits agree, the comets 
may be the same, although the periodic times should 
vary a little ; as that may arise from the attraction of 
the bodies in our system, and which may also alter all 
the other elements a little. We have already observed, 
that the comet which appeared in 1759, had it's 
periodic time increased considerably by the attraction 
oi Jupiter ^nA Saturn. This comet was seen in l682, 
1607, and 1531, all the elements agreeing, except a 
little variation of the periodic time. Th. Halley sus- 
pected the comet in- 168O, to have been the same 
which appeared in II06, 531, and 44 years before 
Christ. He also conjectured, that the comet observed 
by /Jpian, in 1532, was the same as that observed by 
Hevelius, in 1661 ; if so, it ought to have returned in 
^79^i but it has never been observed. But M. 
Mechain having collected all the observations in 1532, 
and calculated the orbit again, found it to be sensibly 
different from that determined by Dr. Halley^ w^hich 
renders it very doubtful whether this was the comet 
which appeared in 1661 ; and this doubt is increased, 
by it's not appearing in 1790- The comet in 177^. 
whose periodic time M. Lexell computed to be five 
years and seven months, has not been observed since. 
There can be no doubt but that the path of this comet, 
for the time it was observed, belonged to an orbit 
whose periodic time was that found by M. Lexell, as 
the computations for such an orbit agreed so very well 
with the observations. But the revolution was proba- 


biy longer before 1770; for as the comet passed very- 
near to Jupiter in 17^7? it's periodic time might be 
sensibly increased by the action of that planet ; and as 
it has not been observed since, we may conjecture, 
with M. Lexell, that having passed in 1772 again into 
the sphere of sensible attraction of Jupiter, a new 
disturbing force might probably take place and destroy 
the effect of the other. According to the above ele- 
ments, the comet would be in conjunction with Jupiter 
on August 23, 1779> and it's distance from Jupiter 
would be only-^i-g- of it's distance from the sun ; conse- 
quently the sun's action would be oniy-^J-^ part of that 
of Jupiter. What a change must this make in the 
orbit ! If the comet returned to it's perihelion in 
March, 177^) it would then not be visible. See 
M. Lexell's account in the Phil. Tram. 1779. The 
elementsof the orbits of the comets, in 1264 and 1556, 
were so nearly the same, that it is very probable it was 
the same comet; if so, it ought to appear again about 
the year 1848. 

On the Nature and Tails of Comets. 

(339.) Comets are not visible till they come into 
the planetary regions. They are surrounded with a 
very dense atmosphere, and from the side opposite to 
the sun they send forth a tail, which increases as the 
comet approaches it's perihelion, immediately after 
which it is longest and most luminous, and then it is 
generally a little bent and convex towards those parts 
to which the comet is moving; the tail then decreases, 
and at last it vanishes. Sometimes the tail is observed 
to put on this figure ^ towards it!s extremity, as that 
did in 1796- The smallest stars arc seen through the 
tail, notwithstanding it's immense thickness, which 
proves that it's matter rnust be extremely rare. The 
opinion of the ancient philosophers, and oi Aristotle 
himself, was, that the tail is a very thin fiery vapour 


arising from the comet. Apian, Cardan, Tycho, and 
others, believed that the sun's rays, being propagated 
through the transparent head of the comet, were re- 
fracted, as in a lens. But the figure of the tail does 
not answer to this ; and, moreover, there should be 
some reflecting substance to render the rays visible, in 
like manner as there must be dust or smoke flving 
about in a dark room, in order that a ray of light 
entering, it may be seen by a spectator standing side- 
ways from it. Kepler supposed, that the rays of the 
sun carry away some of the gross parts of the comet 
which reflect the sun's rays, and give the appearance 
of a tail. Hevel'ms thought that the thinnest parts of 
the atmosphere of a comet are rarified by the force of 
the heat, and driven from the fore part and each side 
of the comet towards the parts turned from the sun. 
Sir /. Newton thinks, that the tail of a comet is a very 
thin vapour, which the head, or nucleus of the comet, 
sends out by reason of it's heat. He supposes, that 
when a comet is descending to it's perihelion, the 
vapours behind the comet, in respect to the sun, being 
rarified by the sun's heat, ascend, and take up with 
them the reflecting particles with which the tail is 
composed, as air rarefied by heat carries up the parti- 
cles of smoke' in a chimney. But as, beyond the 
atmosphere of the comet, the aetherial air [aura 
cetherea) is extremely rare, he attributes something to 
the sun's rays carrying with them the particles of the 
atmosphere of the comet. And when the tail is thus 
formed, it, like the nucleus, gravitates towards the 
sun, and by the projectile force received from the 
comet, it describes an ellipse about the sun, and ac- 
companies the comet. It conduces also to the ascent 
of these vapour,*, that they revolve about the sun, and 
therefore endeavour to recede from it; whilst the at- 
mosphere of the sun is either at rest, or moves with 
su(^h a slow motion as it can acquire from the rotation 
of the sun about it's axis. These are the causes of the 
ascent of the tails in the neighbourhood of the sun. 


where the orbit has a greater curvature, and the comet 
moves in a denser atmosphere of the sun. The tail of 
the comet, therefore, being formed from the heat of 
the sun, will increase till it comes to it's perihelion, and 
decrease afterwards. The atmosphere of the comet is 
diminished as the tail increases, and is least immedi- 
ately after the comet has passed it's perihelion, where it 
sometimes appears covered with a thick black smoke. 
As the vapour receives two motions when it leaves the 
comet, it goes on with the compound motion, and 
therefore the tail will not be turned directly from the 
sun, but decline from it towards those parts which are 
left by the comet ; and meeting with a small resistance 
from the aether, will be a little curved. When the 
spectator, therefore, is in the plane of the comet's 
orbit, the curvature will not appear. The vapour, 
thus rarefied and dilated, may be at last scattered 
through the heavens, and be gathered up by the 
planets, to supply the place of those fluids which are 
spent in vegetation and converted into earth. This is 
the substance of Sir /. Newton's account of the tails of 
comets. Against this opinion, Dr. Hamilton, in his 
Philosophical Essays, observes, that we have no proof 
of the existence of a solar atmosphere ; and if we had, 
that when the comet is moving in it's perihelion in a 
direction at right angles to the direction of it's tail, the 
vapours which then arise, partaking of the great velocity 
of the comet, and being also specifically lighter than 
the medium in which they move, must suflfer a much 
greater resistance than the dense body of the comet 
does, and therefore ought to be left behind, and would 
not appear opposite to the sun ; and afterwards they 
ought to appear towards the sun. Also, if the splen- 
dour of the tails be owing to the reflection and refrac- 
tion of the sun's rays, it ought to diminish the lustre 
of the stars seen through it, which would have their 
light reflected and refracted in like manner, and conse- 
quently their brightness would be diminished. Dr. 
Halleyy in his description of the Aurora Borealis in 


1716', says, " the streams of light so much resembled 
the long tails of comets, that at first sight they might 
well be taken for such." And afterwards, " this light 
seems to have a great affinity to that which the effluvia 
of electric bodies emit in the dark." Phil. Trans. 
N". 347. D. de Mairan also calls the tail of a comet, 
the aiiroi'a borealis of the comet. This opinion Dr, 
Hamilton supports by the following arguments. A 
spectator, at a distance from the earth, would see the 
aurora borealis in the form of a tail opposite to the 
sun, as the tail of a comet lies. The aurora borealis 
has no effect upon the stars seen through it, nor has 
the tail of a comet. The atmosphere is known to 
abound with electric matter, and the appearance of the 
electric matter in vacuo is exactly like the appearance 
of the aurora borealis, which, from it's great altitude, 
may be considered to be in as perfect a vacuum as we 
can make. The electric matter in vacuo suffers the 
rays of light to pass through, without being affected by 
them. The tail of a comet does not expand itself 
sideways, nor does the electric matter. Hence, he 
supposes the tails of comets, the aurora borealis, and 
the electric fluid, to be matter of the same kind. We 
may add, as a further confirmation of this opinion, 
that the comet in 1607 appeared to shoot out at the 
end of it's tail. Le P. Cysat remarked the undula- 
tions of the tail of the comet in 16I8. Hevelms ob- 
served the same in the tails of the comets in l652 and 
1661. M. Pingre took notice of the same appear- 
ance in the comet of 1769' These are circumstances 
exactly similar to the aurora borealis. Dr. Hamilton 
conjectures, that the use of the comets may be to 
bring the electric matter, which continually escapes 
from the planets, back into the planetary regions. The 
arguments are certainly strongly in favour of this 
hypothesis ; and if this be true, we may further add, 
that the tails are hollow ; for if the electric fluid only 
proceed in it's first direction, and do not diverge side- 
ways, the parts directly behind the comet will not be 


filled with it; and this thinness of the tails will ac- 
count for the appearance of the stars through them. 

(340.) In respect to the nature of comets, Sir 
/. Newton observes, that they must be solid bodies 
like the planets; for if they were nothing but vapours, 
they must be dissipated when they come near the sun; 
for the comet in 168O, when it was in it's perihelion, 
was less distant from the sun than one-sixth of the 
sun's diameter, consequently the heat of the comet at 
that time was to the heat of the summer sun as 28000 
to 1. But the heat of boiling water is about three 
times greater than the heat which dry earth acquires 
from the summer's sun ; and the heat of red-hot iron 
about three or four times greater than the heat of 
boiling water. Therefore the heat of dry earth at the 
comet, when in it's perihelion, was about 2000 times 
greater than red-hot iron. By such heat, all vapours 
would be immediately dissipated. 

(341.) This heat of the comet must be retained a 
very long time. For a red-hot globe of iron, of an 
inch diameter, exposed to the o})en air, scarce loses all 
it's heat in an hour; but a greater globe would retain 
it's heat longer, in proportion to it's diameter, because 
the surface, at which it grows cold, varies in that pro- 
portion less than the quantity of hot matter. There- 
fore a globe of red-hot iron, as large as our earth, 
would scarcely cool in 50000 years. 

(342.) The comet in 168O, coming so near to the 
sun, must have been considerably retarded by the 
sun's atmosphere, and therefore, being attracted nearer 
at every revolution, it will at last fall into the sun, and 
be a fresh supply of fuel for what the sun loses by it's 
constant emission of light. In like manner, fixed stars 
which have been gradually wasted, may be supplied 
with fresh fuel, and acquire new splendour, and pass for 
new stars. Of this kind are those fixed stars which 
appear on a sudden, and shine with a wonderful bright- 
ness at first, and afterwards vanish by degrees. Such 
is the conjecture of Sir /. Newton. 


(343.) From the beginning of our sera to this timcj 
it is probable, according to the best accounts, that 
there have appeared about 500 comets. Before that 
time, about 100 others are recorded to have been seen, 
but it is probable that not above half of them were 
comets. And when we consider, that many others 
may not have been perceived, from being too near the 
sun — from appearing in moon-light — from being in 
the other hemisphere — from being too small to be per- 
ceived, or which may not have been recorded, we 
might imagine the whole number to be considerably 
greater; but it is likely, that of the comets which are 
recorded to have been seen, the same may have ap- 
peared several times, and therefore the number may 
be less than is here stated. The comet in 17^6, which 
first appeared on August 1, was discovered by Miss 
Caroline Herschel, a sister of Dr. Herschel ; since 
that time, she has discovered three others. As the 
plan of this work does not permit us to give the 
methods by which the orbits of comets may be com- 
puted, and all the opinions respecting them, if the 
reader wish to see any thing further on the subject, I 
refer him to my Complete System of Astronomy ; or 
to a Treatise, entitled Cometographie, oii Traite His- 
torhjne et Theorique des Cometes, par M. Pingre', 
//. Tom. quarto. Paris, 1784 ; or Sir H. Engle- 
field's Determination of the Orbits of Comets, a 
very valuable work, in which the ingenious Author 
has explained, with great clearness and accuracy, the 
manner of computing the orbits of comets, according 
to the methods of Boscovich andM. de la Place. 


Chap. XXIV, 


(344.) All the heavenly bodies beyond our system, 
are called Fixed Stars, because (except some few) tliey 
do not appear to have any proper motion of their own. 
From their immense distance, they must be bodies of 
very great magnitude, otherwise they could not be 
visible ; and when we consider the weakness of re- 
flected light, there can be no doubt but that the}^ 
shine witli their own light. They are easily known 
from the planets, by their twinkling. The number 
of stars visible at once to the naked eye is about 1000 ; 
but Dr. Herschel, by his improvements of the reflect- 
ing telescope, has discovered that the whole number is 
great, beyond all conception. In that bright tract of 
the heavens, called the Milky IVay^ which, when ex- 
amined by good telescopes, appears to be an immense 
collection of stars which gives that whitish appearance 
to the naked eye, he has, in a quarter of an hour, seen 
116000 stars pass through the field of view of a tele- 
scope of only 15' aperture. Every improvement of 
his telescopes has discovered stars not seen before, so 
that there appears to be no bounds to their number, or 
to the extent of the universe. These stars, which can 
be of no use to us, are probably suns to other systems 
of planets. 

(345.) From an attentive examination of the stars 
with good telescopes, many, which appear only single 
to the naked eye, are found to consist of two, three, or 
more stars. Dr. Maskelyne had observed a Herculis 
to be a double star : Dr. Hormhy had found tt Bootis 



to be double ; M. Cassini, Mr. Mayer, Mr. Plgott, 
and many other Astronomers, had made discoveries of 
the hke kind. But Dr. Herschel, by his improved 
telescopes, has found about 700, of which, not above 
42 had been observed before. We shall here give an 
account of a few of the most remarkable. 

a Herculis, Flam. 64, a beautiful double star ; the 
two stars very unequal, the largest is red, and the 
smallest blue, inclining to green. 

h Lyrce, Flam. 12, double, very unequal, the 
largest red, and smallest dusky ; not easily to be seen 
with a magnifying power of 227. 

a Geminorum, Flam. 6Q, double, a little unequal, 
both white; with a power of 146 ; their distance ap- 
pears equal to the diameter of the smallest. 

6 Lyrce, Flam. 4 and 5, a double-double star; at 
first sight it appears double at a considerable distance, 
and, by a little attention, each will appear double ; one 
set are equal, and both white ; the other unequal, the 
largest white, and the smallest inclined to red. The 
interval of the stars, of the unequal set, is one diameter 
of the largest, with a power of 227. 

7 Andromedce, Flam. 57, double, very unequal, 
the largest reddish white, the smallest a fine bright 
sky-blue, inclining to green. A very beautiful object. 

a Ursce minoris. Flam. 1, double, very unequal, 
the largest white, the smallest red. 

/3 Lyrce, Flam. 10, quadruple, unequal, white, but 
three of them a little inclined to red. 

a Leonis, Flam. 32, double, very unequal, largest 
white, smallest dusky. 

e Bootis, Flam. 36, double, very unequal, largest 
reddish, smallest blue, or rather a faint lilac ; very 

h Draconh, Flam. 39, a very small double star, 
very unequal, the largest white, smallest inclining to 

X Orionis, Flam. 39, quadruple, or rather a double 
star, and has two more at a small distance, the double 


star considerably unequal, the largest white, smallest 
pale rose colour. 

^ Librae y Flam, ultima, double-double, one set 
very unequal, the largest a very fine white. 

(U Cygni, Flam. 78, double, considerably unequal, 
the largest white, the smallest blueish. 

fjL Herculis, Flam. 86, double, very unequal ; the 
small star is not visible with a power of 2/8, but is 
seen very well with one of 46o ; the largest is inclined 
to a pale red, smallest duskish, 

a Capricorni, Flam. 5, double, very unequal, the 
largest white, smallest dusky. 

V Lyroe, Flam. 8, treble, very unequal, the largest 
white, smallest both dusky. 

a Lyroe, Flam. 3, double, very unequal, the largest 
a fine brilliant white, the smallest dusky ; it appears 
with a power of 227. Dr. Herschel measured the 
diameter of this fine star, and found it to be 0",3553. 

(346.) These are a few of the principal double, &o. 
stars mentioned by Dr. Herschel, in his catalogues, 
which he has given us in the Phil. Titans. 1782 and 
178.5. The examination of double stars with a tele- 
scope, is a very excellent and ready method of proving 
it's powers. Dr. Herschel recommends the following 
method. The telescope and the observer having been 
some time in the open air, adjust the focus of the tele- 
scope to some single star of nearly the same magni- 
tude, altitude, and colour of the star to be examined; 
attend to all the phaenomena of the adjusting star as it 
passes through the field of view, whether it be per- 
fectly round and well defined, or affected with little 
appendages playing about the edge, or any other cir- 
cumstances of the like kind. Such deceptions may be 
detected by turning the object glass a little in it's cell, 
when these appendages will turn the same way. Thus 
you will detect the imperfections of the instrument, 
and therefore will not be deceived when you come to 
examine the double star. 

(347.) Several stars, mentioned by ancient Astrono- 
R 2 


mers, are not now to be found, and several are now 
observed, which do not appear in their catalogues. 
The most ancient observation of a new star, is that by 
Hipparchus, about 120 years before J. C. which oc- 
casioned his making a catalogue of the fixed stars, in 
order that future Astronomers might see what altera- 
tions had taken place since his time. We have no 
account where this new star appeared. A new star is 
also said to have appeared in the year 130; another 
in 389; another in the ninth century, in 15° of 
Scorpio ; a fifth in 945 ; and a sixth in 1264 ; but the 
accounts we have of all these are very imperfect. 

(348.) The first new star we have any accurate ac- 
count of, is that which was discovered by Cornelius 
Gemma, on November 8, 1572, in the Chair of 
Cassiopea. It exceeded Sirius in brightness, and 
was seen at mid-day. It first appeared bigger than 
Jupiter f but it gradually decayed, and after sixteen 
months it entirely disappeared. It was observed by 
Ti/cho Brake, who found that it had no sensible 
parallax ; and he concluded that it was a fixed star. 
Some have supposed that this is the same which ap- 
peared in 945 and 1264, the situation of it's place 
favouring this opinion. 

(349.) On August 13, 1596, David Fahricius ob- 
served a new star in the Neck of the Whale, in 25°. 
45' of Aries, with 15°. 54' south latitude. It disap- 
peared after October in the same year. Phocy Hides 
Holwarda discovered it again in 1637, not knowing 
that it had ever been seen before ; and after having 
disappeared for nine months, he saw it come into view 
again. Bullialdus determined the periodic time be- 
tween it's greatest brightness to be 333 days. It's 
greatest brightness is that of a star of the second 
magnitude, and it's least, that of a star of the sixth. 
It's greatest degree of brightness, however, is not always 
the same, nor are the same phases always at the same 

(350.) In the year 160O; IVilliam, Janseniits dis- 


covered a changeable star in the Neck of the Swan. 
It was seen by Kepler, who wrote a Treatise upon it, 
and determined it's place to be 16". 18'x:c^, with 55°. 
30' or 32' north latitude. Ricciolus saw it in l6l6, 
1621, 1624, and 1629. He is positive that it was 
invisible in the last years from l640 to l650. M. 
Cassini saw it again in l655 ; it increased till 1660, 
and then grew less, and at the end of 1 66 1, it disap- 
peared. In November 1665, it appeared again, and 
disappeared in 168I. In 1715 it appeared of the 
sixth magnitude, as it does at present, 

(351.) On June 20, 1670, another changeable star 
was discovered near the Swans Head, by P. Anthelme. 
It disappeared in October, and was seen again on 
March 17, 1671. On September 11, it disappeared. 
It appeared again in March 1672, and disappeared in 
the same month, and has never since been seen. It's 
longitude was r. 52'. 26" of ^ , and it's latitude 47". 
25'. 22"N. The days are here put down for the new 

(352.) In 1686, Kirxhius observed ^ in the Swan 
to be a changeable star; and, from twenty years ob- 
servations, the period of the return of the same phases 
was found to be 405 days ; the variations of it's mag- 
nitude, however, were subject to some irregularity. 

(353.) In the year l604, at the beginning of 
October, Kepler discovered a new star near the heel 
of the right foot of Serpentarius, so very brilliant, that 
it exceeded every fixed star, and even Jupiter, in mag- 
nitifde. It was observed to be every moment changino- 
into some of the colours of the rainbow, except when 
it was near the horizon, when it was generally white. 
It gradually diminished, and disappeared about October 
1605, when it came too near the sun to be visible, and 
was never seen after. It's longitude was 17°. 40'of ^ , 
with l"". 56' north latitude, and was found to have no 

(354.) Montanari discovered two stars in the con- 
stellation of the Ship J marked /3 and 7 by Bayer^ to b<5 


wanting. He saw them in l664, but lost them in 
1668. The star 6 in the tail of the Serpent, reckoned 
by Tycho of the third, was found, by him, of the fifth 
magnitude. The star |0 in Serpe.ntarius did not appear, 
from the time it was observed by him, till 1695. The 
star \// in the Lion, after disappearing, was seen by him 
in 1667. He observed also, that /3 in Medusas Head 
varied in it's magnitude. 

(355.) M. Cassini discovered owe new star of the 
fourth, and two of the fifth magnitude in Cassiopea; 
2\&oJive new stars in the same constellation, of which 
three have disappeared ; two new ones in the beginning 
of the constellation Eridanus, of the fourth and fifth 
magnitude ; and four new ones of the fifth or sixth 
magnitude, near the north pole. He further observed, 
that the star, placed by Bayer near e of the Little 
Bear, is no longer visible ; that the star A of Andi^o- 
meda, which had disappeared, had come into view 
again in 1695 ; that in the same constellation, instead 
of one in the Knee^ marked v, there are two others 
come more northerly; and that ^ is diminished ; that 
the star placed by Tycho, at the end of the Chain of 
Andromeda, as of the fourth magnitude, could then 
scarcely be seen ; and that the star which, in Tycho's 
catalogue, is the twentieth of Pisces, was no longer 

(356.) M. Maraldi observed, that the star k in the 
left leg of Sagittarius, marked by Bayer of the third 
magnitude, appeared of the sixth, in 1671 ; in 1676 
it was found, by Dr. Halley, to be of the third; in 

1693 it could hardly be perceived, but in 1693 and 

1694 it was of the fourth magnitude. In 1704 he 
discovered a star in Hydra to be periodical; it's posi- 
tion is in a right line with those in the tail marked 
TT and 7. The time between it's greatest lustre, which 
is of the fourth magnitude, was about two years ; in 
the intermediate time it disappeared. In 1666, 
Hevelius says, he could not find a star of the fourth 
magnitude in the eastern scale of Libra, observed by 


Tycho and Bayer; but Maraldi, in 1709, says, that 
it had then been seen for 15 years, smaller than one of 
the fourth. 

(357".) t/. Goodrlche, Esq. has determined the pe- 
riodic variation of Algol, or /3 Persei (observed by 
Montanari to be variable) to be about Q,d. 21 h. It's 
greatest brightness is of the second magnitude, and 
least of the fourth. It changes from the second to 
the fourth in about three hours and a half, and back 
again in the same time, and retains it's greatest bright- 
ness for the other part of the time. 

(358.) Mr. Gooflfr/c/ire also discovered, that j3 Lyras 
was subject to a periodic variation. The following is 
the result of his observations. It completes all it's 
phases in 12 days I9 hours, during vvhicli time, it 
undergoes the following changes: — 1. It is of the 
third magnitude for about two days. 2. It diminishes 
in about ].\ days. 3. It is between the fourth and 
fifth magnitude for less than a day. 4. It increases 
in about two days. 5. It is of the third magnitude 
for about three days. 6. It diminishes in about one 
day. 7- It is something larger than the fourth mag- 
nitude for a little less than a day. 8. It increases in 
about one day and three quarters to the first point, 
and so completes a whole period. See the Phil. Trans. 
1785. He has also found, that ^ Cephei is subject to 
a periodic variation of 5d. 8/z. 37'j; during which 
time it undergoes the following changes : I. It is at 
it's greatest brightness about one day thirteen hours. 
2. It's diminution is performed in about one day 
eighteen hours. 3. It is at it's greatest obscuration 
about one day twelve hours. 4. It increases in about 
thirteen hours. It's greatest and least brightness is 
that between the third and fourth, and between the 
fourth and fifth magnitudes. 

(359.) E. Pigott, Esq. has discovered ?/ Antinoi to 
be a variable star, with a period of 'jd. 4h. 38'. The 
changes happen as follows : 1. It is at it's greatest 
brightness 44 + hours. 2. It decreases 62 + hours. 


3. It is at it's least brightness 30 ± hours. 4. It 
increases 36 + hours. When most bright, it is of the 
third or fourth magnitude, and when least, of the fourth 
or fifth. See the Phil. Trans. 1785. 

(36o.) In the Phil. Tram. 1796, Dr. Herschel has 
proposed a method of observing the changes that may 
happen to the fixed stars j with a catalogue of their 
comparative brightness, in order to ascertain the per- 
manency of their lustre. 

(361.) Dr. Herschel, in a Paper of the Phil. Timns. 
1783, upon the proper motion of the solar system, has 
given a large collection of stars which were formerly 
seen, but are now lost; also a catalogue of variable 
stars, and of new stars ; and very justly observes, that 
it is not easy to prove that a star was never seen before ; 
for though it should not be contained in any catalogue 
whatever, yet the argument for it's former non- 
appearance, which is taken from it's not having been 
observed before, is only so far to be regarded, as it can 
be made probable, or almost certain, that a star would 
have been observed, had it been visible. 

(362.) There have been various conjectures to ac- 
count for the appearances of the changeable stars. 
M. Maupertuis supposes, that they may have so quick 
a motion about their axes, that the centrifugal force 
may reduce them to flat oblate spheroids, not much 
unlike a mill-stone; and it's plane may be inclined to 
the plane of the orbits of it's planets, by whose attrac- 
tion the position of the body may be altered, so that 
when it's plane passes through the earth, it may be 
almost or entirely invisible, and then become again 
visible as it's broadside is turned towards us. Others 
have conjectured, that considerable parts of their sur- 
faces are covered with dark spots, so that when, by the 
rotation of the star, these spots are presented to us, the 
stars become almost or entirely invisible. Others 
have supposed, that these stars have very large opaque 
bodies revolving about and very near to them, so as to 
obscure them when they come in conjunction with us. 


The irregularity of the phases of some of them shows 
the cause to be variable, and therefore may, perhaps, 
be best accounted for by supposing that a great part 
of the body of the star is covered with spots, wliich 
appearand disappear hke those on the sun's surfoce. 
The total disappearance of a star may probably be the 
destruction of it's system ; and the appearance of a 
new star, the creation of a new system of planets. 

(363.) The fixed stars are not all evenly spread 
through the heavens, but the greater part of them arc 
collected into clusters, of which it requires a large 
magnifying power, with a great quantity of light, to 
be able to distinguish the stars separately. With a 
small magnifying power and quantity of light, they 
only appear small whitish spots, something like a small 
light cloud, and thence they were called Nehulce. 
There are some nehulce, however, which do not receive 
their light from stars. For in the year 1 656, Huygens 
discovered a nebula in the middle of Orion^s Sword; 
it contains only seven stars, and the other part is a 
bright spot upon a dark ground^ and appears like an 
opening into brighter regions beyond. In l6l2, 
Simon Marius discovered a nebula in the Girdle of 
Andromeda. Dr. Halleif, when he was observing the 
southern stars, discovered one in the Centaur, but this 
is never visible in England. In 1714, he found 
another in Hercules, nearly in a line with ^' and j; of 
Bayer. This shows itself to the naked eye, when the 
sky is clear and the moon absent. M. Cassini dis- 
covered one between the Great Dog and the Ship, 
which he describes as very full of stars, and very 
beautiful, when viewed with a good telescope. There 
are two whitish sjjots near the south pole, called, by 
the sailors, the Magellanic Clouds, which, to the 
naked eye, resemble the milky way, but, through 
telescopes, they appear to be composed of stars. 
M. de la Caille in his catalogue of fixed stars observed 
at the Cape of Good Hope, has remarked 42 nebulae 
which he observed, and which he divided into three 


classes; 14, in which he could not discover the stars; 
14, in which he could see a distinct mass of stars ; and 
14, in which the stars appeared of the sixth magnitude, 
or below, accompanied with white spots, and nebulae 
of the first and third kind. In the Connolssance des 
Temps, for 1783 and 17^4, there is a catalogue of 103 
tiebulae, observed by Messier and Mechain, some of 
which they could resolve, and others they could not. 
But Dr. Herschel has given us a catalogue of 2000 
nebulae and clusters of stars, which he himself has 
discovered. Some of them form a round, compact 
system; others are more irregular, of various forms ; 
and some are long and narrow. The globular systems 
of stars appear thicker in the middle than they would 
do if the stars were all at equal distances from each 
other; they are, therefore, condensed towards the 
center. That the stars should be thus accidentally 
disposed, is too improbable a supposition to be ad- 
mitted; he supposes, therefore, that they are thus 
brought together by their mutual attractions, and that 
the gradual condensation towards the center is a proof 
of a central power of that kind. He further observes, 
that there are some ad(^itional circumstances in the 
appearance of extended clusters and nebulae, that very 
much favour the idea of a power lodged in the brightest 
part. For although the form of them be not globular, 
it is plainly to be seen that there is a tendency towards 
sphericity, by the swell of the dimensions as they draw 
near the most luminous place, denoting, as it were, a 
course, or tide of stars, setting towards a center. As 
the stars in the same nebula must be very nearly all at 
the same relative distance from us, and they appear 
nearly of the same size, their real magnitudes must be 
nearly equal. Granting, therefore, that these nebulae 
and clusters of stars are formed by their mutual attrac- 
tion. Dr. Herschel concludes that we may judge of 
their relative age by the disposition of their component 
parts, those b> ing the oldest which are most com- 
pressed. He supposes the milky way to be a nebula, 


of which our sun is one of it's component parts. See 
the Phil Trans. 1786 and 1789. 

(364.) Dr. Herschel has discovered other phae- 
nomena in the heavens, which he calls Nebulous 
Stars; that is, stars surrounded with a faint luminous 
atmosphere, of a considerable extent. Cloudy or ne- 
bulous stars, he observes, have been mentioned by 
several Astronomers j but this name ought not to be 
applied to the objects which they have pointed out as 
such ; for, on examination, they prove to be either 
clusters of stars, or such appearances as may reason- 
ably be supposed to be occasioned by a multitude of 
stars at a vast distance. He has given an account of 
seventeen of these stars, one of which he has thus de- 
scribed. " November 13, 1790- A most singular 
phaenomenon ; a star of the eighth magnitude, with a 
faint luminous atmosphere, of a circular form, and of 
about 3' diameter. The star is perfectly in the center, 
and the atmosphere is so diluted, faint, and equal 
throughout, that there can be no surmise of it's con- 
sisting of stars ; nor can there be a doubt of the evident 
connexion between the atmosphere and the star. 
Another star not much less in brightness, and in the 
same field of view with the above, was perfectly free 
from any such appearance." Hence, he draws the 
following consequences. Granting the connexion be- 
tween the star and the surrounding nebulosity, if it 
consist of stars very remote which give the nebulous 
appearance, the central star, which is visible, must be 
immensely greater than the rest ; or if the central star 
be not larger than common, how extremely small and 
compressed must be those other luminous points which 
occasion the nebulosity ! As, by the former supposi- 
tion, the luminous central point must far exceed the 
standard of what we call a star, so, in the latter, the 
shining matter about the center will be much too small 
to come under the same denomination ; we therefore 
either have a central body which is not a star, or a star 
which is involved in a shining fluid, of a nature totally 


unknown to us. This last opinion Dr. Herschel 
adopts. The existence of tliis shining matter, he says, 
does not seem to be so essentially connected with the 
central points, that it might not exist without them. 
The great resemblance there is between the chevelure 
of these stars, and the diffused nebulosity there is about 
the constellation Orion, which takes up a space of 
more than 6o square degrees, renders it highly proba- 
ble that they are of the same nature. If this be ad- 
mitted, the separate existence of the luminous matter 
is fully proved. Light reflected from the star could 
not be seen at this distance. And, besides, the out- 
ward parts are nearly as bright as those near the star. 
In further confirmation of this, he observes, that a 
cluster of stars will not so completely account for the 
milkiness, or soft tint of the light of these nebulae, as 
a self-luminous fluid. This luminous matter seems 
more fit to produce a star by it's condensation, than to 
depend on the star for it's existence. There is a tele- 
scopic milky way extending in right ascension from 
bh. 15'. 8" to 5^.3.9'. \" , and in polar distance from 
87**. 46' to 98°. 10'. This, Dr. Herschel thinks, is 
better accounted for, by a luminous matter, than from 
a collection of stars. He observes, that perhaps some 
may account for these nebulous stars, by supposing 
that the nebulosity may be formed by a collection of 
stars at an immense distance, and that the central star 
may be a near star, accidently so placed ; the ap- 
pearance, however, of the luminous part does not, in 
his opinion, at all favour the supposition that it is pro- 
duced by a great number of stars; on the other hand, 
it must be granted that it is extremely difficult to admit 
the other supposition, when w^e know that nothing but 
a solid body is self-luminous, or, at least, that a fixed 
luminary must immediately depend upon such, as the 
flame of a candle upon the candle itself See Dr. 
HerscheVs Account, in the Phil Trans. 1791. 


On the Constellations. 

{SQb.) As soon as Astronomy began to be studied, 
it must have been found necessary to divide the 
heavens into separate parts, and to give some rejjre- 
sentations to them, in order that Astronomers might 
describe and speak of the stars, so as to be understood. 
Accordingly we find that these circumstances took 
place very early. The ancients divided the heavens 
into Constellations, or collections of stars, and repre- 
sented them by animals, and other figures, according 
to the ideas which the dispositions of the stars sug- 
gested. We find some of them mentioned by Job ; 
and although it has been disputed, u hether our trans- 
lation has sometimes given the true meaning to the 
Hebrew words, yet it is agreed, that they signify con- 
stellations. Some of them are mentioned by Homer 
and Hesiod, but Aratus professedly treats of all the 
ancient ones, except three which were invented after 
his time. The number of the ancient constellations 
was 48, but the present nuniber upon a globe is abmit 
70 ; by rectifying which, and setting it to correspond 
with the stars in the heavens, you may, by comparing 
them, very easily get a knowledge of the different con- 
stellations and stars. Those stars which do not come 
into any of the constellations, are called Unformed 
Stars. The stars visible to the naked eye are divided 
into six classes, according to their magnitudes ; the 
largest are called of the first magnitude, the next of 
the second, and so on. Those which cannot be seen 
without telescopes, are called Telescopic Stars. The 
stars are now generally marked upon maps and globes 
with Bayer's letters ; the 1st letter in the Greek 
alphabet being put to the greatest star of each con- 
stellation ; the 2d letter to the next greatest, and so 
on ; and when any more letters are wanted, the Italic 


letters are generally used; this serves as a name to the 
star, by which it may be pointed out. Twelve of 
these constellations lie upon the ecliptic, including a 
space about 15'' broad, called the Zodiac, within 
which all the planets move. The constellation Aries, 
or the Ra?7i, about 2000 years since, lay in the^r,s^ 
sign of the ecliptic ; but, on account of the precession 
of the equinox, it now lies in the second. The follow- 
ing are the names of the constellations, and the number 
of the stars observed in them by different Astronomers. 
Antinoiis was made out of the unformed stars near 
Aquila; and Coma Berenices out of the unformed stars 
near the Lions Tail. They are both mentioned by 
Ptolemy, but as unformed stars. The constellations 
as far as the Triangle, with Coma Berenices, are 
northern ; those after Pisces, are southern. 








Ursa Minor 

The Little Bear 





Ursa Major 

The Great Bear 






The Dragon 

















Corona Boreal i 

s The Northern Crown 






Hercules kneeling 






The Harp 






The Swan 






The Lady in her Chair 1 3 






29 29 




The Waggoner 












The Serpent 






The Arrow 






The Eagle? 






Antinous 5 




The Dolphin 
















Coma Berenices 












Canis Major 

Canis Mmor 








Corona Australis 

Pisces Australis 

The Horse's Head 

The Flying Horse 


The Triangle 

The Ram 

The Bull 

The Twins 

The Crab 

The Lion ) 

Berenice's Hairj 

The Virgin 

The Scales 

The Scorpion 

The Archer 

The Goat 

The Water-bearer 

The Fishes 

The Whale 



The Hare 

The Great Dog 

The Little Dog 

The Ship 

The Hydra 

The Cup 

The Crow 

The Centaur 

The Wolf 

The AUar 

The Southern Crown 

The Southern Fish 




























































































































Columba Noachi 

Robur Carol inum 





Apus, Avis Ind'ica 

Apis, Musca 


Triangulum Australe 

Piscis volans, Passer 

Dorado, Xiphias 



Noah's Dove - - - - 10 

Tlie Royal Oak - - - . 12 

I'he Crane ----- 13 

The Phoenix - - - - 13 

The Indian - - - - 12 

The Peacock - - - - 14 

The Bird of Paradise - - 11 

The Bee, or Fly - - - 4 

The Chameleon - - - 10 

The South Triangle - - 5 

The FlyingFish - - - 8 

The Sword Fish - - - 6 

The American Goose - 9 

The Water Snake - - 10 



Leo Minor 

Asteron and Chara 


Vulpecula and Anser 

Scutum Sobieski 





The Lynx 

The Little Lion 
The Greyhounds 

The Fox and Goose 
Sobieski's Shield 
The Lizard 
The Camelopard 
The Unicorn 
The Sextant 












Besides the letters which are prefixed to the stars, 
many of them have names, as Regulus, Sirins, 
Arcturus, &c. 

(366.) Kepler, who was afterwards, in this conjec- 
ture, followed by Dr. Halley, has made a very inge- 
nious observation upon the magnitudes and distances 


of the fixed stars. He observes, that there can be only 
13 points upon the surface of a sphere as far distant 
from each other as from the center ; and supposing 
the nearest fixed stars to be as far from each other as 
from the sun, he concludes that there can be only 
thirteen stars of the first magnitude. Hence, at twice 
that distance from the sun, there may be placed four 
times as many, or 52; at three times that distance, 
nine times as many, or 1 1/; and so on. These num- 
bers will give, pretty nearly, the number of stars of the 
first, second, third, &c. magnitudes. Dr. Halley 
further remarks, that if the number of stars be finite, 
and occupy only a part of space, the outward stars 
would be continually attracted towards those which are 
within, and, in process of time, they would coalesce 
and unite into one. But if the number be infinite, 
and they occupy an infinite space, all the parts would 
be nearly in equilibrio, and consequently, each fixed 
star being drawn in opposite directions, would keep 
it's place, or move on till it had found an equilibrium. 
Pkil Tram. N°. 364. 

On the Catalogues of the Fixed Stars. 

(367.) At the time o^ Hipparchuso^Bhode^, about 
120 years before J. C. a new star appeared, upon 
which he set about numbering the fixed stars, and re- 
ducing them to a Catalogue^ that posterity might 
know whether any changes had taken place in the 
heavens, Ptolemy^ however, mentions that Tymocharis 
and Arystillus left several observations made 180 years 
before. The catalogue of ///jo/>a re// w a contained 1022 
stars, with' their latitudes and longitudes, which 
Ptolemy published, with the addition of four more. 
These Astronomers made their observations with an 
armillary sphere, placing the armilla, or hoop repre- 
senting the ecliptic, to coincide with the ecliptic in the 
heavens by means of the sun in the day-time, and 


then they determined the place of the moon in respect 
of the sun by a moveable circle of latitude. The next 
night by the help of the moon (whose place before 
found they corrected by allowing for it's motion in the 
interval of time) they placed the hoop in such a situa- 
tion as was agreeable to the present moment of time, 
and then compared, in like manner, the places of the 
stars with the moon. Thus they found the latitudes 
and longitudes of the stars ; it could not, however, be 
done with such an instrument to any very great degree 
of accuracy. Ptolemy adapted his catalogue to the 
year 137 ^ft^'' J.C.; but supposing, with Hipparchns, 
who made the discovery, the precession of the equi- 
noxes to be r in 100 years,instead of about 72 years, lie 
only added 2". 40'. to the numbers in Hipparchus for 
26'5 years (the difference of the epochs) instead of 3". 
42'. 22", according to Dr. Mciskelyne^ Tables. To 
compare his Tables, therefore, with the present, we 
must first increase his numbers by 1°. 2'. 22", and 
then allow for the precession from that time to this. 
The next Astronomer who observed the fixed stars 
a-new, was Uliigh Beigh, the grandson of Tamer- 
lane the Great; he made a catalogue of 1022 stars, 
reduced to the year 1437- WiU'iam, the most illus- 
trious Landgrave of Hesse, made a catalogue of 400 
stars which he observed; he computed their latitudes 
and longitudes from their observed right ascensions and 
declinations. In the year 1610, Tycho Brake's cata- 
logue of '/J'H' stars was published from his own observa- 
tions, made with great care and diligence. It was 
afterwards, in 1627, copied into the Rudolplune 
Tables^ and increased by 223 stars, from other observa- 
tions of Tycho. Instead of a zodiacal armilla, Tycho 
substituted the equatorial arm ilia, by which he ob- 
served the difference of right ascensions, and the de- 
chnations, out of the meridian, the meridian altitude 
being always made use of to confirm the others. From 
thence he computed the latitudes and longitudes. 
Tycho compared Fe?ius with the sun, and then the 


Other stars with Venus, allowing for it's parallax and 
refraction ; and having thus ascertained the places of 
a few stars, he settled the rest from them ; and although 
his instrument was very large, and constructed with 
great accuracy, yet, not having pendulum clocks to 
measure his time, his observations cannot be very ac- 
curate. The next catalogue was that of R. P. Rlcciolus, 
which was taken from Ti/cho's, except 101 stars which 
he himself had observed. //eiT/m^- of Dantzick, in 
1690, published a catalogue of 1930 stars, of which 
950 were known to the ancients ; 603 he calls his 
own, because they had not been accurately observed by 
any one before himself; and 37/ of Dr. Hallei/y 
which were invisible to his hemisphere. Their places 
were fixed for the year 1660. The British Cata- 
logue, which was published by Mr. Flaimtead, con- 
tains 3000 stars, rectified for the year 1689. They 
are distinguished into seven degrees of magnitude (of 
which the seventh degree is telescopic) in their proper 
constellations. This catalogue is more correct than 
any of the others, the observations having been made 
with better instruments. He also published an Atlas 
Ccelestis, or maps of the stars, in which each star is 
laid down in it's true place, and delineated of it's own 
magnitude. Each star is marked with a letter, begin- 
ning with the first letter a of the Greek alphabet for 
the largest star of each constellation, and so on, ac- 
cording to their magnitudes, following, in this respect, 
the charts of the same kind which were published by 
J. Bayer, a German, l603. In the year 1757, M. 
de la Caille published his Fundamenta Asfronomice^ 
in which there is a catalogue of 397 stars ; and in 1763, 
he published a catalogue of 1942 southern stars, from 
the tropic of Capricorn to the south pole, with their 
right ascensions and declinations for 1750. He also 
published a catalogue of zodiacal stars in the Epheme- 
rides from I765 to 1774. Mr. Mayer also published 
a catalogue of 600 zodiacal stars. In the A'autical 
Almanac for 177^? there is published a catalogue of 
s 2 


380 stars observed by Dr. Bi^adley, with their longi- 
tudes and latitudes. In the year 1782, J. E. Bode^ 
Astronomer at Berlin, published a set of Celestial 
Charts, containing a greater number of stars than in 
those of Mr. Flamstead, with many of the double 
stars and nebulae. He also published, in the same 
work, a catalogue of stars, that of Flamstead being the 
foundation, omitting some stars, whose positions were 
left incomplete, and altering the numbers; to which 
he has added stars from Hevelius, M. de la Caille, 
Mayer, and others. In the year 177^'j there was 
published at Berlin, a work entitled Recueil de Tables 
Astronomiques, in which is contained a very large 
catalogue of stars from Hevelius, Flamstead, M.. de la 
Caille, and Dr. Bradley, with their latitudes and 
longitudes for the beginning of 1800; with a catalogue 
of the southern stars of M. de la Caille \ of double 
stars ; of changeable stars, and of nebulous stars. This 
is a very useful work for the practical Astronomer. 
But the most complete catalogue is that published by 
the Rev. Mr. Wollastoyi, F. R. S. in 1789, entitled, 
A Specimen of a General Astronomical Catalogue, 
arranged in Zones of North Polar Distance, and 
adapted to January 1, 1790; containing a Compara- 
tive Fiew of the Mean Positions of Stars, as they 
come out upon Calculation from the Tables of several 
principal Observers. 

On the Proper Motions of the Fixed Stars. 

(368.) Dr. Mashelyne, in the explanation and use 
of his Tables, which he published with the first volume 
of his Observations, observes, that many, if not all 
the fixed stars, have small motions among themselves, 
which are called their Proper Motions ; the cause and 
laws of which are hid, for the present, in almost equal 
obscurity. From comparing his own observations at 
that time, with those of Dr. Bradley, Mr. Flamstead^ 


and Mr. Roemer, he then found the annual proper 
motion of the following stars, in right ascension, to be, 
ofSirius - 0",63, of Castor - o",28, of Procyon - 0",8, 
of Pollux — O" ,^3 , of Regulus — 0'\4l, of Arcturus 
— l",4, and o{ a Aquiloc -f o",57; and o^ Sirius in north 
polar distance r',20, and of Arcturus 2",01, both 
southwards. But since that time he had continued 
his observations, and from a catalogue of right ascen- 
sions of 36 principal stars (which he communicated to 
Mr. fVoUaston, and which is found in his work), it 
appears that 35 of them have z. proper motion m right 

(369.) In the year 1759, M. Mayer observed 80 
stars, and compared them with the observations of 
Roemer in 1706. M. Mayer is of opinion, that (from 
the goodness of the instruments with which the ob- 
servations were made) where the disaojreement is at 
least 10" or 15", it is a very probable indication of a 
proper motion of such a star. He further adds, that 
when the disagreement is so great as he has found it 
in some of the stars, amongst which is Fomahandy 
where the difference was 21" in 50 years, he has no 
doubt of a proper motion. Dr. Herschel following 
Mayers judgement of his own and Roemer's observa- 
tions, has compared the observations, and leaving out 
of his account all those stars which did not show a dis- 
agreement amounting to lO", he found that 56 of 
them had a proper motion. From thence he attempts 
to deduce the motion of the solar system in the fol- 
lowing manner. 

(370.) If the sun be in motion as well as the stars, 
the effects will be altered according to their motion, 
compared with the motion of our sun. Some of them, 
therefore, from their own proper motions, might de- 
stroy, or more than counteract, the effects arising from 
the motion of the sun. In whatever direction our 
system should move, it would be very easy to find 
what effect in latitude and longitude would have taken 
place upon any star, by means of a celestial globe, by 


conceiving the sun to move from the center upon any 
radius directed to the point to which the sun is mov- 
ing. Dr. Herschel describes the effect thus. Let an 
arc of 90° be apphed to the surface of a globe, and 
aUvays passing through that point to which the motion 
of the system is directed. Then whilst one end moves 
along the equator, the other will describe a curve 
passing through it's pole, and returning into itself; 
and the stars in the northern hemisphere, within this 
curve, will appear to move to the north ; and the rest 
will go to the south. A similar curve may be described 
in the southern hemisphere, and like appearances will 
take place. 

(371.) Now Dr. Herschel first takes the seven stars 
before mentioned, whose proper motions had been de- 
termined by Dr. Maskelyne ^ and he finds, that if a 
point be assumed about the 77** of ^'ig^^^ ascension, and 
the sun to move from it, it will account for all the 
motions in right ascension. And if, instead of sup- 
posing the sun to move in the plane of the equator, it 
should ascend to a point near to X HercuUs, it will ac- 
count for the observed change of declination of Sinus 
and Arctiirus. In respect to the quantity of motion, 
of each, that must depend upon their unknown rela- 
tive distances ; he only speaks here of the directions 
of the motions. 

(372.) He next takes twelve stars from the cata- 
logue of 56, whose proper motions have been deter- 
mined from a comparison of the observations oi Roemer 
and Mayer, and adds to them Regulus and Castor \ 
these have all a proper motion in right ascension and 
declination, except Regulus, which has none in decli- 
nation. Of these 27 motions, the above-supposed 
motion of the solar sytem will satisfy 22. There are 
also some remarkable circumstances in the quantities 
of these motions. Arcturus and Sirius being the 
largest, and therefore, probably, the nearest, ought to 
have the greatest apparent motion ; and so we find they 
have. Also, Arcturus is better situated to have a mo- 


tion in right ascension, and it has the greatest motion. 
Several other facts of the same kind are found also to 
take place. But there is a very remarkable circum- 
stance in respect to Castor. Castor is a double star; 
now, how extraordinary must appear the concurrence, 
that two such stars should both have a proper motion 
so exactly alike, that they never have been found to 
vary a second ! This seems to point out the common 
cause, the motion of the solar system. 

(3'/ 3.) Dr. Hersdiel next takes 32 more of the 
same catalogue of 56 stars, and shows that their mo- 
tions agree very well with his supposed motion of the 
solar system. But the motions of the other 12 stars 
cannot be accounted for upon this hypothesis. \n 
these, therefore, he supposes the effect of the solar 
motion has been destroyed and counteracted by their 
own proper motions. The same may be said of I9 
stars out of the 32, which only agree with the solar 
motion one way, and are, as to sense, at rest in the 
other. According to the rules of philosophising, 
therefore, which direct us to refer all phaenomena to as 
few and simple principles as are sufficient to explain 
them, Dr. Herschel thinks we ought to admit the 
motion of the solar system. Perhaps, however, this 
argument cannot be properly applied here, because 
there is no new cause or principle introduced, by sup- 
posing each star to have a proper motion. Admitting 
the doctrine of universal gravitation, the fixed stars 
oufifhtto move as well as the sun. But the sun's mo- 
tion, as here estimated, cannot be owing to the action 
of a body upon it which might give it a rotatory motion 
at the same time, as M. de la Lande conjectures; 
because a body acting on the sun, to give it it's rota- 
tion about it's axis, would not, at the same time, give 
it that progressive motion. See Dr. Herschel' ^ Acconwi 
in the Phil. Trans. 1/83. 

(374.) But it will be proper to consider how far 
this motion of the solar system agrees with the proper 
motion of the 35 stars determined by T)v,,Maskelyn^, 


Now, upon supposition that the sun moves, as conjec- 
tured by Dr. Herschel, that motion will account for 
the motion of 20 of them, so far as regards their di- 
rections ; but the motion of the other 15 is contrary to 
that which ought to arise from this supposition. As 
some of the stars must have a proper motion of their 
own, even upon the hypothesis of a solar motion, and 
which probably arises from their mutual attraction, it 
is very probable that they have all a proper motion 
from the same cause_, but most of them so very small, 
as not yet to have been discovered. And it might also 
happen, that such a motion might be the same as that 
which would arise from the motion of the solar system. 
Yet it must be confessed, that the circumstance of 
Castor, and the motions, both in right ascension and 
declination, of many of the stars being such as arise 
from this hypothesis, with the apparent motion of 
those stars being greatest which are probably nearest, 
form a strong argument in it's favour. 

•^ On the Zodiacal Light. 

(315.) The Zodiacal Light is a pyramid of light 
which sometimes appears in the morning before 
sun-rise, and in the evening after sun-set. It has 
the sun for it's basis, and in appearance resembles 
the Aurora Borealis. It's sides are not straiglit, 
but a little curved, it's figure resembling a lens edge- 
ways. It is generally seen here about October and 
March, that being the time of our shortest twilight; 
for it cannot be seen in the twilight ; and when the 
twilight lasts a considerable time, it is withdrawn 
before the twilight ends. It was observed by M. 
Cassini, in l683, a little before the vernal equinox, in 
the evening, extending along the ecHptic from the sun. 
He thinks, however, that it has appeared formerly, 
and afterwards disappeared, from an observation of 
Mr. J. Children/, in a book published in l66l, en- 
titled, Britannia Baconia. He says, that " in the 


month of February, for several years, about six o'clock 
in the evening, after twilight, he saw a path of light 
tending from the twilight towards the Pleiades^ as it 
were touching them. This is to be seen whenever the 
weather is clear, but best when the moon does not 
shine. I believe this phsenomenon has been formerly, 
and will hereafter appear always at the above-men- 
tioned time of the year. But the cause and nature of 
it I cannot guess at, and therefore leave it to the en- 
quiry of posterity." From this description, there can 
be no doubt but that this was the zodiacal light. He 
suspects also, that this is what the ancients called 
Trabes, which word they used for a meteor, or im- 
pression in the air like a beam. Pliny ^ lib. II. p. 26, 
says, Emicant Trabes, quos ducosvocaut. Des Cartes 
also speaks of a phsenomenon of the same kind. M. 
Fatio de Didlller observed it immediately after the 
discovery by M. Cassini, and suspected that it had 
always appeared. It was soon after observed by M. 
Kirch and Eimmart in Germany. In the 37^ear I/O/? 
on April 3, it was observed by IVIr. Derham, in Essex. 
It appeared in the western part of the heavens, about 
a quarter of an hour after sun-set, in the farm of a py- 
ramid, perpendicular to the horizon. The base of this 
pyramid he judged to be the sun. It's vertex reached 
15° or 20° above the horizon. It was throughout of a 
dusky red colour, and at first appeared pretty vivid 
and strong, but faintest at the top. It grew fainter by 
degrees, and vanished about an hour after sun-set. 
This solar atmosphere has also been seen about the 
sun in a total solar eclipse, a luminous ring appearing 
about the moon at the time when the eclipse was 

(376.) M. Fatio conjectured, that this appearance 
arises from a collection of corpuscles encompassing the 
sun in the form of a lens, reflecting the light of the 
sun. M. Cassini supposed that it might arise from an 
infinite number of planets revolving about the sun ; so 
that this light might owe it's existence to these bodies. 


as the milky way does to an innumerable number of 
fixed stars. It is now, however, generally supposed, 
that it is matter detached from the sun by it's rotation 
about it's axis. The velocity of the equatorial parts 
of the sun being the greatest, would throw the matter 
to the greatest distance, and, on account of the dimi- 
nution of velocity towards it's poles, the height to 
which the matter would there rise would be diminish- 
ed j and as it would probably spread a little sideways, 
it would form an atmosphere about the sun something 
in the form of a lens, whose section perpendicular to 
it's axis would coincide with the sun's equator. And 
this agrees very well with observation. There is, 
however, a difficulty in thus accounting for this phae- 
nomenon. It is very well known, that the centrifugal 
force of a point of the sun's equator is a great many 
times less than it's gravity. It does not appear, there- 
fore, how the sun, from it's rotation, can detach any 
of its gross particles. If they be particles detached 
from the sun, they must be sent off by some other 
unknown force; and in that case they might be sent 
off equally in all directions, which would not agree with 
the observed figure. The cause is probably owing to 
the sun's rotation, although not immediately to the 
centrifugal force arising therefrom. 


Chap. XXV. 


{3YJ.) The situation of any place upon the earth's 
surface is determined from it's latitude and longitude. 
The latitude maybe found from t'ne meridian altitude 
of the sun, or a known fixed star; from two altitudes 
of the sun, and the time between ; and by a variety 
of other methods. These operations are so easy in 
practice, and opportunities are so continually offering 
themselves, that the latitude of a place may generally 
be determined, even under the most unfavourable cir- 
cumstances, to a degree of accuracy sufficient for all 
nautical purposes. But the longitude cannot be so 
readily found. Philip III. King of Spain, was the 
first person who offered a reward for it's discovery ; 
and the States of Holland, soon after, followed his 
example, they being at that time rivals to Spain, as a 
maritime power. During the minority of Louis XV. 
of France, the regent power promised a great reward 
to any person who should discover the longitude at 
sea. In the time of Charles II. about 1675, the 
Sieur de St. Pierre, a Frenchman, proposed a method 
of finding the longitude by the moon. Upon this, a 
commission was granted to Lord Viscount Brounker, 
President of the Royal Society, Mr. Flamstead, and 
several others, to receive his proposals, and give opinion 
respecting it. Mr. Flamstead gave his opinion, that 
if we had Tables of the places of the fixed stars, and 
of the moon's motions, we might find the longitude, 
but not by the method proposed by the Sieur 


de St. Pm-re. Upon this, Mr. Flamsfead was ap- 
pointed Astronomer Royal, and an Observatory was 
built at Greenwich for him ; and the instructions to 
'him and his successors were, " That they should apply 
themselves with the utmost care and diligence to 
rectify the Tables of the motions of the heavens, and 
the places of the fixed stars, in order to find out 
the so- much desired longitude at sea, for the perfecting 
of the art of navigation." 

(378.) In the year 1714, the British Parliament 
offered a reward for the discovery of the longitude ; 
the sum of lOOOO/. if the method determined the 
longitude to 1° of a great circle, or 60 geographical 
miles; of 15000/. if it determined it to 40 miles; and 
of 20000/. if it determined it to 30 miles; with this 
proviso, that if any such method extend no further 
than 80 miles adjoining to the coast, the proposer shall 
have no more than half such rewards*. The Act also 
appoints the First Lord of the Admiralty, the Speaker 
of the House of Commons, the First Commissioner of 
Trade, tlie Admirals of the Red, White, and Blue 
Squadrons, the JVIaster of Trinity-House, the President 
of the Royal Society, the Royal Astronomer at Green- 
wich, the two Savihan Professors at Oxford, and the 
Lucasian and Plumian Professors at Cambridge, with 
several other persons, as Commissioners for the Lono-i- 
tude at Sea. The Lowndian Professor at Cambridge 
was afterwards added. After this Act of Parliament, 
several other Acts passed, in the reigns of George l\. 
and \\l, for the encouragement of finding the longi- 
tude. At last, in the year I774, an Act passed, re- 
pealing all other Acts, and oflfering separate rewards to 
any person who shall discover the longitude, either by 
the lunar method, or by a watch keeping true time, 
within certain limits, or by any other method. The 


* See Widston's account of ih? proceedings to obtain this Act, iri 
the Preface to his Longitude discovered hi/ Jupiter's Planets. 


Act proposes, as a reward for a time-keeper, the sum 
of 5000/. if it determine the longitude to 1°, or 6o geo- 
graphical miles ; the sum of 'JbOOl. if it determine the 
same to 40 miles ; and the sum of lOOOO/. if it deter- 
mine the same to 30 miles, after proper trials specified 
in the act. If the method be by improved solar and 
lunar Tables, constructed upon Sir /. Newton\ theory 
of gravitation, the author shall be entitled to 5000/. if 
such Tables shall show the distance of the moon from 
the sun and stars within 15" of a degree, answering;- to 
about 7' of longitude, after making an allowance of half 
a degree for the errors of observation. And for any 
other method, the same rewards are offered as those 
for the time-keeper, provided it gives the longitude 
true within the same limits, and be practicable at sea. 
The commissioners have also a power of giving smaller 
rewards, as they shall judge proper, to any one who 
shall make any discovery for finding the longitude at 
sea, though not within the above limits. Provided, 
however, that if such person or persons shall afterwards 
make any further discovery so as to come within the 
above-mentioned limits, such sum or sums, as they 
may have received, shall be considered as part of such 
greater reward, and deducted therefrom accordingly. 

(379.) After the decease of Mr. Flatnstead, Dr. 
Halley, who was appointed to succeed him, made a 
series of observations on the moon's transit over the 
meridian, for a complete revolution of the moon's 
apogee, which observations being computed from the 
Tables then extant, he was enabled to correct the 
Tables of the moon's motion. And as Mr. Hadley 
had then invented an instrument by which altitudes 
could be taken at sea, and also the moon's distance 
from the sun or a fixed star, Dr. Halley strongly re- 
commended the method of finding the longitude from 
such observations, having found, from experience, the 
impracticability of all other methods, particularly at 


Tojind the Longitude by the Moon^s Distance from 
the San^ or ajlxed Star. 

(380.) The steps by which the longitude is found 
by this method, are these : 

From the observed altitudes of the moon and the 
sun, or a fixed star, and their observed distance, com- 
pute the moon's true distance from the sun or star. 

From the Nautical Almanac^ find the time at 
Greenwich when the moon was at that distance. 

From the altitude of the sun or star, find the time 
at the place of observation. 

The diffcM'cnce of the times thus found, gives the 
difference of the longitudes. 

(381.) To find t!ie true distance of the moon from 
the sun or star by observation, let Z be the zenith, S 
the apparent place of the sun or a star, s the true place, 


il/the apparent place of the moon, m it's true j)lace; 
then the altitudes of 71/ and s being known, by observa- 
tion, the refractions Ss, Mm are known; also MS is 
known by observation ; hence, in the triangle ZS3I, 
we know S M the apparent distance, SZ, Z M, the 
complements of the apparent altitudes, to find the 
angle Z (Trig. Art. 239) ; ^"d then in the triangle 
sZm, we know the angle Z, and sZ, mZ, the comple- 

UPON THE earth's SURFACE. 28/ 

ments of the true altitudes, to find sm the true 
distance* (Trig. Art. 233). 

Ex. Suppose on June 29, 1793, the sun's apparent 
zenith distance ZS was observed to be 70*^. 56'. 24", 
the moon's apparent zenith distance ZM to be 48°. 
63'. 58", their apparent distance S3I to be 103". 29'. 
27", and the moon's horizontal parallax to be 58'. 35"; 
to find their ttue distance sm. 

The true distance sm, computed by the above 
method, is 103^ 3'. 18". 

(382.) The^rwe distance of the moon from the sun 
or star being thus found, we are next to find the time 
at Greenwich. For this purpose, the sun or such 
fixed stars are chosen, as lie in or very near the moon's 
way, so that, looking upon the moon's motion to be 
uniform for a small time, the moon may be considered 
as approaching to, or receding from, the sun or star 
uniformly. To determine, therefore, the time at 
Greenwich corresponding to any given true distance of 
the moon from the sun or star, the true distance is 
computed in the Nautical Almanac for every three 
hours, for the meridian of Greenwich. Hence, con- 
sidering that distance as varying uniformly, the time 
corresponding to any other true distance may be thus 
computed : Look into the Nautical Almanac, and 
take out two distances, one next greater and the other 
next less than the true distance deduced from observa- 
tion, and the diflference D of these distances gives the 
access of the moon to, or recess from, the sun or star 
in three hours; then take the diflference d between the 
moon's distance at the beginning of that interval and 
the distance deduced from observation, and then say, 
D : d :: 3 hours : the time the moon is acceding to, or 
receding from, the sun or star by the quantity d; which 

* There are shorter methods than this direct one, of compu'ing 
the true distance, as the reader will see in my Complete Sj/stein of 
Astronomy ; but we here purpose only to explain ihe principles' by 
which the longitude qjay be thus found. 


added to the time at the beginning of the interval, gives 
the apparent time at Greenwich, corresponding to the 
given true distance of the moon from the sun or star. 
Ex. On June 29, 1 793, in latitude 52°. 12'. 35", the 
sun's altitude in the morning was, by observation, 19°. 
3'. 36", the moon's altitude was observed to be 41°. 6'. 
2", the sun's declination at that time was 23°. 14'. 4", 
and the moon's horizontal parallax 58'. 35"; to find 
the apparent time at Greenwich. 

True dist. of d from by Art. 381. - 103°. 3'. 18" 
Truedisthy Armt. Jim. on June'2Q,iit3h.l03. 4. 58 
Truedist.byiVfl?«^.^/m.onJune29,at6/z.l01. 26'. 42 

d= O. 1. 40 

D= 1. 38. 16 

Hence, 1°. 38'. 16". : O". 1'. 40" : 3h. :: 0\ 3'. 3", 
which added to 3h. gives 3^. 3'. 3^', the apparent time 
at Greenwich, when the ^r«e distance was 103°. 3'. 18". 
(383.) Find the apparent time at the place of ob- 
servation, by the altitude of the sun (12). Then the 
difference of the times at Greenwich, and at the place 
of observation, is the distance of the meridians in time. 
(384.) Now to find the apparent time at the place 
of observation, we have the sun's declination 23". 14'. 
4", it's altitude 19°. 3'. 36", it's refraction 2'. 44", and 
parallax 8"; hence, it's true altitude was 19''. 1? and 
therefore it's true zenith distance was 70°. 59' ; also, 
the complement of declination was 66°. 45'. 36"; hence, 
by Art. 92 : 

66°. 46'. 56' - - ar. CO. sin. 0,0367325 
37. 47.25 - - ar. CO. sin. 0,2127004 
70. 56. 24 

175. 29. 45 

87. 44. 52 - - - - sin. 9,9996644 
16. 48. 28 . - - - sin. 9,46oi408 




the cosine of 44°. 18'. 52", which doubled gives 88°. 
37'. 44", the hour-angle from apparent noon, which in 
time gives 5h. 54'. 31", the time before apparent noon, 
or 18h. 5'. 29", on June 28. Hence, 

Apparent time at place of observ. June 28, 18''. 5'. 29'' 
. at Greenwich, June 29, 3. 3. 3 

Difference of meridians in time . . - 8. 57. 34 

Which converted into degrees, gives 134°. 23'. 30", 
the longitude of the place of observation west of 

To find the Longitude hy a Time-heeper. 

(385.) Let the time-keeper be well regulated, and 
set to the meridian of Greenwich ; then if it neither 
gain nor lose, it will always show the time at Green- 
wich. Hence, to find the longitude of any other 
place, find the mean time from the sun's altitude by 
Art. 92 ; and observe, at the instant of taking the alti- 
tude, the time by the watch ; and the difference of 
these times, converted into degrees, at the rate of 15*^ 
for an hour, gives the longitude from Greenwich. If, 
for example, the time by the watch, when the altitude 
was taken was Qh. 19', and the mean time deduced 
from that altitude was 9/i. 23', the difference 3h. 4', 
converted into degrees, gives 46° the longitude of the 
place ea^t from Greenwich, because the time at the 
place of observation is for^warder than that at Green- 
wich. Thus the longitude could be very readily 
determined, if you could depend upon the watch. 
But as a watch will always gain or lose, before it is sent 
out, it's gaining or losing every day for some time, a 
month for instance, is observed ; this is called the 
rate of going of the watch, and from thence the mean 
rate ol" going is thus found. 

(386.) Suppose, for instance, I examine the rate of a 
watch for 30 days ; on some of those days I find it has 



gained, and on some it has lost; add together all the 
quantities which it has gained, and suppose they 
amount to 17"; add together all the quantities which 
it has lost, and let the sum be 13"' ; then the difference 
4" is the mean rate of gaining for 30 days, which 
divided by 30, gives 0",133 for a mean daily rate of 
gaining. Or you may get the mean daily rate thus. 
Take the difference between what the watch was too 
fast, or too slow, on the first and last days of observa- 
tion, if it be too fast or too slow on each day j but take 
the sum, if it be too fast on one day and too slow on 
the other, and divide by the number of days between 
the observations*. And to find the time at the place 
of trial at any future period by this watch, you must 
put down, at the end of the trial, how much the watch 
is too fast or too slow; then subtract from the time 
shown by the watch, 0"133 x number of days from 
the end of the trial, being the exact quantity which it 
has gained according to the above mean rate of gaining, 
and you are then supposed to get the true time affected 
with the error at the end of the trial. This would be 
all the error, if the watch had continued to gain ac- 
cording to the above rate; and although, from the 
different temperatures of the air to which the watch 
may be exposed, and from the imperfection of the 
workmanship, this cannot be expected, yet, by taking 
it into consideration, the probable error of the time 
will be diminished. In watches which are under trial 
at the Royal Observatory at Greenwich, as candidates 
for the rewards offered by Parliament for the discovery 
of the longitude, this allowance of a mean rate, to be 
applied in order to get the time, is not granted by the 
Act ofParliament, but it requires that the watch itself 
should go within the limits assigned; the Commission- 
ers, however, are so indulgent as to grant the applica- 

* For further information on this subject, see Mr. Wales's 
Mclhod of finding the Longitude at Sea. 


tion of a mean rate, which is undoubtedly favourable 
to the watches. 

(387.) As the rate of going of a watch is subject to 
vary from so many circumstances, the observer, when- 
ever he goes ashore and has sufficient time, should 
compare his watch, for several days, with the mean 
time deduced from the altitude of the sun or a star, by 
which he will be able to determine it's rate of going. 
And whenever he comes to a place whose longitude is 
known, he may correct his watch, and set it to Green- 
wich time. For instance, if he goto a place known to 
be 30". east longitude from Greenwich, his watch 
should be two hours slower than the time at that place. 
Find therefore, the time at that place by the altitude 
of the sun or a tixed star, and correct it by the equation 
of time, and compare the time so found with the time 
by the watch when the altitude was taken, and if the 
watch be two hours slower than the time deduced from 
observation, it is right ; if not, correct it by the dif- 
ference, and it again gives Greenwich time. 

(388.) In long voyages, unless you have sometimes 
the means of adjusting the watch to Greenwich time, 
it's error will probably be very considerable, and conse- 
quently the longitude deduced from it will be subject 
to a proportional error. In short voyages, a watch is 
undoubtedly very useful, and also in long ones, where 
you have the means of correcting it from time to time. 
It serves to carry on the longitude from one known 
place to another, supposing the interval of time not to 
be very long ; or to keep the longitude from that 
which is deduced from a lunar observation, till you can 
get another observation. Thus the watch may be 
rendered of great service in Navigation^ 

To find the Longitude hij an Eclipse of the Moon, 
and of Jupiter's Satellites. 

(389.) By an eclipse of the moon. This eclipse 
begins when the umbra of the earth first touches the 


moon, and it ends when it leaves the moon. Having 
the times calculated when the eclipse begins and ends 
at Greenwich, observe the times when it begins and 
ends at any other place; the difference of these times 
converted into degrees, gives the difference of longi- 
tudes. For as the phases of the moon in an eclipse 
happen at the same instant to every observer, the dif- 
ference of the times at different places, when any 
phase is observed, will give the difference of the longi- 
tudes. This would be a very ready and accurate 
method, if the time of the first and last contact could 
be accurately observed ; but the darkness of the pe- 
numbra continues to increase till it comes to the 
umbra, so that, until the umbra actually gets upon the 
moon, it is not discovered. The umbra itself is also 
very badly defined. The beginning and end of a lunar 
eclipse cannot, in general, be determined nearer than 
l' of time; and very often not nearer than 2' or 3'. 
Upon these accounts, the longitude, from the observed 
beginning and end of an eclipse, is subject to a con- 
siderable degree of uncertainty. Astronomers, there- 
fore, determine the difference of the longitudes of two 
places by corresponding observations of other phases, 
that is, when the umbra bisects any of the spots upon 
the moon's surface. And this can be determined with, 
a greater degree of accuracy than the beginning and 
end ; because, when the umbra is gotten upon the 
moon's surface, the observer has leisure to consider and 
fix upon the proper line of termination, in which he 
will be assisted by running his eye along the circum- 
ference of the umbra. Thus the coincidence of the 
umbra with the spots may be observed with tolerable 
accuracy. The observer, therefore, should have a 
good map of the moon at hand, that he may not 
mistake. The telescope, to observe a lunar eclipse, 
should have but a small magnifying povi^er with a 
great deal of light. The shadow comes upon the 
moon on the east side, and goes off on the west ; but 
if the telescope invert, the appearances will be contrary. 

UPON THE earth's SURFACE. 293 

(390.) The eclipses o{ Jupiter''?, satellites afford the 
readiest method of determining the longitude of places 
at land. It was also hoped that some method might 
be invented to observe them at sea, and Mr. Irwin 
made a chair to swing for that purpose, for the observer 
to sit in ; but Dr. Maskeli/ne, in a voyage to Barba- 
does, under the direction of the Commissioners of 
Longitude, found it totally impracticable to derive any 
advantage from it ; and he observes, that, '^ consider- 
ing the great power requisite in a telescope for making 
these observations well, and the violence as well as 
irregularities of the motion of a ship, I am afraid the 
complete management of a telescope on ship-board 
will always remain among the desiderata. However, 
I would not be understood to mean to discourage any 
attempt, founded upon good principles, to get over 
this difficulty." The telescopes proper for making 
these observations, are common refracting ones from 
15 to 20 feet, reflecting ones of 18 inches or 2 feet, or 
the 46-inch achromatic with three object glasses, 
which were first made by Mr. Dolland. On account 
of the uncertainty of the theory of the satellites, the 
observer must be settled at his telescope a few minutes 
before the expected time of an immersion ; and if the 
longitude of the place be also uncertain, he must look 
out proportionably sooner. Thus, if the longitude be 
uncertain to 2°, answering to eight minutes of time, he 
must begin to look out eight minutes sooner than is 
mentioned above. However, when he has observed 
one eclipse, and found the error of the Tables, he may 
allow the same correction to the calculations of the 
Ephemeris for several months, which will advertise 
him very nearly of the time of expecting the eclipses 
of the same sateHite, and dispense with his attending 
so long. Before the opposition of Jupiter to the sun, 
the immersions and emersions happen on ,the west side 
of Jupiter, and after opposition, on the east side ; but 
if the telescope invert, the appearance will be the con- 
trary. Before opposition, "the immersions only of the 


first satellite are visible; and after opposition, the 
emersions only. The same is generally the case with 
respect to the second satellite ; but both immersion 
and emersion are frequently observed in the two outer 

(391.) When the observer is waiting for an emer- 
sion, as soon as he susj)ects that he sees it, he should 
look at his watch, and note the second, or begin to 
count the beats of the clock, till he is sure that it is 
the satellite, and then look at the clock, and subtract 
the number of seconds which he has counted from the 
time then observed, and lie vvilHiave the time of emer- 
sion. If Jupiter be 8° above the horizon, and the sun 
as much below, an eclipse will be visible ; this may be 
determined near enough by a common globe. 

(392.) The immersion or emersion of a satellite 
being observed according to apparent time, the longi- 
tude of the place from Greenwich is found, by taking 
the difference between that time and the time set down 
in the Nautical Almanac, which is calculated for ap- 
parent time. 

Ex. Suppose the emersion of a satellite to have been 
observed at the Cape of Good Hope, May 9, IT^/? ^^ 
lOh. 4&. 45" apparent time; now the time in the 
Nautical Almanac is 9//. 33'. 12"; the difference of 
which time is \h. 13'. 33", the longitude of the Cape 
east o{ Greenwich in time, or 18°. 23'. 15". 

(393.) But to find the longitude of a place from an 
observation of an eclipse of a satellite, it is better to 
compare it with an observation made under some well- 
known meridian, than with the calculations in the 
Ephemeris, because of the imperfection of the theory : 
but where a corresponding observation cannot be ob- 
tained, find what correction the calculations of the 
Ephemeris require, by the nearest observations to the 
given time that can be obtained; and this correction, 
applied to the calculation of the given eclipse in the 
Ephemeris, renders it almost equivalent to an actual 
observation. The observer must be careful to regulate 

UPON THE earth's SURFACE. 295 

liis clock or watch by apparent time, or at least to 
know the difference ; this may be done, either by 
equal altitudes of the sun, or of proper stars; or the 
latitude being known, from one altitude at a distance 
from the meridian, the time may be found by Art. 92. 
(3^)4.) In order the better to determine the dif- 
ference of longitudes of two places from corresponding 
observations, the observers should be furnished with 
tlie same kind of telescopes. For at an immersion, as 
the satellite enters the shadow, it grows fainter and 
fainter, till at last the quantity of light is so small that 
it becomes invisible, even before it is immersed in the 
shadow ; the instant, therefore, that it becomes invisi- 
ble will depend upon the quantity of light which the 
telescope receives, and it's magnifying power. The 
instant, therefore, of the disappearance of a satellite 
will be later the better the telescope is, and the sooner 
it will appear at it's emersion. Now the immersion is 
the instant the satellite is wholly gotten into the 
shadow, and the emersion is the instant before it 
begins to emerge from the shadow ; if, therefore, two 
telescopes show the disappearance or appeaiance of the 
satellite at the same distance of time from the immer- 
sion or emersion, the difference of the times will be 
the same as the difference of the true times of their 
immersions and emersions, and therefore will show 
the difference of longitudes accurately. But if the 
observed titne at one place be compared with the 
computed time at another, then we must allow for the 
difference between the apparent and true times of 
immersion or emersion, in order to get the true time 
where the observation was made, to compare with the 
true time from computation at the other place. This 
difference may be found, by observing an eclipse at any 
place whose longitude is known, and ci»mparing it 
with the time by computation. Observers, therefore, 
should settle the difference accurately by the mean of 
a great number of observations thus compared witfi 
the computation, by which means the longitude will 


be ascertained to a much greater accuracy and cer- 
tainty. After all this precaution, however, the different 
statesof the air at different times, and also the different 
states of the eye, will introduce a small degree of un- 
certainty ; the latter case may perhaps, in a great 
measure, be obviated, if the observer will be careful to 
remove himself from all warmth and light for a little 
time before he makes the observation, that the eye 
may be reduced to a proper state ; which precaution 
the observer should also attend to, when he settles 
the difference between the apparent and true times of 
immersion and emersion. Perhaps also the difference 
arising from the different states of the air might, by 
proper observations, be ascertained to a considerable 
degree of accuracy; and as this method of determining 
the longitude is of all others, the most ready, no 
means ought to be left untried to reduce it to the 
greatest certainty. 

( 29r ) 


For converting Degrees, Minutes and Seconds, into 
Time, at the Rate of 36o Degrees for 24 Hours. 







i>ec. o/" 














































































For converting Time into Degrees, Minutes, and 
Seconds, at the Rate of 24 Hours for 36o Degrees. 

Hou . 





Dec. oj 








































































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