Download EQUINOCTIAL vLOBE ·

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EQUINOCTIAL vLOBE�·�
BY
WHICH
THE PRECESSION OF THE EQUINOCTIAL POINTS IS·
EXPLAINED IN A FAMILIAR MANNERy
�N'rl!NDE�
TO HLUSTRATE VARIQUS fASSAGES'
IN
AUTHORS-
THE GREEK AND ROMAN
"
.J.
-
BY THE
R.ELATING
REV.
TO
HOLT
OKES, A. M.
...
Fellow
0/ the Cambridge
.
Philosophical Society.
....
,-LONDON:
PRINTED BY J. F.
FOR
M.
BATES,
R.
DOVE,_ST.
PRIESTLEY,
MATHEMATICAL
,DEIGHTON AND
D. A.
JOHN'S SQ UARE:'
HOLBORN ;
INSTRJJMENT MAKER,
SONS, CAMBRIDGE;
TALBOYS, OXFORD,
1820.
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PRE F ACE.
THE effects
points
globes.
are
resulting from the motion of the equinoctial
imperfectly explained ill the usual treatises on the
equinoctial globe, all the changes
place in the longitude, the declination;
possibly
the times of the rising and setting of the stars, at any latitude,
during the whole revolution of the equinoctial points may be
familiarly shewn, together with the change of position in the
equator, the tropics, and the arctic and antarctic circles.
By the
which
assistance of the
take
can
.
problems relating immediately to the equinoctial globe
few in number, but interesting and important; and will be
easily understood by the student who is conversant with the
usual problems on the celestial sphere.
The
are
Woodford, Essex;
June.9,
1820.
�o.�n.w
f...oIUi1v",)
"f{�3
SECTION
L
THE'
ON
EQUlNOCT1AL
PRECESSION OF THE
POINTS.
THE poles of the earth are not always directed to the sallie
points in the heavens, but describe circles round the poles of
the
ecliptic,
in
a
direction contrary, to the order of the
equinoctial points
in consequence of which the
receding, and the sun returns to the
it returns to the
same
point
are
signs,
continually
equinoctial point before
in the heavens.
Fig. 1. Let P be the pole of the ecliptic CV" $ � vY. N the
pole of the equator CV" �, cutting the ecliptic in <yI and �.
Suppose N to nave moved, contrary to the order of the signs,
to n, and let Nn=30°, then the intersection of the equator with
the ecliptic will be in )(, VJ, i. e. the point usually called the
first point' of Aries, or that point in the ecliptic when the days
and nights are equal, will have receded 30°.
ecliptic, C L. the eclip
tic, N S the poles of the earth. Suppose the poles of the
earth, or its axis" to be directed to the points a and e in the
heavens, in about 6,447 years, the pole N of the earth will
have moved to x, and the pole S to y (the circle S y s lying in
an opposite direction to the circle N x n), consequently the axis
of the earth will be directed to the points b and! in the heavens.
Fig.
2. Let P. p. be the
In twice this time,
or
poles
in about 12,895
earth will be in the direction
d, g, in the
of the
n
s, and
...
'years, the axis of the
consequently
heavens, wilt be vertical to the
B
poles
n,
s,
th-e
points
2
Hipparchus,
B. C. 125,
by
"the
prince of
comparing
his
ancient astronomers, who wrote
own
observations
on
Spica Virginis
and other fixed stars, with those of Timocharis who lived about
170 years before
equinoctial points,
him, first discovered the precession of the
which he determined to be
one
Modern astronomers have determined the
100 years.
to be about
50"� annually, which amounts to a
nearly 14" in 1000 years.
little
degree in
precession
more
than
10 in 72 years, and to
In consequence of the precession of the equinoctial points,
the equator of the earth, the tropics, the arctic and antarctic
circles, and the colures, are gradually referred to new places in
the
heavens, and the declinations, the times of rising and setting
right ascensions, &c. are subject to a
of the fixed stars, their
progressive change. Hence it arises, that the observations re­
specting. the fixed stars, which occur in Hesiod, Aratus, Hippar­
chus, Ptolemy, and other ancient writers, do not correspond
with ihose of the present period.
The revolution of the
25,79i
equinoctial.points
is
completed in
about
years. This perio� is called by some writers the Platonic
year, but must not be confounded with the annus magnus men­
tioned
by
Cicero.
3
TABLE
OF
TJlE
PRECESSION OF THE EQUINOCTIAL POINTS.
.
Signs,
Years.
-----
.
-----
-----
1
..........
..... ,
2
..........
..........
..........
..........
3
4
!\
',
5
6
8
9
10
20
30
'40
50
,25,,791
..........
.....
...........
.........
........
..........
...........
..........
!
..........
...
..........
..........
..........
..........
..........
..
..........
..........
...........
.....
'. ,
•
,
.....
........
..........
·
..........
...•..
....
........
........
�
..
"
•.
..........
,
...
,
..........
,
.
3
4
,.,.
•••
•
.•..••
•
••.••
.
••••.•
.•••••
••••.
..•.
·20
.,
..........
••.•.
..........
•.•..•
.
·,25
••••
.
1
1
•••.••
2
.; •••••. 9
4
.......
••
•.•••••
.
...
••
..
·9
.......
11
....
11
..
12
..
·.;
:
..
.....
"
11
·25
19
··9
.•••••• 18
..
,
.•
l·· ,29,
..
) nearly
7,5
30
52,5
15
37,5
22,.5
45
30
22,5
15
45
30
15
·10
··33
·57
··56
.•••.•
........
,
22
·46
•••••.••
·27
........
....
·58
•••.••
........
.....
..
11
'.13
45
·35
••••••
12
..........
•
··.11
.••..
.."
9
'.15
23
47
·29
.••.•
•...••
•
42
32,25
22,50
····7
.
•
••...•
·58
.
....
.....•
,"
..
.....
.•.•
........
�
..........
......
·
.......
.....
......
,
••••
'
.•••.
.
·5
·6
.•.....• 8
......
..
•
..••••
··2
..........
••
•
···l
..........
11,25
1,50
51,75
••
1
30,7.5
!tl
.•.•.••
·1
..........
50,25
46,5
........
...
......
-----
....
•••
,
•••••••••
�
...
•.••••
.
..........
..........
,
,
1
2
····3
4
.'
·5
.., .•.• ·5
6
••••..•• 7
8
�
16
.,
·25
·33
',,"'"
••.•. ·41
·50
...··
.....
..........
�
900
1,500
1,800
2,000
3,000
4,OQO
5,000
10,000
20,000
25,000
...........
..........
..........
60
70
80
90
1,000
..........
-
7
100
200
250
300
400
500
600
700
800
•••
..........
••••••
-----
....
7
45
30
1.5
30
55
·52
·50
·,·····47
,
·35
,
·10
•.•..•
30
••••••
.••••.
•
30
•.•
....
�
=r
·57
:>:
�g
30
.t)7,75
USE OF THE TABLE ..
1.
PROP.
Given the distance of time, required the precession of the
equinoctial points 1
Example. How many degrees have the equinoctlal points re,
ceded during1820 years?
'
years
,
By
the table
-
-
-
-
1500
=
300
=
20
=
56'
20°
4
15"
11
15
16
45
24
�5,
,
is'2<i"=' 25
y.
Or
by proportion,
-
y.
1 :
50",25 ;:
precession
of the
find the time 1
Ex. In what
2°
�W
24'
15".
2�
PROP.
Given the
lS20 :
e�uinoctial
period, oftime .have
the
points, required
to
equinoctial points moved
55'1
years.
By
the
table
,
�
.
�o
2°
47'
;l0"
7
30
9
5�
-
-s-
By proportion
-
-
�
50"! ::1'
200
209
y.
..
..
PR:OP;
2°
55' : 209.
3.
time, to find what degree of the ecliptic, at that pe­
riod, corresponds with a given degree of the ecliptic at the pre­
Given the
sent time.
degrees corresponding with the time, which,
time be past, subtract from the place of the given degree;
time be future, add to it,
Rule. Find the
if the
if the
5
'Ex. 1. What degree 500 years ago
corresponds
with
n
28° 1
signs
28°
U
By
=
fable 500 years
28°
2
2
Ans,
rr
I'
211)
Ex. 2. What
6
5B
'45
21
1.
15
=
15".'
degree
5000 years ago
corresponds
with ,� 2° 1
s,
t5 2°
B�
table 5000 years
EJJ.
n
=
13
2°
=
2
9
47
30
10
22
12
30
,3. What degree, 500 years hence will correspond with
present ecliptic 1
28· of the
s,
IT
By
table
28°
500 years
-
2
'"'-
3
Ans,
QD
4<:'
_
58'
45"./'
II
28&
6
58
45
4
58
45
6
SECTION
DESCIUPTION AND (lSE OF THE
THIS is
a
celestial
"
globe}
so
It
EQUINOCTIAL G�OBE.
mounted
as
to
have
two
axes
of
motion.
Fig.
3, The axis of the
is not attached to
equator does
not
ecliptic
Pp
perforates
the
globe,
but
the brass meridian, The axis NS of the
penetrate the gloQt'J but, is attached to the
The poles of the axes of motion are connected
by strong .hrass wires. This globe is provided with an equator,
tropics, arctic and antarctic circles of brass, and an hour circle,
which may be fixed, or removed, as occasion may require,"
brass meridian.
4.
PROP.
To
past
rectify
the
globle,
future, and
or
to
the
precession being given for any time
corresponding' position, of the
shew the
circles among the fixed stars.
First, Bring the pole of the ecliptic under the brass
meridian at 66°�) and 8 (i. e. Polus Arcticus, which is marked
Rule.
the
on
sphere)
[The globe
(i.
e.
under Nat
0,
will then have its
for the t date of tlie
position
globe)
coincide with the circles described
tor with the equator described
on
for the present time,
and all the brass circles wilt
on
the
globe, the brass
globe, &c. &c.]
the
equa­
�
1\ globe similarly mounted is described by the Rev. Mr., Coslard ill his
History of Astronomy; it was not, however, provided with the brass eqoator,&c.
and cousequeutly must have been less useful than the present, The Author had
contrived the equinoctial globe before he met with 1\11'. Costard's treatise.
t The dale of the globe in the following problems is 1S00.
7
if the time
Then,
the order
be
past,
turn the
globe 'westward,
or
con­
round the axis P p of the
the
,of
eclip­
signs,
degrees as correspond with the given precession of
the equinoctial points, keeping the pole' of the eeliptic fixed at
66°k under the meridian.
Note. The number of degrees may also be measured by the
motion of Vi westward from the graduated edge of the brass me":
ridian, or by the motion of ry>, or �, from the intersection
which is made by the brass equator 'with the ecliptic, because
the pole of the equator describes a circle parallel to the ecliptic.
to
trary
tic, as
many
The
globe being thus rectified, the brass wires, representing
tropics, arctic and antarctic circles, will trace the
these circles had among the fixed stars, at the
which
position,
period proposed. The brass meridian, will shew the position of
the equator,
solstitial colure, and the wooden horizon, if the poles of the
equator be brought into it, .will point out the position, at the
the
same
.
time, of the equinoctial colure
If the 'precession be
ward,
ecliptic.
or
.
anticipated, the globe must revol�e east­
signs round the axis P p of the
in the order of the
EXAMPLE.
Required to, shew
to
Hipparchus,'
the
dog's tail,
the
position
of the
the solstitial colure
or
globe when, according
passed through the tip of
h Sirius '1
Hipparchus.wrote B.C. 125, which, added to the date of the
globe 1800,=1925::: (by table) 26� 52' 1] \25, or 27° nearly=
difference oflongitude of 1J Sirius at the time of Hipparchus and
the present. Bring the pole of the ecliptic under 66°t and 8
under N. keep the poles P. p of the ecliptic fixed, turn the
globe contrary to the order of the signs 27°, arid it will he rec­
tified, for the brass meridian, which will represent the solstitial
colure, 'will pass over 1J Sirius, as will immediately appear, if,
after the: globe is rectified, the south pole is elevated above the
horizon.
The brass equator will also pass over fl
(Cetus)in the
.
8
hinder
leg of Aries;
.Hipparchus.
which coincides with the account
'0 EV TOt� (J7rtu$-lou; 7roO'tv aUTOU
:
aUTOU Toil
given by
[Arietis] KEi/lf.VOf; dO'T�p
l7r'
iO'1J/lf.PLJlOiJ tpipf.TaL.
Required the principal stars near which the equi­
passed in the time of Meton 1
lived 432 B. C. the precession for which period=
Example
2.
noctial colure
Meton
'310 S' 57".
Rectify the globe as directed for the precession, bring the
'poles of the equator to the edge of the wooden horizon, and the
equinoctial colure may be traced passing near a Aries, a Centau­
rus,&c. &c.
.
The time
star
PROP.
5�
being given, required- the declination"
of
a
given
•
Rectify the globe by Prop.n, bringthestar to thebrass
meridianythe degree marked on- the-meridian will shew the de­
Rule.
'
.
clination.
Ex. Given the latitude of
3000 years,
By
_
required
table the
a
Pegasus
19°
=
24',
.precession
for 3000 years
=
of the
4, the
4P
52', then turn
bring Pegasus to
a
the
the
period 1
52'.
Bring, as
66°� and. 8
of the ecliptic
under
directed, Prop.
pole
ecliptic·
under N, turn the globe westward round the axis
41°
and the time
the declination of this star at that
globe round the axis of the' equator, and
graduated edge' of 'the brass meridian,
.
its declination will- be 45'.
BY· TRIGONOMETRY
P "IY be thesolstitial
•
colure, P the pole of the ecliptie
equinoctial colure, N the pole' of the
equator, Sthe present place of any given star, drawP S B per­
pendicular to' :!!: 'VS',. then P S· == 'eclat, and cy; R::: longitude
.
Let
$' <)0
§
'VS',
of star.
P
<)0
p' the'
ofthe sa,me star, at any given time that' is'
past, draw P s A, then P s :::: P S = eclat, and '<yl A-long. at
the given period. CV" A will be less than- CV" B, the longitude having
Let
be the
s
increased.
noctial
place
AB
Also,
angle A P
or
B
=
precession
of 'the
S, P N
obliquity
equi­
points.
Draw N
P
ecliptic,
Then in the
s.
P N
triangle
colat. of star, and N
s =
=
of
codecln, of star.
s ::::::i
If s.Iie in
Angle s P'N
��oO�g. lon�·o�:
t
.,
tS600-long;s.+ 9().°5
=
S
orr
In the
the
s
Hence
(becam�e
s
0
subtract 10,'
54
,33
1
11,
52
3
10
9
2
3
by .N aut, AIm.·
"
2<f
141°; =
=
20
:2:
9'
1
and 51°; + 990'
II
11,
4tl1'quadrantY
,.�
lies in
from 128,
-!1
,8
=.
By.table 30,00 years
:. Long. s. at given period �
3:'
67.::: 51°' nearly'
67
angle s PN�
<Fig. 4.) Draw N d perpendicular
right-angled triangle N P d.
III
s
P.
produced r
'
X
Cos. N P d
Cot. P N .x t.:p d.
=
10 + COi
.
Cot.
23°
39°
=
19,8906026
28'
=
10;3623894
d
'=.
•••. t. P
...
...
:
'
.
,3a�,
700
=
Long, s' (1815). =:
R
,{5 Qll:l,drlmt
A
are
example proposed,
P
-
4tlj
qa,_isl the complement' 'of"the
by
froin
hence
,Aties';':
� s, may he found;
longitude
P ��.'measured
star's
��t 3d'
2d
S
"
•.
s. s
s
P +
d :
s.
P'd
= s
d
'
89° 14'.
'd ':, 't.
Pd
=
t. N Pd'
=
s.
9,a281132, =.18& 39'
_.
=
P d :: t. 'N P
.
N
s
d.
'9,5048598
'9,90�3692
19,4132290
S.
s
d
=
�,9999611
r, t. N
s
d
=
9,4132679
e
=
14° 31'
,
10
/Cos.
S
d :
CO,S.
P d :: Cos.N
Cos.
s
d
Cos'. P N
s
: Cos. P N.
8,1264710
=
9,9625�'76
=
Cos. Pd
18,0889786
9,9765745
Cos. N
8,1124041
s =
=
890 15'
.
From 90
Take 89
15
45"
=
declination
-�-
found.
required to.,be
being foun�L for any period, of a known
the time of the year. when it rose or set cosmically,
The declination
fixed star,
acronychally, or heliacally, may be found for any given
See Gregory's Astronomy; prob, 25, 26, 27.
latitude.
PROP. 6.
precession
equinoctial points, and the lad.
find
the time of the year, that a
to
place, 'required
or
set, acronychally, (&.Kpovvxwr;) i. e. at sun-set,
rose,
of the
Given the
tude of the
given
star
Rectify the globe for the-precession by Prop. 4. Turn
globe round the axis of the, equator, and bring the star to
the eastern part of the horizon, if, rising; to the western, if set-,
ting. Observe the point of-the ecliptic which is setting. Find
by Prop. 3. the corresponding degree for. the precession, and
Rule.
the
the time of the year will be known.
Ex. Given the latitude of the place
and the time of' Hesiod
940
of the year that Arcturus rose
1800 + 940
(Ascra in Breotia)' 370 45',
required to-find the time
B. Co it is
=
acronychally?
2740
=
380 14' 45"
precession. Rectify the globe for the precession
pole to the iatitude of 'the, place, Bring
Arcturus to the eastern edge of the horizon, the point of the
ecliptic then setting will be :v, 'Ho 30';
(by table)
(Prop. 4.)
=
Elevate the
,
I
1/
From 12
9
30
Take
8
14
45
1
LS
15
1
11
11
acronychally when the sun was in '*
sixty days after the winter solstice,
thedescription of Hesiod.
Arcturus therefore
to 15' 15'� Feb.
according
to
19.
rose
i.
e.
E�T' {iy �' f��KOYTa,
.
/-t.E,Ta Tp07ra{: �EAloLO,
XPJL€pl �KTEMO'rJ �EV� 1JJLaTa, �f] pa T.OT' ltO'T1)p
'ApKTOVpO{: 7rPOAL7rWY LEpew POOy'fiKEaVOtO,
llpWTQY 7raJL¢aivwy I7rLTEAAETaL ltKpOKyE¢aLO{:.
Op. et Dies.
Arcturus rises
�cronychally
at
present April
PROP.
7
Ist
v.
564
.•
..
..
Given the
tude of the
rose or
set
precession of the equinoctial points, and. the .lati,
place, to' find the time of the year that a given star
cosmically, i. e. at sun-rise.
globe for the precession by Prop. 4. and for
place, bring the star, if rising, ·t.o the eastern
of
the
horizon
Observe the
; if setting, to the western.
edge
is
of
the
Find
the
which
point
ecliptic
rising.
point correspond­
ing to the precession by Prop. 3. and the time of the year will
be known!
Rule•.
Rectify
the
the latitude of the
Ex. 1. At what time of the year 50 B. C. did Lucida Yergi�
Iiarum, the bright star in the Pleiades, rise' cosmically, and at
what time of the year did it set
�JO 04'1
cosmically a� Rome,
lat.
=
..
By table the precession for 1850 years = 25° 49' 22",5=26°
nearly, after rectifying the globe as directed for the precession
and the latitude, the point rising with the bright star of the
Pleiades will be � 19° 30', by Prop. 3.
.
� 19° 30'
=
1850 years
=
r
o
the
19°
30'
"
25
49
22,5
23
40,37,5
4.0'. i. e. the
star
13th.
being brought
to the western side of the horizon, the point of the ecliptic then
rising will be m 28°, the corresponding point for which is ttt �o,
corresponding point
rose cosmically April
wi1l therefore be
The
same
('('
23°
star
12
i;
of thePleiades is
cosmical setting
v.
221.-Ante tibi Eoee
Ex. 2. Thales,
Atlantides
according
Shew the
nox.
Pliny, determined the, cosmical set­
be 25 days after the autumnal equi­
"The
Christ.
determination �� the globe.
tr,uth of-this
The latitude of Miletus
abscondantur.
to
of Lucida Pleiadum to
ting
cosmically Oct. 24. ,���
alluded to by Virgil,.qeol'g.,l.
star of the Pleiades set
e-:the bright
37°. -Thales lived 548
=
years before
320 46'
27/1.
precession by
the
and
of
latitude,
precession
point the
the setting of Pleiades is itt 27�. By Prop. 3,
angle' for the
table
=
the globe for the
Rectify
ecliptic rising at
the point at the time
the
eveni�g,
will be 25
of Thales
,� 2/!�
was
which added' to'
seven
�ays
13',
in
or
Oct. 18th, in
the amount
Sept.
days.
For l1t 270
=
7
27
1
2
46
27
6
24,
13
33
'
a,48 years
PROP.
8
.
place, the latitude of a star, and'
th�
the' point of the ecliptic with which it rose cosmically, or set
to determine the I�ngitude of the star, and thence
•
Given
latitude of the
acronychally,
the time when the star
so rose or
set.
TRIGONOMETRICALLY.
zenith, H R the horizon, E Q the equa­
Fig.
tor, cutting H R in q, and the ecliptic elL in x, I the given
point of the ecliptic which rises with s the given star. Draw 10
6. Let Z be the
perpendicular
10
=
decl, of
ecliptic,
I,
x
Q, and'
I
::;::
s
long.
I to C L.
Then
of I.
It is
required' to
at o,
0 x
triangle
I
0 =
0 x
I, right-angled
declination of J.
:.
0
I
x
qI
=
0
I
=
lat. star.
find the
I
==
obliquity
of
may be found.
triangleq 0 I, right-angled at 0,' 0 q 1= colat. of place,
declination-of I /. angle q I 0 may be found, and q I 1=
In the
10
s
of the star s?
longitude
In the
to E
+
0
Ix.
""
See Petavius
� Auot. lib.
i, cap.
v,
13
In the
1
triangle
11,
angle
q
to be
added,
according
1 =.lat star, .�.
s
at
J� 1 Ls
be
:found.
I.. right-angled
s
llmay
•.
=.
supplement
The
arc
II is
subtracted, from the' given point of the ecliptic
or
to the
following' rule.
If the latitude of the star be
north,
Add, when the point rises;
Subtract, when it sets with the star
'If the latitude �{the star be 'south,
.
.
'Sl�btraCt,
when the point rises;
Add. 'when it sets with the shire
Hence the
of the star.
longitude
the time may
being known,
'
be found.
.
Exa"!ple.
Geminus
that Arcturus
required
mentions,
cosmically
rose
to find the
time,
58',' and the latitude
37°
the
on
the
sun was
of the
latitude
of Arcturus
of
authority
when the
=
31°
Euctemon,
in 1IJl. 10°, it is
place (Athens)
=
3'?
triangle x I 0, right-angled' at 0, x is the autumnal equi­
by problem I x = 20°, and 0 x I == 23° 37' (the obli­
of
quity the ecliptic being greater than at present.)
In
nox
...
R X
I
S.
x
s,
0 x
6
x
I
I
cos. 0
t. 10
Cot. I
In
X
=
9,6027278
9,1367795
Cos.
I
0
I
t.
x =
9,1404092
x =
10,4389341
x =
9,5793433
triangle
O.
=
q 01, right-angled at
'= Colat. = 52° 2', 10
R
I.
'
7° &2'.
=
lox Cot.
=
Iq
0
S. 0 x
9,5340517
o.
R X
...
I
s. x
=
10 ='
... S.
Also in
0 =
I
s,
7°
x
J.
67° 42'.
=
52'.
Cos. I q 0 = Cos. lox S. q I
10 + Cos. I q 0 = 19,7890184
X
o,
,
Cos. 10='
...
S.
q I
0 =
9,9938936
: 9,7931248
'
=
38� 23\
-----
hence q I
x:;:::
q 10 +
0
I
x =
.106° a' :.
s
II
=
736 5;,'.
14
In
triangle
Ls l,
at I. Is
right-angled
R
II
X s.
Cot. 11
==
s
=
31° 3'.
x
t. Is.
t. I
s ==
9,7796318
Cot. I I
s ==
9,4598749·
.'. S'. I I ==
9,2395067
9° 58'
=
north, and the point of the
added, :. 1l� 10° + 9°
the longitude of Arcturus = .169° 58', the
Because the latitude of Arcturus is
rises with it, the
ecliptic
50'
=
flJ4 19° 58' :.
present longitude
I) must be
arc
c; 2010 42' :. the difference
50",25
Then 2273--1820
Euctemon
=
: 1
=
31° 44'.
years
year
::
31° 44' : 2273.
453 B. C. the
flourished, according
to
time
of the observation.
Lemprier's Dictionary,
B. C�
present when the sun is in
cosmically
� 13°, the difference between tI.Jl. 10° and zs 13° ;:::: 33°.,
W�r�
the time that Euctemon made the observation computed by this
431.
Arcturus rises
at
difference, it would exceed the
For
50",25
A modern writer
;
1
::
true time
91 years,
by
33° : 2364 years
has, however, given
a
nearly.
solution, upon this prin­
ciple, of several questions which depend upon the precession of
the equinoctial points.
PROP. 9.
The latitude of the
place
and the
find the time of the year when arty
precession being given, to
given star rises or sets he­
liacally.
The rule
given by Ptolemy is,
that
a
star
nitude is visible in the horizon when the
below
it,
of the second
magnitude
of the first mag­
sun
is
depressed
12°
when 13°, of the third mag
..
nitude when 14°, &c.
Rectify the globe for the precession by Prop. 4. Ele­
pole to the latitude of the place. Turn the globe round
the equatorial axis, the pole of the ecliptic remaining fixed, un,
til the given star is at the eastern edge of the horizon, if rising;
Fix the quadrant of altitude on
at the western edge, if setting.
Rule.
vate the
15
the brass meridian at the zenith.
Turn the quadrant to the
rising; to the east, if the star be setting;
and let it cut the ecliptic, above the horizon, in 12°, 13°, or 14-,
according as the star is of the 1st, 2d, or 3d magnitude: the
opposite point of. the ecliptic, thus intersected, is the sun's
place. Find by Prop. 3, the corresponding degree for the pre­
west, if the
star be
cession.
the time of the year that Antares rose
2000 years ago, the latitude of the place being 37° N.
Example. Required
heliacally
By table 2000 years = 28°.
Rectify the globe for the precession and latitude of place. Turn
the, globe round the axis of equator until Antares' is in the
eastern edge .of the horizon: fix the quadrant at the zenith, turn
it to the western side, of the, horizon, and' let it cut the ecliptic
at 12° altitude, the point cut ,in the ecliptic will be n 22°; the
opposite point to which is
t 22°
=
8
28
2000,years
7
Antares, therefore,
rose
22°
heliacally
24
when the
sun was
'in '»t 24°
Nov. 16, at present it rises heliacally Dec. 15. Virgil alludes
to the heliacal rising of Corona Borealis. Georg. I. 222.
"
Gnosiaque
ardentis decedat Stella Coronee."
,16
SECTION
III.
MISCELLANEOUS OBSERVATIONS.
THE poem of Aratus, which describes the phenomena of the
heavens, is formed from the description given by Eudoxusof
sphere of the ancients, which appears' to have been CQIi'"
structed for the latitude of -;l}8, in which latitude is Thessaly the
the' longest
country of Chiron : the poet stating the length
the
Q�
day,
to the
length
of the shortest
night,
to be five to-three,
will also appear to have been made in the age of
It
B. C.
Chiron,
period, the
is
which
.of
the
and
Aratus
given by
position
equator
tropics
amQng the fixed stars, will be fouud, except in a few particulars,
940:
for if the
remarkably
equinoctial globe
correct.
attributed to the
is
rectified
The-variations which
changes
made
for this
occur
by succeeding
may be
justly
astronomers in
the
configuration .of the constellations. Those stars, 'fQr ex­
ample, which form at present the zone of Orion, are represented
both by Aratus and Hipparchus as being in the equator ; yet in
the time of the latter they were several degrees below it, a cir­
cumstance which could not have escaped the notice of SQ accu­
rate an observer,
They were still lower in the time of Aratus
and Eudoxus, It is probable, therefore, that those stars which
now form the zone of Orion
anciently composed the'Sw.ord.
Aratus
represents the equator as passing through part of Aries,
Taurus, the belt of Orion, the flexure of Hydra (CQr
the knees of
Hydree) Crater, Corvus, &c. He describes the heads of the
as coinciding with the summer tropic: in the same curve
he places the knees .of Auriga, and the right arm of Andromeda
above the elbow, the palm of the hand rising above the tropic, &c.
Gemini
17
,
represe�ts the wink,. tropic as passing through the middle
Capricorn us, the feet of Aquari�s, the sting of Scorpio, &c.
He
of
given no description of the co lures, but this defi­
supplied by Hipparchus, who has preserved the ac­
count given by Eudoxus, according to which astronomer the
equinoctial co lure passed through the left hand of Bootes, the
right hand and fore knees of Centaurus, the head and right
hand of Perseus. This description will be found correct by
rectifying the globe, as before, for the time of Chiron,
Aratus has
ciency
is
That
alterations
frequent
were
made in the constellations is
confirmed' by Ptolemy, who pleads the example of ancient astro
The same liberty has been
nomers, for those made by himself.
taken by modern astronomers. Thus the consteliation Hercules,
..
suppliant figure, without a name, but called by
the Greeks, from his position, 'EY'Yo�aO'ty, the man on his knees,
having his hands extended and empty. In the time of Manilius
this figure had notexperienced any change-He thus describes it,
was
anciently
Proxima
a
fulgentes arctos, boreamque rigentem
species genibus, sibi conscia causes.
Nixa venit
Lib. i.
v.
321.
The greater Bear formerly consisted only of seven stars, the
fish near Andromeda had a hawk's head, the cross formed. the
hinder
legs
of the Centaur, and Arcturus
was
of the
one
un­
formed stars.
The two
now form the heads of Castor and Pollux,
Chiron, considerably above the summer tro­
stars which
were, ill the time of
pic, yet they
are
described
by
Aratus
as
lying in
it.
.
possible that anciently the heads of these sons of Leda
represented lower, and that the .two stars w�re .placed
above them, as they frequently appear on ancient coins, .or,
which is more probable, the Poet might consider them sufficiently
So also the
near tel be .introduced as a guide to find' the tropic.
It is
were'
'
D
18
sting of Scorpio
was
Poet considered it
<to to the
north of the winter tropic, but the
near to form a part of his de­
sufficiently
scription.
The arctic and antarctic circles among the ancients were de­
by the latitude of the place, and constituted those
termined
circles which
are
�etual apparition
called
by modern
astronomers
circles of per­
and occultation.
'Ap/CTtKOr,; flf:V o'Jv �O'rt
fL€YLO'TO!;
�EU'pOVJ1,1.vwv KV/cAwv, 0 ecpa7rT6flElIOr,; TOU
/Ca2
/CaO'
€11
SAO!; -D7rep yijv u7roAafl{3av6flElIO!;.
O'r}flEiov,
0p[{OllTO!;
Geminus thus defines the arctic circle.
KtkAo!; 0
TWV
Eudoxus
Alexandria,
€"
{LE2
observes that the star Can opus, which
was
was
visible
not visible above the horizon where he
at
wrote,
UVTtj rfj ucpallfi /cVKA� cpipEO'Out.
Hipparchus censures Eudoxus for this assertion, and mentions
Canopus was sometimes visible at Athens, the latitude of
this city being 37°, and the distance of Canopus from the south
pole 38�0. This was true at the time of Hipparchus and Ex­
doxus, but the globe being rectified for the eera of Chiron, Ca­
nopus will be found to. be only 36° from the south pole, it was,
that
therefore, at that time invisible
visible in the latitude of Thessaly.
The
rising
and
setting
at
Athens, much less
was
it
of the
Aratus, will receive considerable
constellations, according to
illustration; whether the globe
period of Eudoxus or Chi ron, the difference
degrees
being sufficient, in this particular, mate­
affect the description of the Poet.
be rectified for the
of
a
few
rially
to
not
Achilles Tatius, when ·describing the course of the sun in the
heavens, says, that the first point of Cancer is the highest to
which it ascends, but that this solstitial point was differently
placed by astronomers,
in the' Bth,
12th,·
16th
placing it in the beginning, others
degree of Cancer, {3ovXovraL 0€ rp01rijv
some
19
a:hdV
[i.
e.
7TO(fiuSa( oi
ijAtOy]
oyoor}Y p.oipay·
01.
0€, 7Tfpt t{3'·
variation doubtless
originated
p.€y, ,7rfpl ra� &pXc1�· 01, �€ 7T'Ep�
'frEP't t€. Toli lCapKiyov. This
01. 0€
from the observations of the
sum­
solstice being made at different periods, and the place
being in the constellation to which they were referred. Eudoxus
placed the solstitial points in the middle, Aratus in the begin­
ning of the signs .•.
mer
When
astronomy
the solstice
was
withstanding
was
first introduced at Rome, it is
in 8° of the constellation Cancer;
the
change
which
necessarily
took
probable
hence,
place,
in
not­
conse­
quence of the precession, the notion prevailed, even in the time
of Pliny, when the solstitial point was in the constellation Pisces,
that it
"
of Cancer: thus Pliny says,
Caned.'!
Lib. xviii. cap. 28.' It
peragi
parte
Petavius observes, in compliance with the prejudices
was
still in the 8th
Solstitium
was,
as
of the
people,
in
degree
VIII
that in the Roman calendar the solstitial
point
eighth degree of Cancer: this author remarks
the same compliance in the principal Roman writers in this re­
spect. "Columella, Plinius, Ovid ius aliique complures in octavis
partibus asserunt eequinoctia et solstitia confici. Sed neque id
omnem retro vetustatem instituisse "dicunt; neque illos, quo's
percensent astronomos ex animi sui sententia ita preescripsisse ;
sed rusticorum, ac vulgi gratia, significant. +"
was
referred to the
By the assistance of the equinoctial globe, the true times of
rising and setting of the stars may be easily found for any
period and for any latitude; hence it may be ascertained, whether
the
the
obs�rvations
tude in which
in ancient writers agree with the time and lati­
they lived. Pliny has collected a variety of ob­
respectingthe rising and setting of the stars, but as
always mentioned the age of the astronomer from
they were borrowed, nor the latitude of the place where
servations
he has not
whom
lit
See Sir I. Newlon's Chronology, P:
t Ad Auct. lib. iii. cap. 11.
82._;.Hipparchus.
lib. ii, 111.
20
they 'were made;'
much assistance cannot be derived from
arid his accounts" appear
LUCIDA
PLEIADUM
At Rome, B.C. 45.
The age of Virgil.
At Ascra, H. C. 940.
The age of Hesiod.
Rose
cy>
12(1
30'
cy>
Set.
�
20
30
llt
3
Set.
ro
4
8
cy>
18
3
�
11
4
�
28
3.
Rose
From
the heliacal
there were,
C5q
TOt
Jl"UKTa!;
re
Kat
220
��
Ol/}
.:;'8
12
of the.
setting to the heliacal rising
according to Hesiod, forty days.
Ar
them,
occasionally contradictory.s
.
'j"
J ��
'::
¥'
Pleiades,
ijp.ara TEO'O'apCtlC01JTU
KEKpvcpaTUC U'�Tt!; 0€ 7rEpt'1rAop.e1JOV E)ltaLlTOt}
Wa[JlOJlTat, Tn 7rpWTa XcipuoO'OP.€JlOLO O'tC5iJpov.
et Dies.
Op.
cy> 4°
nearly with
S' corresponds
with
.March 23;
At the
of
May 2, period
forty. days.
commenced; and mowing at the rising:
a
Virgil
rising,
mentions two
seasons
of
v.
385.
and � 11° 4'
setting, ploughing
,
the one at the
making honey;
'
the other' at the
Taygete
Pleias,
simul
et
setting
os
of the Pleiades.
terris
ostendit honestum
oceani spretos pede
repulit
amnes,
eadem, sidus fugiens ubi piscis aquosi
Tristior, hybernas coelo descendit in undas.
Aut
Georg. iv. ��2.
""
Pliny notices this difference in the authors to whom he refers. Autoru
prodidere ea, quos prretexuimus volumini huic, raro ullius sententia cum alio con-·
gruente. Occasum matutinum Vergiliarum Hesiodus tradidit fieri, cum eequinoc­
tium autumni conflceretur : 'fhaies xxv, die' ab reqnino'ctio: Anaxhnand�.f ·xxix.
.
Euctemoll xlviii. [xx viii.
forsan.]--Lib.
xviii, cap. �xv.
.
..
2l
-
it
Petavius calls this' passage inexplicable, and with
I
certainly
says,
et
"
cannot
Is situs est in
insequi
videri
justice,
be reconciled to the constellations.
ccelo,
possit."
_
for
Heyne
ut alter
The
piscium eas a tergo urgere
slightest inspection of the globe
will shew the incorrectness of this assertion.
Arcturus, in the time of Sophocles, B. C. 450, rose cosmically:
Athens, August 31, and heliacally September 12. It was
probably to the latter rising that the shepherd alludes:
at
'E7C)"'I'J(1tU�oJl TiioE T' Uyop2 Tplit; 8AO'll{;
'E� �PO{; El{; ApKTOVpOJl {P.fL-YJJlO'll{; XpOJlO'll{;.
<Ed.
By
which the
shepherd
Tyr.
v.
states that he fed the flocks
from the termination of
1157.
on
Mount
until, the
rising of
Elmsley,
Arcturus, period
ifLfLfJ
'JIovt;, adopts the conjectual reading of Schtefer €Kp.f/Vov{;, but it
isdifficult to imagine what rpEir; €KfL�'JIovr; can mean.
Citheeron,
spring
instead of
of three months.
a
...
The. changes which take place ill consequence of the preces­
equinoctial points are exceedingly slow; and for a
scarcely perceptible; yet the results, after a series of
ages, are very important with respect to the rising and setting of
the Bt.,.�S : the following instances mjiY serve as, examples,
sion of the
long
time
,
The circle of
perpetual
occultation is
many. of those stars therefore
the
which,
visible at the
horizon,
longer
others, on the contrary, which
observation,
are
no
some
were
continually varying:
ages ago,
same
rose
above
latitude; while
invisible,
come
under
Aratus has, given a very poetical description of the signs at-.
tending the rising of the Altar. This constellation is, at present,
only, partially visible at Athens, the stars {3 and 'Y not rising
above the horizon; but at the time of Chiron they rose several
degrees above it at Athens, and were visible at the latitude of 40·.
In latitude 27°, Crux, at the present period, just appears above,
_
22
the circle' of' occultation: after the
Iapse.however,
of about
14,380 years,' it will, become visible in the latitude of London,
Homer describes the Greater Bear
by the ocean:
37°, only within 21°
unwashed
as
in his time 'YJ in the tail approached in lat.
of the horizon; at present this star
:._
it at the
dips below
same
latitude: and in about 2000 years hence, the whole of the seven
stars, which originally constituted the' Bear, will rise, and' set in
horizon, together with
the
which is
considerable part of that space.
a
included in the
now
configuration.
and other' poets, have imitated the language
of Homer, but their works will live when their allusion will be
Virgil, Boethius,
unintelligi�le, without a reference
noctial
precession
to the
of the
equi­
points.
-
Sirins
heliacally
rose
at
$
at
present when the
is
freque�tly
sun
heat, the
was
period
called the
London,
no
longer
The
change
Was
sun -was
This
sun
was
in .Q 7°, 43'
rising
:
of Sirius
writers, from the influence
supposed to possess in augmenting
ri�ing to its heliacal setting
sun
at
present, in the latitude
is in 1TJl. 3D-in about 7,060 years it
visible.
in the declination of the stars may be illustrated
the annexed
Fig.
is in .Q 220 30'.
dog. days. Sirius rises,
will be
by
940, when the,
from its heliacal
when the
of
B. C.
noticed· in classical
which this constellation
the
Rome,
27°: in- the time of. Virgil, when ·the
in
diagram.
7. Let P be the
pole
of the
ecliptic,
$
Vj�N the pole
of the equator E Q- n N the path described by the pole of the
equator, round P ai s b, the path of .a star during a semi-revolu-
equinoct'ial points, whose latitude 'is less than the ob­
tion of
the
liquity
of the
declination,
ecliptic.
at
s
Then from bto s the star will have north
the declination: = '0, 'and from
s
to a, the de­
clination will be south .,......Also, if lines -be drawn
•
parallel
to
the
equ(\tor, they
will
through a b,
represent parallels of dedi
..
TABLE
of
the
Rising
and
Setting
From
of
Stars,
at
Rome.
B.
45.
SETTING.
RISING.
NAMES
C.
PET AVIUS.
of
STARS.
Antares
•
Arcturus
.
.
Capella
•
t
Delphini
a
Andromedee
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nation at different
place Z, it
periods.
And if H 0 be the horizon of
will be evident that from
a
a
to x, the star will not rise
above the horizon. In t he same manner, figures may be con­
.structed for· stars of different latitudes and for different places.
In.
x
to
Fig.
a
8. the star appears above the
horizon from b to x; from
is invisible.
study of the .equinoctial globe will present many other
interesting phenomena, and suggest a variety of problems, r�
specting the oulminating., the amplitudes, and asimuths of the
Mars, at different periods, of which it will afford a faroil\"l' and
ready illustration by ocular inspection.
The
at
The annexed Table of the rising and setting of f2ev�t"l &t�:rlt
Rome, B. C. 45. may be used as a key' to a variety Q( E","w.,
ples applicable
to
the .sixth, seventh, and ninth
propositions,
CHRONOLOGICAL
TABLE.
1800.
•
Precession
stronomers.
.
7·5/1
,
Chiron
B.C. 950
Thales
548
32
46
27
57
23'
38°
.
f
Meton
432
31
8
Eudoxus
352
30
2
Aratus
277
28
55
Eratosthenes
194
27
49
58,5
Hipparchus
125
26
52
11,25
26
11
25
7
,
Geminus
;
77
Manilius
Plinius
•
A.D. 79
24
120
23
·140
23
]0
400
19
32
Theon
Ptolemeeus
.
Avienus
11,far Vi rearl tlJl..
8. By Trigonometry, add Fig. 4.
9. line 6,fO)' PN S read PN s,
line 24,/01' (Fig. 4:) read (Fig.
/2..
/9
/rY .);r4,!/Jo,(?, e�'>';' o--d,t.GJL
1. line
-
5.,)
-
.
Printed
,
by 1. F. DOVE, S1. John's Square.
.
.
59,25
30
27
ERRATA.
Page
24
30
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