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Of Circumstances regulated by Ascensions
IN any climate whatever, the magnitude of a given day or night is to be computed by the number of ascensional times proper to that particular climate. For example, the magnitude of the day will be ascertained by numbering the times between the Sun's zodiacal degree and the degree diametrically opposite, in the succession of the signs; and
that of the night, by numbering the times, from the degree diametrically opposite to the Sun, onwards, in the order of the signs, to be degree actually occupied by the Sun: because, by dividing the respective amounts of these times so obtained, by fifteen, the number of equatorial hours belonging to each space will be exhibited; and if the division be made by twelve, instead of fifteen, the result will show the numbers of degrees equivalent to one temporal hour of either of the said spaces respectively. 1
The magnitude of any temporal hour may be, however, more easily found by referring to the annexed Table of Ascensions, and taking the difference between the respective aggregate numbers, inserted therein under the heads of the equinoctial parallel or right sphere, and of any particular climate for which the magnitude of the temporal hour is required; and, if the said hour be a diurnal hour, the aggregate times as stated against the zodiacal degree occupied by the Sun; but, if nocturnal, those stated against the degree diametrically opposite, are to be compared; and the sixth part of the difference between them is to be added, if the said degree be in the northern signs, to the fifteen times of an equatorial hour; but subtracted therefrom, if in the southern signs. The amount thus obtained will be the required number of degrees of the temporal hour in question. 2
And if it be required to reduce the temporal hours of any given day or night, in a certain climate, into equatorial hours, they must be multiplied by their proper horary times, whether diurnal or nocturnal,
as the case may be; the product is then to be divided by fifteen, and the quotient will necessarily be the number of equatorial hours in the climate in question, on the given day or night. 1 On the other hand, equatorial hours are also to be reduced into temporal hours by being multiplied by fifteen, the product of which is to be divided by the horary times proper to the given day or night in the said climate.
The degree ascending in the ecliptic, at any given temporal hour, may also be ascertained by multiplying the number of temporal hours since sunrise, if the given hour be diurnal, but if nocturnal, since sunset, by their proper horary times; and the product is to be added, in the succession of the signs, to the aggregate number (as shown by the ascensions proper to the climate) of the Sun's degree, if the given hour be diurnal, but, if nocturnal, to that of the degree diametrically opposite, and that particular degree of the ecliptic which shall correspond with the total number thus found in the ascensions of the climate will be the degree then ascending. 2
But, in order to ascertain the degree on the meridian above the earth, the number of temporal hours since the preceding noon are also to be multiplied by their proper horary times, and the product is to be added to the aggregate number of the Sun's right ascension; and that degree of the ecliptic, with which the total number as found in the aggregate
times of right ascension shall correspond, will then be on the meridian. 1 The degree on the oriental horizon will, however, also show what degrees occupies the meridian; for, by subtracting 90 times (the amount of the quadrant) from the aggregate number ascribed to the said ascending degree in the Table proper to the climate, the number so reduced will be found, in the aggregate times of the Table of Right Ascension, to correspond with the degree on the meridian. And again, on the other hand, by adding 90 to the aggregate times ascribed by right of ascension to the degree on the meridian above the earth, the degree ascending may be obtained, for it will be that degree which corresponds to that total number, as stated in the Table proper to the climate. 2
The Sun always preserves an equal distance in equatorial hours from all parts of the same meridian; but his distance in equatorial hours from different meridians varies according to the degrees of distance between meridian and meridian.
The foregoing extracts have been made to show the entire agreement between the astronomy of the Tetrabiblos and that of the Almagest. The Tables herein given from the latter work are, of course, now, in some degree, superseded by others of modern calculation, infinitely more complete.
TABLE OF LATITUDES, AS SHOWN BY THE DURATION OF
THE LONGEST DAY
[From the Almagest.]
|
LONGEST DAY. |
LATITUDE. |
LONGEST DAY. |
LATITUDE. | ||||
|
H. |
M. |
D. |
M. |
H. |
M. |
D. |
M. |
|
12 |
0 |
0 |
0 |
16 |
15 |
50 |
15 |
|
12 |
15 |
4 |
15 |
16 |
30 |
251 |
35 |
|
12 |
30 |
8 |
25 |
16 |
45 |
52 |
50 |
|
12 |
45 |
12 |
30 |
17 |
0 |
54 |
I |
|
13 |
0 |
16 |
27 |
17 |
15 |
55 |
0 |
|
13 |
15 |
20 |
14 |
17 |
30 |
56 |
0 |
|
13 |
30 |
23 |
51 |
17 |
45 |
57 |
0 |
|
13 |
45 |
27 |
40 |
18 |
0 |
58 |
0 |
|
14 |
0 |
130 |
22 |
18 |
30 |
59 |
30 |
|
14 |
15 |
33 |
18 |
19 |
0 |
61 |
0 |
|
14 |
30 |
36 |
0 |
19 |
30 |
62 |
0 |
|
14 |
45 |
38 |
35 |
20 |
0 |
63 |
0 |
|
15 |
0 |
40 |
56 |
21 |
0 |
64 |
30 |
|
15 |
15 |
43 |
5 |
22 |
0 |
65 |
30 |
|
15 |
30 |
45 |
1 |
23 |
0 |
66 |
0 |
|
15 |
45 |
46 |
51 |
24 |
0 |
66 |
10 |
|
16 |
0 |
48 |
32 |
|
|
|
|
|
SIGNS. |
Tenth Degree. |
In a Right Sphere under the Equator, Diurnal Arc 12 Hours: |
3rd Climate, thro’ Lower Ægypt, Lat. 30° 22' N. Diurnal Arc 14 Hours. |
8th Climate thro’ Southern Britain, Lat. 51° 30' N. Diurnal Arc 16 Hs. 20 Mts: | |||
|
Times of Ascen. |
Aggregate Times. |
Times of Ascen. |
Aggregate Times. |
Times of Ascen. |
Aggregate Times. | ||
|
|
|
D. M. |
D. M. |
D. M. |
D. M. |
D. M. |
D. M. |
|
Aries |
10 |
9.10 |
9.10 |
6.48 |
6.48 |
4.5 |
4.5 |
|
|
20 |
9.15 |
18.25 |
6.55 |
13.43 |
4.12 |
8.17 |
|
|
30 |
9.25 |
27.50 |
7.10 |
20.53 |
4.31 |
12.48 |
|
Taurus |
10 |
9.40 |
37.30 |
7.33 |
28.26 |
4.56 |
17.44 |
|
|
20 |
9.58 |
47.28 |
8.2 |
36.28 |
5.34 |
23.18 |
|
|
30 |
10.16 |
57.44 |
8.37 |
45.5 |
6.25 |
29.43 |
|
Gemini |
10 |
10.34 |
68.18 |
9.17 |
54.22 |
7.29 |
37.12 |
|
|
20 |
10.47 |
79.5 |
10.0 |
64.22 |
8.49 |
46.1 |
|
|
30 |
10.55 |
90.0 |
10.38 |
75.0 |
10.14 |
56.15 |
|
Cancer |
10 |
10.55 |
100.55 |
11.12 |
86.12 |
11.36 |
67.51 |
|
|
20 |
10.47 |
111.42 |
11.34 |
97.46 |
12.45 |
80.36 |
|
|
30 |
10.34 |
122.16 |
11.51 |
109.37 |
13.39 |
94.15 |
|
Leo |
10 |
10.16 |
132.32 |
11.55 |
121.32 |
14.7 |
108.22 |
|
|
20 |
9.58 |
142.30 |
11.54 |
133.26 |
14.22 |
122.44 |
|
|
30 |
9.40 |
152.10 |
11.47 |
145.13 |
14.24 |
137.8 |
|
Virgo |
10 |
9.25 |
161.35 |
11.40 |
156.53 |
14.19 |
151.27 |
|
|
20 |
9.15 |
170.50 |
11.35 |
168.28 |
14.18 |
165.45 |
|
|
30 |
9.10 |
180.0 |
11.32 |
180.0 |
14.15 |
180.0 |
|
Libra |
10 |
9.10 |
189.10 |
11.32 |
191.32 |
14.15 |
194.15 |
|
|
20 |
9.15 |
198.25 |
11.35 |
203.7 |
14.18 |
208.33 |
|
|
30 |
9.25 |
207.50 |
11.40 |
214.47 |
14.19 |
222.52 |
|
Scorpio |
10 |
9.40 |
217.30 |
11.47 |
226.34 |
14.24 |
237.16 |
|
|
20 |
9.58 |
227.28 |
11.54 |
238.28 |
14.22 |
251.38 |
|
|
30 |
10.16 |
237.44 |
11.55 |
250.23 |
14.7 |
265.45 |
|
Sagittarius |
10 |
10.34 |
248.18 |
11.51 |
262.14 |
13.39 |
279.24 |
|
|
20 |
10.47 |
269.5 |
11.34 |
273.48 |
12.45 |
292.9 |
|
|
30 |
10.55 |
270.0 |
11.12 |
285.0 |
11.36 |
303.45 |
|
Capricornus |
10 |
10.55 |
280.55 |
10.38 |
295.38 |
10.14 |
313.59 |
|
|
20 |
10.47 |
291.42 |
10.0 |
305.38 |
8.49 |
322.48 |
|
|
30 |
10.34 |
302.16 |
9.17 |
314.55 |
7.29 |
330.17 |
|
Aquarius |
10 |
10.16 |
312.32 |
8.37 |
323.32 |
6.25 |
336.42 |
|
|
20 |
9.58 |
322.30 |
8.2 |
331.34 |
5.34 |
342.16 |
|
|
30 |
9.40 |
332.10 |
7.33 |
339.7 |
4.56 |
347.12 |
|
Pisces |
10 |
9.25 |
341.35 |
7.10 |
346.17 |
4.31 |
351.43 |
|
|
20 |
9.15 |
350.50 |
6.55 |
353.12 |
4.12 |
355.55 |
|
|
30 |
9.10 |
360.0 |
6.48 |
360.0 |
4.5 |
360.0 |
148:1 Thus (according to the Table inserted at p. 152), in the climate or latitude of Lower Ægypt, the times of ascension between the first point of Gemini and the first point of Sagittarius, diametrically opposite, are 205° 18', which, being divided by 15, give 13 hours 41 minutes and a fraction of equatorial time, as the length of the day of the first point of Gemini. And the same number of times of ascension, divided by 12, give 17° 6' and a fraction of the equator, as the length of the diurnal temporal hour. In the latitude of Southern Britain, the times of ascension between the same points as above mentioned are 236° 2', which, divided by 15, give 15 hours 44 minutes and a fraction of equatorial time, as the length of the day of the first point of Gemini; and, if divided by 12, they produce 19° 40' and a fraction of the equator, as the length of the diurnal temporal hour.
148:2 Thus, the aggregate times of ascension, in a right sphere, of the first point of Gemini are S7° 44'; and, in the climate of Lower Ægypt, 45° 5': the sixth part of the difference between them is 2° 6' and a fraction, which, added to 15°, again makes the diurnal temporal hour of the first point of Gemini equal to 17° 6' and a fraction of the equator. In the climate of Southern Britain, the aggregate times of ascension of the first point of Gemini are 29° 43': the sixth part of the difference between that sum and 57° 44' of right ascension is 4° 40' and a fraction, which, added to 15°, makes the diurnal temporal hour of the first point of Gemini, in South Britain, equal to 19° 40' and a fraction of the equator, as before shown.
149:1 For example,
|
Diurnal horary times of the first point of Gemini, in the latitude of Alexandria |
17° |
6' |
30" |
|
Number of temporal hours |
|
|
12 |
|
|
15)205 |
18 |
0 |
|
Diurnal equatorial hours of the first point of Gemini in the latitude of Alexandria |
13 |
41 |
12 |
|
Diurnal horary times of the first point of Gemini in the latitude of Southern Britain |
19° |
40' |
10" |
|
Number of temporal hours |
|
|
12 |
|
|
15)236 |
2 |
0 |
|
Diurnal equatorial hours of the first point of Gemini in the latitude of Southern Britain |
15 |
44 |
8 |
149:2 Let the first point of Gemini be on the meridian above the earth; the number of temporal hours since sunrise will then be 6, by which 17° 6' 30" are to multiplied. The product will be 102° 39': this, added to 45° 5', the aggregate number of the first point of Gemini in the latitude of Alexandria, will give 147° 44', which, in the ascensions of the climate in question, will correspond to the 3d degree of Virgo, and show that to be the degree ascending. In the latitude of Southern Britain the total number would still amount to the same, viz. 147° 44', but it would show 7° and about 30' of Virgo to be ascending.
150:1 Let the first point of Gemini be three temporal hours past the meridian; these hours reduced to degrees, in the latitude of Alexandria, will give 51° 19', which, added to the right ascension of the first point of Gemini, make 109° 3', showing the 18th degree of Cancer on the meridian. In the latitude of Southern Britain, these hours would produce 59°, which, added to the right ascension, would make 116° 44', and show the 25th degree of Cancer on the meridian.
150:2 Thus, in the latitude of Alexandria, when the first point of Gemini is three temporal hours past the meridian, the 16th degree of Libra will be on the ascendant, and the aggregate times of ascension of that degree in the said latitude are 109° 3': by subtracting 90 from this sum, the remainder will be 19° 3', the right ascension of the mid-heaven answering to the 18th degree of Cancer. In the latitude of Southern Britain, the 18th degree of Libra would be on the ascendant, of which degree the aggregate times of ascension in that latitude are 206° 44', from which, if 90 be subtracted, the remainder will be 116° 44', the right ascension of the mid-heaven answering to the 25th degree of Cancer. The converse of these operations seems too obvious to need explanation.
151:1 Alexandria.
151:2 Southern Britain.