Primary Directions: the story of calculations (vol. 1)

I wrote two posts on story of primary directions. It was couple years ago, and now these posts are buried deeply in the archives of Angelicus Merlin Group. Therefore I decided to repeat that stuff and perhaps to develop the subject.

First of all, because primary directions are based on daily rotation of celestial sphere, we must have a mathematical model of this rotation to calculate directions. The first model of this rotation was made in Babylon, but astrologers of Hellenistic period mostly call it “according to Egyptians”.

The complete description of this system we can find in the text of Antiochus of Athens (as it was quoted by another astrologer, Rhetorius the Egyptian). See Catalogus Codicum Astrologorum Graecorum (aka CCAG), I, p.163.

How One Calculates the Increment of the Signs And the Hours of the Climes.

For example, in the third clime, when the greatest of the times, i.e. from Cancer to Sagittarius, become 210 years, and the least time of it is from Capricorn to Gemini, 150 years. These [years], then, if you divide them by 15, they are equinoctial horary times, [and] there is the greatest day of 14 equinoctial hours, the least day of 10 equinoctial hours. We shall take then the sixth pf each of the greatest and least times: they become then of the greatest time, of the 210, the sixth is 35; of the least, [of the 150], 25. The difference of these is 10; of this, the third is 3 1/3, which is the increment of each sign. By this same method you will know [the increment] in the other climes. (The trasnslation by James H. Holden)

It is not so easy to understand, isn’t it? But it is easy enough, in fact. We will build this system step by step to understand what it is and how it was made. But first I would like to note one detail: the usage of Greek word ´έτος – a year (210 years, 150 years…). Hellenistic authors often use years for degrees, when they talk about times of ascension in primary directions, because the equation 1 degree of ascension = 1 year of life was quite common. Therefore we must understand these 210 years and 150 years as 210 degrees and 150 degrees.

Now we are going to reinvent this model. We have a rotating celestial sphere, and we must make a mathematical model, which describes this rotation. Fortunately, we have no interest in the whole sphere, but we are mostly interested in Zodiacal belt, because planets and lights are there. Therefore, our aim is a model of the rotation of Zodiacal belt, i.e. of Zodiacal signs. Fortunately, Babylonian and Hellenistic astrologers made the problem even easier. They had a lot of problems with calculation and usage of Zodiacal latitudes, therefore they just simplified their life and used not the planets, but their projections on the ecliptic. So we have to describe the rotation of the ecliptic, i.e. to describe how the parts of ecliptic (belonging to every Zodiacal sign) ascend. To say more easy, we have to find the times of ascensions of these parts (for shortness, hereinafter I will call these parts ’signs’).

We don’t need tables, developed instruments, or trigonometry, neither plane nor spherical. The only thing we need is to know the duration of the longest day of a year (i.e. the day of summer solstice) in Alexandria, because we will make this model for Alexandria (as they say, for the third clime).

The duration of the day of summer solstice in Alexandria is about 14 hours. The Sun is in 0 of Cancer at this day. The Sun rises when the Ascandant is in this point, then the Ascendant goes thru Leo, Virgo… etc. upto Sagittarius, and when it comes to the end of Sagittarius and enters into Capricorn, the Sun sets. I.e. all signs of long ascension (Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius) ascend during this day, i.e during 14 hours. 210 degrees of equator are passed thru the Meridian during this time (210 = 14 x 15, because 15 degrees of equator passed thru the Meridian per one hour). Hellenistic astrologers were expressing an ascensional time in these degrees, i.e. the time of ascension of the signs of long ascension is 210 degrees. Thence, the time of ascension of the signs of short ascension is 150 degrees (i.e. 360° – 210°).

To understand the future explanation better, you can draw the Zodiacal circle divided into 12 signs and follow the explanation looking on this circle.
Now, the ascensional time of one half of the circle, of Cancer, Leo, Virgo, Libra, Scorpio, and Sagittarius, i.e. of signs of long ascension, is 210°00′. The ascensional time of the second half of the circle, of Capricorn, Aquarius, Pisces, Aries, Taurus, and Gemini, i.e. of signs of short ascension, is 150°00′.

The celestial sphere rotates evenly, therefore, if we start from the border between signs of long and short ascensions (i.e. from the beginning of Capricorn), the time of ascension of a sign must first diminish to some minimal value, then incrase upto transfer to the signs of long ascension. In the signs of long ascension, the time of ascension of a sign increases upto some maximal value, then diminish upto transfer to the signs of short ascension. Therefore, the extremal values of ascensional time must be in the middles of zones of signs of long and short ascensions. Hence, Babilonian astrologers gave the minimal time of ascension to Pisces and Aries (the signs placed in the middle of the zone of short ascension), and the maximal time of ascension to Virgo and Libra (the signs placed in the middle of the zone of long ascension).
Then they gave the middle times among signs of short and long ascension to the signs alongside to those. I.e. Aquarius (alongside to Pisces) and Taurus (alongside to Aries) got the middle time of ascension among the signs of short ascension. Leo (alongside to Virgo) and Scorpio (alongside to Libra) got the middle time of ascension among the signs of long ascension.
The rest signs became, as it were, transitional. I.e. Capricorn and Gemini are transitional from the signs of short ascension to the signs of long ascension. Cancer and Sagittarius are transitional from the signs of long ascension to the signs of short ascension.

Now we have such characteristics of the Zodiacal signs in relation to their times of ascension:
Aries – extremal of short ascension.
Taurus – middle of short ascension.
Gemini – transitional of short ascension.
Cancer – transitional of long ascension.
Leo – middle of long ascension.
Virgo – extremal of long ascension.
Libra – extremal of long ascension.
Scorpio – middle of long ascension.
Sagittarius – transitional of long ascension.
Capricorn – transitional of short ascension.
Aquarius – middle of short ascension.
Pisces – extremal of short ascension.

Well, now we have to find all these values. First of all, we find the values of the mean times of short and long ascensions. We just divide times of ascensions of the signs of long ascension (210°00′, as you can see above) and of the signes of short ascension (150°00′) by 6 (because there are 6 signs of long ascension and 6 signs of short ascension):
210°00′ / 6 = 35°00′ – ascensional time of Scorpio and Leo (middle of long ascension).
150°00′ / 6 = 25°00′ – ascensional time of Aquarius and Taurus (middle of short ascension).

Now we find the difference between the mean long ascension and mean short ascension:
35°00′ – 25°00′ = 10°00′.

Now, because the celestial sphere rotates evenly, this difference must be distributed among signs uniformly. I.e. this 10 degrees must be uniformly distributed among the signs placed between, for example, Scorpio and Aquarius (middle signs of long and short ascension). As we can see, we have 3 transitions from sign to sign in this zone: from Scorpio to Sagittarius (1), from Sagittarius to Capricorn (2), from Capricorn to Aquarius (3). Therefore we divide 10 degres by 3:
10°00′ / 3 = 3°20′.
I.e. we have got the difference between ascensional times of two adjacent signs, and therefore we can find the ascensional time of every sign.

The middle of short ascension (as we marked it above), has a mean time of short ascension – 25°00′.
The extremal of short ascension, its ascensional time must be shoter than the mean one. And because the difference between adjacent signs is 3°20′, as we found, the ascensional time of the extremal is
25°00′ – 3°20′ = 21°40′.
The transitional of short ascension, its ascensional time must be longer than the mean one. Therefore, the ascensional time of the transitional is
25°00′ + 3°20′ = 28°20′.

We have found the ascensional times of middle, extremal and transitional signs of short ascension.
I.e. Aries and Pisces (extremal of short ascension) – 21°40′.
Taurus and Aquarius (middle of short ascension) – 25°00′.
Gemini and Capricorn (transitional of short ascension) – 28°20′.

We can do the analogous procedure for the signs of long ascension. And then we will have:
Virgo and Libra (extremal of long ascension) – 38°20′.
Leo and Scorpio (middle of long ascension) – 35°00′.
Cancer and Sagittarius (transitional of long ascension) – 31°40′.

Thus we have got the Babylonian system of ascensions, which was in use among astrologers of Hellenistic epoch. Usually they call this system “according to Egyptians”. But now sometimes this system also is called “arithmetical”, because the ascensional times of the signs can be described with an arithmetical progression (in our case, it is 6.5 – 7.5 – 8.5 – 9.5 – 10.5 – 11.5 (3°20′)) and because all calculations are made with four rules of arithmetic.

This system is absolutely wrong from mathematical and astronomical point of view. It is easy to see, especially for hight geographical latitudes. But even for Alexandria:

Aries and Pisces
this system – 21°40′
in fact – 20°57′

Taurus and Aquarius
this system – 25°00′
in fact – 24°13′

Gemini and Capricorn
this system – 28°20′
in fact – 29°50′

Cancer and Sagittarius
this system – 31°40′
in fact – 34°32′

Leo and Scorpio
this system – 35°00′
in fact – 35°36′

Virgo and Libra
this system – 38°20′
in fact – 34°52′

As you can see, the errors are not only in values, but in the dinamics of changing of ascensional time from sign to sign. As you can see, there is no constant difference of ascensional times of adjacent signs, and the most long ascending are Leo and Scorpio, not Virgo and Libra.

Babylonian-Egyptian-arithmetical model is wrong. The right model is based on trigonometry, it was known in Hellenistic epoch. But athough it was known even for some astrologers too, but most of them can’t use it, because trigonometry was too new and dark subject for them.

When Hipparhus (II c. BC) solved a problem with chord, and so set up trigonometry, he (according to Strabon) applied this to the sky, i.e. made trigonometrical system of ascensions and tables of ascensions. These tables (if they were) did not survive. Ptolemy in his Almagest gives trigonometrical system, too. More over, he wrote a special book, so called Handy Tables, with tables of ascensions according to trigonometrical system.

Many post-ptolemean Hellenistic astrologers were understanding, that Ptolemean system is much better than Babylonian, but they were not skilled and educated enough to use it. Most of them were common people with more or less common level of education, and trigonometry in that time was a toy for great scholars.

For example, Paulus of Alexandria says in the introduction to his book, that ascensions of Ptolemy are much better than “according to Egiptians”, and Ptolemean ascensions should be used. But then, over all his book, he never uses this Ptolemean ascensions, he always uses Babylonian system. The same is even in Rhetorius, though he writes in 400 years after Ptolemy. Trigonometry becomes standard astrological tool only in Arabian epoch.

Next time we consider Hellenistic method of primary directions.

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