Primary Directions: the story of calculations (vol. 2)

I examined the system of ascensions of the Zodiacal signs used by Hellenistic astrologers. You can find it in the previous post of this line. This time we consider Hellenistic method of primary directions. Let’s consider an example of calculation of primary directions from the text of Paulus Alexandrinus.

Paulus gives this example in the chapter about crises. There are Sun in 15 Scorpio, Mars in 23 Leo, and Moon in 6 Sagittarius. His calculations are for Alexandria, i.e. for the third clime, therefore we can use the ascensional times from the previous post. Paulus calculates two directions – Sun to the square of Mars and Sun to the body of the Moon.

1) Sun to the square of Mars.
Mars is in 23 Leo, its square is in 23 Scorpio. Therefore Paulus directs Sun to 23 Scorpio. He calculates the ascensional time of the arc from 15 Scorpio (the place of the Sun) to 23 Scorpio (the place of the square of Mars). It is the arc of direction. Paulus does it in the following way:
First of all, he divides the ascensional time of Scorpio by 30. As we found in the previous post, the ascensional time of Scorpio is 35°00′. Therefore:
35°00′ / 30 = 1°10′
It is the ascensional time of one degree of Scorpio. But there are 8 such degrees from 15 Scorpio to 23 Scorpio. Therefore Paulus multiplies the ascensional time of one degree (i.e. 1°10′) by 8 to get the arc of direction:
1°10′ x 8 = 9°20′
So, the arc of direction is 9°20′. Then he converts this arc of direction into years of life (1 degree = 1 year), and gets 9 years and 4 months.

2) Sun to the body of the Moon.
The Moon is in 6 Sagittarius, and Paulus directs the Sun to this degree, i.e. from 15 Scorpio to 6 Sagittarius.
First of all, he divides this arc in two parts – from 15 Scorpio to the end of Scorpio (i.e. 15 degrees in Scorpio), and from the beginning of Sagittarius to 6 Sagittarius (i.e. 6 degrees in Sagittarius).
He calculates the ascensional time of the part in Scorpio. The ascensional time of one degree of Scorpio is 1°10′, as we found just before. But there are 15 degrees in Scorpio, therefore he multiplies it by 15:
1°10′ x 15 = 17°30′
After this, he calculates the ascensional time of the part in Sagittarius. He calculates the ascensional time of one degree of Sagittarius. The ascensional time of Sagittarius is 31°40′, as we found in the previous post. Paulus divides this by 30 to find the ascensional time of one degree of Sagittarius:
31°40′ / 30 = 1°03′20”
There are 6 degrees in Sagittarius, therefore:
1°03′20” x 6 = 6°20′
Then he sums up this two ascensional times to get the arc of direction:
17°30′ + 6°20′ = 23°50′
The arc of direction is 23°50′. He converts this into time:
23°50′ = 23 years and 10 months.

As you can see, the algorithm of calculation is unpretentious. One of the most interested and important point in this algorithm is when they divide an ascensional time of a sign by 30 to get an ascensional time of one degree of that sign. This mathematical operation is valid only in one case, viz. if a speed of ascension of a sign is constant. I.e. according to this algorithm, a sign ascends with some constant speed, then there is a sudden change of this speed on the sign border, then the next sign ascends with its constant speed, then again a sudden change etc. I.e. the celestial sphere rotates unevenly. And it is the obvious inadequacy of the model.

More over, if in some place on the Earth at some moment an Ascendant is on a sign border, and there must be that sudden change of speed of ascension; at
the same time, but in some other place on the Earth, an Ascendant is somewhere in the middle of a sign, and there must be nothing change of speed. So, we can’t have one celestial sphere with one rotation according to this Hellenistic model.

Most Hellenistic astrologers directed this way. The most important exception is Ptolemy. Ptolemean method of directions is much more advanced in respect
to mathematics and astronomy. But his method wasn’t spread among Hellenistic astrologers, and becomes a standard way of directing only in Arabian epoch.
Actually, common Arabian method of directions is Ptolemean method described in Tetrabiblos III,10 and in Almagest II,8,9.

Now we should note some important traits of Hellenistic directions:

• they are based on wrong ascensional times (on Babylonian arithmetical system of ascensions);
• they use wrong model of rotation of celestial sphere;
• not planets are directed, but degrees of ecliptic, i.e. ecliptical projections of planets;
• these degrees of ecliptic are directed to other ecliptical degrees, not to planets (if we direct to bodily conjunction), and aspects of planets are on the ecliptic (to direct to some aspect is to direct to respective ecliptical degree).

The previous post on this subject:

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

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