Primary Directions: the story of calculations (vol. 3)

I considered the Hellenistic method of primary directions in two previous posts Primary Directions: the story of calculations (vol.1 and vol.2). Most Hellenistic astrologers directed this way. But there is a very important exception. This exception is Ptolemy.

Ptolemy was an eminent scholar of his time, the level of his knowledges and abilities was much higher than the level of a common astrologer of that time. Ptolemaic method of directions is much more advanced with respect to mathematics and astronomy, than the Hellenistic method I considered before, but his method wasn’t spread among Hellenistic astrologers.

Let’s consider the Ptolemaic method of primary directions as it is described in Tetrabiblos III,10.

First of all, Ptolemy uses quite correct model of the celestial shpere rotation. It makes him the exception among other astrologers of his epoch.
Ptolemy directs an Ascendant and points placed on it according to oblique ascension; Meridian and points on it according to right ascension; and Descendant and points on it according to oblique descension. As he says:

Whenever the prorogative and preceding place is actually on the eastern horizon, we should take the times of ascension of the degrees up to the meeting-place; for after this number of equinoctial periods the destructive planet comes to the place of the prorogator, that is, to the eastern horizon. But when it is actually at the mid-heaven, we should take the ascensions on the right sphere in which the segment in each case passes mid-heaven; and when it is on the western horizon, the number in which each of the degrees of the interval descends, that is, the number in which those directly opposite them ascend. (Tetrabiblos, III.10. Robbins’s translation)

And he uses quite correct algorithm for directions of other points with a view to the true rotation of the celestial sphere.

The primary directions are based on the diurnal rotation of the celestial sphere. This is the rotation revolving on celestial axis. The points of the celestial sphere circumscribe circles as a result of  this rotation. These circles are called diurnal circles or diurnal paths of the points. And this diurnal paths never cross one another. We can see two points of the celestial sphere (point A and point B) and their diurnal paths in the following picture. These paths don’t cross each other.

Therefore we have a problem – what must we take as the meeting of these points, their conjunction? Obviously, we must assign some point (A’) on the diurnal path of one point (B) to represent another point (A). The point of the diurnal path of the point that has the same position relative to rising, culmination, setting, anti-culmination as the position of another point, this is the assigned point on the diurnal path of the first point. Look at the picture:

We find point А’ on the diurnal path of point B that corresponds to the position of point A relative to rising, culmination, setting, anti-culmination. When point B comes to point A’, it is the conjunction of points B and  A. The distance from B to A’ expressed in degrees of the celestial equator, i.e. in degrees of right ascension, is the arc of direction.

That is what we have to do. We have the significator (point А) and promittor (point B). We must find a point (point A’) on the diurnal path of the promittor that corresponds to the significator. And then we must find the distance between promittor and this point in right ascension. This distance is the arc of direction.

Let’s see how Ptolemy solves this problem. Ptolemy explains how to calculate primary directions and gives an example in Tetrabiblos III,10. He gives an example of such a latitude, where the duration of the most long day of a year is 14 hours. This is the lattitude of so called third clime. Alexandria, the city where Ptolemy lived and worked is placed in this clime. Therefore we will use the geographical latitude of Alexandria (31n12) for our calculations to check the algorithm of Ptolemy.

Ptolemy uses two point in his example – the beginning of Aries (the significator) and the beginning of Gemini (the promittor). His calculations show absolutelly clear that he uses points of the ecliptic, not the real bodies of the planets. The significator and the promittor are both points of the ecliptic. Therefore, Ptolemaic directions are obviously zodiacal, not mundane.
And the MC in this example is in 18th degree of Taurus.

Ptolemy describes his calculations in the following way:

ὑποκείσθω τοίνυν ἐπὶ μηδενὸς μὲν οὖσα τῶν κέντρων ἡ ἀρχὴ τοῦ Κριοῦ, ἀπέχουσα δέ, λόγου ἕνεκεν εἰς τὰ προηγούμενα τῆς μεσημβρίας καιρικὰς ὥρας τρεῖς, ἵνα μεσουρανῇ μὲν ἡ τοῦ Ταύρου μοῖρα ὀκτωκαιδεκάτη, ἀπέχῃ δὲ κατὰ τὴν πρώτην θέσιν ἡ τῶν Διδύμων ἀρχὴ τοῦ ὑπὲρ γῆν μεσουρανήματος εἰς τὰ ἑπόμενα χρόνους ἰσημερινοὺς ιγ× ἐὰν οὖν πάλιν τοὺς ιζ χρόνους ἐπὶ τὰς γ ὥρας πολλαπλασιάσωμεν, ἀφέξει μὲν καὶ κατὰ τὴν δευτέραν θέσιν ἡ τῶν Διδύμων ἀρχὴ τῆς μεσημβρίας εἰς τὰ προηγούμενα χρόνους να, τοὺς δὲ πάντας ποιήσει χρόνους ξδ.

Now let it be assumed that the beginning of Aries is not on any of the angles, but removed, for example, three ordinary hours from the meridian in the direction of the leading signs, so that the 18th degree of Taurus is at mid-heaven, and in its first position the beginning of Gemini is 13 equinoctial times removed from the mid-heaven above the earth in the order of the following signs. If, then, again we multiply 17 equinoctial times into the three hours, the beginning of Gemini will at its second position be distant from mid-heaven in the direction of the leading signs 51 equinoctial times, and it will make in all 64 times. (Tetrabiblos, III.10. Robbins’s translation)

17 is the diurnal temporal hour of 0° of Gemini (i.e. of the point B), as Ptolemy says above. We can verify it with my tools you can find in this site.
First off all, we need right ascensions and declinations of the mentioned points, i.e 0° of Aries, 0° of Gemini and 18th degree of Taurus. We can get it with The Convertor Of Coordinates.
Let’s input 150 in Year (i.e. the middle of II century AD, times of Ptolemy).
Ecliptical longitude 60°0′ (0° of Gemini).
Ecliptical latitude 0°0′ (because it is the point of the ecliptic).
And we get the right ascension 57°46′.
Declination is 20n21.

Ptolemy says that MC is in the 18th degree of Taurus, he doesn’t mentions the exact degrees and minutes. Let’s take 17°30′ of Taurus for our calculations.
We input the ecliptical longitude 17°30′.
All others remain the same.
We get the right ascension 44°59′.
Declination is 17n13.

We can use the Converter for 0° of Aries as well. But its right ascension and declination are well-known, they both are 0.

So, we have the right ascensions and the declinations of all three points.

Now we must turn to the Hour Distance Calculator to get the diurnal temporal hour of the 0° of Gemini and the hour distance of 0° of Aries.
We input the geographical latitude of 31n12 (i.e. the geographical latitude of Alexandria, as stated above).
Right ascension of MC 44°59′.
Both points A and B are above the horizon.
We input the rigth ascension of the point B 57°46′.
The declination of the point B 20n21.

And we get the diurnal temporal hour 17.1635°. Ptolemy gives 17°, it is close enough.

Now we input the right ascension of the point A 0°0′.
Its declination 0°0′.

And we get the hour distance 2.9989. Ptolemy gives 3, it is very close as well.

Ptolemy calculates distance from the point B to MC according to right ascension. Let’s do the same. We must subtract  rigth ascension of MC from the rigth ascension of the point B
57°46′ – 44°59′ = 12°47′
Ptolemy gives 13°, the difference is quite acceptable.

Then Ptolemy multiplies the temporal hour of the point B by the hour distance of the point A. And we do the same
17.1635 х 2.9989 = 51.47162015
If we convert it into degrees and minutes, we get 51°28′.
Ptolemy multiplied 17 by 3 and got 51°.

This multiplication is the most important part of the calculation. Ptolemy determines the point A’ with this operation. He finds the distance of the point A’ (the point on the diurnal path of the point B that corresponds to the point A) from the MC according to right ascension.

Now we can easily find the distance from B to A’ according to right ascension, i.e. the arc of direction. This distance is divided in two parts. The first part is the distance from B to MC. It is 12°47′ as we calculated above (13° according to Ptolemy). The second part is the distance from MC to A’. It is 51°28′ (51° according to Ptolemy). We must sum up these two parts to get the complete distance.
12°47′ + 51°28′ = 64°15′ (or 13° + 51° = 64° according to Ptolemy)
We have the arc of direction, and we can convert it into the age of life with the Age <-> Arc Converter.
We input the arc 64°15′ and get the age of 64 years 91 days (according to the key of Ptolemy 1° = 1 year).

Let’s note some important traits of Ptolemaic directions:

• they are based on quite correct model of the celestial shpere rotation;
• 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).

This Ptolemaic method of primary directions wasn’t spread among Hellenistic astrologers, and becomes a standard way of directing only in Arabian epoch. Though the medieval algorithm of calculations is different from the described in Tetrabiblos – new times, new tools.

The previous posts on this subject:

Primary Directions: the story of calculations (vol. 1)
Primary Directions: the story of calculations (vol. 2)

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