Primary Directions: the story of calculations (vol. 4)

In this episode of the serial I’m going to consider medieval directions. Or to be more precise, I begin to consider medieval directions.

The mainstream of medieval primary directions is based on Ptolemy. In fact, it is nothing but the variant of calculations described by Ptolemy in Tetrabiblos. Let’s consider this medieval algorithm as it is described by Alcabitius (the English translation is quoted from Al-Quabisi: The Introduction to Astrology; Warburg Institute, 2004).

Alcabitius directs an Ascendant and points placed on it according to oblique ascension, Meridian and points on it according to right ascension, etc. exactly as Ptolemy:

If you want this and the indicator which you want to move to a degree of the ecliptic is in the ascendant, you subtract the rising-times of the degree of that which you want to move from the rising-times of the degree to which you want to move it. What remains are the degrees of the motion (tasyīr).

If the indicator is in the degree of the seventh [place] (i.e. descendant), you subtract the rising-times of the degree diametrically opposite to that in which the indicator is from the rising-times in oblique ascension of the degree diametrically opposite to that to which you want to move [it].

If I the indicator is in the midheaven or in the degree of the cardine [under] the earth, you subtract the rising-times of the degree of the indicator from the rising-times in right ascension of the degree to which you want to move [it]. What remains are the degrees of the motion (tasyīr).

The indicator is significator, and the degree of the ecliptic to which you want to move is promittor.

Then Alcabitius proceeds to the directions of a significator placed not on the horizon nor on the meridian.

If the indicator which you want to move is in  other than these four cardines, you look at its distance from one of the two cardines, i.e. the midheaven and the cardine [under] the earth. This [means] that you look and if the indicator is between the ascendant and the midheaven, you subtract the rising-times in right ascension of the degree of the midheaven from the rising-times of the degree of the indicator, and if it is between the seventh [place] and the midheaven, you subtract the rising-times in right ascension of the indicator from the rising-times of the degree of the midheaven. Whatever remains, in either of these two [cases], you divide it by the degrees of the hours of the day of the degree in which the indicator is. What results are the hours of the distance from the cardine.

If [the indicator] is between the ascendant and the cardine [under] the earth, you subtract the rising-times in right ascension of the degree of the indicator from the rising-times of the degree of the cardine [under] the earth. If it is between the cardine [under] the earth and the seventh [place], you subtract the rising-times in right ascension of the degree of the cardine [under] the earth from the rising-times of the degree of the indicator. Whatever remains, in either of these two [cases], you divide it by the time-degrees of the hours of the day of the degree diametrically opposite to that of the indicator. What results are the hours of the distance from the cardine.

I.e. first of all, he finds the hour distance of the significator. We can easy do it with one of my tools – Hour Distance Calculator.

When you know the hours of the distance from the cardine and you want to move the indicator to a point on the ecliptic and the indicator is in the eastern hemisphere, which is from midheaven to the cardine [under] the earth in what follows the ascendant, subtract the rising-times in right ascension of the degree in which the indicator which you want to move is, from the rising-times in right ascension of the degree to which you want to move. What remains is the indicator in right ascension. Keep it [in mind]. Then subtract the rising-times in oblique ascension of the degree in which the indicator is from the rising-times in oblique ascension of the degree to which you want to move it. What remains is the indicator in oblique ascension. Then you take the excess of what is between the indicator in right ascension and the indicator in oblique ascension. Then you take its sixth [part] and multiply it by the hours of the distance from the cardine. What results is the equation. If the indicator in right ascension is less than the indicator in oblique ascension, you add the equation to the indicator in right ascension. If the indicator in right ascension is more, you subtract the equation from it. What remains is the degrees of the motion (tasyīr).

And if a significator is in the western hemisphere, we operate the same way with the opposite points (i.e. opposite to significator and promittor) – medieval astrologers still don’t use oblique descensions.

This is the core of the algorithm, the gist of the medieval way of directing. Let’s follow the  reasoning of medieval astrologers to understand what and why they do.

They knew that the points of horizon must be directed with oblique ascernsions (or discensions), the point of meridian must be directed with right ascensions. But how to direct the points between horizon and meridian? with what ascensions? Obviously, not oblique, nor right. But perhaps with some ascensions between oblique and right. This ascensions must be less oblique than oblique and less right than right. Or, in other words, it is more right than oblique ascensions, and more oblique than right ascensions. It is something between them, some mixing of them… Those they came to the idea of another kind of ascensions – mixed ascensions.
As Haly (Abu’l Hasan Ali ibn Ridwan Al-Misri) says in his comments to Tetrabiblos:

And therefore direction in this case must be done with ascensons mixed of ascensions of the place, where [the native] was born, and of ascensions of the right circle.

Apparently, the nearer a point to the horizon the more oblique its ascension, the nearer to the meridian the more right its ascension. But how to represent it mathematically? how to calculate it? Fortunately, they already had the solution. They found it in Tetrabiblos.
Ptolemy analyses his example of calculation considered in the previous post of this series (Robbins’s translation):

But it made 46 times by the same procedure when the prorogative place was rising, 58 when it was in mid-heaven, and 70 when it was setting.

I.e. if the significator (0 of Aries) would be on the Ascendant, then we calculate it with oblique ascension, and the arc of direction is 46 degrees. If the significator would be on the Mid-Heaven, we direct it with right ascension, and the arc of direction is 58 degrees. If the significator would be on the Descendant, we direct it with oblique descension, and the arc of direction in this case is 70 degrees. But the significator is between Mid-Heaven and the Descendant, and the arc of direction is 64 degrees, i.e. between the arcs of last two cases (58 degrees and 70 degrees).

Hence the number of equinoctial times at the position between mid-heaven and the occident differs from each of the others. For it is 64, and the difference is proportional to the excess of three hours, since this was 12 equinoctial times in the case of the other quadrants at the centres, but 6 equinoctial times in the case of the distance of three hours.

But the significator is distant from the Mid-Heaven in 3 hours. There are 6 hours between the Mid-Heaven and the Descendant, i.e. the significator is in the middle of this distance (3 is a half of 6). From the other hand, the difference of the arcs of direction in the cases when the significator would be on the Mid-Heaven and on the Descendant is:
70 – 58 = 12
The one half of 12 is 6.
And if we add this 6 to 58, we get the actual arc of direction, i.e. 64 degrees.
There is a proportion, as we can see! This is the solution. Ptolemy proceeds:

And inasmuch as in all cases approximately the same proportion is observed, it will be possible to use the method in this simpler way. For again, when the precedent degree is at rising, we shall employ the ascensions up to the subsequent; if it is at mid-heaven, the degrees on the right sphere; and if it is setting, the descensions. But when it is between these points, for example, at the aforesaid interval from Aries, we shall take first the equinoctial times corresponding to each of the surrounding angles, and we shall find, since the beginning of Aries was assumed to be beyond the mid‑heaven above the earth, between mid-heaven and the occident, that the corresponding equinoctial times up to the first of Gemini from mid-heaven are 58 and from the occident 70. Next let us ascertain, as was set forth above, how many ordinary hours the precedent section is removed from either of the angles, and whatever fraction they may be of the six ordinary hours of the quadrant, that fraction of the difference between both sums we shall add to or subtract from the angle with which comparison is made. For example, since the difference between the above mentioned 70 and 58 is 12 times, and it was assumed that the precedent place was removed by an equal number of ordinary hours, three, from each of the angles, which are one half of the six hours, then taking also one-half of the 12 equinoctial times and either adding them to the 58 or subtracting them from the 70, we shall find the result to be 64 times.

And this is exactly the same that Alcabitius suggests to do.

Let’s calculate the same Ptolemaic example with Alcabitian algorithm. I would remind that the significator in that example is in 0 of Aries, the promittor is in 0 of Gemini, Mid-Heaven is in 17°30′ of Taurus. Gegraphical lattitude of the place is 31n12 (Alexandria), and the year is 150AD.

The significator is placed between the Mid-Heaven and the Descendant, i.e. in the western hemisphere. We can use oblique descensions in this case, but to be as medieval as possible, let’s follow Alcabitius and operate with the opposite points. I.e. 0 of Libra for the significator, and 0 of Sagittarius for the promittor.

First of all, we must find the oblique and right ascensions of 0 of Libra and 0 of Sagittarius. We can make it easily with Ascensions And Descensions Calculator.

0 of Libra
oblique ascension is 180°0′
right ascension is 180°0′

0 of Sagittarius
oblique ascension is 250°45′
right ascension is 237°46′

Now we must find the differences between their right and oblique ascensions to get the indicator in right ascension and the indicator in oblique ascension.

237°46′ – 180°0′ = 57°46′
250°45′ – 180°0′ = 70°45′

We must find the difference between these two indicators.

70°45′ – 57°46′ = 12°59′

Now we have to find what we must to add to the difference in right ascensions (i.e. to the indicator in right ascension) to get the difference in mixed ascensions, i.e. the arc of direction. We will do it by means of proportion. We calculated in the last post that the hour distance of the significator is 2.9989 (or we can calculate it again with Hour Distance Calculator). Therefore we must devide the difference between indicators in 6, and multiply the result by 2.9989.

12°59′ : 6 = 2°9′50”
2°9′50” x 2.9989 = 6°29′21”

We add it to the indicator in right ascension:
57°46′ + 6°29′21” = 64°15′21”

And when we calculated this direction in the last post according to the first algorithm of Ptolemy, we get the same 64°15′.

So we can conclude that the common medieval way to calculate primary directions is nothing but the second Ptolemaic algorithm from Tetrabiblos.

The previous posts on this subject:

Primary Directions: the story of calculations (vol. 1)
Primary Directions: the story of calculations (vol. 2)
Primary Directions: the story of calculations (vol. 3)

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