Primary Direction in Regiomontanus System

You already know that primary directions are a form of an applying aspect between two planets on the celestial sphere. The number of degrees to the exact aspect - the length of the direction arc - determines the year of life in which the event will occur.

This article will discuss the algorithm for calculating the directional arc in the Regiomontanus house system.

Mundane Conjunction

We say that two objects are conjunct when in the same circle of positions. It means they have the same mundane positions.

Primary Direction to Conjunction

We say that point $P_2$ moves to mundane conjunction with $P_1$ when $P_2$ moves with a rotating celestial sphere toward the circle of the position of $P_1$, which is fixed.

The path the pont $P_2$ travels till circle of position of the point $P_1$ is called a primary direction

Historical reference

The original idea of Regiomontanus was the following. First, he assumed the horizon of the position of the stationary planet as an actual horizon. He rotated the celestial sphere at a certain angle $\alpha$ along prime vertical so that the position horizon was parallel to the observer's gaze, as shown in the figure below.

He called the deviation of the pole from that new horizon the pole's altitude above the circle of position. And in this view, the primary direction was a simple difference of oblique ascensions of two planets.

As it is clear from figure 1 of the previous article, angle $\beta$ is $90° - alt$. We already evaluated

$$ \tan \beta = \frac{1}{\tan\phi \cos(OA_{ASC} - RA_M)} $$

where $RA_M$ - the right ascension of the mundane position of the planet.

Since pole's altitude above the circle of position is $alt = 90° - \beta$, and $OA_{ASC} = RAMC + 90°$

$$ \tan alt = \tan\phi \sin(RA_M - RAMC) $$

Direct and converse directions

In the direct or following direction, the promittor is a moving point, while the significator is fixed together with its circle of position.

The significator and its horizon are the moving points in the converse or preceding direction. We rotate the celestial sphere with the significator and its circle of position to the number of degrees corresponding to the current year of life. All the other planets and stars are fixed. Whatever conjuncts this rotated circle of position is the promittor (if any, for the chosen year).

Technically you can think of a converse direction as the reverse direction in time. It starts at a particular moment after birth. Then the promittor moves back to the birth date and the circle of the position of the significator.

In both following and preceding directions, we know how to get the coordinates of the mundane position for a given right ascension and declination of the point.

What we don't know is how to get the right ascension of the intersection of the directional arc on a given declination with the circle of the position of another planet.

Let's derive the formula, which defines the right ascension $RA(RA_M, D)$ as a function of declination $D$ of the directional arc and the mundane position of another planet $RA_M$.

RA for a Given Mundane Position

Lets expand $\tan RA_M$ in equation (1) of mundane position from a previous article and consider formulas for the sine of two angles.

We will get

Here

• $RA_M$ - mundane coordinates of the circle of position toward which the arc is directed.
• $D$ - declination of the promittor/significator, directed to the circle of position
• $\phi$ - geographical latitude of the observer
• $RA_p$ - the coordinate of the end-point of the directional arc.

Direction of the Significator to the Aspect of the Promittor

According to tradition, we say "direction of the significator to the body/aspect point of the promittor." But in fact, it is the promittor who is moved (directed) to a fixed significator in the direct direction.

From the theory of primary direction promittor can be a planet or its aspect. Promittor's aspects belong to the aspect circle.

Here is the most general algorithm for calculating a primary direction.

1. We take a planet whose aspect will be a promittor. It is a red point in the figure above.
2. We calculate its aspect along the circle of aspects (shown in orange) according to the formulas of aspects given in this article. After finding that point's ecliptical longitude and latitude, we convert them to the equatorial right ascension and declination. It will be the coordinates of the promittor (a point P in the figure, which is a quadrature of the planet)
3. Then, we calculate the significator's mundane position according to this equation. It is the blue point in the figure above.
4. Finally, we find the coordinates of the end-point of the directional arc with the formula (1) of that article. After that, we find the length of the arc by simple subtraction of the right ascension of the promittor from the right ascension of the end-point.

In the converse direction, we slightly change the sequence.

1. First, we calculate the equatorial coordinates of the promittor (which is the aspect of the planet)
2. Then, we calculate the promittor's mundane position.
3. Finally, we calculate the end-point of the directional arc from the significator to the promittor. And thus, we find the length of the directional arc in the converse direction.