Primary Direction in Placidus System

As you know, primary directions refer to the applying aspect between two planets on the celestial sphere. The length of the direction arc is the number of degrees to the exact aspect, which determines the year when the event will occur.

This article will cover the method for calculating the direction arc in the Placidus house system. We will use a mundane position formula for the Placidus system that we derived earlier.

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Mundane Conjunction

We say that two objects are conjunct when they belong to the same S-curve. It means they have the same mundane positions or they have the same ratio of meridian distance to the diurnal/nocturnal semi-arc.

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 curve of the position of $P_1$, which is fixed.

The path the point $P_2$ travels until it meets the curve of position of the point $P_1$ is called a primary direction.

Direct and converse directions

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

The significator and its S-curve are the moving points in the converse or preceding direction. We take an arc toward a primary motion corresponding to the current year in question and then draw an S-curve of the position of a significator through its end. Whatever conjuncts this curve is the promittor.

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.

The 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 in the Placidus system.

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 given in this article. After finding that point's elliptical longitude and latitude, we convert them to the equatorial right ascension and declination, as described here. It will be the coordinates of the promittor (a point P in the figure, which is a quadrature of the planet)
2. Then, we calculate the significator's ratio of meridian distance to the semi-arc according to the left part of the equation (1).
3. Finally, we find the coordinates of the end-point of the directional arc with the formula (2). 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 meridian distance ratio to the semi-arc.
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.