Crown of Treatises: Chapter on Mathematics

Proof / Rationale

When the Nonagesimal the point of the ecliptic 90 degrees behind the ascendant is at the zenith, the ecliptic takes the form of the vertical circle; therefore, the mean parallax is itself the apparent parallax. Now, in the situation where the Nonagesimal is inclined from the zenith, the calculation of parallax is performed by another method. The vertical circle of the Nonagesimal is called the Drkshepa-mandala the vertical circle of the nonagesimal and its sine is called the Drkshepa the ecliptic zenith distance.

In the diagram described below, the north-south difference along the pole-star string the secondary to the ecliptic from the Moon's position to the disk is the minutes of latitude (nati), and from there to the Sun's disk along the vertical circle, the mean east-west difference is the minutes of parallax (lambana). The square root of the difference of their squares applying the Pythagorean theorem is the apparent parallax in the east-west direction on the original ecliptic; this is the opinion of the ancients. In reality, this parallax occurs on a small circle of the ecliptic produced by the cosine of the latitude. By the proportion: "If this is the parallax in the cosine of the latitude, what would it be in the radius of the circle?", the apparent parallax is converted to the ecliptic; this is the opinion of the moderns. Here, the Teacher has written extensively in the original text regarding the means of achieving the "vertical latitude" (drng-nati). That vertical latitude is equal to the square root of the difference between the squares of the Drkshepa and the sine of the Sun's zenith distance. However, it is not easily seen in the diagram as it stands. The sine of the difference between the Nonagesimal and the Sun, converted into the radius of the Nonagesimal's gnomon, takes the form of the vertical latitude. This should be understood through measurement. The Drkshepa is the base, the sine of the Sun's zenith distance is the hypotenuse, and the vertical latitude is the upright side.

The diagram illustrates the intersection of the Ecliptic (Kranti-vrtta), the Vertical Circle (Drng-mandala), and the secondary circles used to calculate the displacement of a celestial body due to the observer's position on Earth.

Planetary Mathematics (Grahaganita): The branch of Indian astronomy dealing with the calculation of planetary positions.
Nonagesimal (Vitribha Lagna): The point on the ecliptic 90 degrees behind the rising point; the highest point of the ecliptic above the horizon.
Parallax (Lambana): The apparent displacement of a celestial body when viewed from different positions.
Latitude (Nati): In this context, the celestial latitude or the north-south component of parallax.
Ecliptic Zenith Distance (Drkshepa): The sine of the arc between the zenith and the highest point of the ecliptic.