Elucidation of the Five Treatises

Page 75

The sum of the latitude and the declination becomes the zenith distance—knowing this forward movement, its placement is established. Let it be assumed that:
Line SV is the Equator original: viṣuvadrekhā,
Line SR is the Ecliptic original: krāntivṛttarekhā.
The angle ∠VSR = 24° The traditional Indian value for the maximum obliquity of the ecliptic. Here, the perpendicular line upon the Equator is the Ecliptic, and the sine of declination is established. Then, in the right-angled triangle SVR, by trigonometry:
SR = VR × sine 60° / sine ∠VSR
Therefore, the knowledge of the sine of the base the side of the triangle occurs automatically when the position of the Sun is known. From the degrees of that arc, the Sun's longitudinal degrees are found, and from there, the knowledge of the direction based on the quadrant is easily understood.

12-13 Now, he describes the knowledge of the lunar day Tithi: a lunar day, defined as the time it takes for the longitudinal angle between the Sun and Moon to increase by 12 degrees through observation. "From the center, by the sight-staff," etc.
From the observation of two sight-staffs placed at the center, each equal to the radius, the degrees of interval that appear between the Sun and the Moon are the "Sun-Moon degrees." One-twelfth of that is the true Tithi known from the calculation. For example, from the observation of the connection of the two sight-staffs at the center, the degrees of the arc between the Sun (R) and Moon (C) are known. As stated before, one-twelfth of this position is the Tithi.
In this way, from this interval, another Tithi should be known on the second day.
Thus, in these degrees of interval between the Sun and Moon, by using a cutting instrument a mathematical or physical divider and adding the Sun (Bhāskara) previously known by declination observation, the Moon (Niśākara) is found at that time using the same instrument.
The proof here is very simple: if the degrees of interval are added to the Sun, the Moon is found; if subtracted from the Moon, the Sun is found.

14-16 Now he explains the knowledge of the shadow-rotation line. "From the center, the shadow of the gnomon," etc.
From the center, at the middle of the directions, the calculator should mark the shadow of the gnomon Śaṅku: a vertical pillar or gnomon used to measure time and direction by its shadow three times in a day. With those three points of the shadow's tip, one should produce two "fishes" Matsya: a fish-shaped geometric figure formed by the intersection of two circles, used to find a perpendicular bisector. From the intersection where the two lines emerging from the mouths of those "fishes" meet, a circle is drawn touching the three points. That circle is the path the shadow follows on that day without leaving the circumference. A line from the gnomon's base to the center of the circle is the North-South line.

Commentary

The northern distance or gap between that circle and the base of the gnomon is the shadow at midday noon on that day. To produce the circle, it is assumed:
The three shadows of the gnomon standing in the center of the directions on one day are named 1st, 2nd, and 3rd. From points 1 and 2, a "fish" is made; from points 2 and 3, another "fish" is made. The intersection of the ropes from the tails and mouths of these "fishes" is at point Y. The circle drawn through the three points from that center is the path the shadow travels on that day. This is the ancient saying. In reality, if the Sun's diurnal circle is assumed to be fixed for one day, the form of the shadow-rotation line is circular only at the North Pole Meru and not elsewhere In other latitudes, the path is actually a conic section, usually a hyperbola.