The Universal System of Astronomy

Siddhānta Sārvabhauma Page 7 7

Verse 41–44

The portion of the arc known as the "sky-number" khisaṃ-jñā: likely a specific numerical parameter used in the calculation of the sine should be calculated using the radius trijyā: the sine of a 90-degree arc, used as the standard radius in Indian trigonometry. From this, the sine of the solar degrees is determined. 41

By utilizing the cord rajju: the suspension string of the instrument, the board instrument phalaka-yantra, and the latitude akṣa, one calculates the "power of the gaze" dṛg-balaya: the vertical circle or the force of the observed altitude. 42 At the moment of sunrise, the light originates at the center; as time passes, the shadow moves accordingly across the earth. Thus, the tip of the shadow always lies in the direction opposite to the Sun. 43

The interval between the shadow's tip and the solar mark on the circle indicates the degrees of solar altitude above the horizon kṣitijonnatāṃśāḥ. Their sine is the gnomon's sine śaṅku-jyā, and the "final sine" antyā: a technical term for the versed sine of the hour angle multiplied by the day-radius is measured by the length of the gnomon-staff yaṣṭi. 44

Verse 45–48

If the gnomon-staff is twelve digits dvādaśāṅgula: the standard measure for a gnomon in Indian astronomy, roughly 9 inches in height, then what is the hypotenuse of the shadow? In the same way, the desired "final sine" is produced from the tip of the shadow. 45

The straight line drawn from the degree-mark on the board to the length-line dairghya-rekhā: the vertical axis on the instrument board is established through calculation. Here, the expert should place the plate paṭṭikā according to the mathematical demonstration upapatti. 46

From the base of the plate, one should measure a number of digits equal to the Sun's position; the point where this falls on the circle is the shadow-tip. From there, one should mark a sign up to the length-line; that distance is exactly the "final sine" antyā. 47

At noon, the observed "final sine" is calculated. By subtracting the remaining portion, the versed sine utkrama-jyā is found. According to the celestial hemisphere gola, the "ascensional difference" carajyā: the difference between the local and equatorial rising times is added or subtracted to find the midday "final sine." 48

Verse 49–53

Because of the latitude, the midday sine and the ascensional difference are used in sequence. There, on a specific part of the line, the "midday final sine" is clearly seen. 49 Below this, the line measured by the desired "final sine" becomes the versed sine. Within that circle, the sine is produced; from its tip, the elapsed units of time nāḍyāḥ: units of 24 minutes are determined. 50

Therefore, with ease, the "final sine" is marked on the line via the gnomon-point. Following the order of the celestial sphere as previously taught, the line situated at the end of the length-line becomes the "full sine" pūrṇā-jīvā: the standard chord or sine. 51

In this way, the measurement of the width and length of the instrument is perfectly achieved through the power of the "final sine" antyā marked on the plate. 52

Thus, the "Solar Board" Phalakārka: another name for the Phalaka-yantra when used for solar observations has been described by me in detail, following the teachings of Bhāskara original: "Bhāskara-uktataḥ"; referring to Bhāskara II (1114–1185 CE), author of the Siddhānta Śiromaṇi. This instrument should be used to calculate the time elapsed in the night and other such matters. 53

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