The Surya Siddhanta · Chapter 5 of 8

#The gnomon, precession and the ascendant

Chapter III, "Of direction, place and time", is the most practical chapter of the Surya Siddhanta: it tells the astronomer how to find the directions, the latitude, the time — and the ascendant.

#The cardinal directions (III.1–6)

On a stony surface, made water-level, or upon hard plaster, made level, there draw an even circle ... At its centre set up the gnomon, of twelve digits ... and where the extremity of its shadow touches the circle in the former and after parts of the day, there fixing two points upon the circle, and calling them the forenoon and afternoon points, draw midway between them, by means of a fish-figure, a north and south line (III.1–3).

The method is elegant: the shadow-tip touches the circle once in the morning and once in the afternoon at the same length; the line midway between the two points runs north–south. A "fish-figure" (two intersecting arcs) bisects it; the perpendicular gives east–west (III.4). The east–west line is the prime vertical and equinoctial line (III.6).

#The shadow triangle (III.7–8)

The square root of the sum of the squares of the gnomon and shadow is the hypotenuse; if from the square of the latter the square of the gnomon be subtracted, the square root of the remainder is the shadow (III.8).

The gnomon (12 digits), its shadow and the hypotenuse form a right triangle from which the Sun's altitude follows.

#Precession (III.9–12)

In an Age, the circle of the asterisms falls back eastward thirty score of revolutions. Of the result obtained after multiplying the sum of days by this number, and dividing by the number of natural days in an Age, take the part which determines the sine, multiply it by three, and divide by ten; thus are found the degrees called those of the precession (ayana) (III.9–10).

This is the Surya Siddhanta's account of precession (Chapter 2 of the course): not a steady motion all the way round, but a swinging back and forth (libration) of up to 27° either side, 600 times in a great age. Modern astronomy knows precession as a steady circuit; the idea of a libration (also known in medieval Europe as "trepidation") was a different theory.

What matters most is the next instruction:

The circle, as thus corrected, accords with its observed place at the solstice and at either equinox; it has moved eastward, when the longitude of the Sun, as obtained by calculation, is less than that derived from the shadow, by the number of degrees of the difference; then, turning back, it has moved westward by the amount of difference, when the calculated longitude is greater (III.11–12).

That is: check the calculation against the gnomon's shadow at the solstice or equinox, and correct by the difference. Observation decides.

#Latitude (III.13–17)

The noon shadow of the gnomon at the equinox (the "equinoctial shadow") gives the sine and cosine of the latitude (III.13–14); on other days, the Sun's zenith distance at noon, combined with its declination, gives the latitude (III.15–17).

#Time from the shadow, and the shadow from time (III.18–41)

The middle of the chapter gives the trigonometry for finding the time of day from the gnomon's shadow, and the shadow from the time, at any declination and latitude, and for tracing the path of the shadow's tip through the day (III.18–42).

#The rising times of the signs (III.43–45)

The signs take different times to rise. At the equator (Lanka), the first three signs rise in:

(III.43–44; the three add up to 5,400 pranas, six hours.) At any other latitude, these are corrected by the ascensional difference: shortened for Aries–Gemini, lengthened for Cancer–Virgo, and reversed for the other six (III.44–45). This is why, at Indian latitudes, Pisces and Aries rise quickly and Virgo and Libra slowly.

#Finding the ascendant (III.46–51)

From the Sun's longitude and the time since sunrise, the text gives the method for the lagna, the point of the ecliptic on the eastern horizon:

1. Find how long the remaining part of the Sun's sign takes to rise (III.46).
2. From the time elapsed since sunrise (in respirations), subtract that, and then the rising times of the following signs one by one, as long as possible (III.47).
3. The remainder, multiplied by 30 and divided by the rising time of the sign not wholly risen, gives the degrees of that sign: "the result is the point of the ecliptic (lagna) which is then on the horizon" (III.48).

The midheaven is found the same way, from the Sun's hour-angle and the rising times at the equator (III.49). And if the lagna's longitude is less than the Sun's, it is the latter part of the night; if greater, day (III.51).

This is the method by which every horoscope in classical India was cast.

#Summary

• The cardinal directions from the equal-length shadow points on a circle (III.1–6); the shadow triangle (III.8).
• Precession as a libration of up to 27°, to be corrected by observing the Sun with the gnomon at the solstices and equinoxes (III.9–12).
• Latitude from the noon shadow (III.13–17); time from shadow and shadow from time (III.18–42).
• Rising times of the signs: 1,670, 1,795 and 1,935 pranas at the equator, corrected for latitude (III.43–45).
• The ascendant from the Sun's longitude and the time since sunrise (III.46–51).

#Exercises

1. How does the gnomon give the north–south line?
2. How does the Surya Siddhanta describe precession, and how does it tell the astronomer to correct it?
3. A gnomon of 12 digits casts a shadow of 5 digits. What is the hypotenuse?
4. Why do Aries and Pisces rise faster than Virgo and Libra in India?
5. Outline the steps for finding the ascendant.

Answers: 1. The shadow-tip touches a circle round the gnomon at two points, morning and afternoon; the line midway between them is north–south (III.1–3). 2. As a swing back and forth of up to 27°, 600 times in an age; by comparing the calculated Sun with the gnomon's shadow at a solstice or equinox and correcting by the difference (III.9–12). 3. √(144 + 25) = 13. 4. Because their equatorial rising times are shortened by the ascensional difference at northern latitudes, while those of Virgo and Libra are lengthened (III.44–45). 5. Find the rising time of the rest of the Sun's sign; subtract it and the following signs' rising times from the time since sunrise; turn the remainder into degrees of the next sign (III.46–48).