ĀRKThe gnomon
A stick, and what it knows
Chapter 3 of the Sūrya-siddhānta asks you to level a patch of ground and stand a rod on it 12 aṅgulasAṅgula · a finger’s breadthA finger’s width — the unit the gnomon and its shadow are measured in. tall. From the shadow it throws, the chapter derives where you are standing, how long today will last, where on the horizon the Sun will come up, and which way is north. No instrument, no table, no sky to read — just the stick.
The rod below is that rod. It throws two shadows, because the chapter works from anobliquityObliquity · the tiltHow far the Earth’s axis leans from upright — the reason there are seasons. of 24° and the measured value is 23.4393°. Both are on the floor at the same time. Move the year to an equinox and they merge into one; move it to a solstice and they part.
The reasoning
- 01
Why the two shadows meet at an equinox
Not approximately — exactly. The Sun’s declination is the obliquity multiplied by the sine of its place on the ecliptic, and at an equinox that sine is zero. Zero times either obliquity is zero, so on those two days the disagreement cannot express itself at all. Every other day of the year it can, and the further the Sun is from the equator the more it does.
- 02
What the gap is worth on the ground
The two obliquities differ by 0.5607°, which is about 62 kilometres of latitude. The text’s Tropic of Cancer sits that far north of the real one. At Ujjain at midwinter the two shadows differ by about a third of an aṅgula on a shadow of seventeen — small on paper, and visible from across a courtyard.
- 03
Finding north without knowing north
Mark the shadow’s tip in the morning, and again in the afternoon at the moment it is the same length. The line bisecting those two marks is the meridian, exactly — at every latitude, in every season. Naming which end of it is north needs one more fact: the noon shadow falls away from the Sun, so it points north wherever you stand further from the equator than the Sun does that day, which in India is almost always.
The construction has one blind spot, and it is a good one. On the equator at an equinox the Sun rises due east, passes straight overhead and sets due west — so both shadows fall on the same straight line and there is no bisector to take. The method fails precisely at Laṅkā, on the day the text uses Laṅkā to define.
- 04
What this page does not do
The Sun’s place on the ecliptic is a slider here, not a derivation. Deriving it needs chapter 2’s mandaManda · the slow correctionThe correction for a body moving faster at one end of its orbit than the other. correction for the Sun’s varying speed, which this site attempted, measured, found worse than the mean, and reverted — the note is in lib/sky/reckoning.ts and it still stands.
Day lengths are geometric: no refraction, and the Sun treated as a point rather than a disc. Both of those make a real sunrise a few minutes earlier than this page says. That is deliberate — the page is measuring the text’s rule, not the atmosphere, and the test asserts the geometric figure so it stays that way. Past the polar circles the functions return nothing at all rather than a rounded-off number, because a day with no sunset is the arithmetic noticing something true.