Tuesday, September 10, 2019

The obliquity of the ecliptic

On IndiaFacts, Anil Narayanan makes the case that the astronomers who wrote the Surya Siddhanta measured the obliquity of the ecliptic to be arcsin(1397/3438) = 23.975° and not 24° as has been translated by Whitney et. al. since 1858; since 24° would be expressed as arcsin(1398/3438) or arcsin(1399/3438) and never arcsin(1397/3438).
This is an important observation which bears repeating: The precision of the Indian R-Sine is 1/3438.

This puts the Surya Siddhanta to some 3000 BC.  Anil Narayanan promises more to support this date.
We currently know of at least 3 other items in Indian astronomy that point to 3000 BC, or thereabouts.

1) The value of the Sun’s equation-of-center given in the SS indicates a time range of 3000 BC or older;

2) The ubiquitously mentioned pole-star in Indian astronomy and literature, namely Dhruva (modern name Thuban), indicates a period about 3000 BC;

3) It is mentioned in the Satapatha Brahmana text that the Krittika Nakshatra rises exactly in the East, which occurred only in ancient times, around 3000 BC. Nowadays Krittika rises between East and North-East.

We will discuss these in other articles.
If Anil Narayanan is right, then he is also right about this:
The misconception, which has to do with the tilt, or obliquity, of the earth’s axis, also ranks among the most clever and successful obfuscations in Indian astronomy carried out by the European scholars of yesteryear. They skillfully achieved the difficult task of hiding the treasure in plain sight, so to speak.
The two questions that I have are - how was this angle measured or inferred, and what is the origin of the Indian standard radius of 3438?

Answer to 3438 - it is the approximate radius of the circle in minutes (the exact value is  3437.74677078...).

My criticism of Anil Narayanan's article - see Aryabhatta's sine table on Wiki.  1397 comes from a linear interpolation for 24° between rows 6 and 7 of that table.

That is,
22° 30' = 1350' has jya = 1315
26° 15' = 1575' has jya = 1520
What is the angle whose jya = 1397?
The linear interpolation answer is 1350' + (1397 - 1315) * (1575' - 1350')/(1520 - 1315)
= 1350' + 82 * 225'/ 205 = 1440' (exactly!)
1440' = 24°.

This is probably how Whitney et. al. came to 24°.   The question then was linear interpolation the method of calculation used? e.g., see the same Wiki article.  I think to establish the point made Anil Narayana's article, we have to know how intermediate values in the R-sine table were computed.

Or, following Anil Narayanan's philosophy, that the precision of the Indian R-Sine is 1/3438, the obliquity of the ecliptic in the Surya Siddhanta is not an approximate 24°:
According to Mr. Bentley, the Hindu astronomers (unless in cases where extraordinary accuracy is required) make it a rule, in observing, to take the nearest round numbers, rejecting fractional quantities: so that we have only to suppose that the observer who fixed the obliquity of the ecliptic at 24 degrees, actually found it to be 23 and 1/2.
 Rather the measured obliquity of the ecliptic is bounded by arcsin(1396/3438) and arcsin(1398/3438), i.e, between 23° 59' and 24° 1'  (23.983° and 24.017°).

Using the formula here: http://glossary.ametsoc.org/wiki/Obliquity_of_the_ecliptic

or a more exactly formula one can estimate the range of times of that observation.