How did the authors of Surya Siddhanta find the diameters of other planets in the solar system?
The Surya Siddhanta, "a Sanskrit treatise in Indian astronomy from the late 4th-century or early 5th-century CE" is truly a great work.
But how was it possible for the writers to find the exact values of the diameters of different planets and the distance between the sun and the earth?
Also, are there any mistakes in this book?
If there's any problem in my question please inform me.
This site works better asking about very specific problems you might encounter in your day to day work or study. Speculating about supernatural influence in a religious work is outside the scope of this site, and asking users to review an entire work to list all mistakes or errata isn't really a great fit for this type of Q&A.
It looks like an interesting book, but I also don't understand what exactly you ask. The Wikipedia article (your link) explains how the astronomical knowledge in the book is based on ancient Greek astronomy. The table in the article lists the parameters from the Surya Siddhanta together with Greek and modern value. It looks to me that the the Surya Siddhanta has similar accuracy as the Greek. This knowledge developed over centuries of observations, and nothing in that Wikipedia article seems unusual to me. Can you be more specific which facts surprise you?
@StephanMatthiesen I edited it to clarify my question, you can have look
Ah, I think I understand now. Perhaps you should change the title and ask specifically "how did the Surya Siddhanta know the diameter of planets?" Your question was (before you edited it) still hard to understand because there are many different astronomical data in the text. Precise value for the period of planets are NOT surprising because you can easily observe them, but values for the diameter are difficult, so that is a fair question. It might be better suited in the history of science stackexchange.
The authors assume a geocentric universe (first thing that is wrong). They then assume that the planet Mars has the same apparent diameter as a globe 30 yojana in diameter (about 150 miles) in the same orbit as the moon, from a perspective at the centre of the Earth. This is just stated, and appears to be supposition. It is an incorrect figure. It is conceivable that some kind of system for sighting through a pinhole could be used for this estimate. This method tends to over estimate the apparent size of planets.
They next claim that the other planets have apparent sizes that would be the same as globes with diameter 37.5 (Saturn), 45 (Mercury) 52.5 (Jupiter) and 60 yojana (Venus). These form an arithmetic sequence with a common difference of 7.5, but apart from that, these values seem to be arbitrary (and are wrong). They might have been based on some kind of pinhole observations, but rounded to a simple sequence for the sake of the poetry or for easier memorisation.
There is uncertainty about the ancient value of the yojana. A different conversion to miles will give different values throughout.
Using this assumption and the distance already calculated to the moon they calculate the apparent angular diameter of each planet. The actual values they get are wrong, In fact, the angular diameter varies as the planet's distance varies.
Planet calculated value modern observed value Mars 2 arcmin (actually 0.06 - 0.39) Saturn 2.5 arcmin (0.25 - 0.34) Mercury 3 arcmin (0.08 - 0.17) Jupiter 3.5 arcmin (0.51 - 0.83) Venus 4 arcmin (0.16 - 1.05)
As you see the calculated values are much too large (ie wrong). They are comparable with values supposed to have been found by Hipparchus, but without a known observational basis. Tycho Brahe also gave similar values, found by sighting through a pinhole. This simple method doesn't give a good estimate of the angular size. The simple pattern in the calculated values is due to the original assumption that the globes were in an arithmetic sequence.
There is then a calculation of the geocentric orbital radii of the planets. They assume that Venus, Mercury and the Sun all orbit at a distance of 3.4 million miles. This is a significant underestimate, especially for the sun (this is wrong). The distances to Mars, Jupiter and Saturn were also underestimated.
One can then combine the apparent angular diameter (which is too large) with the calculated distance (which was too small) to obtain an estimate of the planetary diameter. When this is done, it gives exceedingly good estimates of the diameter of Mercury and Saturn (within 1%). A good estimate for the diameter of Mars (about 10% out), and a poor estimate for Venus and Jupiter (about 50% of the accepted modern values).
It is clear that there is much that is wrong. A combination of overestimates of some values and underestimates of others cancels out to give some impressive looking figures for two out of five planets. But do note that the estimates for Jupiter and Venus were a long way out, and all the numbers which the calculations were based on were wrong. In total there are 15 values calcuated here 5 apparent diameters, 5 geocentric distances and 5 actual diameters. Of these only 2 values are impressively accurate.
They did better than other pre-telescopic observers, but it isn't clear that they obtained values by anything other than luck.
As a sidebar, is it atmospheric scattering that makes planets appear larger than they are to the historic scientists in your answer, or is that not really a factor? (no good way to measure things that small might be the entire answer). Perhaps that's a new question but it seems related enough to ask here.
Not certain, as I can't find details of Brahe's experiment. It is probably something like "set up pin hole. Measure time for which planet is visible through pinhole as Earth rotates, convert time to angular size." A 1arcmin disc would be visible for about 4 seconds. There are lots of inaccuracies: non-zero size of pinhole, diffraction at pinhole, moving one's head slightly. These all tend to overestimate size.