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.
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.
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