Planetary Diameters in Surya Siddhanta
Surya Siddhanta also estimates the diameters of the planets. The estimate for the diameter of Mercury is 3,008 miles, an error of less than 1% from the currently accepted diameter of 3,032 miles.
It also estimates the diameter of Saturn as 73,882 miles, which again has an error of less than 1% from the currently accepted diameter of 74,580.
Its estimate for the diameter of Mars is 3,772 miles, which has an error within 11% of the currently accepted diameter of 4,218 miles.
It also estimated the diameter of Venus as 4,011 miles and Jupiter as 41,624 miles, which are roughly half the currently accepted values, 7,523 miles and 88,748 miles, respectively.
Trigonometry in Surya Siddhanta
Surya Siddhanta contains the roots of modern trigonometry. It uses sine (jya), cosine (kojya or “perpendicular sine”) and inverse sine (otkram jya) for the first time, and also contains the earliest use of the tangent and secant when discussing the shadow cast by a gnomon in verses 21–22 of Chapter 3:
Of [the sun’s meridian zenith distance] find the jya (“base sine”) and kojya (cosine or “perpendicular sine”). If then the jya and radius be multiplied respectively by the measure of the gnomon in digits, and divided by the kojya, the results are the shadow and hypotenuse at mid-day.
In modern notation, this gives the shadow of the gnomon at midday as :
Even today many astrologers in India use Surya Siddhanta as base to compute their Panchangs (Almanacs) in many languages.
The diagram below shows a variation of the standard position triangle that first appeared in Sanskrit astronomical texts some time before the Twelfth Century. The most famous of these is the सूर्यसिद्धान्त (Surya Siddhanta).
The Sanskrit word for bow is चाप (ca̅pa). It is also the name given to an arc of a circle. The Sanskrit word for bowstring is ज्या (jya̅). It is also the name given to a chord of a circle. At some point, Indian astronomers found that knowing the size of half a chord was more useful than knowing the size of a whole chord. Half a chord in Sanskrit is ज्या अर्ध (jya̅ ardh). This term became so popular that the modifier अर्ध (ardh) was dropped and the the word ज्या(jya̅) or the similar word जीव (ji̅va) came to mean half a chord all by itself.
Arab scholars transliterated जीव (ji̅va) to جيب(jiba). They basically made up a new word in Arabic. European scholars didn’t know it was a new word, so they read it as an already existing one. This kind of thing is easy to do since vowels are never written in Arabic. The letters جيب might represent jiba (a word they probably didn’t know) or they might represent jaib (a word they probably did know). The Latin scholars went with the latter. The Arabic word جيب (jaib) is the English word bosom is the Latin word sinus. Bowstring becomes bosom becomes sine. You can’t make up a story like that. It must be true.