Aryabhata · Mathematics
Aryabhata’s table of sine-differences - the first true sine table in history - and the strange journey of his word <em>jya</em>, the half-chord, which travelled through Arabic mistransliteration to become the ‘sine’ every student now learns.
To predict where a planet will stand, an astronomer must convert between angles and lengths - between the arc a body sweeps and the straight-line distances that follow from it. The Greeks did this with a table of chords: for each arc, the length of the straight line joining its two endpoints across the circle. Aryabhata (drawing on an emerging Indian practice) made a small change with vast consequences. Instead of the full chord of an arc, he tabulated the half-chord of half the arc - the length we would now draw as the vertical line from one end of an arc down to the diameter. He called it the jya, the ‘bowstring’ or half-chord. This half-chord is, in modern terms, the sine. It is a more natural and more powerful quantity than the full chord: it plugs directly into the right-triangle relationships that dominate astronomy. With that one adjustment, Aryabhata took a decisive step from the geometry of chords toward trigonometry as we know it.
Aryabhata’s table, packed into a single verse of the Aryabhatiya, is a list of twenty-four numbers. He divides a quarter of the circle - ninety degrees - into twenty-four equal steps of 3 degrees 45 minutes each, and for each step he records not the sine itself but the difference between successive sines, the small increments by which the half-chord grows as the arc lengthens. From these twenty-four sine-differences the whole table of sines can be rebuilt by addition. He works with a circle whose radius is 3,438 units - a number chosen because it is very nearly the number of minutes of arc in a radian, so that for small angles an arc and its sine come out almost equal, a clever convenience. The result is the earliest sine table known to history, compact enough to memorize and accurate enough to drive real astronomy. It is a monument of applied mathematics hiding inside a line of verse.
The history of the word ‘sine’ is one of the great accidents of scholarship. Aryabhata’s half-chord was the jya (also jiva), the ‘bowstring.’ When Indian astronomy was translated into Arabic, the scholars simply borrowed the sound, writing it as jiba - a nonsense word in Arabic, kept only as a technical term. But Arabic script often omits short vowels, so the written form j-b could be read more than one way, and later readers pronounced it jaib, an ordinary Arabic word meaning ‘bosom,’ ‘fold,’ or ‘bay.’ When these Arabic texts reached twelfth-century Europe and were translated into Latin, the translators rendered jaib by its meaning, choosing the Latin word for a fold or bay: sinus. From sinus came ‘sine.’ So the term every student of trigonometry learns is a fossil of a mistake - Aryabhata’s ‘bowstring,’ transliterated into a meaningless Arabic word, misread as ‘bay,’ and translated into Latin as though it had always meant a fold of cloth. The mathematics passed faithfully; the word arrived scrambled.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned that Aryabhata replaced the Greek full-chord with the half -chord ( jya ), built the first table of sine-differences at intervals across a quarter-circle, and that jya became Arabic jiba , misread as jaib (‘bay, fold’), then Latin sinus , then ‘sine.’ Explain the mathematics and the etymology.
Leads to Al-Khwarizmi.
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