Jya: The Bowstring That Became Sine

How Indian astronomy invented trigonometry and a mistranslation carried it west

Long before Europe spoke of sines, Indian astronomers at Kusumapura had tabulated the half-chord and called it jya, the bowstring. This lesson traces how that Sanskrit word became the Latin sinus through a chain of three languages, and why the half-chord is the object that makes modern trigonometry work.

The Slate at Kusumapura

In 499 CE, in the astronomical college at Kusumapura on the banks of the Ganges near modern Patna, a twenty-three-year-old scholar named Āryabhaṭa sat down to compose two lines of Sanskrit verse. He had in front of him a palm-leaf folio, a reed pen, and a problem. The Gupta empire, the political order under which the college had been built, was visibly fraying. Libraries burned. Manuscripts rotted in the monsoon. Whatever he wrote down had to survive not by being stored but by being memorized.

He wrote makhi bhakhi phakhi dhakhi ṇakhi ñakhi ṅakhi. Decoded through the alphanumeric cipher he had stated three verses earlier in the same chapter, those seven syllables read 225, 224, 222, 219, 215, 210, 205. They are the first seven sine-differences at intervals of 225 minutes of arc across a quadrant of a circle, and Āryabhaṭa had just compressed the earliest surviving trigonometric table in world mathematics into a single couplet. The word he used for the object being tabulated was jyā, bowstring. Fifteen hundred years later, after the word had been mistransliterated into Arabic and then mistranslated into Latin, the same object would be called sine. This lesson is the story of how a piece of archery vocabulary became the beating heart of modern science.

The Greek Chord and the Indian Half-Chord

The Greek astronomers had trigonometry, after a fashion. Hipparchus of Nicaea, around 150 BCE, compiled what is traditionally named as the first Western table of chords. Three centuries later, Ptolemy published an elaborate chord table in his Almagest. A chord is the straight line joining the two ends of an arc. If you imagine an archer drawing a bow, the arc is the curved wood and the chord is the taut string stretched between the two tips. The Greek word for this line was chordē, which originally meant, yes, a bowstring.