Jyā





Jyā, or jyārdha, or R Sine was a mathematical quantity used by ancient Indian astronomers. It would later evolve into the modern 'sine' function. It was used in many astronomical computations, such as finding right ascenscions, declination, polar latitude, length of day, etc.

The Jyā is half of an arc's chord on a circle. If two lines are drawn from the center of the circle to the two ends of the jyā, a right-triangle is formed. Thus the mathematics of right triangles can be used to understand the movements of the heavens. In terms of the modern sine function, jyā is the radius multiplied by the sine of the angle formed by the arc. This is also called the R sine. Whereas sine takes an angle as it's argument, and returns a ratio, jyā takes an arc and returns a length.

Āryabhaṭa I created tables of jyā for a circle with a radius of 3438 units. Varāhamihira created tables with a radius of 120 units. The tables presented the value of jyā for each 3°45' of arc.

The word jyā itself comes from the bow. The full chord together with the arc of the circle resemble the shape of a bow and bowstring. The word Jyā is literally 'bow string' in Sanskrit. Eventually 'ardhajyā' (half jyā) was simply abbreviated as 'jyā' in the texts. The term later was mangled during translation from Arabic to Latin, and became 'sine' in Europe.

Lalla describes many uses of jyā in his description of an armillary sphere. There are measures, for example, like the natajyā, dṛgjyā, and trijyā, which had specific meaning regarding the altitudes and position of planets.