A bow, its string, an arrow
Every term on this page is archery. To the Indian astronomers an arc of a circle looked like a strung bow — cāpa or dhanus. The chord stretched across it was its string: jyā, from jiyā, "bowstring". And the short segment pinned between the middle of the string and the back of the bow was the arrow lying ready — śara.
Rotate the picture and you have the diagram above: the violet arc is the bow, the rose half-chord the string, the golden versine the arrow. Drag the point on the circle and watch the bow draw and relax.
Why half a chord?
Greek astronomers — Hipparchus, then Ptolemy — tabulated the full chord of each arc. Indian astronomy made a small change with large consequences: tabulate half the chord of twice the arc. In the diagram the dashed line is that full chord (samasta-jyā) spanning the double arc 2θ; the bright jyā is its half.
The half-chord is one leg of a right triangle whose hypotenuse is the radius, so it obeys the Pythagorean rule directly — jyā² + koṭijyā² = R² — and slots straight into planetary computation without factor-of-two bookkeeping. That convenience is why every table since — Arabic, Latin, modern — is a table of sines, not chords. Our sin θ is, quite literally, half the chord of 2θ.
From jyā to “sine” — a word’s journey
When Indian astronomy travelled west, jyā — in its variant form jīvā — was carried into Arabic as a loanword. Written without short vowels, its consonants were later re-read as an ordinary Arabic word meaning “fold” or “pocket of a garment”, and the 12th-century Latin translators of Toledo faithfully translated that. Every sine in every textbook still carries, garbled in transmission, the Sanskrit word for a bowstring.
The radius of 3438
Āryabhaṭa measured arcs in minutes: a full circle is 360° × 60′ = 21,600′. Choosing R = 21600 ⁄ 2π ≈ 3438′ makes the radius itself an arc length — string and bow are measured with the same ruler. Two elegant consequences follow: for small arcs jyā ≈ cāpa (the first entry of his table is 225′ for an arc of 225′), and the radius earns its own name, trijyā — “the jyā of three signs”, i.e. of 90°, the largest sine of all.
Other schools chose other rulers: Ptolemy’s chord table uses R = 60, the Romaka-siddhānta 150, the Pauliśa-siddhānta 3270. Switch R with the chips above — the geometry never changes, only the ruler.
Āryabhaṭa’s 24 sines · आर्यभटीय, c. 499 CE
Verse 12 of the Āryabhaṭīya’s Gītikāpāda compresses a whole sine table into one line of syllables — makhi bhakhi phakhi dhakhi… — twenty-four sine differences, one every 3°45′ from 0° to 90°. Accumulate them and you get the jyā values below, almost everywhere within one arc-minute of the modern value. Tap a row to load it into the diagram.
| # | Cāpa (arc) | Jyā | 3438 sin θ | Δ′ |
|---|
| # | Cāpa (arc) | Jyā | 3438 sin θ | Δ′ |
|---|
Note the first and last rows: jyā(3°45′) = 225 — for a small arc the string hugs the bow — and jyā(90°) = 3438 = R, the trijyā. Row 8 is exact by geometry alone: jyā(30°) = R⁄2 = 1719.