From chords to the sine, the story of a word

The sine we now read as the ordinate of a point on the unit circle has a long and surprising history, one that crosses Greece, India and the Islamic world before reaching medieval Europe.

The Greek tables of chords

To predict the positions of the planets, Greek astronomers needed to relate angles to lengths. They did not have the sine, however: they used the chord of an arc, that is, the distance between the two endpoints of the arc subtended by a central angle. Hipparchus of Nicaea (2nd century BC) is traditionally regarded as the author of the first table of chords, now lost; the surviving witness is the great table of Claudius Ptolemy in the Almagest (2nd century AD), computed in steps of half a degree (Boyer). Chord and sine are linked by a simple relation: if the radius is $1$, $\operatorname{crd}\alpha = 2\sin\frac{\alpha}{2}.$ The chord is thus, in essence, twice the sine of a halved angle.

The Indian turn: the half-chord

It was the Indian mathematicians, between the 5th and 6th centuries, who took the decisive step. In the Aryabhatiya (499 AD) Aryabhata and his school tabulated no longer the full chord but the half-chord, precisely our function $\sin\alpha$: far more convenient for astronomical computation. They called it jya (or ardha-jya, “half-chord”), a Sanskrit word meaning “bowstring”, the chord of an archer’s bow (Katz).

How jya became sinus

The word then made a curious journey. Passing into Arabic, jya was transliterated phonetically as jiba, but in Arabic writing, which usually omits short vowels, only the consonant cluster jb remained. In the Islamic world astronomers, among them al-Battani (9th–10th century), made wide use of the half-chord in their tables. When in the 12th century European translators rendered these texts into Latin, they read jb as jaib, an Arabic word meaning “bosom”, “fold of a garment”, “bay”; and they translated it with the Latin sinus, which has the same meaning (Boyer). From this come the English sine, the Italian seno and their symbols. Thus a word that originally denoted the chord of an arc reached us transformed, through a translation, into the name of a bay or the fold of a cloak.

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Topics: Trigonometry
Concepts: Sine · Unit circle
People: Hipparchus · Claudius Ptolemy · Aryabhata · al-Battani