origin & evolution of the trigonometric term ‘sine’ also etymologically traceable to aryabhata

04/01/2020

Excerpted from the really delightful book of Eli Maor, called ‘Trigonometric Delights‘ published by Princeton University Press, 1998. (p 35-37)

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…An early Hindu work on astronomy, the Surya Siddhanta (ca. 400 CE), gives a table of half-chords based on Ptolemy’s table (fig. 15). But the first work to refer explicitly to the sine as a function of an angle is the Aryabhatiya of Aryabhata (ca. 510), considered the earliest Hindu treatise on pure mathematics. In this work Aryabhata (also known as Aryabhata the elder; born 475 or 476, died ca. 550 CE) uses the word ardha-jya for the halfchord, which he sometimes turns around to jya-ardha (“chordhalf”); in due time he shortens it to jya or jiva.

Now begins an interesting etymological evolution that would finally lead to our modern word “sine.” When the Arabs translated the Aryabhatiya into their own language, they retained the word jiva without translating its meaning. In Arabic—as also in Hebrew—words consist mostly of consonants, the pronunciation of the missing vowels being understood through common usage.

Thus jiva could also be pronounced as jiba or jaib, and jaib in Arabic means bosom, fold, or bay. When the Arabic version was translated into Latin, jaib was translated into sinus, which means bosom, bay, or curve (on lunar maps regions resembling bays are still described as sinus).

We find the word sinus in the writings of Gherardo of Cremona (ca. 1114–1187), who translated many of the old Greek works, including the Almagest, from Arabic into Latin. Other writers followed, and soon the word sinus—or sine in its English version—became common in mathematical texts throughout Europe.

The abbreviated notation sin was first used by Edmund Gunter (1581–1626), an English minister who later became professor of astronomy at Gresham College in London. In 1624 he invented a mechanical device, the “Gunter scale,” for computing with logarithms—a forerunner of the familiar slide rule—and the notation sin (as well as tan) first appeared in a drawing describing his invention.

Mathematical notation often takes unexpected turns. Just as Leibniz objected to William Ougthred’s use of the symbol “×” for multiplication (on account of its similarity to the letter x), so did Carl Friedrich Gauss (1777–1855) object to the notation Sin²φ for the square of sinφ:

sin²φ is odious to me, even though Laplace made use of it; should it be feared that sin²φ might become ambiguous, which would perhaps never occur…well then, let us write sinφ‘2, but not sin²φ, which by analogy should signify sin sinφ‘.

Notwithstanding Gauss’s objection, the notation sin²φ has survived, but his concern for confusing it with sinsinφ‘ was not without reason: today the repeated application of a function to different initial values is the subject of active research, and expressions like sinsinsinsinφ‘‘‘ appear routinely in the mathematical literature.

The remaining five trigonometric functions have a more recent history. The cosine function, which we regard today as equal in importance to the sine, first arose from the need to compute the sine of the complementary angle.

Aryabhata called it kotijya and used it in much the same way as trigonometric tables of modern vintage did (until the hand-held calculator made them obsolete), by tabulating in the same column the sines of angles from 0◦ to 45◦ and the cosines of the complementary angles. The name cosinus originated with Edmund Gunter: he wrote co.sinus, which was modified to cosinus by John Newton (1622–1678), a teacher and author of mathematics textbooks (he is unrelated to Isaac Newton) in 1658. The abbreviated notation Cos was first used in 1674 by Sir Jonas Moore (1617–1679), an English mathematician and surveyor.

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#excerptise provided by, um, the well known excerpt on anything & everything, yeah.

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