Thursday 13 October 2022

#20 Sine...... from old Math Terms notes

Sine The name sine came to us from the Latin sinus, a term related to a curve, fold, or hollow. It is often interpreted as the fold of a garment, which was used as we would use a pocket today. The use in mathematics probably comes about through the incorrect translation of a Sanskrit word. The actual first use seems to be a source of some disagreement. Here are the details according to the "Earliest Known Uses of Some of the Words of Mathematics " web site of Jeff Miller

Aryabhata the Elder (476-550) used the word jya for sine in Aryabhatiya, which was finished in 499.

According to Cajori (1906), the Latin term sinus was introduced in a translation of the astronomy of Al Battani by Plato of Tivoli (or Plato Tiburtinus).

According to some sources, sinus first appears in Latin in a translation of the Algebra of al-Khowarizmi by Gherard of Cremona (1114-1187). For example, Eves (page 177) writes:
The origin of the word sine is curious. Aryabhata called in ardha-jya ("half-chord") and also jya-ardha ("chord-half"), and then abbreviated the term by simply using jya ("chord").

From jya the Arabs phonetically derived jiba, which, following Arabian practice of omitting vowels, was written as jb. Now jiba, aside from its technical significance, is a meaningless word in Arabic. Later writers, coming across jb as an abbreviation for the meaningless jiba, substituted jaib instead, which contains the same letters and is a good Arabic word meaning "cove" or "bay." Still later, Gherardo of Cremona (ca. 1150), when he made his translations from the Arabic, replaced the Arabian jaib by its Latin equivalent, sinus, whence came our present word sine. However, Boyer (page 278) places the first appearance of sinus in a translation of 1145. He writes:
It was Robert of Chester's translation from the Arabic that resulted in our word "sine." The Hindus had given the name jiva to the half chord in trigonometry, and the Arabs had taken this over as jiba. In the Arabic language there is also a word jaib meaning "bay" or "inlet." When Robert of Chester came to translate the technical word jiba, he seems to have confused this with the word jaib (perhaps because vowels were omitted); hence he used the word sinus, the Latin word for "bay" or "inlet." Sometimes the more specific phrase sinus rectus, or "vertical sine," was used; hence the phrase sinus versus, or our "versed sine," was applied to the "sagitta," or the "sine turned on its side."

Smith (vol. 1, page 202) writes that the Latin sinus "was probably first used in Robert of Chester's revision of the tables of al-Khowarizmi."

Fibonacci used the term sinus rectus arcus.(Sinus rectus is also the name for an area of the skull below the brain.)

Regiomontanus (1436-1476) used sinus, sinus rectus, and sinus versus (now called versine, short for versed, or turned sine, it is 1-cos) in De triangulis omnimodis (On triangles of all kinds; Nuremberg, 1533) [James A. Landau].

Copernicus and Rheticus did not use the term sine (DSB).

The earliest known use of sine in English is by Thomas Fale in 1593:

It was when Leonardo de Fibonacci used the term in his writing, it became permanent. According to Carl Boyar's "A History of Mathematics", the idea of the sine of an angle came from an Indian book written around the year 400. The early use of sine referred to a length of the chord in a circle. It was not until the 1700's and Leonid Euler [pronounced Oiler] that it became common to use the sine as a ratio.