What is the etymology behind sine, cosine, tangent, etc.?

I heard somewhere that it was actually a mistake in translation. What's the correct story?

• The word "tangent" comes from the Latin tangere, 'to touch'. Oct 6, 2016 at 13:25

Victor Katz is not a linguist and a lot of what he says in the quoted extract is wrong: for example that “Arabic is written without vowels” and that the word in question is spelt “jb”. In fact it is written jyb جيب (as mobileink has pointed out). But the decisive error from the viewpoint of the history of science is his failure to remark that Sanskrit jyā and jīva mean “bow string” and are thus translations of the Greek khordē “gut string”, which was borrowed into Latin as chorda (which in turn is the source of both “chord” and “cord” in English).

As a mathematical term khordē (and chord) means a line connecting two points on an arc A and B; if we draw lines from A and B to the centre of the circle (O) the length of the chord AB is defined as the chord of the angle AOB. The Indians borrowed Greek trigonometry and translated khordē literally as jyā, but they used this word not (usually) for a chord but for a half-chord, what we call a sine.

The Arabic astronomers inherited both the Greek and the Indian systems of trigonometry and the Sanstrit word jyā (jīva) was transcribed (not translated) as jīb. In the 12th century the famous Arabic-Latin translator Gerard of Cremona misread the latter as jayb “breast pocket” and rendered this with its Latin equivalent sinus. In Arabic script jīb and jayb are both written جيب .

So: this is a Greek word, first translated into Sanskrit, then transcribed into Arabic, then mistranslated into Latin.

• Ernest Klein, "Klein's Comprehensive Etymological Dictionary of the English Language", Elvevier 1971, which is reputed to be particular thorough for English words with Semitic roots, has this: As a mathematical term, L. sinus was first used by Gherardo of Cremona in his translation from the Arabic (about the year 1150) to render Arab. jayb, 'chord of an arc; sine' (fr. OI. jīva-, 'bowstring') which he confused with Arab. jayb, 'fold of garment'. Oct 8, 2016 at 17:13
• Klein was a rabbi, not a mathematician. I think the answers to this question nicely demonstrate the problems with factual accuracy that crop up when mathematicians speak to linguistic issues, and language experts talk about mathematics. Oct 8, 2016 at 18:14
• @mobileink. It begins with the Yavanajātaka ("Ionian treatise") which introduced Greek astrology to the Indians in the 3rd century CE.
– fdb
Oct 10, 2016 at 19:42
• @mobileink. No, 3rd century Christian Era is 600 years after Alexander. I think that if an Indian text explicitly says that it is a translation from "Ionian" then that should be taken seriously.
– fdb
Oct 10, 2016 at 20:06
• @mobileink Here. There were Greek-culture-dominated Northern Indian kingdoms which lasted for centuries. Oct 22, 2021 at 14:43

For sinus, see :

The English word “sine” comes from a series of mistranslations of the Sanskrit jya-ardha (chord-half). Aryabhata frequently abbreviated this term to jya or its synonym jiva. When some of the Hindu works were later translated into Arabic, the word was simply transcribed phonetically into an otherwise meaningless Arabicword jiba. But since Arabic is written without vowels, later writers interpreted the consonants jb as jaib, which means bosom or breast.

In the twelfth century, when an Arabic trigonometry work was translated into Latin, the translator used the equivalent Latin word sinus, which also meant bosom, and by extension, fold (as in a toga over a breast), or a bay or gulf. This Latin word has now become our English “sine.”

For cosine :

The prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's Canon triangulorum (1620), which defines the cosinus as an abbreviation for the sinus complementi (sine of the complementary angle) and proceeds to define the cotangens similarly.

co.sinus was suggested by the English mathematician Edmund Gunter (1581-1626) in his Canon triangulorum, sive, Tabulae sinuum et tangentium artificialium ad radium 100000.0000. & ad scrupula prima quadrantis (1620). According to Smith (page 619), the term was "soon modified by John Newton (Trigonometria Britanica, 1658) into co-sines, a word which was thereafter received with general favor."

• minor nit: jyb refers to the opening in clothing near the neck or bosom. that's where they kept stuff, hence modern "pocket". see lex.sibawayhi.org/html/pi/جيب Oct 6, 2016 at 20:31