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

Question

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

Answer 1

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.

Answer 2

For sinus, see :

• Victor Katz, A History of Mathematics (3rd edition, 2008), apge 253 :

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.”

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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.

According to Earliest Known Uses of Some of the Words of Mathematics :

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."