# Talk:History of trigonometry

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## Disputed

This article makes the claim that Indian mathematicians were in the 5th century BC, before Hipparchus. In actuality, they seem to have been in the 5th century AD (and were influenced by the work of Ptolemy if I remember the book by Boyer correctly)!

Overall, I'm concerned that the article goes beyond acknowledging the Indian and Arabic contributions to trigonometry, to actually minimizing the Greek and European contributions. In this it seems to follow the "Crest of the Peacock" book cited as a reference, and which seems to have come under heavy criticism in academic circles. Indeed, its author admitted that he intentionally left out the Greek contributions from his book, preferring to focus on the non-European roots. More serious is the criticism made in the Pingree review linked above that the Peacock author "has no particular expertise" in the field (and in particular relies on secondary sources because he does not know the languages) and his accounting of Indian and Arab contributions especially "abound[s] in inaccuracies of dating, of names, and of historical facts" and that the "misleading and just plain wrong statements in the book seriously affect the persuasiveness of the author's arguments."

I'm not a professional historian myself, although I've read a couple of books (by Boyer and Maor) on the history of mathematics and trigonometry which give a quite different picture from the one in this article. It doesn't seem in keeping with WP:NPOV to (apparently) base this article so strongly on a single iconoclastic source. Not to mention mixing up BC with AD!

—Steven G. Johnson 18:05, 31 October 2006 (UTC)

## UNDISPUTABLE

Steven this article was indeed incorrect but the page is been restructured & now it seems to be correct.Just i want to say to u that what ever u have said regarding only this article is correct.But i want to say u tht your view regarding the Indian mathematics is incorrect.It is true that Hipparchus & ptolemy were the first to bring about developement in Trigonometry After that Indian came to bring developement in trigonometry which was significant & mainly was not influenced by ptolemy's work.Just to make your info more correct that Ptolemy's & greek mathematicians works were on chord functions & Indian mathematician's works were based on sine functions.So it cannot be said that Indian mathematicians were influenced by ptolemy.Indian mathematicians prepared their own sine table & world knows that very well.Because sine is the oldest trigonometric functions & were originated by Indian mathematicians.Just make more research on this Origin of sine functions u will know the truth.One more time truely speaking "Hipparchus" is the creator of Trigonometry Because the earliest one is the father.Hipparchus is truely said as the father of Trigonometry.I THINK NOW WE SHOULD REMOVE THIS FROM DISPUTATIONS.

202.179.64.9 16:27, 4 November 2006 (UTC)Aaditya D.Singh

Thanks for fixing the BC/AD mistake. I changed the disputed tag to a POV tag, since I don't see any obvious factual errors (although I'm skeptical of any claims sourced to the Peacock book because of the criticisms cited above), but the article still seems woefully unbalanced.
I didn't say that Indian and Arab mathematicians didn't make significant contributions (e.g. it's clear that our modern definition of sine as the half-chord instead of the chord, and even the name "sine", comes from India...and in fact I was the first person to add this information to Wikipedia a year or two ago), just that the contributions of Greek and (later) European mathematicians are currently grossly understated in the article.
You appear to have no evidence to support your assertion that there were no significant Hellenistic influences on the Indian work 600 years later; as far as I can tell, professional historians of mathematics seem to disagree. (Note the article still makes the biased, and arguably incorrect claim that "the first significant developments of trigonometry were in India".)
By the way, if "u" take a little more time to write like a literate adult, it will be easier to take you seriously. —Steven G. Johnson 19:21, 4 November 2006 (UTC)
PS. What is the evidence that Bhaskara II treated trigonometry as a subject in its own right, not as an adjunct to astronomy etc.? According to the Boyer reference, the first person to do so was Regiomontanus and that's what the Wikipedia article originally said, and a user later moved this statement to apply to Bhaskara without citing any source. Euler's contributions are also understated by the deprecating words "in Europe" added to his contributions, as he seems to be the first person anywhere to define trig functions by their infinite series and to analytically continue them to the complex plane (this is what is meant by "analytic" treatment of a function, in case you don't know math). The way that this article has evolved, with sourced statements about one person moved to unsourced statements about other people, does not inspire trust. —Steven G. Johnson 19:31, 4 November 2006 (UTC)

## OK NOW

Steven,I dont know who had edited this article.But its correct now.Yes sentence "the first significant developement of trigonometry was in India" has biased view.But its correct now.Rest i do not see any disputable sentences in article.If there is any, then comment on it,We will make this page as a #REDIRECTWP:NPOV.Regarding ptolemys influence on Indian mathematicians i think this link will answer your question better then me-http://www.trigonometry-help.net/history-of-trigonometry.php. This site is authorised & clearly used related for info regarding trigonometry.It mentions clearly that Indian mathematicians worked on sine functions & not on half-chord functions as greeks & europeans did. 202.179.64.9 13:01, 5 November 2006 (UTC)Aaditya D.Singh

## The Sulba Sutras

I am removing an unsourced claim in the history section about the Sulba Sutras containing trigonometric functions. There is no evidence of this. Since the claim was made in other WP pages as well, I decided to probe it more and realized that the source provided was G. G. Joseph's book The Crest of the Peacock: The Non-European Roots of Mathematics (p. 232). However what is provided in Joseph's book is a modern-day proof of some results stated in the Sulba sutras, and that proof uses $\sin \theta$, (and that too a little redundantly since the angle is 45 degrees and he is really talking about the diagonal of a square). There is no indication in Joseph's book anywhere that sine, cosine, or anything resembling trigonometric functions are mentioned in the Sulbasutras. What is mentioned is the following line in Sankrit verse: "Divide the diameter of a circle into 15 equal part and take 13 of them to be the side of the square," (for "squaring the circle"). The Sulbasutras say that and nothing else (and no indication is given of how the result was discovered.) That is not evidence for knowledge of trigonometric functions. Fowler&fowler«Talk» 14:20, 22 February 2007 (UTC)

However what is provided in Joseph's book is a modern-day proof of some results stated in the Sulba sutras, and that proof uses $\sin \theta$, (and that too a little redundantly since the angle is 45 degrees and he is really talking about the diagonal of a square).

I was about to suggest OR violation as are apparent by constant vandalism you have caused in Indian mathematics related articles but then "The whole of Indian geometry and trignometry is dominated by the theorum of the suqare and the diagonal." (Geometry in Ancient and Mediaeval India By T.A. Sarasvati Amma page 58). 21:10, 22 February 2007 (UTC)

Yes? But what does your quote have to do with the Sulba Sutras? There was a lot of great trigonometry in India in the first millennium CE. However, in the Sulba Sutras, no trigonometry is present. There was knowledge of Pythagoras's Theorem, but no trigonometry. Computing the ratio of the side of a square to its diagonal doesn't mean that you have also computed $\sin(\frac{\pi}{4})$ and therefore you know about trigonometric functions! Fowler&fowler«Talk» 09:52, 24 February 2007 (UTC)

## Rename article "History of trigonometry"

Hello,

Considering how it has been around two weeks since I've posted this topic, I will now proceed to change the title of this article from "History of trigonometric functions" to "History of trigonometry". selfwormTalk) 02:22, 12 August 2007 (UTC)

## Early Trigonometry

Need fix on text. Broken - can't read right. —Preceding unsigned comment added by 121.1.55.86 (talk) 13:10, 18 June 2010 (UTC)

## جب

I doubt that Arabic jiba and jaib are written the same way. جب would be either jib or jab, but hardly jaib, and jiba only if the final a is desinential. This needs verification. --dab (𒁳) 12:18, 1 July 2010 (UTC)

I have just consulted Lane, and I find no evidence that جب means either "bay" or "chord". Something is wrong here. --dab (𒁳) 12:33, 1 July 2010 (UTC)

As so often, it turns out that these claims are based on nothing at all. Our source here is this, a 1996 online article which does not cite its own sources, but which more significantly doesn't match our claims here. Connor and Robertson say that Arabic jiba and jaib are words for "chord" and "fold" respectively. That's it. No word on homography, and no claim that either word is spelled jb جب . I just wasted ten minutes because people will insist on making stuff up based on no evidence. --dab (𒁳) 12:39, 1 July 2010 (UTC)

I am dismayed to see that this piece of misinformation has travelled through time from 2004. Worse, the 2004 version still offers as speculation that jiba may have been abbreviated (not "spelled") as jb. This was then "fixed" to the irritating "Arabic writes no vowels" by some well-meaning but clueless editor in 2005. For the next five years, this artefact of one Wikipedian with a clue writing speculation being "improved" by another Wikipedian without a clue, was carried along without being touched or questioned. Seriously, I prefer people inserting "penis" in random places in articles, as that's at least easy to catch and doesn't give anybody a false impression. --dab (𒁳) 13:00, 1 July 2010 (UTC)

Here is another source: Boyer, History of Mathematics, page 252:
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 the 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."
And here is another source: Mao, Trigonometric Delights, chapter 3:
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.
(Note that the above two sources were provided back in 2004, both in the article and in the discussion in the Talk page when someone asked about this specific topic. In 2004, unfortunately, MediaWiki did not support the ref tag, so associating information in the text with the corresponding sources was difficult. You are correct that additional unsourced discussion of the Arabic language was later added, so I've just trimmed these additions in the current article.) — Steven G. Johnson (talk) 15:34, 1 July 2010 (UTC)

You note, of course, that the two sources you cite are contradictory? The first one says that jiba was misidentified as jayb ("jaib")yby the translator into Latin (Robert of Chester), while the second claims that the move from jiba to jayb happened in Arabic tradition itself. The second version is what I also found in O'Connor (1996). But your second source (Mao) is clearly unaware of Arabic spelling. By consulting any Arabic dictionary, you will find that jayb is spelled with yod. There is little point in trying to uphold an apparent mistake in a treatise on Trigonometric Delights on an incontrovertible point of Arabic orthography ((Lane: jayb (p. 492): "opening at the neck and bosom [of a garment]", "the heart or bosom", "the place of entrance of the land")). I also presume that jiba is spelled with ta marbouta, but I couldn't find the word. It would be ever so helpful if the authors who have already researched this stuff could bring themselves to provide the actual Arabic spelling.

The mistake that happens here is that some authors correctly state that jb may have been the abbreviation for jiba , much like sin is the abbreviation for sinus (but they fail to make clear whether this is ad hoc speculation, or whether we have positive evidence that this abbreviation was in use in medieval Arabic mathematical literature), and then other authors assume that this is due to the "fact" that "Arabic doesn't write vowels", unaware that there is no other way of spelling "jaib" than jyb.

What we still don't know is whether jayb is an artefact of the Latin translators, or whether jayb became the pronunciation of the abbreviation jb even in Arabic. I suggest that we treat Boyer's account as more authoritative, although we should phrase it carefully, since we have two other accounts of lesser reliability that say otherwise. --dab (𒁳) 08:58, 2 July 2010 (UTC)

Further light is shed on this by ar:دوال مثلثية. It turns out that جيب is the name of "sine" in Modern Arabic. I can only assume it is pronounced jayb, not jīb, although this needs verification. In the light of this, and seeing that none of your references supports a spelling jb جب , I strongly recommend that we do not restore this spelling unless we can clearly explain what we are claiming was spelled this way, based on what evidence. --dab (𒁳) 09:25, 2 July 2010 (UTC)

The Maor reference seems ambiguous to me regarding whether the change happened in Arabic or not; it only says that it could be pronounced jaib in Arabic, not that it was. Since the Maor is ambiguous, and Boyer is not, I don't think that you should delete Boyer's argument, nor should you delete the references.
In any case, I agree that the Arabic spelling itself, which was a later unsourced addition to the article, should be removed. (It's not like جب is comprehensible to an English reader anyway.) — Steven G. Johnson (talk) 15:33, 2 July 2010 (UTC)

For what it's worth, here's another source (Kim Plofker's Mathematics in India, and Plofker is a careful scholar). This is what it says, p. 257:

The standard Arabic word for Sine, jayb (literally "cavity", "pocket"), is apparently a misinterpretation of an earlier word jība using the same consonants j-y-b. This term jība, being meaningless in Arabic, was read as the more familiar word jayb. (It is the literal sense of jayb as "pocket" or "fold" that was later translated into Latin as "sinus", whence our "sine.")

She goes on to point out that jība comes from Sanskrit jīvā a synonym of the standard jyā, and quotes al-Biruni:

[…] call the half-Chords juyūb [plural of jayb], for the name of the Chord in the Indian [language] was jībā, and [the name] of its half jībārd. But since the Indians use only the half-Chords, they applied the name of the full [Chord] to the half, for ease of expression

She also has a footnote on "This well-known story", so presumably she has checked multiple sources. And the transition happened in the Arabic tradition itself, because the former word was meaningless. Shreevatsa (talk) 21:32, 2 July 2010 (UTC)

So, it sounds like we have different reputable sources that disagree on this point, so we clearly should report both. — Steven G. Johnson (talk) 22:30, 2 July 2010 (UTC)

## System of notation

I find it astonishing that we gloss over the system of numeric notation in use in the various cultures at various times, as these various systems, by themselves, determine the overall development of mathematics. Greek numerical notation was little more than hash marks (Attic, Ionian, etc.). Roman numerals were better. The Greeks developed algebra and other symbolic notations because their actual numbers were so unwieldy. Which is why they used chords - and other clever dodges - to begin with.

Note the Indian contribution comes immediately after their development of what we now call "Arabic" numbers around the 4th century, but also note the strange "accurate to four decimal places" attributed to Indian mathematicians c.450 AD. I am informed Islamic mathematicians developed decimals several centuries later. Islam got "Arabic" numbers from India (date uncertain), which Fibonacci introduced to Europe in 1202. Which is a date that ought to be engraved in every 6 year old's head, as it is one of the most significant dates in all history.

Lack of an effective number system also explains the slowness of the Chinese to take up trig, as their number systems (various) were perhaps the most unwieldy of all, and which were in common use into the 20th century. This is not to say that one positively, absolutely could not perform trig in Attic or Chinese numbers, but that the process was so difficult as to be impractical.

The abacus was not a Chinese invention. It had been in widespread use in Europe, before the advent of Arabic numbers. Dave of Maryland (talk) 22:25, 31 August 2012 (UTC)