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WikiProject Mathematics (Rated C-class, High-importance)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
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 Field: Geometry
One of the 500 most frequently viewed mathematics articles.


WOW! How lame!! The first exposure I had (and I believe most have) to tangent is with respect to triangles. Nowhere to be found, here. What a bunch of rubbish. I fail to see any excuse. I went to the disambiguation page and found no resolution. Miserable. Please consider a more gentle approach that includes the current pedagogical evolution of the term you are writing on. No lame excuses that you refer people to geometry. Oh wait, geometry is a separate branch of mathematics, is it? rolf. —Preceding unsigned comment added by (talk) 22:35, 15 February 2009 (UTC)

Of course it is not only lame, but absurd to talk about tangent lines when referring to non-smooth geometric objects. Tangent lines are the core part of the reason curves are smooth. How ridiculous to even consider a tangent line to a triangle vertex. More stupidity from ignorant modern mathematicians. (talk) 15:39, 11 June 2012 (UTC)

For above you're thinking of a Tangent function: —Preceding unsigned comment added by (talk) 08:10, 15 March 2009 (UTC)

The definition of "tangent line" is incorrect. Who will replace it?

Certainly the original was grotesquely incorrect. But the present one is distinctly inferior to the one that Axel Boldt wrote earlier. I fear that any good account of the matter will have to make clear not only a correct definition, but also why some of the inferior characterizations are inferior. Michael Hardy 18:30 28 May 2003 (UTC)

With the addition of the "vertical tangent" comment, this article's silliness continues. "Tangent" should first be defined in a context in which "slope" makes no sense, and only later set in the narrower context that refers to slopes. Unfortunately, I think this is being written by people whose knowledge of the subject comes only from calculus courses. Michael Hardy 20:06 3 Jun 2003 (UTC)

And now I've reverted. It makes no sense to speak of slope except when there is a coordinate system, but "tangent" makes sense regardless of the choice of coordinate system. Admittedly, the article has room for improvement, but at least the silliness of writing about slope outside of the context of a choice of coordinate system has been expunged. Probably something about that, in its proper context, should be added. Michael Hardy 21:35 3 Jun 2003 (UTC)]

I should think you should either take the time to re-write the offending "silliness" or leave it there for somebody else to fix. Pizza Puzzle

Would User:Michael Hardy explain why he chose to delete:

Pizza Puzzle

Related meaning[edit]

To "go off on a tangent" means to go off topic, likely derived from the geometric meaning of a tangent.

I removed this piece of folk etymology because Wikipedia is not a dictionary.
Herbee 21:10, 16 May 2006 (UTC)

Redirect to Trig function?[edit]

Should this page be redirected to the trigonometric function page just like the sine and cosine pages?

Sure it should be redirected. Both meanings are placed on one page just because the name is identical. In many languages (eg. in Polish) there are two different words, not relating each other. It came from Latin: "tango" means "to touch". If we continue such practices, someone will eventually place here information about tango dance... 11:12, 7 September 2005 (UTC)

I think "tango" means "I touch" and "tangere" means "to touch". Michael Hardy 21:12, 7 September 2005 (UTC)
I concur, tangere means to touch, and is a form of the verb tango.

The quality of this article has grown progressively worse. Also note that semicolons are not commas.

Tangent Line Problem[edit]

Um, I noticed that there was not an article nor a redirect at Tangent line problem, so I created a redirect to here from there. Seeing as how this article is most relevant. I was kind of surprised there was no article about that particular problem... --Comrade4·2 02:59, 12 September 2006 (UTC)

More confusing redirects[edit]

There is also a redirect from "tangent problem", as in "Gottfried Leibniz gave a complete solution to the tangent problem.". There is no clear explanation on this page of why this redirects here, what it refers to, or what Leibniz did. (talk) 23:14, 2 February 2011 (UTC)


"And I dare say that this is not only the most useful and general [concept] in geometry, that I know, but even that I ever desire to know." Descartes (1637)"

this sentence makes no actual sense -- 16:05, 2 October 2006 (UTC)

  • This quote should be removed until it can be verified. I can't find it elsewhere on the internet. Can whoever added provide a reference? It's also grammatically incorrect, no? Tnek46 16:42, 9 October 2006 (UTC)
I tried to find the original (French or latin) citation but couldn't find it. If someone finds it, I might be able to make a better translation--Yitscar (talk) 10:20, 20 November 2007 (UTC)
A facsimile of the original French is in ISBN 0486600688, but I can't get it online, and I couldn't translate it anyway. I added two cites to the translated quote. Tom Harrison Talk 13:35, 20 November 2007 (UTC)
Whether Descartes said it or not, it makes a lot of sense. Tangents are the reason a given continuous curve is smooth. Remove smoothness and you have no natural averages. In other words, no calculus - John Gabriel. Descartes was the greatest of all French mathematicians in my opinion. (talk) 11:19, 11 June 2012 (UTC)


Shouldn't the section derivative, integral, and power series be subsections of the trigonometry sections since they refer to that particular concept? Ricardo sandoval 20:44, 8 May 2007 (UTC)

No, whereas they can be interpreted to be somehow related to tangent lines, it is mostly related to limits.--A 22:58, 26 October 2007 (UTC)


I have removed the obsolete section about the trigonometric function tangent — the topic is much better covered at the trigonometric functions and doesn't have much in common with the tangents in geometry. This was discussed at WT:WPM and didn't raise objections. Unfortunately, no one came forward willing to fix the incoming links (a program like AWB should make it a very straighforward even if tedious task), so I'd like to repeat the request here.

The article requires major expansion, it presently contains little beyond the definition and a description of the approach to tangent lines based on calculus. Arcfrk (talk) 05:50, 1 March 2008 (UTC)

...and that definition as as muddly as the vast majority of Wikipedia math. The section that describes the derivative-based equation of some aspect of a tangent line on a Cartesian plane first applies the constraint that x = a, and then states the equation as follows (not neglecting the smörgåsbord of different ways to denote a derivative in mathematical notation):
Given x = a, this simplifies to:
Perhaps, as all Wikipedia mathematicians, the original author was speaking figuratively when s/he wrote x = a. (talk) 10:47, 16 August 2008 (UTC)

Merge with page[edit]

The page about local linearity is the same topic. Please discuss at Talk:Local_linearity#Merge

Derivative motivation sections[edit]

I'd like to remove the sections 'Intuitive description', 'More rigorous description' and 'How the method can fail' from the article and replace them with a short summary and a link to derivative. We already have a description of how the tangent line is used to motivate definition of the derivative in that article and we don't need a redundant explanation here. The material in these sections is mostly about the definition of the derivative and so is off topic for this article. I would have proposed merging the material into the other article but that article's treatment is already fairly complete.--RDBury (talk) 07:00, 19 September 2010 (UTC)

D.Lazard's edits[edit]

It is one thing for a mathematician to believe what he wants but quite another thing regarding the factual validity of what he believes. I have corrected this article because a tangent line never crosses a curve at the POINT of Tangency. Inflection points DO NOT have tangents according to the original definition of tangent. This is a misconception amongst ignorant mathematicians. (talk) 12:07, 9 June 2012 (UTC)

Again wp:personal attacks, like in Talk:Inflection point‎‎. About your assertion, all text books who define tangents give a definition which implies that an inflection point have a tangent. Thus, if I follow you, all the mathematicians are ignorants. D.Lazard (talk) 13:14, 9 June 2012 (UTC)
I do not care about what textbooks say or don't say. Just because something is printed in a book, does not mean it is correct. Here is a link to the definition that existed long before any of the textbooks you refer to were printed (1594) -

In fact, this definition existed long before Newton was born. If you followed me, then you would not be one of the ignorant mathematicians. An inflection point cannot have a tangent. In fact, a derivative does not exist at an inflection point, unless you are using Cauchy's flawed definition of derivative. (talk) 22:02, 9 June 2012 (UTC)

May be you do not care what textbook say, but the Wikipedia policy is to rely on well published reliable sources. For mathematics, the main such sources are textbooks. Thus Wikipedia care about what textbooks says.
On the other hand, it seems that you believe that mathematical notions have only one definition which may not change when mathematics evolves. This is not the case. In the case of "tangent", you are right on a point. The definition of this notion which is in use nowadays, is not the one which were given before Leibniz. I have edited the history section to point this fact. But nowadays the definition of a tangent relies on the derivative and the definition of the derivative is that of Cauchy or a variant of it. If there are "ignorant mathematicians", they are those who ignore that. D.Lazard (talk) 09:07, 10 June 2012 (UTC)
You are not correct in a few of your assertions. The derivative relies on the definition of the tangent and not the other way round. The problem however, is that mathematicians have been using Cauchy's wrong calculus for so long, that a change to the original definition was attempted to justify contradictions in his calculus. For example, the inflection point on the cubic at which it is impossible by the original definition to construct a tangent. To say that the tangent relies on the derivative is not only ridiculous, it is absurd. In fact, Cauchy's definition of derivative relies (like every other definition) on the fact that a curve is continuous and smooth. In order for a curve to be continuous and smooth, it must have no disjoint paths and it must be possible to construct only ONE finite tangent line with DEFINED gradient at every point in a given interval except for inflection points. If the derivative replaces the tangent, then it is possible to construct infinitely many tangent lines to the cubic at the origin. All one has to do, is let the tangent pass through the origin. Again, absurd! If your response is that Cauchy's derivative corresponds to that tangent line which has zero gradient, then in the case of the cubic and according to the original definition of tangent, there are NO tangent lines at the origin, so what does the Cauchy derivative correspond to then?

Wikipedia has a description of how the tangent line is used to motivate definition of the derivative in the article on derivative.

I suggest you decide which definition you will use first. You are very confused. (talk) 11:18, 10 June 2012 (UTC), please keep in mind that you may only add text to wikipedia which is referenced from published sources. Whatever you write, if it is contested, you must support by footnotes. Also please don't re-add contested content intio article without reaching consensus in talk page. Staszek Lem (talk) 18:34, 11 June 2012 (UTC)
There was a reference (Merriam Webster's). Reaching consensus with whom? There are so many things wrong with this article. Another statement that is absolute rubbish is: "The existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as differentiability." The derivative is based on the tangent line. Its sole reason for existence is a tangent line. Also, differentiability depends on smoothness (not the other way round) and smoothness depends on the tangent line. A planar function is smooth if and only if it is continuous and exactly one tangent line (not more than one) can be constructed at every point in a given interval except at points of inflection. (talk) 13:19, 13 June 2012 (UTC)
Webster's dictionary is quoted in the historical section. It does not support any of your assertions. Moreover, a general purpose dictionary is not a reliable reference for mathematical questions. D.Lazard (talk) 13:57, 13 June 2012 (UTC)
Firstly, Lazard, your English is extremely poor and so I suggest you have someone check your grammar before you edit "anything". Secondly, nothing is "dismissed" by anyone whether alive or dead. Thirdly, check your sources. The link you provided is not Webster's. Fourthly, Pierre Fermat did not produce anything which redefines the tangent. Adequality is not accepted by mathematicians in general. In fact, adequality is nonsense. I have removed these from the history section and expect you NOT to change this section. Finally, Staszek, I am still waiting for your response as to whom one should gain consensus from. This is new to me. I was under the impression that anyone can edit Wikipedia. So why should one need consensus from anyone else? (talk) 19:10, 13 June 2012 (UTC)
If you continue behaving in this way, you will discover may other things new to you. Like being blocked for disruptive editing, if you don't stop reinserting disputed text into the article without reaching a compromise with those who object and continue insulting your opponents. The answer to your last sentence is in your next to last one. If you want to be taken serious, don't be belligerent, respect people who were here before you (and probably will be here after you). Staszek Lem (talk) 23:32, 13 June 2012 (UTC)
What an arrogant fool you are Staszek. As if the previous edits were reached with the "compromise" of others. This is why Wikipedia is not taken seriously by educators, that is, incompetent admins and sysops like yourself run the show. Mathematics is not about compromise, it is about correctness. You can leave it wrong as it is for all I care. (talk) 13:52, 14 June 2012 (UTC)
You have to supply proof it is wrong, by citing modern scholarly sources. The fact that previous edits were possibly not very well discussed does not mean that subsequent edits must be just as careless. Wikipedia is taken seriously by many educators which are not pompous and arrogant. In fact many educators actively contribute to wikipedia and encourage their students to do so. Admins and sysops (who are not me) run the show only to the extent of enforcing wikipedia rules and policies. Admins who abuse their power, e.g., in order to enforce their personal point of view in articles, are punished. No admin has more power with respect to content. Only solid arguments, supported by modern respectable publications, are taken into account. So far you provided close to none. After all, we are talking mathematics here. If you claim something is incorrect, just prove it mathematically. Staszek Lem (talk) 16:22, 14 June 2012 (UTC)
To the IP user: Please read carefully WP:no personal attack and WP:Civility. If you continue personal attacks like in your last posts, you may be blocked from editing. This will not be done by Staszek Lem nor by me, but by an administrator. D.Lazard (talk) 16:55, 14 June 2012 (UTC)

Old definition dismissed?[edit]

The History section has:

"The first definition of a tangent was "a right line which touches a curve, but which when produced, does not cut it".[1] This old definition prevents inflection points from having any tangent. It has been dismissed and the modern definitions are equivalent to those of Leibniz. "

Where is it published that the old definition was dismissed? That's not true to my knowledge. Inflection points do not have tangent lines. (talk) 09:08, 9 June 2014 (UTC)

In every modern text book, tangent are defined by using derivatives. Inflexion points are properties of second derivatives. Therefore, when considering tangents, is is unnecessary complicated to consider that inflexion points do not have tangent. For example, in mechanics, the tangent to the trajectory of a mobile point is defined as the derivative of the velocity. The direction of the tangent is the instantaneous direction of the movement. It would be a nonsense to consider that a vehicle making zig-zags hat not any instantaneous direction at some points. Therefore, in modern mathematics, an inflexion point has a tangent. D.Lazard (talk) 09:27, 9 June 2014 (UTC)
There is not a single modern text book that dismissed the original definition of tangent line. Also, tangent lines are not defined using derivatives, but the other way round. The derivative does not exist without the tangent line.

Unless you can back up your false claim with a reference, you must remove everything the articles says about "dismissing" anything. As usual, Wikipedia is not encyclopedic, it is the view of mainstream academia, who run and control the site. (talk) 18:39, 9 June 2014 (UTC)

I find it quite funny that no one can edit Wikipedia pages anymore. The Wikipedia Moronica is the encyclopedia of mainstream ideas. Still waiting for a reference that states the definition of tangent line was 'dismissed'. Absolutely Hilarious! Not a single user on this site would be able to dismiss anything. (talk) 07:45, 12 June 2014 (UTC)

history of tangents[edit]

The only reference to tangents that MacTutor makes is the following:

In 1649 Queen Christina of Sweden persuaded Descartes to go to Stockholm. However the Queen wanted to draw tangents at 5 a.m. and Descartes broke the habit of his lifetime of getting up at 11 o'clock. After only a few months in the cold northern climate, walking to the palace for 5 o'clock every morning, he died of pneumonia.

This does not seem to support the claims just added to the history section on Descartes and Fermat by User:Salix alba. Is there any source for these claims? Tkuvho (talk) 18:58, 1 June 2015 (UTC)

Whoops got the wrong reference, I was in a rush and though I got it from a different website. It should have been from History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball[1].--Salix alba (talk): 21:50, 1 June 2015 (UTC)
Rouse Ball claims that
Descartes proposed to substitute for this a statement equivalent to the assertion that the tangent is the limiting position of the secant; Fermat, and at a later date Maclaurin and Lagrange, adopted this definition.
However, Rouse Ball does not give any source for such a claim on Fermat. I have not seen the definition via secants in Fermat though it does appear in his contemporary Beaugrand. No modern historian that I have seen follows Rouse Ball on this with the exception of Barner, and Barner himself admits he does not understand why Fermat used what Barner describes as an "inconsistent procedure" rather than what Barner claims was the secant method he should really have been using, and Barner concludes by admitting that he "does not know" (see adequality). Tkuvho (talk) 07:03, 2 June 2015 (UTC)
Note also that Descartes' book La Géométrie dates from 1637. Fermat's method of adequality that in particular allowed Fermat to find tangent lines dates from before that, to his work at Bordeaux in 1629 as well as manuscripts distributed in the mid-1630s. Rouse Ball's claim that Fermat followed Descartes on tangents is historically untenable. Tkuvho (talk) 07:32, 2 June 2015 (UTC)
Note also the WP:weasel wording of Rouse Ball's comment: he does not claim that Descartes gave the secant definition, but only that he had a "statement equivalent to the assertion". Anything can be equivalent to that assertion. Tkuvho (talk) 07:34, 2 June 2015 (UTC)