Talk:Adequality

Vote for deletion

This article is completely non-noteworthy. What Fermat supposedly did, is exactly what you still do today in calculus.

  0 ~ {b(x+e)-(x+e)^2-(bx-x^2)}/e
  0 ~ b - 2x - e    {same as f'(x)=0~b-2x-dx}
  b ~ 2x + e        {same as b = 2x }

166.249.130.7 (talk) 00:01, 15 June 2012 (UTC)

This issue has been resolved. A majority of the editors are not in favor of article deletion. The material on a possible new version of the article has been moved to the subsection Talk:Adequality#New version below. Tkuvho (talk) 13:17, 20 February 2013 (UTC)

february thread

This seems dubious to me. Adequality has been interpreted by some scholars to mean "approximate equality". is a red flag: in maths, things mean something or they don't William M. Connolley (talk) 19:11, 7 February 2011 (UTC)

A controversy exists concerning the meaning of Fermat's adequality. Weil is probably the most famous author to have written about it, and his views are reflected in this article. I hope to add additional references eventually. You can do it, as well :) Tkuvho (talk) 19:16, 7 February 2011 (UTC)

Unfortunately, I don't know what it means. Would it be better if expressed as "In the history of mathematics..."? William M. Connolley (talk) 19:18, 7 February 2011 (UTC)

One view is that it represents "approximate equality" in the sense of "equality up to an infinitesimal". Thus, 2x+dx adequals 2x, which allows us to compute the derivative of x². Tkuvho (talk) 19:20, 7 February 2011 (UTC)

I just found this article and find in mystifying. What is 'd' in '2x+dx'? it is not explained. Algr (talk) 19:08, 14 April 2011 (UTC)

Fermat uses two different latin words in place of the the equals sign = , namely: aequabitur and adaequabitur. Aequabitur is used when the equals sign stands between two constants or when the equation is universally valid. And he uses adaequabitur when the equation describes a relation between two variables which is not universally valid. Typical examples are (x+e)(b-x-e) aequabitur bx-x²-2ex+be-e², and bx-x² adaequabitur bx-x²-2ex+be-e²; or x³+y³ aequabitur (x+y)(x²-xy+y²) and x³+y³ adaequabitur 3xy. I have shown this in my paper Barner, Klaus: Fermats <<adaequare>> - und kein Ende?, Mathematische Semesterberichte 58 (2011), 13-45, which is unfortunately written in German. Fermat's works about maxima and minima and about tangents were written when he was very young (ca. 21 years old) and neither Descartes' nor Fermat's own work on analytic geometry was already created at that time. In Viete's work all equations between different terms (of one or more variables) are universally valid. They never describe curves or other relations where the variables are not independent. The young Pierre Fermat (1607-1665) broke new ground when he started his mathematical career in Bordeaux in 1628. There are only equations and nothing else (like pseudo-equations or counterfactual equations) in his work on maxima and minima or tangents. Klaus Barner (talk) 11:41, 4 February 2013 (UTC)

Here "dx" should not be thought of as a product but rather as a single notation. This notation was developed by Leibniz, see Leibniz notation. It denotes an infinitesimal change in the x-variable. For example, if position s depends on time t, one would calculate the velocity by using a tiny change dt in time, calculate the corresponding change ds in position, and form the quotient ds/dt to get the velocity. In each case, the "d" alludes to the change in the corresponding variable. Tkuvho (talk) 04:50, 15 April 2011 (UTC)

It is mentioned above that there is a controversy about what adequality means, and perhaps we should do a bit better to cover both sides and discuss the controversy itself. I have added some references that take a different point of view from Stillwell and Katz&Katz, but probably they are not worked in as well as could be. Thenub314 (talk) 07:06, 30 May 2011 (UTC)

You should read the article from 1968 by Stromholm, where he preventively demolishes I. Kleiner and N. Movshovitz-Hadar. Kleiner is no historian. You have not yet answered my query at the bottom of the page. Tkuvho (talk) 13:22, 30 May 2011 (UTC)

I will take a look at it, but even if he does, it is not a reason not to report on what they say. Also, more recent papers have continued to agree and disagree with what Stromholm said. Overall we should not say that Fermat did or didn't did anticipate the work of Robinson, we should say that the experts disagree on this point, and summarize the various positions. I thought the question below was rhetorical, but I will give you my opinion as soon as I have a bit of time to. Thenub314 (talk) 13:56, 30 May 2011 (UTC)

Copyright violation

I just removed part of a section which was a copyright violation, being copied directly from the source with maybe one word changed.--JohnBlackburne^words_deeds 19:30, 16 April 2011 (UTC)

Brief quotations are fine. Is this going to be a redux of A. H. Lightstone? Tkuvho (talk) 19:43, 16 April 2011 (UTC)

To make it clear, quotations are allowed but they should be brief relative to the size of the articel, should be clearly marked as quotes (with attribution and source), should quoted verbatim and should only be used where encyclopaedic content won't do: where they express a particular POV, for example. They should not be used in place of proper encyclopaedic content.--JohnBlackburne^words_deeds 19:49, 16 April 2011 (UTC)

It is true that the article is currently short. Why don't you add to it instead of artificially creating problems? Tkuvho (talk) 04:27, 17 April 2011 (UTC)

I am simply removing a clear copyright violation, and restoring and ISBN I added which you keep removing for no good reason. Please stop adding this: it is not a proper quote, or being used as a quotation – see WP:Quotations for the policy on quotes. Two editors have told you this now, so there is clearly also a consensus for its removal (not that there needs to be for a copyright violation).--JohnBlackburne^words_deeds 08:21, 17 April 2011 (UTC)

I agree that it was a COPYVIO; quotes of this length are OK but only if marked as quotes with the source given. On the other hand, it should probably have been reworded rather than just removed.--RDBury (talk) 17:48, 17 April 2011 (UTC)

I might have if I understood the topic well, but I was looking at the source because I couldn't understand it – it seemed to be missing quite a lot of detail – while the rest of the article is even less clear to me. So I had no choice but to remove it. With what's there now being unclear and what was removed a copyvio it could do with rewriting and expanding by someone familiar with the topic.--JohnBlackburne^words_deeds 18:01, 17 April 2011 (UTC)

Kleiner

Do you have a more precise quote from Kleiner? Tkuvho (talk) 04:06, 27 May 2011 (UTC)

Sure. I should note he doesn't use the phrase adequality, but use 'approximately equal'.

Fermat's method was severely criticized by some of his contemporaries. They objected to his introduction and subsequent suppression of the mysterious e. Dividing by e meant regarding it as not zero. Discarding e implied treating it as zero. This is inadmissible, they rightly claimed. In a somewhat different context, but with equal justification, Bishop Berkeley in the 18th century would refer to such e's as "the ghosts of departed quantities," arguing that "by virtue of a twofold mistake. . . [one] arrive[d], though not at a science yet at the truth"([13], p. 428).

Thanks, Thenub. I am interested in the phrase "Discarding e implied treating it as zero. This is inadmissible". If that's the case, wouldn't the standard part function solution to the definition of the derivative, also be inadmissible? What "st" does is, in effect, to discard the "e". Tkuvho (talk) 07:22, 27 May 2011 (UTC)

I don't see how this relates to the standard part precisely. To begin with Fermat never claimed e was infinitesimal, or even small, so we already already making a big assumption about the past to assume this is what he meant. Even if we accept it you need a reason to discard an infinitesimal quantity. Why is not the answer he gets before he discards e the actual maximum, etc. There is a big difference between applying the standard part function because it is part of your definitions, and discarding unwanted terms. Thenub314 (talk) 04:14, 31 May 2011 (UTC)

The issue is not the nature of Fermat's e. The issue is Kleiner's claim that to drop this e amounts to setting it equal to zero. This is rejected in strong terms already in the 1968 paper mentioned above. Tkuvho (talk) 04:42, 31 May 2011 (UTC)

deletion of stillwell reference

Why did user thenub delete the reference by Stillwell? He is a respected and widely-read author. Tkuvho (talk) 07:43, 3 June 2011 (UTC)

Same reason I removed Kleiner, we had other references that said more or less the same thing. Specifically, Stillwell was being cited with regaurds to the derivative of y=x². When I decided to follow some of the other books I was looking and and take Fermat's own example, we weren't using Stillwell to support that part anymore. But I have added it back, I have no particular qualms with the reference. Thenub314 (talk) 08:08, 3 June 2011 (UTC)

Thanks for this. I would add Kleiner back in, as well. We might as well comment on what Stillwell wrote. I would summarize it as follows: Fermat's technique was well ahead of his time. In fact, it was not fully implemented until the advent of non-standard analysis (which Stillwell mentions explicitly in his comment on adequality). Since you have recently reverted a number of my edits, I prefer to discuss this here rather than engaging in an edit war. Tkuvho (talk) 08:17, 3 June 2011 (UTC)

Snell's law

If as you claim Grabiner claims that he used adequality to prove Snell's law by a variational argument, wouldn't this imply that he thought of the increment e as a variation? Tkuvho (talk) 07:55, 3 June 2011 (UTC)

I don't think I would go that far, she says exactly: "Though the considerations that led Fermat to his method may seem surprising to us, he did devise a method of finding extrema that worked, and it gave results that were far from trivial. For instance, Fermat applied his method to optics. Assuming that a ray of light which goes from one medium to another always takes the quickest path (what we now call the Fermat least-time principle), he used his method to compute the path taking minimal time. Thus he showed that his least-time principle yields Snell's law of refraction [7] [12, pp. 387-390]" I can double check what the reference and see what he was actually doing, but if it were variational methods, that would have put all doubt aside as to what his method was about a long time ago. Thenub314 (talk) 08:14, 3 June 2011 (UTC)

I can't see how it could possibly be anything other than passing from an energy or length extremality to the corresponding Euler-Lagrange equation. On the other hand, I don't think this would put all doubt aside. What did you mean exactly? Why would an Euler-Lagrange interpretation out all doubts aside? Tkuvho (talk) 08:19, 3 June 2011 (UTC)

Which work by Grabiner are you citing? The reference you added to the article has the page range 195–206 , which is not compatible with the pages you mentioned above. Tkuvho (talk) 08:21, 3 June 2011 (UTC)

Sorry to be unclear, those are the page numbers for the reference she is citing. (Which is Mahoney in this case.) I am quoting from page 197.

If Mahoney discusses Fermat's application of adequality to obtain a variational principle, how coherent is it to assert that he thinks adequality purely algebraic? The source of adequality is algebraic, but that's different from what you wrote. Tkuvho (talk) 08:29, 3 June 2011 (UTC)

I checked out Mahoney, there is nothing even approaching variational arguments. The version of snell's law involved is that of a boundary between two materials with a constant index of refraction. That is the diagrams should look something like the image on the right.

Knowing that the diagram looks like this you can find an algebraic expression for the time it would take light to travel along that path as a function of the intersection point along the interface. You can then minimize this expression. Mahoney is quite clear in several places that (and this is his opinion mind you, not mine) Fermat was not doing calculus either with infinitesimals or limits (and yes some sources claim limits!). If people see calculus problems that people see reading his work, it is because they know calculus. Thenub314 (talk) 15:39, 3 June 2011 (UTC)

Misrepresentation of Mahoney

Thenub's edit misrepresent Mahoney's position, without even bothering to provide a page number. Tkuvho (talk) 08:08, 3 June 2011 (UTC)

I added a quote and a page number. Let me know if that is sufficient. Thenub314 (talk) 08:23, 3 June 2011 (UTC)

Fermat himself said it comes from the theory of equations, namely Diophantus. But he applied it in a different context. As I already mentioned, Mahoney also wrote that approximate equality is one of the meanings of the term, so your presentation is misleading. Tkuvho (talk) 08:25, 3 June 2011 (UTC)

Well Mahoney has a someone longer version involves a few more ingredients then just Diophantus, but your point is taken. But to listen to Mahoney describe it, the method is still primarily about double roots of polynomials, at least that was the impression I got. I noticed the part about approximate equality, but I thought the more contested part of Mahoney was that he was clearly stated a few times that Fermat's method was not infinitesimal/limit based. He has this opinion that it is much closer to the classical Method of false position which is more or less (he claims) what Diophantus was doing. No particular smallness reasoning required. Thenub314 (talk) 08:43, 3 June 2011 (UTC)

maxima

"Maxima" seems to be the generally accepted plural. Do they use "maximums" in England? Tkuvho (talk) 08:27, 3 June 2011 (UTC)

Don't know, never lived there. According to Webster's dictionary either form is acceptable, but feel free to change it to maxima if you prefer. (Ironically the dictionary in Firefox recognizes maximums but not maxima, which surprises even me.). Thenub314 (talk) 08:46, 3 June 2011 (UTC)

Syntax

Not sure if this sentence is correct 'his method with sufficient clarity of completeness to determine' shouldn't that be 'sufficient clarity [or] completeness'? SamCardioNgo (talk) 12:54, 30 October 2012 (UTC)

This section could be improved, too. Give it a try! Tkuvho (talk) 10:38, 31 October 2012 (UTC)

Dubious Scholarly controversy content

I just removed the following from the article, as it has a number of issues. It is unsourced and any controversy should be reliably sourced to properly establish the holders of the viewpoints. It's seems to be rather than reporting the controversy putting one particular point of view, describing certain views as 'misleading' and 'nonsense', so is both point of view and unencyclopaedic. And articles should never be signed: signed statements belong on the talk page.--JohnBlackburne^words_deeds 20:50, 7 February 2013 (UTC)

H. Breger however holds the view that Fermat used the word adaequare in the sense of "to put equal" and has to be represented by the usual equals sign =. K. Barner shares Breger's view and shows that Fermat uses "adaequabitur" in place of "aequabitur" when the equation describes a relation between two variables which is not a universally valid formula. The misleading introduction of the symbol $\backsim$ by Paul Tannery had the consequence that many authors tried to give this symbol a proper name: almost equal, approximately equal, pseudo-equal, counterfactually equal, compared by adégalité, as nearly equal as possible. That, however, is simply all wrong. The meaning is equal. Klaus Barner (talk) 20:13, 7 February 2013 (UTC)

This is one interpretation found in the recent literature. Giusti's recent article relies exclusively on Latin sources (rather than Tannery's translation and its \backsim) and does not seem to subscribe to the "=" interpretation. Tkuvho (talk) 16:16, 10 February 2013 (UTC)

I was aware of this. But Giusti is definitely wrong. He shares Mahoney's absurd opinion. I discuss all this in my paper Fermats <<adaequare>> - und kein Ende? I am astonished that so many historians of mathematics are not able to read (not to write or to speak) German which was one of the most important languages in the history of science. I am German, however, I read English, French, Italian, Spanisch, Latin and classical Greek, and sometimes, whith the help of my wife, Russian. I could not do research in that field without being able to do so. If you send me your mail address I will send you the pdf-file of my paper. My address is klaus.barner@uni-kassel.de. Klaus Barner (talk) 18:59, 10 February 2013 (UTC)

I continue the dubious scholarly controversy: Fermat borrowed from Bachet de Meziriac's Latin translation of Diophantus’ Arithmetika the noun adaequalitas and the verb adaequare where they have a meaning in a completely different context, namely solving a special class of arithmetical problems by the method of false position. Fermat, however, uses those words (and the word adaequabitur) in his method of determining maxima and minima of algebraic terms and of computing tangents of conic sections and other algebraic curves. Fermat’s method is purely algebraic. There is no "infinitesimal error term". Fermat uses two different Latin words for "equals": aequabitur and adaequabitur. He uses aequabitur when the equation describes the identity of two constants or is used to determine a solution or represents a universally valid formula. Example:

$x^{3}+y^{3}=(x+y)(x^{2}-xy+y^{2})$.

And he uses the word adaequabitur ($\doteq$) when the equation describes a relation which is no valid formula. Example (Descartes' folium):

$x^{3}+y^{3}\doteq 3xy$.

Typical examples are:

$(x+e)(b-x-e)=bx-x^{2}-2ex+be-e^{2}$

and

$be\doteq 2ex+e^{2}$.

Equations which describe relations between two variables (x and y, x and e) were unknown to Vieta. So the 21-year-old Fermat felt compelled to introduce a new concept of equality and called it adaequalitas. Later, when he created his analytic geometry he abandoned this special terminology of his younger days. See my paper Barner, Klaus: Fermats <<adaequare>> - und kein Ende? Math. Semesterber. (2011) 58, 13-45, unfortunately written in German. Klaus Barner (talk) 17:33, 11 February 2013 (UTC)

I am dubious with the preceding posts. Firstly, as far as I remember my old Latin courses, the prefix "ad" means "to tend toward". Thus the difference between aequabitur and adaequabitur is exactly the same as the difference, in modern mathematics between the = of an identity and the = of an equation (in the latter, things are not equal, but one want to make them equal). Trying to extract more meaning to this difference of terms is simply silly or anachronistic (at that time, mathematics were much less formalized as now).

Secondly, the assertion that "Fermat’s method is purely algebraic" is a pure anachronism: at that time, mathematicians did not try to classify the methods they use into algebraic, geometric and analytic. The adjectives algebraic and analytic were not even introduced.

Thirdly, the method that Fermat did use (as described in the article) is exactly the one that is thought in every calculus course to prove the formulas for the derivative. Is that purely algebraic?

Fourth, at Fermat's time, it was highly common to use geometry for effective numerical computations (graphical abaci, by the way it is strange that WP is almost empty about these historical important tools). It is thus sure that Fermat did use his geometrical intuition to find his method, that is that a tangent is a "line passing through two infinitely close (very close) points". IMHO, the very new idea of Fermat was to transcribe this intuition (which was certainly not new, because abacists did draw many tangents) into a computational method. To decide if this is infinitesimal computation or not is a matter of taste, nothing more.

All the content of the article is dubious for the same reasons. D.Lazard (talk) 18:07, 13 February 2013 (UTC)

Dear User Lazard, I think that your arguments are worth a reply. Unfortunately I am German and my English is weak. There is much misunderstanding behind your arguments.

Firstly, you are completely right interpreting the prefix "ad" concerning the semantic meaning in the Latin language. But now, here ist your error. It should read: "Thus the difference between aequabitur and adaequabitur is exactly the same as the difference, in modern mathematics between the = of an identity and the = of an equation which describes a relation between to variables (x ad e or, in the case of Descartes' folium, x and y), which is no identity. The algebraic terms $x^{3}+y^{3}$ and $nxy$ are definitely not equal. However, if you put them equal you get an algebraic curve, the folium cartesii. Fermat considers this equation and writes "Il faudra comparer, par adéquation, le deux cubes CF, FE avec le solide compris sous N, FC, FE." OEuvres, Vol. II, p. 156. However, this equation describes a famous curve. And Fermat succeeded in computing tangents at this curve (Descartes' challenge) which Descartes failed to do. And if you would check all examples where he applies his method of maxima and minima or of computing tangents in Fermat's work you would see: at that moment where he changes from a valid formula to a relation between two variables (A and E) which is not universally valid he changes from aequabitur to adaequabitur. And when he finally arrives at the solution which is a equation between two constants he again changes back to aequabitur. Read Fermat's work in Latin, not Tannery's translation, you will see that I am right. You will probably not believe me until you have seen it with your own eyes.

Secondly, you blame me calling Fermat's method purely algebraic being "a pure anachronism". That is absurd. I do not claime that Fermat himself calls his method purely algebraic. Of course, he did not so. It is the result of my own analysis formulated with my own words using modern terminology. If you blame this then, in your eyes, analysis of historical mathematical texts with modern methods and words is impossible. Let us consider Fermat's usual introductory example. For shortness I will use the symbol $=$ for aequabitur and the symbol $\doteq$ for adequabitur. Fermat wants to determine the maximum of the term (the concept of function is unknown to him) $x(b-x)$ (i.e. $bx-x^{2}$) in the interval $0\leq x\leq b$. He replaces $x$ by $x+e$ and computes:

$(x+e)(b-x-e)=bx-x^{2}-2ex-be-e^{2}$,

which is a valid mathematical formula. Next he puts the terms $bx-x^{2}$ and $bx-x^{2}ex-be-e^{2}$ equal:

$bx-x^{2}\doteq bx-x^{2}-2ex-be-e^{2}$.

This is a relation between the variables $x$ and $e$ which is no valid formula. Substracting $bx-x^{2}$ on both sides he arrives at:

$be\doteq 2ex+e^{2}$.

Now cancelling the common factor $e$ Fermat arrives at the relation

$b\doteq 2x+e$.

Now follows Fermat's final step: he puts $e=0$ (=nihil) which gives the determining equation

$b=2x$.

Each step is purely algebraic. It is always the same with the applications of Fermat's method, it follows always the same scheme. It is a purely algebraic method.

Thirdly, "the method that Fermat did use (as described in the article) is exactly the one that is thought(?) in every calculus course to prove the formulas for the derivative." Indeed, this claim is a pure anachronism.

Fourth, if one wants to prove the validity of Fermats "method" (especially the last step) it works in all cases with Ulisse Dini's implicit function theorem. But of course: Fermat does not use this theorem. --Klaus Barner (talk) 20:12, 14 February 2013 (UTC)

I am not a native English speaker. In my preceding post, I have badly expressed my opinion. I'll try to be clearer. Firstly, I am interested in the mathematics of Fermat, not in the philosophy of his mathematics (epistemology). I am not sure that historical epistemology is a field that is sufficiently established to deserve Wikipedia articles. Similarly, the Fermat's thoughts about his methods could be interesting but will never been known. Thus, I'll strictly concentrate on his computations, because his method is a computing method. Also, I'll describe his method in the same way that I describe the content of badly written mathematical articles, which I have frequently to review.

If one transcripts the computation made by Fermat into the modern language of functions (I agree that Fermat did not have it), we get the following .

Starting from a function f, we get two functions g and h such that

$f(x+e)-f(x)=eg(x)+e^{2}h(x,e)$

(This is a variant of Taylor's theorem for analytic functions.) Then we divide by e to get

$g(x)+eh(x,e).$

Then, there are two ways to describe the last step: either one simply put e=0 to get g(x); or one makes e tend to 0 to get also g(x).

IMHO, all the discussion about the difference between aequabitur and adaequabitur consists in trying to decide if Fermat did meant "putting e=0" or "making e tend to 0". For me this discussion is irrelevant, because the important thing is what Fermat did, not what he thought (which, in any case, may not be decided). What I see, is that he has introduced a systematic method to apply Taylor's theorem on various examples, more than 50 years before Taylor. D.Lazard (talk) 18:36, 15 February 2013 (UTC)

Dear Lazard, I definitely like this discussion with you. But sometimes I wish you would more carefully consider my arguments. Fermat had no conception of the continuum which is familar to us since Cantor, Dedekind and Weierstrass. He had no conception of function, of convergence, limit, and continuity. The concept of the local slope of a curve and derivative are far from his world of ideas. There is no infinitesimal in all his treatises and letters on maxima and minima and tangents. So the answer to your question is clear: Fermat ment "putting e=0". He gave a very clear account of how he developed his "method" in his paper Methodus de maxima et minima (OEuvres, vol. I, pp.147-153). His ideas, developed during his study of Viète, are purely algebraic.

When Fermat did his work on maxima and minima and tangents in 1628 he was about 21 years old and was in Bordeaux, far from Paris, far from the center of scientific activities. He had carefully studied Viète's works in the famous library of his friend Étienne d'Espagnet. However, in Viète's mathematics there are only three kinds of equations: equations where the "aequabitur" (throughout used by Viète) is between two constants, determining equations (for instance quadratic equations), and universally valid equations (Bourbaki would say théorèmes) like

$x^{3}+y^{3}=(x+y)(x^{2}-xy+y^{2}).$

Fermat however was confronted with equations which were not universally valid but relations describing curves like

$x^{3}+y^{3}\doteq n\,x\,y,$

the folium cartesii. Fermat never met an equation like this in Viète's work and did not dare to write aequabitur in this case. So he used the verb adaequare and the noun adaequalitas (and the corresponding French words). Consequently Fermat wrote: "Il faudra comparer, par adéquation, le deux cubes CF, FE avec le solide compris sous N, FC, FE. "Making e tend to 0"??? Of course, also equations like

$b\,e+2\,b\,x\doteq 3\,x\,e+3\,x^{2}+e^{2}$

are relations of two variables (x and e) and no universally valid formulas. Of course, we would like "making e tend to 0". But Fermat puts e=0. Dear Lazard, don't read the experts, read the original: Fermat! --Klaus Barner (talk) 13:37, 16 February 2013 (UTC)

Some comments:

I have never pretended that Fermat did have the notion of function. I have used function in my description of what Fermat did because it was the simplest way to describe his general method without considering explicitly each case of application. In fact, I would have better to use "expression" depending on some variables, which is exactly what Fermat consider.

IMO, comparer par adéquation should be understood as "finding the condition for having equality" or, in other words "solving the equation in term of an expression". I agree that this was really new for the time of Fermat.

By the way, you use improperly the word "equation", which means (in modern mathematics) "equality to be solved". Therefore the three kinds of "Viète's equations" are in modern terminology, the equalities between constants, the equations to be solved in term of a constant and the identities (they are not at all theorem, they are only parts of theorems that specifies the set of values for which they are true). Using improperly the mathematical terminology is not the right way to explain Fermat's work. In this case, the correct modern terminology to describe what Fermat did appear to be "solving an equation in term of an expression"

Putting e=0 or letting e tend to 0: As far as I know (I have not read Fermat, only the article and your comments), Fermat did describe a method of computation, but did not explain nor prove why the result is correct. For the computation, it is clear that he (and anybody else) puts e=0. For the explanation he certainly had in mind, I can not imagine how he could justify his method without any notion of limit. I agree that the notions of limit, derivative, function, ... were totally ignored by the algebraists (by opposition to geometers) of that time. But, the notions of slope and limit were certainly known by the graphical abacists that constructed machines to draw elaborated curves. Before to assert that Fermat had no notion of limit, one should examine his knowledge about abaci and his relations with abacicts. Is there an historian that has considered this aspect of Fermat's biography?

To conclude, I have to say, that I am not at all an historian of mathematics, but a mathematician more interested in effective computation than in proof of abstract general theorems (what happens in that case rather than what can and can not happen). For this reason, I feel closer from mathematician of XVIIIth century and earlier than from Bourbaki. This the reason why, when I read the description of Fermat method I understand it as a method of computation rather than a proof of theorem, which it is certainly not. I should also confess that my knowledge in history of mathematics is extremely weak. For Fermat's time, it comes mainly from the French habilitation of Dominique Tournès about the graphical computation about which I was a reviewer. The main part of the habilitation was about a more recent period than Fermat's, but he emphasized that the tradition of graphical abaci date from the Middle Age, were it was used to obviate the difficulty of computing with Latin numeration. It appear also that the importance of this tradition and its development until the beginning of XXth century has been strongly underestimated by the historians. D.Lazard (talk) 16:20, 16 February 2013 (UTC)

Dear Lazard, you have not read Fermat. How can you say all this? I will not comment it at all. I am a historian of mathematics (and a number theorist). I worked on Fermat's life and work since more than 12 years. I have published a series of papers on Fermat's life. Some of them are reviewed in Mathematical Reviews and Zentralblatt. My paper "Fermats <<adaequare>> - und kein Ende?" Math. Semesterber. 58 (2011), 13-45, was checked and aknowledged before handing in for publication by four experts, Professors Herbert Breger (Hannover), Menso Folkerts and Ivo Schneider (both Munich), Michael von Renteln (Karlsruhe). The paper was then peer-reviewed by two experts. One of them, Prof. Niels Jahnke (Bochum), later outed himself to me and wrote: "In the meantime I have read your revised version and find it very compact and stringent. It would be fine if the "thoughtless" descriptions of Fermat's method now would become fewer. Though, I do not expect that." And the leading Fermat expert, Prof. Roshdi Rashed (Paris), wrote to me: "Je tenais à vous remercier pour ce travail de très grande qualité." Unfortunately I feel unable to convince you. --Klaus Barner (talk) 20:00, 16 February 2013 (UTC)

Editors should keep in mind that work on wiki pages is primarily based on secondary sources, rather than primary sources. According to the rules, User:D.Lazard is entitled to work on the article even if he never studied Fermat in the original. Tkuvho (talk) 14:03, 20 February 2013 (UTC)

That is formally correct, but not very encouraging for me to work for Wikipedia. I spent much time answering User:D.Lazard and explaining the subject to him. And I took his arguments seriously. --Klaus Barner (talk) 16:30, 20 February 2013 (UTC)

There might be good reasons for this. The Math. Semesterber. piece suffers from serious weaknesses which make it less convincing. For example, the author's interpretation forces him to assume that Fermat made a systematic mistake in confusing secant and tangent lines in his presentation. Other interpretations do not suffer from such a weakness. Tkuvho (talk) 18:06, 20 February 2013 (UTC)

O Mikhail! Si tacuisses, philosophus mansisses. Boethius --Klaus Barner (talk) 21:53, 1 March 2013 (UTC)

New version

I plan to write a completely new text und temporarily "park" some citations here.

Heinrich Wieleitner (1929): "Fermat ersetzt A durch A+E. Er setzt dann den neuen Ausdruck dem alten angenähert gleich, nimmt die gleichen Glieder auf beiden Seiten weg, dividiert durch die höchstmögliche Potenz von E, lässt dann alle mit E behafteten Glieder weg und setzt dann das, was übrig bleibt, wirklich gleich. Daraus ergibt sich A. Auch dass E möglichst klein sein soll, wird nirgends gesagt, und ist höchstens durch den Ausdruck "adaequalitas" ("adegalité") zum Ausdruck gebracht." Wieleitner uses the symbol $\sim$.

Max Miller (1934): "Sodann setze man, wie Diophant sagt, die beiden das Maximum oder Minimum ausdrückenden Gleichungen einander näherungsweise gleich". Miller uses the symbol $\approx$.

Jean Itard (1948): "On sait que l'expression <<adégaler>> est reprise par Fermat à Diophante, traduit par Xylander et par Bachet. Il s'agit d'une égalité approximative". Itard uses the symbol $\backsim$.

Joseph Ehrenfried Hofmann (1963): "Fermat wählt eine hinreichend klein gedachte Grösse h und setzt f(x+h) ungefähr gleich f(x). Sein Fachausdruck ist adaequare." Hofmann uses the symbol $\approx$.

Peer Stromholm (1968): "The basis of Fermat's approach was the comparition of two expressions which, though they had the same form, were not exactly equal. This part of the process he called "comparare par adaequalitatem" or "comparer per adaequalitatem", and it implied that the otherwise strict identity between the two sides of the "equation" was destroyed by the modification of the variable by a small amount:

$\scriptstyle f(A){\sim }f(A+E)$.

This, I believe, was the real significance of his use of Diophantos' párison, stressing the smallness of the variation. The ordinary translation of 'adaequalitas' seems to be "approximate equality", but I much prefer "pseudo-equality" to present Fermat's thought at this point."

Claus Jensen (1969) Moreover, in applying the notion of adégalité - which constitutes the basis of Fermat's general method of constructing tangents, and by which is meant a comparition of two magnitudes as if they were equal, although they are in fact not - I will employ the nowdays mor usual symbol $\approx$.

Ivo Schneider (1971) Der von Fermat als Vorschrift bezeichnete Algorithmus besteht nun aus mehreren Schritten der Umformung einer Art von Gleicheit bzw. Gleichung, die Fermat mit dem Terminus adaequalitas beschreibt. Die begriffliche Scheidung in aequalitas und adaequalitas mußte Fermat vornehmen, weil die einzelnen Schritte seiner Vorschrift Umformungen verlangen, die mit den bekannten für Gleichungen nicht verträglich sind.

Michael Sean Mahoney (1971) "Fermat's Method of maxima and minima, which is clearly applicable to any polynomial P(x), originally rested on purely finitistic algebraic foundations. It assumed, counterfactually, the inequality of two equal roots in order to determine, by Viete's theory of equations, a relation between those roots and one of the coefficients of the polynomial, a relation that was fully general. This relation then led to an extreme-value solution when Fermat removed his counterfactual assumption and set the roots equal. Borrowing a term from Diophantus, Fermat called this counterfactual equality "adequality".

Charles Henry Edwards, Jr. (1979): For example, in order to determine how to subdivide a segment of length $\scriptstyle b$ into two segments $\scriptstyle x$ and $\scriptstyle b-x$ whose product $\scriptstyle x(b-x)=bx-x^{2}$ is maximal, that is to find the rectangle with perimeter $\scriptstyle 2b$ that has the maximal area, he [Fermat] proceeds as follows. First he substituted $\scriptstyle x+e$ (he used A, E instead of x, e) for the unknown x, and then wrote down the following "pseudo-equality" to compare the resulting expression with the original one:

$\scriptstyle b(x+e)-(x+e)^{2}=bx+be-x^{2}-2xe-e^{2}\;\sim \;bx-x^{2}.$

After canceling terms, he divided through by e to obtain

$\scriptstyle 2\,x+b\;\sim \;b.$

Finally he discarded the remaining term containg e, transforming the pseudo-equality into the true equality

$\scriptstyle x={\frac {b}{2}}$

that gives the value of x which makes $\scriptstyle bx-x^{2}$ maximal. Unfortunately, Fermat never explained the logical basis for this method with sufficient clarity or completeness to prevent disagreements between historical scholars as to precisely what he meant or intended.

Kirsti Anderson (1980): The two expressions of the maximum or minimum are made "adequal", which means something like"as nearly equal as possible".

Herbert Breger (1993): I want to put forward my hypothesis: Fermat used the word "adaequare" in the sense of "to put equal" ... In a mathematical context, the only difference between "aequare" and "adaequare" seems to be that the latter gives more stress on the fact that the equality is achieved. --Klaus Barner (talk) 13:02, 18 February 2013 (UTC)

Thanks for your input. These quotations are very interesting. The ones that are not in English would have to be translated before inclusion in the article. This is because a wikipedia page is for the general public rather than for historians of mathematics. Any changes to the article should be incremental. Thus, any new material should be added in small increments. Any specific deletions should be discussed at this talkpage first. An attempt to replace the adequality page with an entirely different text may not be helpful, nor necessarily accepted by other editors. Editors should be careful not to impose their preferred interpretation on the topic of the page, but rather seek broader coverage. The page cannot undergo a complete overhaul every time a new research article on Fermat is published. Furthermore, K. Barner's article does not seem to be the last one published on the topic. Tkuvho

Of course, The quotations that are not in English will be translated. I plan to develop the whole article and to put it forward for discussion here. Before it is not generally accepted I will not copy it into the main page. - Indeed, my article is not the last published on the topic. Mikhail Katz asked me for a copy and I sent him the pdf-file. But he was unable to read it because it is written in German. So I read his paper, however, he did not read mine. The same happened with Enrico Giusti. I read his French and Italian papers, but he could not read my article. I am very astonished that historians of mathematics who work on the mathematics of the 17th century are not in the position to read the language of Gauss, Dirichlet, Riemann, Cantor, Dedekind, and Weierstrass. One should at least have some knowledge of English, French, German, and Latin. Classical Greek, Italian, and Spanish would be fine. Can you read German? If so, have a look at my article about Fermat's life (Leben) in the German Wikipedia! --Klaus Barner (talk) 11:50, 19 February 2013 (UTC)

From some of your comments I have the impression that you are not entirely familiar with the ways wiki works. Note that new comments should be placed at the bottom of the talk page. The easiest way of doing this is by clicking on the "new section" button at the top of the page. Following this, you are asked to define the title of a new section.

Furthermore, I think it would be preferable to work with the current version of the article rather than developing an entirely new version. Developing an entirely new version implicitly assumes that there is something irreparably wrong with the current one, which may not be the opinion of the other editors. Granted, in the assumption that the truth about Fermat's adequality is expressed in the 2011 article in Semesterber. Math., the current version is unacceptable. However, I don't think that working on the basis of such an assumption is an appropriate framework for the article. Therefore my suggestion would be to work on the present version, delete material in small doses following discussion on this talkpage, and, similarly, add material in small increments. Tkuvho (talk) 13:14, 20 February 2013 (UTC)

Dear Tkuvho, this suggestion is not applicable in my case. I understand you well. However, the existing article is extremely short and meaningless. I proposed that I will write a complete new article and put it in here for discussion. Then there are two possibilities, either the article is acceptable (after improvements and corrections following the discussion) or my work was in vain. --Klaus Barner (talk) 16:49, 20 February 2013 (UTC)

Discussions on talk pages should not be viewed as an attempt by an editor to convince fellow editors of the truth of his way, but rather as a way of developing consensus. A take-it-or-leave-it-attitude is not likely to produce results. Tkuvho (talk) 17:44, 20 February 2013 (UTC)

I wrote "after improvements and corrections following the discussion". That means, that I will accept corrections and criticism. I do not change my mind. And I will not longer continue this fruitless debate. The following would have been the beginning of my text. However I will not wast my time any more. --Klaus Barner (talk) 20:24, 20 February 2013 (UTC)

Behind the word adequality (Latin: adaequalitas) hides a long story of scholarly interpretations of a technical term used by Pierre de Fermat in his seminal work of extreme values and tangents.

The Latin noun adaequalitas and the accompanying Latin verb adaequare are used by Pierre de Fermat in his treatises and letters about his method of determining maxima and minima of algebraic terms and of computing tangents to algebraic curves (like conic sections). It is a technical term within this method and does not mean the method itself.

The verb adaequare and the noun adaequalitas are used by Fermat in different idioms like adaequentur, adaequabuntur, comparare per adaequalitatem, debet adaequare, fiat per adaequalitatem, and, in his French correspondence, comparer par adéquation, however, most frequently in the form adaequabitur (third person singular passive future) which means "it will be (made) equal" (indefinite tens).

The verb adaequare is synonymous to the verb aequare (Thesaurus Linguae Latinae, Vol. I, p.562), which was used by Viète exclusively to denote equality in place of the symbol =. This symbol was invented by Robert Recorde in 1557. Viète wrote most frequently aequabitur, and Fermat adopted it from him. So Fermat used both words, aequabitur and adaequabitur, in place of the equals sign, obviously systematically and in a subtle difference of meanings. And this is the cause of a long scholarly debate which still goes on. For the purpose of this article we will denote aequabitur by $\scriptstyle =$ and adaequabitur by $\scriptstyle \doteq$. --Klaus Barner (talk) 20:29, 20 February 2013 (UTC)

The focus on the Latin term and its meaning in Latin is somewhat misplaced because Fermat himself makes the connection with Diophantus's parisotes. For the same reason the meaning of adaequalitat in Viete is of marginal relevance here. Tkuvho (talk) 21:47, 21 February 2013 (UTC)

Breger on Bachet

The current version states that "Breger questions the correctness of Bachet’s translation of those Greek words". Does User:Klaus.Barner have a source for this claim concerning Breger? Where and in what terms did Breger question the correctness of Bachet's translation? This seems to refer to translating "parisotes" as "adaequalitat" by Bachet. Tkuvho (talk) 09:20, 1 March 2013 (UTC)

That is not possible in a short answer here. Bregers whole paper (of 27 pages) has only one objective: to show that Fermat's "adaequare" means "to put equal" (that "adaequabitur" means "equals") and that "adaequalitas" and "parisótes" have completely different meanings. "Parisótes", of course, means "approximative equality" and "adaequalitas" means "equality". There is a paragraph Nr.5, The "Diophantus Passage" (pages 199-202), in Breger's paper, where he discusses the completely different context of Diophantus' "parisótes" and Fermat's "adaequalitas". When I was prepairing my paper I visited Herbert Breger in Hannover, and we had a fruitful discussion about this topic. I had understood him well, and he was convinced that my solution of the old riddle of the meaning of "adaequare" would end the discussion. However, this is an error. There is the barrier of the German language. I would be glad if you (or anybody else) would correct my mistakes concerning the English language in my article for Wikipedia. My English ist rather weak. --Klaus Barner (talk) 14:41, 1 March 2013 (UTC)