Adequality

Adequality is a mathematical term introduced by Pierre de Fermat, which, roughly speaking, means equating things that are different (the exact meaning is controversial, see below). As a mathematical concept, this term had been used by Fermat and his followers, but fell into disuse from the 18th century onward.

Adequality has been reintroduced by modern historians of mathematics to name Fermat's method of computing maxima, minima and tangents, and to discuss the exact meaning of the word in Fermat's work.

Fermat said he borrowed the term from Diophantus. Diophantus coined the term παρισὀτης to refer to an approximate equality. The term was rendered as adaequalitas in Bachet's Latin translation of Diophantus, and adégalité in French.

Fermat's method

Fermat used adequality first to find maxima of functions, and then adapted it to find tangent lines to curves.

To find the maximum of a term $p(x)$, Fermat did equate (or more precisely adequate) $p(x)$ and $p(x+e)$ and after doing algebra he could divide by e, and then discard any remaining terms involving e. To illustrate the method by Fermat's own example, consider the problem of finding the maximum of $p(x)=bx-x^{2}$. Fermat adequated $bx-x^{2}$ with $b(x+e)-(x+e)^{2}=bx-x^{2}+be-2ex-e^{2}$. That is (using the notation $\backsim$ to denote adequality, introduced by Paul Tannery):

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

Canceling terms and dividing by $e$ Fermat arrived at

$b\backsim 2x+e.$

Removing the terms that contained $e$ Fermat arrived at the desired result that the maximum occurred when $x=b/2$.

Fermat also used his principle to give a mathematical derivation of Snell's laws of refraction directly from the principle that light takes the quickest path.

Descartes' criticism

Fermat's method was highly criticized by his contemporaries, particularly Descartes. V. Katz suggests this is because Descartes had independently discovered the same new mathematics, known as his method of normals, and Descartes was quite proud of his discovery. He also notes that while Fermat's methods were closer to the future developments in calculus, Descartes methods had a more immediate impact on the development.

Scholarly controversy

Both Newton and Leibniz referred to Fermat's work as an antecedent of infinitesimal calculus. Nevertheless, there is disagreement amongst modern scholars about the exact meaning of Fermat's adequality. Edwards explains this is because Fermat never described his method with sufficient clarity or completeness to determine precisely what he intended. Fermat never explained whether e was supposed to be taken to be small, infinitesimal, or if he was taking a limit. Depending on how one reads Fermat's work, he either found an algebraic method for computing maxima of polynomials, or he began the field of infinitesimal calculus. Michael Sean Mahoney writes that Fermat's methods were essentially algebraic and not an introduction to limits or infinitesimals. On p. 164, end of footnote 46, Mahoney notes that one of the meanings of adequality is approximate equality or equality in the limiting case. Katz & Katz wrote that Fermat provided the seeds of the solution to the infinitesimal puzzle a century before George Berkeley ever lifted up his pen to write The Analyst. Katz, Schaps and Shnider develop a formalisation of Fermat's technique of adequality in modern mathematics as the standard part function sending a finite hyperreal number to the real number infinitely close to it. John Stillwell wrote:

Fermat introduced the idea of adequality in 1630s but he was ahead of his time. His successors were unwilling to give up the convenience of ordinary equations, preferring to use equality loosely rather than to use adequality accurately. The idea of adequality was revived only in the twentieth century, in the so-called non-standard analysis (Stillwell 2006 p. 91).

Analysis of "adequality" meaning in Fermat's work

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 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 with the verb aequare (Thesaurus Linguae Latinae, Vol. I, p.562). The latter word was used by Viète nearly exclusively (with very few exceptions) 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. However, Fermat used both words, aequabitur and adaequabitur, in place of the equals sign, obviously systematically and in a subtle difference of meanings.

In Fermat’s work about maxima and minima and tangents there is a place where he says that he adopted the word “adaequentur” from Diophantus. In his treatise “’’Methodus ad disquirendam maximam et minimam’’” (Fermat, Œuvres, Vol. I, pp.133-136) Fermat writes: ‘’Adaequentur, ut loquitur Diophantus, duo homogenia maxima aut minima aequalia …’’ (We make, as Diophantus says, two expressions of maximum or minimum in like manner equal.) But there are more questions than answers what that exactly means. This topic is taken up further below.

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 (and its equivalent idioms) by $\scriptstyle \doteq$.

Fermat’s example Nr. 1. (Œuvres, Vol. I, p.134) To divide the line $\scriptstyle AB$ by the point $\scriptstyle E$ this way that the product of the two segments $\scriptstyle AE$ and $\scriptstyle EB$ becomes a maximum.

Fermat’s solution: Let $\scriptstyle b={\overline {AB}}$. Next let $\scriptstyle x$ be one of the two segments. Then the other one will be $\scriptstyle b-x$. And the product, which shall become a maximum, then equals $\scriptstyle b\,x-x^{2}$. However, if the first segment is $\scriptstyle x+e$ then the second one will be $\scriptstyle b-x-e$. The product of the segments is then

$\scriptstyle (x+e)(x-b-e)\,=\,b\,x-x^{2}-2\,e\,x+b\,e-e^{2}.$.

Next Fermat puts the two terms equal:

$\scriptstyle b\,x-x^{2}\,\doteq \,b\,x-x^{2}-2\,e\,x+b\,e-e^{2}.$.

Subtracting common terms on both sides yields

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

Now Fermat [tacitly assuming that $\scriptstyle e$ is unequal zero] cancels the common factor $\scriptstyle e$ and gets

$\scriptstyle b\,\doteq \,2\,x+e.$

Finally Fermat deletes $\scriptstyle e$ and yields the solution

$\scriptstyle b\,\,=\,2\,x.$

Fermat then remarks: “The question is answered if we take

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

Nec potest generalior dari methodus.” (It is impossible to give a more general method.)

There are several questions one would like to ask. What is Fermat’s idea behind his method? Did he explain how he found it? Has Fermat proven the validity of his proceeding? And, if not, is it possible to do this by means of modern analysis? And, above all: what is the precise meaning of “adaequabitur” ($\scriptstyle \doteq$)?

Let us first hear the voices of the scholarly experts! When Paul Tannery in 1896 published his French translation of Fermat’s Latin treatises on maxima and minima (Fermat, Œuvres, Vol. III, pp. 121-156) he was obviously insecure of the meaning of adaequare and aequalitas and avoided a translation of these words from Latin into modern French. He invented instead the new French verb “adégaler” and adopted Fermat’s “adéquation”. His introduction of the sign $\scriptstyle \backsim$ for “adaequabitur” in mathematical formulas, however, had the most serious consequences.

Heinrich Wieleitner (1929): “Fermat replaces A with A+E. Then he puts the new expression roughly equal ( angenähert gleich) to the old one, cancels equal terms on both sides, and divides by the highest possible power of E. He then cancels all terms which contain E and puts that, what remains, really equal. From that A results. That E should be as small as possible is nowhere said and is at best expressed by the word "adaequalitas". (Wieleitner uses the symbol $\scriptstyle \sim$.)

Max Miller (1934): "Thereupon one should put the both terms, which express the maximum and the minimum, approximately equal (näherungsweise gleich), as Diophantus says." (Miller uses the symbol $\scriptstyle \approx$.)

Jean Itard (1948): "One knows that the expression <<adégaler>> is adopted by Fermat from Diophantus, translated by Xylander and by Bachet. It is about an approximate equality (égalité approximative) ". (Itard uses the symbol $\scriptstyle \backsim$.)

Joseph Ehrenfried Hofmann (1963): "Fermat chooses a quantity h, thought as sufficiently small, and puts f(x+h) roughly equal (ungefähr gleich) to f(x). His technical term is adaequare." (Hofmann uses the symbol $\scriptstyle \approx$.)

Peer Strømholm (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' πἀρισον, 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 $\scriptstyle \approx$.

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". (Mahoney uses the symbol $\scriptstyle \approx$.)

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 containing 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." (Mrs. Anderson uses the symbol $\scriptstyle \approx$.)

After reading these quotations one must admit that Edwards very precisely describes the state of the art of the research about the meaning of “adaequare” around 1980. No-one of the authors, including Edwards himself, has a well-founded conception what Fermat meant with “adaequare” and “adaequalitas”. And Edwards holds Fermat’s vagueness responsible for that.

The first author, who resolutely contradicts this opinion, is Herbert Breger. “Taking into account that brilliant mathematicians usually are not so very confused when talking about their own central mathematical ideas” he wrote a long paper and proposed a solution of the problem: “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.” (Page 197f. of his paper.)

Breger’s central considerations are as follows. As mentioned above, Fermat wrote in his treatise “’’Methodus ad disquirendam maximam et minimam’’” (Fermat, Œuvres, Vol. I, pp.133-136): ‘’Adaequentur, ut loquitur Diophantus, duo homogenia maxima aut minima aequalia …’’ (We make, as Diophantus says, two expressions of maximum or minimum in like manner equal.) Indeed, one finds in Diophantus’ Arithmetica several places where Diophantus uses the words παρισὀτης and πἀρισον which Bachet de Méziriac translates as ‘’adaequalitas’’ respectively as ‘’adaequale’’. Both words, which occur in the 11th and 14th problems of the fifth book of the Arithmetica, are used in context of the regula falsi which Diophantus uses to solve a certain class of Diophantine equations. The word παρισὀτης is invented by Diophantus to label this method. Fermat himself does not use these words. The meaning of πἀρισον is “almost equal” and the meaning of παρισὀτης is something like “approximation of the solution” which makes sense in the context of the regula falsi. However, Breger questions the correctness of Bachet’s translation of those Greek words. (Bachet had adopted his own translation from Xylander.) Paul Tannery translates πἀρισον with proximum and παρισὀτης with appropinquatio, which is much better.

The verbs aequare and adaequare are synonymous and mean “to make equal”. All great Latin dictionaries say unanimously: both verbs mean “to make equal”. And Breger concludes that both words, aequabitur and adaequabitur, mean “equals”. He provides many strong arguments for this claim. And he is right, as one will see below. However, he fails to answer the question: why should Fermat use two synonymous verbs obviously and systematically in a different sense? Breger’s answer to this question is completely unsatisfactory.

The only way to resolve this paradoxical situation is to analyze the meaning and the different functions of the equals sign $\scriptstyle =$. There are hundreds of books about logic and/or set theory, but almost all of them say no word about the equals sign, they simply use it. The notable exception is the work of Fermat’s virtual compatriot Nicolas Bourbaki. In his Elements of Mathematics he devotes a paragraph (of 3 pages) to the equals sign and its proper use. The following quotations are from Bourbaki (Theory of Sets’’. Addison Wesley, Reading Mass.1968, §5.1).

“An equalitarian theory is a theory Ƭ which has a relational sign of weight 2, written $\scriptstyle =$ (read “equals”).

If $\scriptstyle T$ and $\scriptstyle U$ are terms in Ƭ, the assembly $\scriptstyle =\,T\,U$ is a relation in Ƭ (called the relation of equality). - In practice it is denoted by $\scriptstyle T\,=\,U$ or $\scriptstyle (T)\,=\,(U)$.”

“The negation of the relation $\scriptstyle =\,T\,U$ is denoted by $\scriptstyle T\,\neq \,U$ or $\scriptstyle (T)\,\neq \,(U)$ (where the sign $\scriptstyle \neq$ is read is different from).”

“When a relation of the form $\scriptstyle T\,=\,U$ has been proved in a theory Ƭ, it is often said (by abuse of language) that $\scriptstyle T$ and $\scriptstyle U$ are the same or are identical.”

“Likewise, when $\scriptstyle T\,\neq \,U$ is true in Ƭ, we say that that $\scriptstyle T$ and $\scriptstyle U$ are distinct in place of saying that that $\scriptstyle T$ is different from $\scriptstyle U$.”

Now one should consider a set of relations of equality, one example is from Euler, one from Viète, and the rest is Fermat’s.

Relation of equality Nr.1. (Euler)

$\scriptstyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}\,=\,{\frac {\pi ^{2}}{6}}.$

This beautiful relation of equality is due to Euler. The equals sign = stands between two constants. They look very different, but they represent the same real number. That has been proven by Euler. So, according to Bourbaki, the two constants in the relation of equality above are the same or identical. If the term $\scriptstyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}$ is denoted by $\scriptstyle T$ and the term $\scriptstyle {\frac {\pi ^{2}}{6}}$ is denoted by $\scriptstyle U$ then $\scriptstyle T\;=\;U$ is true.

Relation of equality Nr.2. (Fermat)

$\scriptstyle b\,x\,-\,x^{2}\;=\;{\frac {1}{8}}\,b^{2}.$

This relation of equality is considered by Fermat (Œuvres, Vol. I, p.155). The letter $\scriptstyle b$ denotes a constant, the length of a line. This relation may be called a conditional equation. It is neither true nor false. When it has been solved, for instance by the positive solution

$\scriptstyle x_{0}\;=\;\left(1+{\frac {1}{\sqrt {2}}}\right){\frac {b}{2}},$

and $\scriptstyle x$ is replaced by the solution $\scriptstyle x_{0}$ the relation of equality above is proved and, according to Bourbaki, the constants on both sides of the relation are the same or identical.

Relation of equality Nr.3. (Fermat)

$\scriptstyle (x+e)^{2}(b-x-e)\,=\,b\,x^{2}+b\,e^{2}+2\,b\,x\,e-x^{3}-3\,x\,e^{2}-3\,x^{2}\,e-e^{3}$.

This relation of equality is from Fermat (Œuvres, Vol. I, p.140). Again, $\scriptstyle b$ denotes the positive length of a line. This relation is easily proved by the rules of algebra. The relation is universally valid. According to Bourbaki the terms on both sides of the equals sign are the same or identical. If the term $\scriptstyle (x+e)^{2}(b-x-e)$ is denoted by $\scriptstyle T$ and the term $\scriptstyle b\,x^{2}+b\,e^{2}+2\,b\,x\,e-x^{3}-3\,x\,e^{2}-3\,x^{2}\,e-e^{3}$ is denoted by $\scriptstyle U$ then $\scriptstyle T\;=\;U$ is true.

Relation of equality Nr.4. (Fermat)

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

This relation of equality is on the same page as the relation Nr. 3. However, this relation is not universally valid. It is easy to find a counterexample. For instance $\scriptstyle x={\frac {b}{2}}$ and $\scriptstyle e={\frac {b}{4}}$ yields $\scriptstyle {\frac {b}{64}}=0$. This proves, according to Bourbaki, that the two terms on both sides of the relation are distinct. If the term $\scriptstyle b\,x^{2}-x^{3}$ is denoted by $\scriptstyle T$ and the term $\scriptstyle b\,x^{2}+b\,e^{2}+2\,b\,x\,e-x^{3}-3\,x\,e^{2}-3\,x^{2}\,e-e^{3}$ is denoted by $\scriptstyle U$ then $\scriptstyle T\;\neq \;U$.

The relations of equality Nr.5 and Nr.6 follow further below.

Fermat’s example Nr. 2. (Œuvres, Vol. I, p.140) To divide the line $\scriptstyle AC$ by the point $\scriptstyle B$ this way that the product of the square over $\scriptstyle AB$ and the line $\scriptstyle BC$ is maximal.

Fermat’s solution: Let $\scriptstyle b\,=\,{\overline {AC}}$ and $\scriptstyle x\,=\,{\overline {AB}}$ so that $\scriptstyle {\overline {BC}}\;=\;b-x$. If now we replace $\scriptstyle x$ by $\scriptstyle x+e$ then the product, which results by multiplication of the square of $\scriptstyle x+e$ and $\scriptstyle b-x-e$, is

$\scriptstyle (x+e)^{2}(b-x-e)\;=\;b\,x^{2}\,+\,b\,e^{2}+2\,b\,x\,e\,-\,x^{3}-3\,x\,e^{2}-3\,x^{2}e\,-e^{3}.$

Now Fermat puts this last product equal to the first one:

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

word for word explaining: “Id comparo primo solido $\scriptstyle b\,x^{2}-\,x^{3}$ tanquam essent aequalia , licet revera non sint, et hujus modi comparationem vocavi adaequalitatem.” (I compare this [product] with the first product $\scriptstyle b\,x^{2}-\,x^{3}$ as if they were equal, even though, in reality, they are not, and I called this kind of comparison adaequalitas.) [It is exactly that case, where Bourbaki writes $\scriptstyle T=U$ though $\scriptstyle T\neq U$ is true. It is a relation of equality which describes a relation of two variables, x and e, which are not independent.] After that Fermat cancels those terms which both sides have in common, to wit $\scriptstyle b\,x^{2}-x^{3}$, in which on one side nothing remains, however, on the other side

$\scriptstyle b\,e^{2}+2\,b\,x\,e\,-\,3\,x\,e^{2}-\,3\,x^{2}e\,-\,e^{3}.$

Fermat prefers to separate the terms with a minus sign from those with a plus sign (comparando sunt ergo homogenea notata signo + cum iis quae notantur signo - ) yielding the equation

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

He then cancels the factor $\scriptstyle e$:

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

Fermat deletes all terms, which still contain the factor $\scriptstyle e$:

$\scriptstyle 2\,b\,x\;=\;3\,x^{3}.$

Finally, tacitly assuming that $\scriptstyle x\;=\;0$ is no sensible solution, Fermat cancels the factor $\scriptstyle x$. And he gets

$\scriptstyle 2\,b\;=\;3\,x.$

Therefore the line $\scriptstyle AC$ must be divided that way that $\scriptstyle {\overline {AC}}\,:\,{\overline {AB}}\;=\;3\,:\,2.$

Fermat does not always give complete proofs of his results concerning the determination of maxima and minima or of computing tangents. However, in those cases, where he does it, he follows always exactly the same scheme of his “method”. First Fermat prepares the problem for the application of his method. That depends on the specific conditions and assumptions of the problem. The outcome always consists of two different terms of similar structure, one containing only the variable $\scriptstyle x$ and the other one containing $\scriptstyle x\,+\,e$. The terms may be polynomials, quotients of polynomials, and algebraic terms containing roots. These two different terms are than set equal ($\scriptstyle \doteq$). If necessary, this equation is then transformed by algebraic operations until Fermat gets a polynomial equation. And, a sheer miracle, all summands of the equation contain the factor $\scriptstyle e$ or a power of it. Then this factor (the highest common power of $\scriptstyle e$) is cancelled, and finally those remaining terms, which still contain $\scriptstyle e$ as a factor, are deleted. The result is a conditional equation, which gives, if solved, the solution of the problem. The last equation is the first one where Fermat changes back from adaequabitur to aequabitur. All operations are purely algebraic.

It would be interesting to know, how the adherents of counterfactual equality or almost equality would justify Fermat’s algebraic operations (like cross-multiplying or squaring out roots). However, these guys never give up. But there will be a lucky punch.

Relation of equality Nr.5. (Viète)

$\scriptstyle x^{3}+y^{3}\;=\;(x+y)\left(x^{2}-\,x\,y\,+\,y^{2}\right).$

Of course, this is a special case of Viète’s more general formula for the factorization of the term $\scriptstyle x^{2m+1}+y^{2m+1}$. Let $\scriptstyle T$ be the term $\scriptstyle x^{3}+y^{3}$ and $\scriptstyle U$ the term $\scriptstyle (x+y)\left(x^{2}-\,x\,y\,+\,y^{2}\right).$. Then the relation of equality $\scriptstyle T\,=\,U$ is true. That is easily proved (on the basis of the axioms of a commutative ring) and may be noted by means of Bourbaki’s quantifiers as

$\scriptstyle (\forall _{\mathbb {R}}x)(\forall _{\mathbb {R}}y)\left(x^{3}+y^{3}\;=\;(x+y)\left(x^{2}-\,x\,y\,+\,y^{2}\right)\right).$

We now turn to the story of Fermat and Descartes’ folium cartesii, given by the relation of equality

$\scriptstyle x^{3}+y^{3}\;=\;3\,x\,y.$

Relation of equality Nr.6. (Fermat)

This relation of equality is found in Fermat’s Œuvres (Vol. II, p.156). It describes a curve which was proposed to him by René Descartes. Let, again, $\scriptstyle T$ be the term $\scriptstyle x^{3}+y^{3}$, and $\scriptstyle U$ the term $\scriptstyle 3\,x\,y.$. We want to show that $\scriptstyle T\,\neq \,U$ is true. The relation of equality above is not universally true. If one chooses $\scriptstyle x\,=\,1$ and $\scriptstyle y\,=\,1$ one gets the false equation $\scriptstyle 2\,=\,3.$. Therefore

$\scriptstyle (\exists _{\mathbb {R}}x)(\exists _{\mathbb {R}}y)\left(x^{3}+y^{3}\;\neq \;3\,x\,y\right).$

Applying the rule $\scriptstyle \neg (\forall x)R\iff (\exists x)\neg R$ twice we get

$\scriptstyle \neg (\forall _{\mathbb {R}}x)(\forall _{\mathbb {R}}y)\left(x^{3}+y^{3}\;=\;3\,x\,y\right),$

which proves that $\scriptstyle T\,\neq \,U$ is true.

Fermat’s example Nr. 3. (The lucky punch.) In his letter to Mersenne of 18 January 1638 (Fermat, Œuvres, Vol. III, pp. 126-132) Descartes challenges Fermat to compute the tangent of a new curve, which he had detected and which is nowadays called folium cartesii. On page 129/130 Descartes writes:

“Then, apart from this, his [Fermat’s] alleged rule is not that universal as it seems to him. And it is impossible to extend it to problems which are a little more difficult, but only to the easiest. So he could test it, after having better understood it. He should use it to find, for instance the tangents to the curve $\scriptstyle B\,D\,N$ which I propose to be the following. At an arbitrary place of the curve take the point $\scriptstyle B$. Having drawn the perpendicular $\scriptstyle B\,C$ the two cubes of side lengths $\scriptstyle B\,C$ and $\scriptstyle C\,D,$ are, together, equal (égaux) to the rectangular solid formed of the same sides $\scriptstyle B\,C$ and $\scriptstyle C\,D,$ and the length $\scriptstyle P$ of a given side.”

If one puts $\scriptstyle x\,=\,{\overline {BC}}$, $\scriptstyle y\,=\,{\overline {CD}}$, and $\scriptstyle p\,=\,P$, then one gets the equation of the folium cartesii:

$\scriptstyle x^{3}+\,y^{3}\;=\;p\,x\,y.$

One usually chooses $\scriptstyle p\,=\,3$ which makes the curve looking nice and the computations easier. Mersenne hesitated for a long time sending a copy of Descartes’ letter to Fermat. But finally he did. And Fermat answered in June 1638 (Œuvres, Vol. II, p.156) and gave the solution of Descartes’ challenge. Fermat formulates the description of the curve, which Descartes had proposed, as follows: “Soit la courbe $\scriptstyle C\,A$, de laquelle la propriété est telle que, quelque point qu’on prenne sur la dite courbe, comme $\scriptstyle A$, tirant la perpendiculaire $\scriptstyle A\,B$, les deux cubes $\scriptstyle C\,B$ et $\scriptstyle B\,A$ soient égaux au parallélépipède compris sous une ligne droite donnée, comme $\scriptstyle N$, et les deux lignes $\scriptstyle C\,B$ et $\scriptstyle B\,A$.“ Fermat then introduces two variable auxiliary points $\scriptstyle E$ and $\scriptstyle F$, needed for his “method”. He then starts with his “method” and writes : “Il faudra comparer, par adéquation, les deux cubes $\scriptstyle C\,F$ et $\scriptstyle F\,E$ avec le solide compris sous $\scriptstyle N,\;FC,\;FE.$. (one must compare, by adequality, the two cubes … ). That means

$\scriptstyle {\overline {CF}}^{3}+{\overline {FE}}^{3}\;\doteq \;N\cdot {\overline {CF}}\cdot {\overline {FE}}.$

What may this adequality mean? Approximately equal? Counterfactually equal? As nearly equal as possible? Pseudo-equal? Almost equal? Or put equal? The reader should decide it himself.

Why did Fermat not just write aequabitur instead of adaequabitur? The answer should include some biographical considerations. When Fermat (born November 1607) in 1628 created his “method” he was very young, about 20 or 21 years old. He lived as an âvocat at the parlement de Bordeaux where his friend Etienne d’Espagnet was a conseiller at the same parlement. D’Espagnet possessed a very valuable scientific library which he had inherited from his father, a friend of Viète. In this library Fermat studied the writings of the ancients and Viète’s work. Viète used most frequently the verb aequare and, on some rare occasions, the verb adaequare (8 times) and the noun adaequalitas (once), instead of Recorde’s equals sign. (François Viète, Opera Mathematica. Georg Olms, New York 1970, pp. 80, 134, 135, 137, 141, and 143). In all these cases the meaning of adaequare is to put equal. At that time Fermat had still not studied Diophantus. So he borrowed the verb adaequare and the noun adaequalitas from his admired “example” Viète. Later, in his correspondence with Mersenne and Descartes, he referred to Bachet de Meziriac’s Diophantus. And that was very misleading because Bachet’s translation is false. Viète exclusively dealt with conditional equations and universally valid equations (formulas, theorems), never, however, with relations of equality which describe a relation between two (or more) variables, which are not independent and, occasionally, describe a curve. So Fermat felt that he should give these new relations of equality a different name: adaequabitur. However, when Fermat starts applying his “method” and puts two different terms equal for the first time he prefers to write adaequentur, adaequabuntur, comparare per adaequalitatem, debet adaequare, or fiat per adaequalitatem, and, in his French correspondence, comparer par adéquation.

How did Fermat hit upon his method? Fermat explained that very explicitly in his treatise Methodus de maxima et minima (Œuvres , Vol. I, pp.147-153) which has been preserved as Mersenne’s copy. The following is a concise account in modern mathematical language, nevertheless trying to repeat Fermat’s idea as anadulterated as possible. Let $\scriptstyle t(x)$ be a term, for instance a polynomial term as $\scriptstyle b\,x\,-\,x^{2}$ like that in the first example. The maximum of this term is to be determined, for instance in the Interval $\scriptstyle 0\leq x\leq b.$. Now Fermat argues, picking up a remark of Pappus of Alexandria, that the place $\scriptstyle x_{0}\;\left(={\frac {b}{2}}\right)$, where the maximum $\scriptstyle t_{max}\;\left(={\frac {b^{2}}{4}}\right)$ is attained, that is $\scriptstyle t(x_{0})\;=\;t_{max}$, is singular (plural: unicae et singulares). That means that the value $\scriptstyle t_{max}$ is not attained at another place of the interval $\scriptstyle [0,b]$ again. Because, if one chooses a value $\scriptstyle c$ somewhat smaller than $\scriptstyle t_{max}$ then there are two different places in the interval, say $\scriptstyle x$ and $\scriptstyle x+e$ with $\scriptstyle 0<x<x_{0}<x+e<b$, for which holds

$\scriptstyle t(x)\,=\,t(x+e)\;(=\,c).$

If one again chooses a constant $\scriptstyle c^{\prime }$ with $\scriptstyle c<c^{\prime }<t_{max}$, so to this constant will belong $\scriptstyle x^{\prime }$ and $\scriptstyle e^{\prime }$ of that kind, that

$\scriptstyle t(x^{\prime })\;=\;t(x^{\prime }+e^{\prime })\;(=\,c^{\prime }),$

and $\scriptstyle x<x^{\prime }<x_{0}<x^{\prime }+e^{\prime }<x+e$ and $\scriptstyle 0<e^{\prime }<e.$ That means that the nearer $\scriptstyle c$ comes to $\scriptstyle t_{max}$ the closer $\scriptstyle e$ will be to $\scriptstyle 0$. Only for the place $\scriptstyle x_{0}$, where $\scriptstyle t(x)$ attains its maximum, there exists no other place in the interval where this value is again attained, which corresponds to $\scriptstyle e\,=\,0.$

This gave Fermat the idea of the following method for computing $\scriptstyle x_{0}.$ He puts

$\scriptstyle t(x+e)\;\doteq \;t(x),$

and substracting common terms on both sides of this relation of equality all remaining terms contain $\scriptstyle e$ (or a power of it) as factor. Tacitly assuming $\scriptstyle e\,\neq 0$ he cancels the highest common factor of $\scriptstyle e$ (that may well be $\scriptstyle e^{2}$ if $\scriptstyle t(x)$ contains a square root) . In the remaining equation Fermat then puts $\scriptstyle e\,=\,0$ (equals nihil), which gives him a conditional equation for $\scriptstyle x_{0}$.

Did Fermat prove the validity of his “method”? If this question means: “gave Fermat a proof which fulfills the standard of modern analysis?” then the answer is – of course – no. Fermat did not possess an idea of the continuum which is familiar to us since Cantor, Dedekind, and Weierstraß. His concept of number could possibly be described by naïve constructivism: if one possesses, for a given quantity, an algorithm which allows him to approximate this quantity arbitrarily precisely by rational numbers then this quantity responds to a number. Fermat had no concept of function, of continuity, of convergence, and of limit. He never considers the local gradient of a curve. (When Fermat determines a tangent he always computes, as a second point of the tangent, the point of intersection of the tangent with an obviously chosen reference line, most often the axis of symmetry of the curve.) Fermat had explained how he hit upon the idea of his “method”. And this method always produced the correct solution. He had, admittedly, some trouble with his attempt to explain to his contemporaries (especially Descartes) his proceeding to compute tangents. However, this had another reason.

See also

• Fermat's principle
• Transcendental Law of Homogeneity

Bibliography

• Edwards, C. H. Jr. (1994), The Historical Development of the Calculus, Springer
• Grabiner, Judith V. (1983), "The Changing Concept of Change: The Derivative from Fermat to Weierstrass", Mathematics Magazine, 56 (4): 195–206 {{citation}}: Unknown parameter |month= ignored (help)
• Katz, V. (2008), A History of Mathematics: An Introduction, Addison Wesley

• Barner, K. (2011) "Fermats <<adaequare>> - und kein Ende?" Mathematische Semesterberichte (58), pp. 13-45
• Breger, H. (1994) "The mysteries of adaequare: a vindication of Fermat", Archive for History of Exact Sciences 46(3):193–219.
• Giusti, E. (2009) "Les méthodes des maxima et minima de Fermat", Ann. Fac. Sci. Toulouse Math. (6) 18, Fascicule Special, 59–85.
• Stillwell, J.(2006) Yearning for the impossible. The surprising truths of mathematics, page 91, A K Peters, Ltd., Wellesley, MA.