kb/data/en.wikipedia.org/wiki/Adequality-0.md

---
title: "Adequality"
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source: "https://en.wikipedia.org/wiki/Adequality"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T07:22:58.075678+00:00"
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---

Adequality is a technique developed by Pierre de Fermat in his treatise Methodus ad disquirendam maximam et minimam (a Latin treatise circulated in France c. 1636 ) to calculate maxima and minima of functions, tangents to curves, area, center of mass, least action, and other problems in calculus. According to André Weil, Fermat "introduces the technical term adaequalitas, adaequare, etc., which he says he has borrowed from Diophantus. As Diophantus V.11 shows, it means an approximate equality, and this is indeed how Fermat explains the word in one of his later writings." (Weil 1973). Diophantus coined the word παρισότης (parisotēs) to refer to an approximate equality. Claude Gaspard Bachet de Méziriac translated Diophantus's Greek word into Latin as adaequalitas. Paul Tannery's French translation of Fermat's Latin treatises on maxima and minima used the words adéquation and adégaler.

== 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
        )
      
    
    {\displaystyle p(x)}
  
, Fermat equated (or more precisely adequated) 
  
    
      
        p
        (
        x
        )
      
    
    {\displaystyle p(x)}
  
 and 
  
    
      
        p
        (
        x
        +
        e
        )
      
    
    {\displaystyle p(x+e)}
  
 and after doing algebra he could cancel out a factor of 
  
    
      
        e
        ,
      
    
    {\displaystyle e,}
  
 and then discard any remaining terms involving 
  
    
      
        e
        .
      
    
    {\displaystyle e.}
  
 To illustrate the method by Fermat's own example, consider the problem of finding the maximum of 
  
    
      
        p
        (
        x
        )
        =
        b
        x
        −
        
          x
          
            2
          
        
      
    
    {\displaystyle p(x)=bx-x^{2}}
  
 (in Fermat's words, it is to divide a line of length 
  
    
      
        b
      
    
    {\displaystyle b}
  
 at a point 
  
    
      
        x
      
    
    {\displaystyle x}
  
, such that the product of the two resulting parts be a maximum). Fermat adequated 
  
    
      
        b
        x
        −
        
          x
          
            2
          
        
      
    
    {\displaystyle bx-x^{2}}
  
 with 
  
    
      
        b
        (
        x
        +
        e
        )
        −
        (
        x
        +
        e
        
          )
          
            2
          
        
        =
        b
        x
        −
        
          x
          
            2
          
        
        +
        b
        e
        −
        2
        e
        x
        −
        
          e
          
            2
          
        
      
    
    {\displaystyle b(x+e)-(x+e)^{2}=bx-x^{2}+be-2ex-e^{2}}
  
. That is (using the notation 
  
    
      
        ∽
      
    
    {\displaystyle \backsim }
  
 to denote adequality, introduced by Paul Tannery):

  
    
      
        b
        x
        −
        
          x
          
            2
          
        
        ∽
        b
        x
        −
        
          x
          
            2
          
        
        +
        b
        e
        −
        2
        e
        x
        −
        
          e
          
            2
          
        
        .
      
    
    {\displaystyle bx-x^{2}\backsim bx-x^{2}+be-2ex-e^{2}.}
  

Canceling terms and dividing by 
  
    
      
        e
      
    
    {\displaystyle e}
  
 Fermat arrived at

  
    
      
        b
        ∽
        2
        x
        +
        e
        .
      
    
    {\displaystyle b\backsim 2x+e.}
  

Removing the terms that contained 
  
    
      
        e
      
    
    {\displaystyle e}
  
 Fermat arrived at the desired result that the maximum occurred when 
  
    
      
        x
        =
        b
        
          /
        
        2
      
    
    {\displaystyle 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. Victor 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. Katz 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. Fermat's adequality was analyzed in a number of scholarly studies.  In 1896, Paul Tannery published a French translation of Fermat's Latin treatises on maxima and minima (Fermat, Œuvres, Vol. III, pp. 121–156). Tannery translated Fermat's term as “adégaler” and adopted Fermat's “adéquation”. Tannery also introduced the symbol 
  
    
      
        ∽
      
    
    {\displaystyle \backsim }
  
 for adequality in mathematical formulas.

Heinrich Wieleitner (1929) wrote:Fermat replaces A with A+E. Then he sets 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 sets those that remain equal to each other. From that [the required] 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 
  
    
      
        
          ∼
        
      
    
    {\displaystyle \scriptstyle \sim }
  
.)

Max Miller (1934) wrote: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 
  
    
      
        
          ≈
        
      
    
    {\displaystyle \scriptstyle \approx }
  
.)

Jean Itard (1948) wrote: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 
  
    
      
        
          ∽
        
      
    
    {\displaystyle \scriptstyle \backsim }
  
.)

Joseph Ehrenfried Hofmann (1963) wrote: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 
  
    
      
        
          ≈
        
      
    
    {\displaystyle \scriptstyle \approx }
  
.)

Peer Strømholm (1968) wrote: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:

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