Parabolic geometry (differential geometry)

Adequality is a term introduced by historians of mathematics to discuss the mathematical meaning of the Latin word adaequalitas, as it was used by Pierre de Fermat in his work on finding maxima, minima and tangents. André Weil wrote: "[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 term παρισὀτης (parisotes) to refer to an approximate equality. The term was rendered as adaequalitas in Claude Gaspard Bachet de Méziriac's Latin translation of Diophantus, and adéquation and adégaler in Paul Tannery's French translation of Fermat’s Latin treatises on maxima and minima and related problems.

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.Template:Sfn

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.Template:Sfn

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 $\backsim$ 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 $\sim$.)

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 $\approx$.)

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 $\backsim$.)

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 $\approx$.)

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:

$f(A)\sim f(A+E)$.

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." He further notes that "there was never in M1 (Method 1) any question of the variation E being put equal to zero. The words Fermat used to express the process of suppressing terms containing E was 'elido', 'deleo', and 'expungo', and in French 'i'efface' and 'i'ôte'. We can hardly believe that a sane man wishing to express his meaning and searching for words, would constantly hit upon such tortuous ways of imparting the simple fact that the terms vanished because E was zero." (p. 51)

Claus Jensen (1969) wrote: "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 ("tamquam essent aequalia, licet revera aequalia non sint") - I will employ the nowadays more usual symbol $\approx$." The Latin quotation comes from Tannery's 1891 edition of Fermat, volume 1, page 140.

Michael Sean Mahoney (1971) wrote: "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 $\approx$.) 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.

Charles Henry Edwards, Jr. (1979) wrote: "For example, in order to determine how to subdivide a segment of length $b$ into two segments $x$ and $b-x$ whose product $x(b-x)=bx-x^2$ is maximal, that is to find the rectangle with perimeter $2b$ that has the maximal area, he [Fermat] proceeds as follows. First he substituted $x+e$ (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:

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

After canceling terms, he divided through by e to obtain $2x+b\sim b.$ Finally he discarded the remaining term containing e, transforming the pseudo-equality into the true equality $x=\frac{b}{2}$ that gives the value of x which makes $bx-x^2$ 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 Andersen (1980) wrote: "The two expressions of the maximum or minimum are made "adequal", which means something like as nearly equal as possible." (Anderson uses the symbol $\approx$.)

Herbert Breger (1994) wrote: “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.)

John Stillwell (Stillwell 2006 p. 91) 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."

Enrico Giusti (2009) cites Fermat's letter to Marin Mersenne where Fermat wrote: "Cette comparaison par adégalité produit deux termes inégaux qui enfin produisent l'égalité (selon ma méthode) qui nous donne la solution de la question." Giusti notes in a footnote that this letter seems to have escaped Breger's notice.

Klaus Barner (2011) asserts that Fermat uses two different Latin words (aequabitur and adaequabitur) to replace the nowadays usual equals sign, aequabitur when the equation concerns a valid identity between two constants, a universally valid (proved) formula, or a conditional equation, adaequabitur, however, when the equation describes a relation between two variables, which are not independent (and the equation is no valid formula). On page 36, Barner writes: "Why did Fermat continually repeat his inconsistent procedure for all his examples for the method of tangents? Why did he never mention the secant, with which he in fact operated? I do not know."

Katz, Schaps, Shnider (2013) argue that Fermat's application of the technique to transcendental curves such as the cycloid shows that adequality goes beyond a purely algebraic algorithm, and that, contrary to Breger's interpretation, the terms parisotes and adaequalitas mean "approximate equality". They 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.

See also

• Fermat's principle
• Transcendental Law of Homogeneity

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Bibliography

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

• 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.
• Weil, A., Book Review: The mathematical career of Pierre de Fermat. Bull. Amer. Math. Soc. 79 (1973), no. 6, 1138–1149.