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Adequality
Adequality is a technique developed by Pierre de Fermat in his treatise ''Methodus ad disquirendam maximam et minimam''''METHOD FOR THE STUDY OF MAXIMA AND MINIMA'' English translation of Fermat's treatise ''Methodus ad disquirendam maximam et minimam''. (a treatise circulated in France c. 1636) to calculate of functions, s to curves, |
Pierre De Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of Diophantus' '' Arithmetica''. He was also a lawyer at the '' Parlement'' of Toulouse, France. Biography Fermat was born in 1607 in Beaumont-de-Lomagne, France—the late 15th-century mansion where Fermat was born is now a museum. He was from Gascony, where his father, Domin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transcendental Law Of Homogeneity
In mathematics, the transcendental law of homogeneity (TLH) is a heuristic principle enunciated by Gottfried Wilhelm Leibniz most clearly in a 1710 text entitled ''Symbolismus memorabilis calculi algebraici et infinitesimalis in comparatione potentiarum et differentiarum, et de lege homogeneorum transcendentali''. Henk J. M. Bos describes it as the principle to the effect that in a sum involving infinitesimals of different orders, only the lowest-order term must be retained, and the remainder discarded. Thus, if a is finite and dx is infinitesimal, then one sets :a+dx=a. Similarly, :u\,dv+v\,du+du\,dv=u\,dv+v\,du, where the higher-order term ''du'' ''dv'' is discarded in accordance with the TLH. A recent study argues that Leibniz's TLH was a precursor of the standard part function over the hyperreals. See also *Law of continuity *Adequality Adequality is a technique developed by Pierre de Fermat in his treatise ''Methodus ad disquirendam maximam et minimam'' [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Principle
Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the path that can be traveled in the least time. In order to be true in all cases, this statement must be weakened by replacing the "least" time with a time that is " stationary" with respect to variations of the path — so that a deviation in the path causes, at most, a ''second-order'' change in the traversal time. To put it loosely, a ray path is surrounded by close paths that can be traversed in ''very'' close times. It can be shown that this technical definition corresponds to more intuitive notions of a ray, such as a line of sight or the path of a narrow beam. First proposed by the French mathematician Pierre de Fermat in 1662, as a means of explaining the ordinary law of refraction of light (Fig.1), Fermat's principle was initiall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Part Function
In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such hyperreal x, the unique real x_0 infinitely close to it, i.e. x-x_0 is infinitesimal. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de Fermat,Karin Usadi Katz and Mikhail G. Katz (2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. Foundations of Science.Searxiv The authors refer to the Fermat-Robinson standard part. as well as Leibniz's Transcendental law of homogeneity. The standard part function was first defined by Abraham Robinson who used the notation ^x for the standard part of a hyperreal x (see Robinson 1974). This concept plays a key role in defining the concepts of the calculus, such as continuity, the derivati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantus
Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the author of a series of books called '' Arithmetica'', many of which are now lost. His texts deal with solving algebraic equations. Diophantine equations ("Diophantine geometry") and Diophantine approximations are important areas of mathematical research. Diophantus coined the term παρισότης (parisotes) to refer to an approximate equality. This term was rendered as ''adaequalitas'' in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, Diophantine equations are usually algebraic equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinitesimal Calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Later work, including codifying the idea of limits, put these developments on a more solid conceptual footing. Today, calculus has widespread uses in scien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archive For History Of Exact Sciences
''Archive for History of Exact Sciences'' is a peer-reviewed academic journal currently published bimonthly by Springer Science+Business Media, covering the history of mathematics and of astronomy observations and techniques, epistemology of science, and philosophy of science from Antiquity until now. It was established in 1960 and the current editors-in-chief are Jed Z. Buchwald and Jeremy Gray. Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... of 0.594. References External links * History of science journals Springer Science+Business Media academic journals Bimonthly journals English-language journals Publications ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperreal Number
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form :1 + 1 + \cdots + 1 (for any finite number of terms). Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948. The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic law of continuity. The transfer principle states that true first-order statements about R are also valid in *R. For example, the commutative law of addition, , holds for the hyperreals just as it does for the reals; since R is a real closed field, so is *R. Since \sin()=0 for all integers ''n'', one also has \sin()=0 for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Łoś' theorem of 1955. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marin Mersenne
Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for Mersenne prime numbers, those which can be written in the form for some integer . He also developed Mersenne's laws, which describe the harmonics of a vibrating string (such as may be found on guitars and pianos), and his seminal work on music theory, ''Harmonie universelle'', for which he is referred to as the "father of acoustics". Mersenne, an ordained Catholic priest, had many contacts in the scientific world and has been called "the center of the world of science and mathematics during the first half of the 1600s" and, because of his ability to make connections between people and ideas, "the post-box of Europe". He was also a member of the Minim religious order and wrote and lectured on theology and philosophy. Life Mersenne was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enrico Giusti
Enrico Giusti (born Priverno, 1940), is an Italian mathematician mainly known for his contributions to the fields of calculus of variations, regularity theory of partial differential equations, minimal surfaces and history of mathematics. He has been professor of mathematics at the Università di Firenze; he also taught and conducted research at the Australian National University at Canberra, at the Stanford University and at the University of California, Berkeley. After retirement, he devoted himself to the managing of the "Giardino di Archimede", a museum entirely dedicated to mathematics and its applications. Giusti is also the editor-in-chief of the international journal, dedicated to the history of mathematics "Bollettino di storia delle scienze matematiche". One of the most famous results of Giusti, is the one obtained with Enrico Bombieri and Ennio De Giorgi, concerning the minimality of Simons' cones, and allowing to disprove the validity of Bernstein's theorem in dimens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-standard Analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Nonstandard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers. Nonstandard analysis originated in the early 1960s by the mathematician Abraham Robinson. He wrote: ... the idea of infinitely small or ''infinitesimal'' quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection ... that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
