Infinitesimal

infinitesimalsinfinitely closeinfinitesimally1/∞infinitely smallvery smalldifferentialinfinitely small quantitiesinfinitesinfinitesimal number
In mathematics, infinitesimals are things so small that there is no way to measure them.wikipedia
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Archimedes

Archimedes of SyracuseArchimedeanArchimedes Heat Ray
Archimedes used what eventually came to be known as the method of indivisibles in his work The Method of Mechanical Theorems to find areas of regions and volumes of solids.
Generally considered the greatest mathematician of antiquity and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying concepts of infinitesimals and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including the area of a circle, the surface area and volume of a sphere, and the area under a parabola.

Infinity

infiniteinfinitely
The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinity-th" item in a sequence.
In 1655, John Wallis first used the notation \infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."

Derivative

differentiationdifferentiablefirst derivative
To give it a meaning, it usually must be compared to another infinitesimal object in the same context (as in a derivative).
In Leibniz's notation, an infinitesimal change in

John Wallis

WallisWallis, JohnArithmetica Infinitorum
John Wallis's infinitesimals differed from indivisibles in that he would decompose geometrical figures into infinitely thin building blocks of the same dimension as the figure, preparing the ground for general methods of the integral calculus.
He similarly used 1/∞ for an infinitesimal.

Non-standard analysis

nonstandard analysisnon-standardNonstandard
A mathematical implementation of both the law of continuity and infinitesimals was achieved by Abraham Robinson in 1961, who developed non-standard analysis based on earlier work by Edwin Hewitt in 1948 and Jerzy Łoś in 1955.
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers.

Abraham Robinson

RobinsonA. RobinsonRobinson, Abraham
A mathematical implementation of both the law of continuity and infinitesimals was achieved by Abraham Robinson in 1961, who developed non-standard analysis based on earlier work by Edwin Hewitt in 1948 and Jerzy Łoś in 1955.
Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorporated into modern mathematics.

Cavalieri's principle

method of indivisiblesindivisiblesa similar method
Archimedes used what eventually came to be known as the method of indivisibles in his work The Method of Mechanical Theorems to find areas of regions and volumes of solids.
In the other direction, Cavalieri's principle grew out of the ancient Greek method of exhaustion, which used limits but did not use infinitesimals.

Hyperreal number

hyperreal numbershyperrealhyperreals
The hyperreals implement an infinitesimal-enriched continuum and the transfer principle implements Leibniz's law of continuity.
The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities.

Law of Continuity

continuityprinciple of continuity
Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity.
Leibniz used the principle to extend concepts such as arithmetic operations, from ordinary numbers to infinitesimals, laying the groundwork for infinitesimal calculus.

The Method of Mechanical Theorems

Archimedes' use of infinitesimalsdeveloped this idea furtherThe Method
Archimedes used what eventually came to be known as the method of indivisibles in his work The Method of Mechanical Theorems to find areas of regions and volumes of solids.
The Method takes the form of a letter from Archimedes to Eratosthenes, the chief librarian at the Library of Alexandria, and contains the first attested explicit use of indivisibles (sometimes referred to as infinitesimals).

Dirac delta function

Dirac deltadelta functionimpulse
Augustin-Louis Cauchy exploited infinitesimals both in defining continuity in his Cours d'Analyse, and in defining an early form of a Dirac delta function.
An infinitesimal formula for an infinitely tall, unit impulse delta function (infinitesimal version of Cauchy distribution) explicitly appears in an 1827 text of Augustin Louis Cauchy.

Transcendental law of homogeneity

Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity.
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.

Integral

integrationintegral calculusdefinite integral
Infinitely many infinitesimals are summed to produce an integral.
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

Standard part function

standard part standard partshadow
The standard part function implements Fermat's adequality.
It associates to every such hyperreal x, the unique real x_0 infinitely close to it, i.e. x-x_0 is infinitesimal.

Leonhard Euler

EulerLeonard EulerEuler, Leonhard
The 18th century saw routine use of infinitesimals by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange.
Euler worked in almost all areas of mathematics, such as geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics.

Gottfried Wilhelm Leibniz

LeibnizGottfried LeibnizGottfried Wilhelm von Leibniz
Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. When Newton and Leibniz invented the calculus, they made use of infinitesimals, Newton's fluxions and Leibniz' differential.
Leibniz exploited infinitesimals in developing calculus, manipulating them in ways suggesting that they had paradoxical algebraic properties.

Calculus

infinitesimal calculusdifferential and integral calculusclassical calculus
Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. When Newton and Leibniz invented the calculus, they made use of infinitesimals, Newton's fluxions and Leibniz' differential.
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.

Continuous function

continuouscontinuitycontinuous map
Augustin-Louis Cauchy exploited infinitesimals both in defining continuity in his Cours d'Analyse, and in defining an early form of a Dirac delta function.
Cauchy defined continuity of a function in the following intuitive terms: an infinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable (see Cours d'analyse, page 34).

Differential (infinitesimal)

differentialdifferential elementdifferentials
When Newton and Leibniz invented the calculus, they made use of infinitesimals, Newton's fluxions and Leibniz' differential.
The term differential is used in calculus to refer to an infinitesimal (infinitely small) change in some varying quantity.

Transfer principle

transfer theorem
The hyperreals implement an infinitesimal-enriched continuum and the transfer principle implements Leibniz's law of continuity.
Here infinitesimals are expected to have the "same" properties as appreciable numbers.

Method of exhaustion

exhaustionexhaustion methodExhaustion, method of
In his formal published treatises, Archimedes solved the same problem using the method of exhaustion.
An important alternative approach was Cavalieri's principle, also termed the "method of indivisibles", which eventually evolved into the infinitesimal calculus of Roberval, Torricelli, Wallis, Leibniz, and others.

Cours d'Analyse

Augustin-Louis Cauchy exploited infinitesimals both in defining continuity in his Cours d'Analyse, and in defining an early form of a Dirac delta function.
On page 1 of the Introduction, Cauchy writes: "In speaking of the continuity of functions, I could not dispense with a treatment of the principal properties of infinitely small quantities, properties which serve as the foundation of the infinitesimal calculus."

Archimedean property

Archimedeannon-Archimedean fieldnon-Archimedean
His Archimedean property defines a number x as infinite if it satisfies the conditions |x|>1, |x|>1+1, |x|>1+1+1, ..., and infinitesimal if x≠0 and a similar set of conditions holds for x and the reciprocals of the positive integers.
An algebraic structure in which any two non-zero elements are comparable, in the sense that neither of them is infinitesimal with respect to the other, is said to be Archimedean.

The Analyst

ghosts of departed quantitiesanalyst
The use of infinitesimals was attacked as incorrect by Bishop Berkeley in his work The Analyst.
The Analyst was a direct attack on the foundations of calculus, specifically on Newton's notion of fluxions and on Leibniz's notion of infinitesimal change.

(ε, δ)-definition of limit

epsilon-deltaepsilon-delta definition(''ε'', ''δ'')-definition of limit
In the second half of the nineteenth century, the calculus was reformulated by Augustin-Louis Cauchy, Bernard Bolzano, Karl Weierstrass, Cantor, Dedekind, and others using the -definition of limit and set theory.
Quantities such as E are called infinitesimals.