Calculus, known in its early history as infinitesimal
calculus, is a mathematical discipline focused
on limits, functions, derivatives, integrals,
and infinite series. Isaac Newton and Gottfried
Wilhelm Leibniz independently discovered calculus
in the mid-17th century. However, both inventors
claimed that the other had stolen his work,
and the Leibniz-Newton calculus controversy
continued until the end of their lives.
== Pioneers of calculus ==
=== 
Ancient ===
The ancient period introduced some of the
ideas that led to integral calculus, but does
not seem to have developed these ideas in
a rigorous and systematic way. Calculations
of volumes and areas, one goal of integral
calculus, can be found in the Egyptian Moscow
papyrus (c. 1820 BC), but the formulas are
only given for concrete numbers, some are
only approximately true, and they are not
derived by deductive reasoning. Babylonians
may have discovered the trapezoidal rule while
doing astronomical observations of Jupiter.From
the age of Greek mathematics, Eudoxus (c.
408−355 BC) used the method of exhaustion,
which foreshadows the concept of the limit,
to calculate areas and volumes, while Archimedes
(c. 287−212 BC) developed this idea further,
inventing heuristics which resemble the methods
of integral calculus. Greek mathematicians
are also credited with a significant use of
infinitesimals. Democritus is the first person
recorded to consider seriously the division
of objects into an infinite number of cross-sections,
but his inability to rationalize discrete
cross-sections with a cone's smooth slope
prevented him from accepting the idea. At
approximately the same time, Zeno of Elea
discredited infinitesimals further by his
articulation of the paradoxes which they create.
Archimedes developed this method further,
while also inventing heuristic methods which
resemble modern day concepts somewhat in his
The Quadrature of the Parabola, The Method,
and On the Sphere and Cylinder. It should
not be thought that infinitesimals were put
on a rigorous footing during this time, however.
Only when it was supplemented by a proper
geometric proof would Greek mathematicians
accept a proposition as true. It was not until
the 17th century that the method was formalized
by Cavalieri as the method of Indivisibles
and eventually incorporated by Newton into
a general framework of integral calculus.
Archimedes was the first to find the tangent
to a curve other than a circle, in a method
akin to differential calculus. While studying
the spiral, he separated a point's motion
into two components, one radial motion component
and one circular motion component, and then
continued to add the two component motions
together, thereby finding the tangent to the
curve. The pioneers of the calculus such as
Isaac Barrow and Johann Bernoulli were diligent
students of Archimedes; see for instance C.
S. Roero (1983).
=== Medieval ===
The method of exhaustion was reinvented in
China by Liu Hui in the 4th century AD in
order to find the area of a circle. In the
5th century AD, Zu Chongzhi established a
method that would later be called Cavalieri's
principle to find the volume of a sphere.
In the Middle East, Alhazen derived a formula
for the sum of fourth powers. He used the
results to carry out what would now be called
an integration, where the formulas for the
sums of integral squares and fourth powers
allowed him to calculate the volume of a paraboloid.
In the 14th century, Indian mathematician
Madhava of Sangamagrama and the Kerala school
of astronomy and mathematics stated components
of calculus such as the Taylor series and
infinite series approximations. However, they
were not able to combine many differing ideas
under the two unifying themes of the derivative
and the integral, show the connection between
the two, and turn calculus into the powerful
problem-solving tool we have today.The mathematical
study of continuity was revived in the 14th
century by the Oxford Calculators and French
collaborators such as Nicole Oresme. They
proved the "Merton mean speed theorem": that
a uniformly accelerated body travels the same
distance as a body with uniform speed whose
speed is half the final velocity of the accelerated
body.
=== Modern ===
In the 17th century, European mathematicians
Isaac Barrow, René Descartes, Pierre de Fermat,
Blaise Pascal, John Wallis and others discussed
the idea of a derivative. In particular, in
Methodus ad disquirendam maximam et minima
and in De tangentibus linearum curvarum, Fermat
developed an adequality method for determining
maxima, minima, and tangents to various curves
that was closely related to differentiation.
Isaac Newton would later write that his own
early ideas about calculus came directly from
"Fermat's way of drawing tangents."On the
integral side, Cavalieri developed his method
of indivisibles in the 1630s and 1640s, providing
a more modern form of the ancient Greek method
of exhaustion, and computing Cavalieri's quadrature
formula, the area under the curves xn of higher
degree, which had previously only been computed
for the parabola, by Archimedes. Torricelli
extended this work to other curves such as
the cycloid, and then the formula was generalized
to fractional and negative powers by Wallis
in 1656. In a 1659 treatise, Fermat is credited
with an ingenious trick for evaluating the
integral of any power function directly. Fermat
also obtained a technique for finding the
centers of gravity of various plane and solid
figures, which influenced further work in
quadrature. James Gregory, influenced by Fermat's
contributions both to tangency and to quadrature,
was then able to prove a restricted version
of the second fundamental theorem of calculus
in the mid-17th century. The first full proof
of the fundamental theorem of calculus was
given by Isaac Barrow.One prerequisite to
the establishment of a calculus of functions
of a real variable involved finding an antiderivative
for the rational function
f
(
x
)
=
1
x
.
{\displaystyle f(x)\ =\ {\frac {1}{x}}.}
This problem can be phrased as quadrature
of the rectangular hyperbola xy = 1. In 1647
Gregoire de Saint-Vincent noted that the required
function F satisfied
F
(
s
t
)
=
F
(
s
)
+
F
(
t
)
,
{\displaystyle F(st)=F(s)+F(t),}
so that a geometric sequence became, under
F, an arithmetic sequence. A. A. de Sarasa
associated this feature with contemporary
algorithms called logarithms that economized
arithmetic by rendering multiplications into
additions. So F was first known as the "hyperbolic
logarithm". After Euler exploited e = 2.71828...,
and F was identified as the inverse function
of the exponential function, it became the
natural logarithm, satisfying
d
F
d
x
=
1
x
.
{\displaystyle {\frac {dF}{dx}}\ =\ {\frac
{1}{x}}.}
The first proof of Rolle's theorem was given
by Michel Rolle in 1691 using methods developed
by the Dutch mathematician Johann van Waveren
Hudde. The mean value theorem in its modern
form was stated by Bernard Bolzano and Augustin-Louis
Cauchy (1789–1857) also after the founding
of modern calculus. Important contributions
were also made by Barrow, Huygens, and many
others.
=== Newton and Leibniz ===
Before Newton and Leibniz, the word “calculus”
referred to any body of mathematics, but in
the following years, "calculus" became a popular
term for a field of mathematics based upon
their insights. Newton and Leibniz, building
on this work, independently developed the
surrounding theory of infinitesimal calculus
in the late 17th century. Also, Leibniz did
a great deal of work with developing consistent
and useful notation and concepts. Newton provided
some of the most important applications to
physics, especially of integral calculus.
The purpose of this section is to examine
Newton and Leibniz’s investigations into
the developing field of infinitesimal calculus.
Specific importance will be put on the justification
and descriptive terms which they used in an
attempt to understand calculus as they themselves
conceived it.
By the middle of the 17th century, European
mathematics had changed its primary repository
of knowledge. In comparison to the last century
which maintained Hellenistic mathematics as
the starting point for research, Newton, Leibniz
and their contemporaries increasingly looked
towards the works of more modern thinkers.
Europe had become home to a burgeoning mathematical
community and with the advent of enhanced
institutional and organizational bases a new
level of organization and academic integration
was being achieved. Importantly, however,
the community lacked formalism; instead it
consisted of a disordered mass of various
methods, techniques, notations, theories,
and paradoxes.
Newton came to calculus as part of his investigations
in physics and geometry. He viewed calculus
as the scientific description of the generation
of motion and magnitudes. In comparison, Leibniz
focused on the tangent problem and came to
believe that calculus was a metaphysical explanation
of change. Importantly, the core of their
insight was the formalization of the inverse
properties between the integral and the differential
of a function. This insight had been anticipated
by their predecessors, but they were the first
to conceive calculus as a system in which
new rhetoric and descriptive terms were created.
Their unique discoveries lay not only in their
imagination, but also in their ability to
synthesize the insights around them into a
universal algorithmic process, thereby forming
a new mathematical system.
==== Newton ====
Newton completed no definitive publication
formalizing his fluxional calculus; rather,
many of his mathematical discoveries were
transmitted through correspondence, smaller
papers or as embedded aspects in his other
definitive compilations, such as the Principia
and Opticks. Newton would begin his mathematical
training as the chosen heir of Isaac Barrow
in Cambridge. His aptitude was recognized
early and he quickly learned the current theories.
By 1664 Newton had made his first important
contribution by advancing the binomial theorem,
which he had extended to include fractional
and negative exponents. Newton succeeded in
expanding the applicability of the binomial
theorem by applying the algebra of finite
quantities in an analysis of infinite series.
He showed a willingness to view infinite series
not only as approximate devices, but also
as alternative forms of expressing a term.Many
of Newton's critical insights occurred during
the plague years of 1665–1666 which he later
described as, "the prime of my age for invention
and minded mathematics and [natural] philosophy
more than at any time since." It was during
his plague-induced isolation that the first
written conception of fluxionary calculus
was recorded in the unpublished De Analysi
per Aequationes Numero Terminorum Infinitas.
In this paper, Newton determined the area
under a curve by first calculating a momentary
rate of change and then extrapolating the
total area. He began by reasoning about an
indefinitely small triangle whose area is
a function of x and y. He then reasoned that
the infinitesimal increase in the abscissa
will create a new formula where x = x + o
(importantly, o is the letter, not the digit
0). He then recalculated the area with the
aid of the binomial theorem, removed all quantities
containing the letter o and re-formed an algebraic
expression for the area. Significantly, Newton
would then “blot out” the quantities containing
o because terms "multiplied by it will be
nothing in respect to the rest".
At this point Newton had begun to realize
the central property of inversion. He had
created an expression for the area under a
curve by considering a momentary increase
at a point. In effect, the fundamental theorem
of calculus was built into his calculations.
While his new formulation offered incredible
potential, Newton was well aware of its logical
limitations at the time. He admits that "errors
are not to be disregarded in mathematics,
no matter how small" and that what he had
achieved was “shortly explained rather than
accurately demonstrated."
In an effort to give calculus a more rigorous
explication and framework, Newton compiled
in 1671 the Methodus Fluxionum et Serierum
Infinitarum. In this book, Newton's strict
empiricism shaped and defined his fluxional
calculus. He exploited instantaneous motion
and infinitesimals informally. He used math
as a methodological tool to explain the physical
world. The base of Newton’s revised calculus
became continuity; as such he redefined his
calculations in terms of continual flowing
motion. For Newton, variable magnitudes are
not aggregates of infinitesimal elements,
but are generated by the indisputable fact
of motion. As with many of his works, Newton
delayed publication. Methodus Fluxionum was
not published until 1736.Newton attempted
to avoid the use of the infinitesimal by forming
calculations based on ratios of changes. In
the Methodus Fluxionum he defined the rate
of generated change as a fluxion, which he
represented by a dotted letter, and the quantity
generated he defined as a fluent. For example,
if
x
{\displaystyle {x}}
and
y
{\displaystyle {y}}
are fluents, then
x
˙
{\displaystyle {\dot {x}}}
and
y
˙
{\displaystyle {\dot {y}}}
are their respective fluxions. This revised
calculus of ratios continued to be developed
and was maturely stated in the 1676 text De
Quadratura Curvarum where Newton came to define
the present day derivative as the ultimate
ratio of change, which he defined as the ratio
between evanescent increments (the ratio of
fluxions) purely at the moment in question.
Essentially, the ultimate ratio is the ratio
as the increments vanish into nothingness.
Importantly, Newton explained the existence
of the ultimate ratio by appealing to motion;
“For by the ultimate velocity is meant that,
with which the body is moved, neither before
it arrives at its last place, when the motion
ceases nor after but at the very instant when
it arrives... the ultimate ratio of evanescent
quantities is to be understood, the ratio
of quantities not before they vanish, not
after, but with which they vanish”Newton
developed his fluxional calculus in an attempt
to evade the informal use of infinitesimals
in his calculations.
==== Leibniz ====
While Newton began development of his fluxional
calculus in 1665–1666 his findings did not
become widely circulated until later. In the
intervening years Leibniz also strove to create
his calculus. In comparison to Newton who
came to math at an early age, Leibniz began
his rigorous math studies with a mature intellect.
He was a polymath, and his intellectual interests
and achievements involved metaphysics, law,
economics, politics, logic, and mathematics.
In order to understand Leibniz’s reasoning
in calculus his background should be kept
in mind. Particularly, his metaphysics which
described the universe as a Monadology, and
his plans of creating a precise formal logic
whereby, "a general method in which all truths
of the reason would be reduced to a kind of
calculation."
In 1672 Leibniz met the mathematician Huygens
who convinced Leibniz to dedicate significant
time to the study of mathematics. By 1673
he had progressed to reading Pascal’s Traité
des Sinus du Quarte Cercle and it was during
his largely autodidactic research that Leibniz
said "a light turned on". Like Newton, Leibniz,
saw the tangent as a ratio but declared it
as simply the ratio between ordinates and
abscissas. He continued this reasoning to
argue that the integral was in fact the sum
of the ordinates for infinitesimal intervals
in the abscissa; in effect, the sum of an
infinite number of rectangles. From these
definitions the inverse relationship or differential
became clear and Leibniz quickly realized
the potential to form a whole new system of
mathematics. Where Newton over the course
of his career used several approaches in addition
to an approach using infinitesimals, Leibniz
made this the cornerstone of his notation
and calculus.
In the manuscripts of 25 October to 11 November
1675, Leibniz recorded his discoveries and
experiments with various forms of notation.
He was acutely aware of the notational terms
used and his earlier plans to form a precise
logical symbolism became evident. Eventually,
Leibniz denoted the infinitesimal increments
of abscissas and ordinates dx and dy, and
the summation of infinitely many infinitesimally
thin rectangles as a long s (∫ ), which
became the present integral symbol
∫
{\displaystyle \scriptstyle \int }
.
While Leibniz's notation is used by modern
mathematics, his logical base was different
from our current one. Leibniz embraced infinitesimals
and wrote extensively so as, “not to make
of the infinitely small a mystery, as had
Pascal.” According to Gilles Deleuze, Leibniz's
zeroes "are nothings, but they are not absolute
nothings, they are nothings respectively"
(quoting Leibniz' text "Justification of the
calculus of infinitesimals by the calculus
of ordinary algebra"). Alternatively, he defines
them as, “less than any given quantity.”
For Leibniz, the world was an aggregate of
infinitesimal points and the lack of scientific
proof for their existence did not trouble
him. Infinitesimals to Leibniz were ideal
quantities of a different type from appreciable
numbers. The truth of continuity was proven
by existence itself. For Leibniz the principle
of continuity and thus the validity of his
calculus was assured. Three hundred years
after Leibniz's work, Abraham Robinson showed
that using infinitesimal quantities in calculus
could be given a solid foundation.
==== Legacy ====
The rise of calculus stands out as a unique
moment in mathematics. Calculus is the mathematics
of motion and change, and as such, its invention
required the creation of a new mathematical
system. Importantly, Newton and Leibniz did
not create the same calculus and they did
not conceive of modern calculus. While they
were both involved in the process of creating
a mathematical system to deal with variable
quantities their elementary base was different.
For Newton, change was a variable quantity
over time and for Leibniz it was the difference
ranging over a sequence of infinitely close
values. Notably, the descriptive terms each
system created to describe change was different.
Historically, there was much debate over whether
it was Newton or Leibniz who first "invented"
calculus. This argument, the Leibniz and Newton
calculus controversy, involving Leibniz, who
was German, and the Englishman Newton, led
to a rift in the European mathematical community
lasting over a century. Leibniz was the first
to publish his investigations; however, it
is well established that Newton had started
his work several years prior to Leibniz and
had already developed a theory of tangents
by the time Leibniz became interested in the
question.
It is not known how much this may have influenced
Leibniz. The initial accusations were made
by students and supporters of the two great
scientists at the turn of the century, but
after 1711 both of them became personally
involved, accusing each other of plagiarism.
The priority dispute had an effect of separating
English-speaking mathematicians from those
in the continental Europe for many years.
Only in the 1820s, due to the efforts of the
Analytical Society, did Leibnizian analytical
calculus become accepted in England. Today,
both Newton and Leibniz are given credit for
independently developing the basics of calculus.
It is Leibniz, however, who is credited with
giving the new discipline the name it is known
by today: "calculus". Newton's name for it
was "the science of fluents and fluxions".
The work of both Newton and Leibniz is reflected
in the notation used today. Newton introduced
the notation
f
˙
{\displaystyle {\dot {f}}}
for the derivative of a function f. Leibniz
introduced the symbol
∫
{\displaystyle \int }
for the integral and wrote the derivative
of a function y of the variable x as
d
y
d
x
{\displaystyle {\frac {dy}{dx}}}
, both of which are still in use.
Since the time of Leibniz and Newton, many
mathematicians have contributed to the continuing
development of calculus. One of the first
and most complete works on both infinitesimal
and integral calculus was written in 1748
by Maria Gaetana Agnesi.
=== Operational methods ===
Antoine Arbogast (1800) was the first to separate
the symbol of operation from that of quantity
in a differential equation. Francois-Joseph
Servois (1814) seem to have been the first
to give correct rules on the subject. Charles
James Hargreave (1848) applied these methods
in his memoir on differential equations, and
George Boole freely employed them. Hermann
Grassmann and Hermann Hankel made great use
of the theory, the former in studying equations,
the latter in his theory of complex numbers.
=== Calculus of variations ===
The calculus of variations may be said to
begin with a problem of Johann Bernoulli (1696).
It immediately occupied the attention of Jakob
Bernoulli but Leonhard Euler first elaborated
the subject. His contributions began in 1733,
and his Elementa Calculi Variationum gave
to the science its name. Joseph Louis Lagrange
contributed extensively to the theory, and
Adrien-Marie Legendre (1786) laid down a method,
not entirely satisfactory, for the discrimination
of maxima and minima. To this discrimination
Brunacci (1810), Carl Friedrich Gauss (1829),
Siméon Denis Poisson (1831), Mikhail Vasilievich
Ostrogradsky (1834), and Carl Gustav Jakob
Jacobi (1837) have been among the contributors.
An important general work is that of Sarrus
(1842) which was condensed and improved by
Augustin Louis Cauchy (1844). Other valuable
treatises and memoirs have been written by
Strauch (1849), Jellett (1850), Otto Hesse
(1857), Alfred Clebsch (1858), and Carll (1885),
but perhaps the most important work of the
century is that of Karl Weierstrass. His course
on the theory may be asserted to be the first
to place calculus on a firm and rigorous foundation.
== Integrals ==
Niels Henrik Abel seems to have been the first
to consider in a general way the question
as to what differential equations can be integrated
in a finite form by the aid of ordinary functions,
an investigation extended by Liouville. Cauchy
early undertook the general theory of determining
definite integrals, and the subject has been
prominent during the 19th century. Frullani
integral, David Bierens de Haan's work on
the theory and his elaborate tables, Lejeune
Dirichlet's lectures embodied in Meyer's treatise,
and numerous memoirs of Legendre, Poisson,
Plana, Raabe, Sohncke, Schlömilch, Elliott,
Leudesdorf and Kronecker are among the noteworthy
contributions.
Eulerian integrals were first studied by Euler
and afterwards investigated by Legendre, by
whom they were classed as Eulerian integrals
of the first and second species, as follows:
∫
0
1
x
n
−
1
(
1
−
x
)
n
−
1
d
x
{\displaystyle \int _{0}^{1}x^{n-1}(1-x)^{n-1}\,dx}
∫
0
∞
e
−
x
x
n
−
1
d
x
{\displaystyle \int _{0}^{\infty }e^{-x}x^{n-1}\,dx}
although these were not the exact forms of
Euler's study.
If n is a positive integer, it follows that:
∫
0
∞
e
−
x
x
n
−
1
d
x
=
(
n
−
1
)
!
,
{\displaystyle \int _{0}^{\infty }e^{-x}x^{n-1}dx=(n-1)!,}
but the integral converges for all positive
real
n
{\displaystyle n}
and defines an analytic continuation of the
factorial function to all of the complex plane
except for poles at zero and the negative
integers. To it Legendre assigned the symbol
Γ
{\displaystyle \Gamma }
, and it is now called the gamma function.
Besides being analytic over positive reals
ℝ+,
Γ
{\displaystyle \Gamma }
also enjoys the uniquely defining property
that
log
⁡
Γ
{\displaystyle \log \Gamma }
is convex, which aesthetically justifies this
analytic continuation of the factorial function
over any other analytic continuation. To the
subject Lejeune Dirichlet has contributed
an important theorem (Liouville, 1839), which
has been elaborated by Liouville, Catalan,
Leslie Ellis, and others. On the evaluation
of
Γ
(
x
)
{\displaystyle \Gamma (x)}
and
log
⁡
Γ
(
x
)
{\displaystyle \log \Gamma (x)}
Raabe (1843–44), Bauer (1859), and Gudermann
(1845) have written. Legendre's great table
appeared in 1816.
== Applications ==
The application of the infinitesimal calculus
to problems in physics and astronomy was contemporary
with the origin of the science. All through
the 18th century these applications were multiplied,
until at its close Laplace and Lagrange had
brought the whole range of the study of forces
into the realm of analysis. To Lagrange (1773)
we owe the introduction of the theory of the
potential into dynamics, although the name
"potential function" and the fundamental memoir
of the subject are due to Green (1827, printed
in 1828). The name "potential" is due to Gauss
(1840), and the distinction between potential
and potential function to Clausius. With its
development are connected the names of Lejeune
Dirichlet, Riemann, von Neumann, Heine, Kronecker,
Lipschitz, Christoffel, Kirchhoff, Beltrami,
and many of the leading physicists of the
century.
It is impossible in this place to enter into
the great variety of other applications of
analysis to physical problems. Among them
are the investigations of Euler on vibrating
chords; Sophie Germain on elastic membranes;
Poisson, Lamé, Saint-Venant, and Clebsch
on the elasticity of three-dimensional bodies;
Fourier on heat diffusion; Fresnel on light;
Maxwell, Helmholtz, and Hertz on electricity;
Hansen, Hill, and Gyldén on astronomy; Maxwell
on spherical harmonics; Lord Rayleigh on acoustics;
and the contributions of Lejeune Dirichlet,
Weber, Kirchhoff, F. Neumann, Lord Kelvin,
Clausius, Bjerknes, MacCullagh, and Fuhrmann
to physics in general. The labors of Helmholtz
should be especially mentioned, since he contributed
to the theories of dynamics, electricity,
etc., and brought his great analytical powers
to bear on the fundamental axioms of mechanics
as well as on those of pure mathematics.
Furthermore, infinitesimal calculus was introduced
into the social sciences, starting with Neoclassical
economics. Today, it is a valuable tool in
mainstream economics.
== See also ==
Analytic geometry
History of logarithms
History of mathematics
Non-Newtonian calculus
Non-standard calculus
== Notes ==
== Further reading ==
Roero, C.S. (2005). "Gottfried Wilhelm Leibniz,
first three papers on the calculus (1684,
1686, 1693)". In Grattan-Guinness, I. (ed.).
Landmark writings in Western mathematics 1640–1940.
Elsevier. pp. 46–58. ISBN 978-0-444-50871-3.
Roero, C.S. (1983). "Jakob Bernoulli, attentive
student of the work of Archimedes: marginal
notes to the edition of Barrow". Boll. Storia
Sci. Mat. 3 (1): 77–125.
Boyer, Carl (1959). The History of the Calculus
and its Conceptual Development. New York:
Dover Publications. Republication of a 1939
book (2nd printing in 1949) with a different
title.
Calinger, Ronald (1999). A Contextual History
of Mathematics. Toronto: Prentice-Hall. ISBN
978-0-02-318285-3.
Reyes, Mitchell (2004). "The Rhetoric in Mathematics:
Newton, Leibniz, the Calculus, and the Rhetorical
Force of the Infinitesimal". Quarterly Journal
of Speech. 90 (2): 159–184. doi:10.1080/0033563042000227427.
Grattan-Guinness, Ivor. The Rainbow of Mathematics:
A History of the Mathematical Sciences, Chapters
5 and 6, W. W. Norton & Company, 2000.
Hoffman, Ruth Irene, "On the development and
use of the concepts of the infinitesimal calculus
before Newton and Leibniz", Thesis (M.A.),
University of Colorado, 1937
== External links ==
A history of the calculus in The MacTutor
History of Mathematics archive, 1996.
Earliest Known Uses of Some of the Words of
Mathematics: Calculus & Analysis
Newton Papers, Cambridge University Digital
Library
(in English) (in Arabic) The Excursion of
Calculus, 1772
