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A very short history of Calculus presentation for MATH 1037 by Alex Karassev


Qi Pi can be made arbitrary small. Hence Pi approximate C(R) arbitrarily closely. Elementary ... Pi(R) : Ri (R') = R2:R'2. Suppose that C(R):C(R') R2:R'2 ... – PowerPoint PPT presentation

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Title: A very short history of Calculus presentation for MATH 1037 by Alex Karassev

A very short history of Calculus presentation
for MATH 1037 by Alex Karassev
  • Irrational numbers in Greek math
  • Theory of Proportion
  • The Method of Exhaustion
  • The Area of a Parabolic Segment
  • What is Calculus?
  • Early Results on Areas and Volumes
  • Maxima, Minima, and Tangents
  • The Arithmetica Infinitorum of Wallis
  • Newtons Calculus of Series
  • The Calculus of Leibniz
  • Biographical Notes Archimedes, Wallis, Newton,
    and Leibniz

Irrational numbers in Greek math
  • Discovery of irrational numbers
  • Greeks tried to avoid the use of irrationals
  • The infinity was understood as potential for
    continuation of a process but not as actual
    infinity (static and completed)
  • Examples
  • 1,2, 3,... but not the set 1,2,3,…
  • sequence x1, x2, x3,… but not the limit x lim
  • Paradoxes of Zeno ( 450 BCE) the Dichotomy
  • there is no motion because that which is moved
    must arrive at the middle before it arrives at
    the end
  • Approximation of v2 by the sequence of rational

4.2 Eudoxus Theory of Proportions
  • Eudoxus (around 400 350 BCE)
  • The theory was designed to deal with (irrational)
    lengths using only rational numbers
  • Length ? is determined by rational lengths less
    than and greater than ?
  • Then ?1 ?2 if for any rational r is rational r ?1 )
  • Note the theory of proportions can be used to
    define irrational numbers Dedekind (1872)
    defined v2 as the pair of two sets of positive
    rationals Lv2 r r22
    (Dedekind cut)

The Method of Exhaustion
  • was designed to find areas and volumes of
    complicated objects (circles, pyramids, spheres)
  • approximations by simple objects (rectangles,
    trianlges, prisms) having known areas (or
  • the Theory of Proportions

Approximating the pyramid
Approximating the circle
Example Area enclosed by a Circle
  • Let C(R) denote area of the circle of radius R
  • We show that C(R) is proportional to R2
  • Inner polygons P1
  • Outer polygons Q1 Q2 Q3 …
  • Qi Pi can be made arbitrary small
  • Hence Pi approximate C(R) arbitrarily closely
  • Elementary geometry shows that Pi is proportional
    to R2 . Therefore, for two circles with radii R
    and R' we get Pi(R) Ri (R) R2R2
  • Suppose that C(R)C(R)
  • Then (since Pi approximates C(R)) we can find i
    such that Pi (R) Pi (R) contradicts 5)

Thus Pi(R) Ri (R) R2R2
4.4 The area of a Parabolic Segment Archimedes
(287 212 BCE)
  • Triangles ?1 , ?2 , ?3 , ?4,…
  • Note that ?2 ?3 1/4 ?1
  • Similarly ?4 ?5 ?6 ?7 1/16 ?1 and so on

Thus A ?1 (11/4 (1/4)2…) 4/3 ?1
What is Calculus?
  • Calculus appeared in 17th century as a system of
    shortcuts to results obtained by the method of
  • Calculus derives rules for calculations
  • Problems, solved by calculus include finding
    areas, volumes (integral calculus), tangents,
    normals and curvatures (differential calculus)
    and summing of infinite series
  • This makes calculus applicable in a wide variety
    of areas inside and outside mathematics
  • In traditional approach (method of exhaustions)
    areas and volumes were computed using subtle
    geometric arguments
  • In calculus this was replaced by the set of rules
    for calculations

17th century calculus
  • Differentiation and integration of powers of x
    (including fractional powers) and implicit
    differentiation of polynomials in x and y
  • Together with analytic geometry this made
    possible to find tangents, maxima and minima of
    all algebraic curves p (x,y) 0
  • Newtons calculus of infinite series (1660s)
    allowed for differentiation and integration of
    all functions expressible as power series
  • Culmination of 17th century calculus discovery
    of the Fundamental Theorem of Calculus by Newton
    and Leibniz (independently)
  • Features of 17th century calculus
  • the concept of limit was not introduced yet
  • use of indivisibles or infinitesimals
  • strong opposition of some well-known philosophers
    of that time (e.g. Thomas Hobbes)
  • very often new results were conjectured by
    analogy with previously discovered formulas and
    were not rigorously proved

Early Results on Areas and Volumes
  • Area (1/n)k (2/n)k … (n/n)k(1/n)
  • ? sum 1k 2k … nk

y xk
Volume of the solid of revolution area of
cross-section is p r2 and therefore it is
required to compute sum 12k 22k 32k … n2k
n/n 1
  • First results Greek mathematicians (method of
    exhaustion, Archimedes)
  • Arab mathematician al-Haytham (10th -11th
    centuries) summed the series 1k 2k … nk
    for k 1, 2, 3, 4 and used the result to find
    the volume of the solid obtained by rotating the
    parabola about its base
  • Cavalieri (1635) up to k 9 and conjectured the
    formula for positive integers k
  • Another advance made by Cavalieri was
    introduction of indivisibles which considered
    areas divided into infinitely thin strips and
    volumes divided into infinitely thin slices
  • It was preceded by the work of Kepler on the
    volumes of solids of revolution (New Stereometry
    of wine barrels, 1615)
  • Fermat, Descartes and Roberval (1630s) proved the
    formula for integration of xk (even for
    fractional values of k)
  • Torricelly the solid obtained by rotating y 1
    / x about the x-axis from 1 to infinity has
    finite volume!
  • Thomas Hobbes (1672) to understand this
    result for sense, it is not required that a man
    should be a geometrician or logician, but that he
    should be mad

Maxima, Minima, and Tangents
  • The idea of differentiation appeared later than
    that one of integration
  • First result construction of tangent line to
    spiral r a? by Archimedes
  • No other results until works of Fermat (1629)

modern approach
Fermats approach (tangent to y x2)
  • E small or infinitesimal element which is
    set equal to zero at the end of all computations
  • Thus at all steps E ? 0 and at the end E 0
  • Philosophers of that time did not like such

  • Fermats method worked well with all polynomials
  • Moreover, Fermat extended this approach to curves
    given by p(x,y) 0
  • Completely the latter problem was solved by Sluse
    (1655) and Hudde (1657)
  • The formula is equivalent to the use of implicit

The Arithmetica Infinitorum of Wallis (1655)
  • An attempt to arithmetize the theory of areas and
  • Wallis found that ?01 xpdx 1/(p1) for positive
    integers p (which was already known)
  • Another achievement formula for ?01 xm/ndx
  • Wallis calculated ?01 x1/2dx, ?01 x1/3dx,…, using
    geometric arguments, and conjectured the general
    formula for fractional p
  • Note observing a pattern for p 1,2,3, Wallis
    claimed a formula for all positive p by
    induction and for fractional p by
    interpolation (lack of rigour but a great deal
    of analogy, intuition and ingenuity)

y x2
?01 x1/2dx 1 - 1/3 2/3
?01 x2dx 1/3
  • Wallis formula
  • Expansion of p as infinite product was known to
    Viète (before Wallis discovery)
  • Nevertheless Wallis formula relates p to the
    integers through a sequence of rational
  • Moreover, basing on the formula for p Wallis
    found a sequence of fractions he called
    hypergeometric, which as it had been found
    later occur as coefficients in series expansions
    of many functions (which led to the class of
    hypergeometric functions)

Other formulas for p related to Wallis formula
Continued fraction (Brouncker)
Series expansion discovered by 15th century
Indian mathematicians and rediscovered by Newton,
Gregory and Leibniz
sub. x 1
Newtons Calculus of Series
  • Isaac Newton
  • Most important discoveries in 1665/6
  • Before he studied the works of Descartes, Viète
    and Wallis
  • Contributions to differential calculus (e.g. the
    chain rule)
  • Most significant contributions are related to the
    theory of infinite series
  • Newton used term-by-term integration and
    differentiation to find power series
    representation of many of classical functions,
    such as tan-1x or log (x1)
  • Moreover, Newton developed a method of inverting
    infinite power series to find inverses of
    functions (e.g ex from log (x1))
  • Unfortunately, Newtons works were rejected for
    publication by Royal Society and Cambridge
    University Press

The Calculus of Leibniz
  • The first published paper on calculus was
    by Gottfried Wilhelm Leibniz (1684)
  • Leibniz discovered calculus independently
  • He had better notations than Newtons
  • Leibniz was a librarian, a philosopher and a
  • Nova methodus (1864)
  • sum, product and quotient rules
  • notation dy / dx
  • dy / dx was understood by Leibniz literally as a
    quotient of infinitesimals dy and dx
  • dy and dx were viewed as increments of x and y

The Fundamental Theorem of Calculus
  • In De geometria (1686) Leibniz introduced the
    integral sign ?
  • Note that ? f(x) dx meant (for Leibniz) a sum of
    terms representing infinitesimal areas of height
    f(x) and width dx
  • If one applies the difference operator d to such
    sum it yields the last term f(x) dx
  • Dividing by dx we obtain the Fundamental Theorem
    of Caculus

  • Leibniz introduced the word function
  • He preferred closed-form expressions to
    infinite series
  • Evaluation of integral ? f(x) dx was for Leibniz
    the problem of finding a known function whose
    derivative is f(x)
  • The search for closed forms led to
  • the problem of factorization of polynomials and
    eventually to the Fundamental Theorem of Algebra
    (integration of rational functions)
  • the theory of elliptic functions (attempts to
    integrate 1/v1-x4 )

Biographical Notes
  • Archimedes
  • Wallis
  • Newton
  • Leibniz

  • Was born and worked in Syracuse (Greek city in
    Sicily) 287 BCE and died in 212 BCE
  • Friend of King Hieron II
  • Eureka! (discovery of hydrostatic law)
  • Invented many mechanisms, some of which were used
    for the defence of Syracuse
  • Other achievements in mechanics usually
    attributed to Archimedes (the law of the lever,
    center of mass, equilibrium, hydrostatic
  • Used the method of exhaustions to show that the
    volume of sphere is 2/3 that of the enveloping
  • According to a legend, his last words were Stay
    away from my diagram!, address to a soldier who
    was about to kill him

John Wallis Born 23 Nov 1616 (Ashford, Kent,
England) Died 28 Oct 1703 (Oxford, England)
  • went to school in Ashford
  • Wallis academic talent was recognized very early
  • 14 years old he was sent to Felsted, Essex to
    attend the school
  • He became proficient in Latin, Greek and Hebrew
  • Mathematics was not considered important in the
    best schools
  • Wallis learned rules of arithmetic from his
  • That time mathematics was not consider as a
    pure science in the Western culture
  • In 1632 he entered Emmanuel College in Cambridge
  • bachelor of arts degree (topics studied included
    ethics, metaphysics, geography, astronomy,
    medicine and anatomy)
  • Wallis received his Master's Degree in 1640

  • Between 1642 and 1644 he was chaplain at
    Hedingham, Essex and in London
  • Wallis became a fellow of Queens College,
  • He relinquished the fellowship when he married in
  • Wallis was interested in cryptography
  • Civil War between the Royalists and
    Parliamentarians began in 1642
  • Wallis used his skills in cryptography in
    decoding Royalist messages for the
  • Since the appointment to the Savilian Chair in
    Geometry of Oxford in 1649 by Cromwell Wallis
    actively worked in mathematics

Sir Isaac Newton Born 4 Jan 1643 (Woolsthorpe,
Lincolnshire, England) Died 31 March 1727
(London, England)
  • A family of farmers
  • Newtons father (also Isaac Newton) was a wealthy
    but completely illiterate man who even could not
    sign his own name
  • He died three months before his son was born
  • Young Newton was abandoned by his mother at the
    age of three and was left in the care of his
  • Newtons childhood was not happy at all
  • Newton entered Trinity College (Cambridge) in 1661

  • Newton entered Trinity College (Cambridge) in
    1661 to pursue a law degree
  • Despite the fact that his mother was a wealthy
    lady he entered as a sizar
  • He studied philosophy of Aristotle
  • Newton was impressed by works of Descartes
  • In his notes Quaestiones quaedam philosophicae
    1664 (Certain philosophical questions) Newton
    recorded his thoughts related to mechanics,
    optics, and the physiology of vision

  • The years 1664 66 were the most important in
    Newtons mathematical development
  • By 1664 he became familiar with mathematical
    works of Descartes, Viète and Wallis and began
    his own investigations
  • He received his bachelor's degree in 1665
  • When the University was closed in the summer of
    1665 because of the plague in England, Newton had
    to return to Lincolnshire
  • At that time Newton completely devoted himself to

  • Newtons fundamental works on calculus A
    treatise of the methods of series and fluxions
    (1671) (or De methodis) and On analysis by
    equations unlimited in their number of terms
    (1669) (or De analysis) were rejected for
  • Nevertheless some people recognized his genius
  • Isaac Barrow resigned the Lucasian Chair
    (Cambridge) in 1669 and recommended that Newton
    be appointed in his place
  • Newton's first work as Lucasian Prof. was on
  • In particular, using a glass prism Newton
    discovered the spectrum of white light

  • 1665 Newton discovered inverse square law of
  • 1687 Philosophiae naturalis principia
    mathematica (Mathematical principles of natural
  • In this work, Newton developed mathematical
    foundation of the theory of gravitation
  • This book was published by Royal Society (with
    the strong support from Edmund Halley)
  • In 1693 Newton had a nervous breakdown
  • In 1696 he left Cambridge and accepted a
    government position in London where he became
    master of the Mint in 1699
  • In 1703 he was elected president of the Royal
    Society and was re-elected each year until his
  • Newton was knighted in 1705 by Queen Anne

Gottfried Wilhelm von Leibniz Born 1 July 1646
(Leipzig, Saxony (now Germany) Died 14 Nov 1716
(Hannover, Hanover (now Germany)
  • An academic family
  • From the age of six Leibniz was given free access
    to his fathers library
  • At the age of seven he entered school in Leipzig
  • In school he studied Latin
  • Leibniz had taught himself Latin and Greek by the
    age of 12
  • He also studied Aristotle's logic at school
  • In 1661 Leibniz entered the University of Leipzig
  • He studied philosophy and mathematics
  • In 1663 he received a bachelor of law degree for
    a thesis De Principio Individui (On the
    Principle of the Individual)
  • The beginning of the concept of monad
  • He continued work towards doctorate
  • Leibniz received a doctorate degree from
    University of Altdorf (1666)

  • During his visit to the University of Jena (1663)
    Leibniz learned a little of Euclid
  • Leibniz idea was to create some universal logic
  • After receiving his degree Leibniz commenced a
    legal career
  • From 1672 to 1676 Leibniz developed his ideas
    related to calculus and obtained the fundamental
  • Leibniz was interested in summation of infinite
    series by investigation of the differences
    between successive terms
  • He also used term-by term integration to discover
    series representation of p
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