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Eulers Introductio of 1748


Euler's Calculus Books. 1748 Introductio in analysin infinitorum. 399. 402 ... He remarks that this 'can be confirmed by infinitesimal calculus' ... – PowerPoint PPT presentation

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Title: Eulers Introductio of 1748

Eulers Introductio of 1748
  • V. Frederick Rickey
  • West Point
  • AMS San Francisco, April 29, 2006

(No Transcript)
Eulers Life
  • Basel 1707-1727 20
  • Petersburg I 1727-1741 14
  • Berlin 1741-1766 25
  • Petersburg II 1766-1783 17
  • ____
  • 76

  • The time had come in which to assemble in a
    systematic and contained work the entire body of
    the important discoveries that Mr. Euler had made
    in infinitesimal analysis . . . it became
    necessary prior to its execution to prepare the
    world so that it might be able to understand
    these sublime lessons with a preliminary work
    where one would find all the necessary notions
    that this study demands. To this effect he
    prepared his Introductio . . . into which he
    mined the entire doctrine of functions, either
    algebraic, or transcendental while showing their
    transformation, their resolution and their

  • He gathered together everything that he found to
    be useful and interesting concerning the
    properties of infinite series and their
    summations He opened a new road in which to
    treat exponential quantities and he deduced the
    way in which to furnish a more concise and
    fulsome way for logarithms and their usage. He
    showed a new algorithm which he found for
    circular quantities, for which its introduction
    provided for an entire revolution in the science
    of calculations, and after having found the
    utility in the calculus of sine, for which he is
    truly the author, and the recurrent series . . .
  • Eulogy by Nicolas Fuss, 1783

Eulers Calculus Books
  • 1748 Introductio in analysin infinitorum
  • 399
  • 402
  • 1755 Institutiones calculi differentialis
  • 676
  • 1768 Institutiones calculi integralis
  • 462
  • 542
  • 508
  • _____
  • 2982

Euler was prolific
  • I Mathematics 29 volumes
  • II Mechanics, astronomy 31
  • III Physics, misc. 12
  • IVa Correspondence 8
  • IVb Manuscripts 7
  • 87
  • One paper per fortnight, 1736-1783
  • Half of all math-sci work, 1725-1800

Euler about 1737, age 30
  • Painting by J. Brucker
  • Mezzotint of 1737
  • Black below and above right eye
  • Fluid around eye is infected
  • Eye will shrink and become a raisin
  • Ask your opthamologist
  • Thanks to Florence Fasanelli

(No Transcript)
Often I have considered the fact that most of the
difficulties which block the progress of students
trying to learn analysis stem from this that
although they understand little of ordinary
algebra, still they attempt this more subtle art.
From the preface
Chapter 1 Functions
  • A change of Ontology
  • Study functions
  • not curves

VI Exponentials and Logarithms
  • A masterful development
  • Uses infinitesimals

Euler understood convergence
  • But it is difficult to see how this can be since
    the terms of the series continually grow larger
    and the sum does not seem to approach any limit.
    We will soon have an answer to this paradox.

How series converge
  • Log(1x/1-x) is strongly convergent
  • Sin(mp/2n) converges quickly
  • Leibniz series for p/4 hardly converges
  • Another form converges much more rapidly

Two problems using logarithms
  • If the population in a certain region increases
    annually by one thirtieth and at one time there
    were 100,000 inhabitants, we would like to know
    the population after 100 years.
  • People could not believe he population of Berlin
    was over a million.

  • Since after the flood all men descended from a
    population of six, if we suppose the population
    after two hundred years was 1,000,000, we would
    like to find the annual rate of growth.
  • Euler was deeply religious
  • Yet had a sense of humor After 400 years the
    population becomes 166,666,666,666

VIII Trig Functions
  • Sinus totus 1
  • p is clearly irrational
  • Value of p from de Lagny
  • Note error in 113th decimal place
  • scribam p
  • W. W. Rouse Ball discovered (1894) the use of p
    in Wm Jones 1706.
  • Arcs not angles
  • Notation sin. A. z

  • Here you can find
  • The original Latin of 1748 (1967 reprint)
  • Opera omnia edition of 1922
  • French translation of 1796 (1987 reprint)
  • Recherche
  • Télécharger

XIII Recurrent Series
  • Problem When you expand a function into a
    series, find a formula for the general term.

XIII Recurrent Series
  • Problem When you expand a function into a
    series, find a formula for the general term.

XIII Recurrent Series
  • Problem When you expand a function into a
    series, find a formula for the general term.

XIII Recurrent Series
  • Problem When you expand a function into a
    series, find a formula for the general term.

A recursive relation a(0) 1 a(1)
0 a(n) a(n-1) 2 a(n-2)
xvIII On Continued Fractions
  • He develops the theory for finding the
    convergents of a continued fraction, but is
    hampered by a lack of subscript notation
  • He shows how to develop an alternating series
    into a continued fraction

Lots of examples
  • He starts with a numerical value for e
  • He notes the geometric progression
  • He remarks that this can be confirmed by
    infinitesimal calculus
  • But, he does not say that e is irrational

Continued Fractions and Calendars
  • The solar year is 365 days, 48 minutes, and 55
  • Convergents are 0/1, 1/4, 7/29, 8/33, 55/227, . .
  • Excess h-m-s over 365d is about 1 day in 4 years,
    yielding the Julian calendar.
  • More exact is 8 days in 33 years or 181 days in
    747 years. So in 400 years there are 97 extra
    days, while Julian gives 100. Thus the Gregorian
    calendar converts three leap years to ordinary.

Read Euler, read Euler, he is our teacher in
  • Laplace
  • as quoted by Libri, 1846

Lisez Euler, lisez Euler, c'est notre maître
à tous.
  • Laplace
  • as quoted by Libri, 1846

  • A Readers Guide to Eulers Introductio
  • Errata in Blantons 1988 English translation