Eulers Introductio of 1748

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Eulers Introductio of 1748

• V. Frederick Rickey
• West Point
• AMS San Francisco, April 29, 2006

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(No Transcript)
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Eulers Life

• Basel 1707-1727 20
• Petersburg I 1727-1741 14
• Berlin 1741-1766 25
• Petersburg II 1766-1783 17
• ____
• 76

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• 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
  development.

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• 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

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Eulers Calculus Books

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

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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

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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

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(No Transcript)
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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
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Chapter 1 Functions

• A change of Ontology
• Study functions
• not curves

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VI Exponentials and Logarithms

• A masterful development
• Uses infinitesimals

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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.

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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

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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.

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• 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

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VIII Trig Functions
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• 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

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gallica.bnf.fr

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

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XIII Recurrent Series

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

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XIII Recurrent Series

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

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XIII Recurrent Series

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

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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)
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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

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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

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Continued Fractions and Calendars

• The solar year is 365 days, 48 minutes, and 55
  seconds
• 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.

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Read Euler, read Euler, he is our teacher in
everything.

• Laplace
• as quoted by Libri, 1846

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Lisez Euler, lisez Euler, c'est notre maître
à tous.

• Laplace
• as quoted by Libri, 1846

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www.dean.usma.edu/departments/math/people/rickey/
hm/

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