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## Eulers Introductio of 1748

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### 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

1
Eulers Introductio of 1748
• V. Frederick Rickey
• West Point
• AMS San Francisco, April 29, 2006

2
(No Transcript)
3
Eulers Life
• Basel 1707-1727 20
• Petersburg I 1727-1741 14
• Berlin 1741-1766 25
• Petersburg II 1766-1783 17
• ____
• 76

4
• The time had come in which to assemble in a
systematic and contained work the entire body of
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.

5
• 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

6
Eulers Calculus Books
• 1748 Introductio in analysin infinitorum
• 399
• 402
• 1755 Institutiones calculi differentialis
• 676
• 1768 Institutiones calculi integralis
• 462
• 542
• 508
• _____
• 2982

7
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

8
• 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
• Thanks to Florence Fasanelli

9
(No Transcript)
10
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
11
Chapter 1 Functions
• A change of Ontology
• Study functions
• not curves

12
VI Exponentials and Logarithms
• A masterful development
• Uses infinitesimals

13
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.

14
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

15
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.

16
• 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

17
VIII Trig Functions
18
• 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

19
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

20
XIII Recurrent Series
• Problem When you expand a function into a
series, find a formula for the general term.

21
XIII Recurrent Series
• Problem When you expand a function into a
series, find a formula for the general term.

22
XIII Recurrent Series
• Problem When you expand a function into a
series, find a formula for the general term.

23
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)
24
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

25
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

26
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.

27
everything.
• Laplace
• as quoted by Libri, 1846

28
Lisez Euler, lisez Euler, c'est notre maître
à tous.
• Laplace
• as quoted by Libri, 1846

29
www.dean.usma.edu/departments/math/people/rickey/
hm/
• A Readers Guide to Eulers Introductio
• Errata in Blantons 1988 English translation