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Eulers Introductio of 1748

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

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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

- 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.

- 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

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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

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

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

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

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.

Read Euler, read Euler, he is our teacher in

everything.

- Laplace
- as quoted by Libri, 1846

Lisez Euler, lisez Euler, c'est notre maître

à tous.

- Laplace
- as quoted by Libri, 1846

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

hm/

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