Euler’s Introductio of 1748

V. Frederick Rickey

West Point

AMS San Francisco, April 29, 2006

Euler’s 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

Euler’s Calculus Books

• 1748 Introductio in analysin infinitorum399

402

• 1755 Institutiones calculi differentialis676

• 1768 Institutiones calculi integralis462

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

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.

log1 z z z2

2z3

3z4

4z5

5z6

6 . . .

2.30258 9

192

293

394

495

596

6 . . .

How series converge

• Log(1+x/1-x) is strongly convergent

• Sin(mπ/2n) converges quickly

• Leibniz series for π/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• π is “clearly” irrational• Value of π from de

Lagny• Note error in 113th

decimal place• “scribam π”• W. W. Rouse Ball

discovered (1894) the use of π 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.

1 z

1 z 2z2 1 2 z2 2 z3 6 z4 10 z5 22 z6 . . .

XIII Recurrent Series• Problem: When you expand a function into a

series, find a formula for the general term.

1 z

1 z 2z2 1 2 z2 2 z3 6 z4 10 z5 22 z6 . . .

1

3

1

1 2z1

31 2 z 4 z2 8 z3 16 z4 32 z5 64 z6 . . .

1

3

2

1 z1

32 2 z 2 z2 2 z3 2 z4 2 z5 2 z6 . . .

XIII Recurrent Series• Problem: When you expand a function into a

series, find a formula for the general term.

1 z

1 z 2z2 1 2 z2 2 z3 6 z4 10 z5 22 z6 . . .

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

e 1

2

1

1 1

6 1

10 1

14 1

18 122...

• 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 Reader’s Guide to Euler’s Introductio

• Errata in Blanton’s 1988 English translation