Talk given at the Consulate of Switzerland on April 16th, 2007; celebrating the life and legacy of Leonhard Euler and overview of the Euler Archive (http://www.eulerarchive.org)

1. The Euler Archive:
Illuminating the Life and Times
of Leonhard Euler
http://www.EulerArchive.org
Lee Stemkoski
Adelphi University
April 16, 2007

2. Leonhard Euler
 A Leading Scholar
of the 18th Century
(1707 – 1783)
– Quantity of work (850+)
– Breadth of topics
– Depth of investigation
– Presentation style

3. Outline of Topics
 A history of publishing Euler’s works
 The Euler Archive: history, layout
 Life in Basel, St. Petersburg, Berlin
 Some of Euler’s major contributions
 Euler’s continuing influence

4. Publishing Euler’s works
 1830: St. Petersburg Academy
“finishes” backlog of Euler’s works
 1843: P.H. Fuss creates index
 Fuss-Jacobi correspondence
 1849: Commentationes Arithmeticae
 1883: F. Rudio delivers biographical talk
 1896: Hagen creates updated index

5. The Euler Commission
 April, 1908: International Mathematical
Congress discusses publication
 Previously discussed in 1907:
- International Association of Academies
- Swiss Society of Natural Sciences
 August, 1908: Euler Commission created
 1910: Eneström creates updated index
 1911: Opera Omnia publication begins

6. Worldwide Support
Subscriptions
 Academies of Paris, Berlin, and
St. Petersburg each order 40 sets
Donations
 German Association of Mathematicians
 American Mathematical Society

7. Euler’s Opera Omnia
 Series I: Mathematics (29/29)
 Series II: Mechanics, Astronomy (29/31)
 Series III: Physics, Miscellaneous (12/12)
 Series IVA: Correspondence (4/10)
 Series IVB: Manuscripts and Notebooks

8. The Euler Archive: History
 2001: The Euler Society
 Dr. Edward Sandifer, talk at Dartmouth
 Two graduate students: Klyve, Stemkoski
 2002: Conference presentation
 2003: Euler Archive, version 1.0
 Primarily a response to inaccessibility of Euler’s
works (cost and location)
 Marianne Gerber, Deputy Consul, SHARE

9. The Euler Archive: Layout
 One webpage for each of Euler’s works
 Organization:
Subject, Date, Journal
 Context:
History, Biography, Academies,
Contemporaries, Miscellaneous
 Features:
Translations, Letters, Resources

10. The Life and Times of
Leonhard Euler
 Primary Sources:
– Autobiography
– Eulogy by Nicolaus Fuss
– Eulogy by Marquis de Condorcet
– Correspondence

11. The Basel Period
 Born: April 15, 1707 - Basel, Switzerland
 Parents, Religion, and Mathematics
 1720: enters University of Basel
 1722: prima laurea (~ Bachelor of Arts)
 1723: magister (~ Master of Arts)
 Johann Bernoulli: no lessons, only questions
 Friends: Daniel and Nicolaus Bernoulli,
Johann Kaspar Wettstein

12. Search for Employment
 1725: young Bernoulli brothers leave for
the new St. Petersburg Academy;
Euler offered professorship of medicine
 1726: applies for professor of physics at
University of Basel
 1727: leaves for St. Petersburg
 Paris Prize Competition – essay on
masting of ships wins honorable mention

13. The St. Petersburg years
 Death of Empress Catherine I
 Christian Goldbach and Number Theory
 1733: Succeeds Daniel Bernoulli as
professor of mathematics
 Jan. 1734: marries Katherina Gsell
 Nov. 1734: birth of Johann Albrecht

14. The St. Petersburg years:
early mathematics
 1732: Fermat numbers [E-26]
3, 5, 17, 257, 65537, 4294967297
 1734: Mixed partial derivatives [E-44]
 1734: Functional notation f(x) [E-45]

15. The Basel Problem
 In 1735: [E-41]
The sum of the series
1/1 + 1/4 + 1/9 + 1/16 + 1/25 + …
is exactly equal to
π2 / 6 .

16. Bridges of Königsberg
 In 1735: [E-53]
Can you arrange a route that crosses
every bridge exactly once?

17. Academy Duties
 Examination board member of
academic gymnasium
 Weights and measures commission
 Development of hydraulic machines

18. Cartography
 1735: Euler and Delisle direct the
academy’s Department of Geography
 1753: Preface to
Geographic Atlas
 1775: on the
mathematics
of projections

19. Paris Prize Competition
 Euler: 18 total entries (12 wins)
 Sample topics:
– 1727: masting of ships
– 1738: properties of fire
– 1740: tides
– 1747: determining time from observations
– 1748: magnets
– 1748: movement of Jupiter and Saturn
– 1753: movement of ships without wind

20. Major Works during first
St. Petersburg period
 1736: Mechanics [E-15][E-16]
 1739: Music Theory [E-33]
 1738/1740: Arithmetic [E-17][E-35]

21. Move to Berlin (1741)
 Political tensions in St. Petersburg
 Growing Family: Johann Albrecht, Karl,
Katharina, Christoph, Charlotte, …
 Correspondence continues
 Russian Lodgers/Students:
Razumovsky, Kotelnikov, Rumovsky

22. Major Works in Berlin
 1744: Calculus of Variations
 1745: New Principles of Gunnery
 1748: Introduction to the Analysis
of the Infinite
 1749: Theory of Ships
 1753: Theory of Lunar Motion
 1755: Differential Calculus
 1762: Construction of Achromatic Lenses
 1765: Theory of the Motion of Solid Bodies

23. Introductio
 1748 [E-101]
– Variables and functions
– Polynomials and algebraic functions
– Exponents and logarithms
– Trigonometry
– Series

24. Polyhedral Formula
1750-1751 [E-230]
V = # vertices (corners)
E = # edges
F = # faces
Then: V – E + F = 2

25. Euler-Fermat Theorem
1760 [E-271]
Choose integer a and prime number p
ap – 1 (mod p) = 1
Replace p by arbitrary number n
aφ(n) (mod n) = 1
Basis of many encryption algorithms

26. End of Berlin period
 Fredrick the Great
 Maupertuis
 Return to St. Petersburg (1766)

27. Major Works from Euler’s
second St. Petersburg period
 1768: Letters to a German Princess
 1768: Integral Calculus
 1769: Optics
 1770: Algebra
 1772: Theory of Lunar Motion
 1773: Complete Theory on the
Construction and Maneuvering of Ships

28. Letters to a German Princess
 First published in 1768 (234 letters)
 General science: gravity, cosmos, tides
 Philosophy: the spirit, religion,
morality, foundations of knowledge
 Physics: electricity, magnetism,
telescopes, microscopes
 Over 40 editions by 1840

29. Euler’s final day
 Assistants Fuss and Lexell present
 Discussions about
– height of hot-air balloons
– orbit of planet Uranus
 Suffered a stroke in the early evening

30. Euler’s continuing influence
 Disciples / Students
– J.A. Euler, Fuss, Lexell, Rumovsky, …
 Style of presentation
 Legacy of publications

31. Conclusion
 From the Eulogy written by Fuss:
“What is there left … except our tender
and eternal gratitude … [Euler] was
worthy of admiration as much for his
rare virtues as for the astonishing
force of his genius.”