# The Early Mathematics of Leonhard Euler ### by C. Edward Sandifer

Print ISBN: 978-0-88385-559-1
416 pp., Hardbound, 2007
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Series: Spectrum

The Early Mathematics of Leonhard Euler describes Euler’s early mathematical works: the 50 or so mathematical articles he wrote before he left St. Petersburg in 1741 to join the Academy of Frederick the Great in Berlin. These pieces contain some of Euler’s greatest mathematics; the Konigsberg bridge problem, his solution to the Basel problem, and his first proof of the Euler-Fermat theorem. It also presents important results that we seldom realize are due to Euler: that mixed partial derivatives are (usually) equal, our $f(x)$ notation, and the integrating factor in differential equations.

Preface
1. Construction of isochronal curves in any kind of resistant
2. Method of finding reciprocal algebraic trajectories
3. Solution to problems of reciprocal trajectories
4. A new method of reducing innumerable differential equations of the second degree to equations of the first degree
5. On transcendental progressions, or those for which the general term cannot be given algebraically
6. On the shortest curve on a surface that joins any two given points
7. On the summation of innumerably many progressions
8. General methods for summing progressions
9. Observations on theorems that Fermat and others have looked at about prime numbers
10. An account of the solution of isoperimetric problems in the broadest sense
11. Construction of differential equations which do not admit separation of variables
12. Example of the solution of a differential equation without separation of variables
13. On the solution of problems of Diophantus about integer numbers
14. Inferences on the forms of roots of equations and of their orders
15. Solution of the differential equation $asndx=dy+y2dx$
16. On curves of fastest descent in a resistant medium
17. Observations on harmonic progressions
18. On an inquiry of curves of a given kind, or a method of finding equations for an infinity of curves of a given kind
19. Additions to the dissertation on infinitely many curves of a given kind
20. Investigation of two curves, the abscissas of which are corresponding arcs and the sum of which is algebraic
21. On sums of series of reciprocals
22. A universal method for finding sums which approximate convergent series
23. Finding the sum of a series from a given general term
24. On the solution of equations from the motion of pulling and other equations pertaining to the method of inverse tangents
25. Solution of a problem requiring the rectification of an ellipse
26. Solution of a problem relating to the geometry of position
27. Proof of some theorems about looking at prime numbers
28. Further universal methods for summing series
29. A new and easy way of finding curves enjoying properties of maximum or minimum
30. On the solution of equations
31. An Essay on Continued Fractions
32. Various observations about infinite series
33. Solution to a geometric problem about lunes formed by circles
34. On rectifiable algebraic curves and algebraic reciprocal trajectories
35. On various ways of closely approximating numbers for the quadrature of the circle
36. On differential equations which sometimes can be integrated
37. Proofs of some theorems of arithmetic
38. Solution of some problems that were posed by the celebrated Daniel Bernoulli
39. On products arising from infinitely many factors
40. Observations on continued fractions
41. Consideration of some progressions appropriate for finding the quadrature of the circle
42. An easy method for computing sines and tangents of angles both natural and articial
43. Investigation of curves which produce evolutes that are similar to themselves
44. Considerations about certain series
45. Solution of problems in arithmetic of finding a number, which, when divided by given numbers leaves given remainders
46. On the extraction of roots of irrational quantities
47. Proof of the sum of this series $1 + \frac 14 + \frac 19 + \frac {1}{16} + \frac {1}{25} + \frac{1}{36} +$ etc.
48. Several analytic observations on combinations
49. On the utility of higher mathematics
Topically Related Articles
Index

### About the Author

C. Edward Sandifer received his A.B. from Dartmouth College 1973. And his M.A., Ph.D., from the University of Massachusetts, Amherst 1975, 1980. He was Chair of the Northeastern Section of the MAA, 1998-2000, and Secretary of The Euler Society, 2002-present. He is also a member of the American Mathematical Society, the Canadian Society for History and Philosophy of Mathematics, and the British Society for the History of Mathematics. He is one of the founding member of The Euler Society, and on the charter committee at the founding of the History of Mathematics Special Interest Group of the MAA (HOMSIGMAA). He has run 34 consecutive Boston Marathons and won the 1984 Northeastern (USA) Regional Marathon Championship.

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