*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.

### Table of Contents

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

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Index

About the Author

### 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.