The Early Mathematics of Leonhard Euler

Preface; Part I. 1725–1727: 1. Construction of isochronal curves in any kind of resistant; 2. Method of finding reciprocal algebraic trajectories; Part II. 1728: 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: Integrating factor; Part III. 1729–1731: 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; Part IV. 1732: 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; Part V. 1733: 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 axn dx = dy + y2dx; Part VI. 1734: 16. On curves of fastest descent in a resistant medium; 17. Observations on harmonic progressions; 18. On an infinity 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; Part VII. 1735: 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; Part VIII. 1736: 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; Part IX. 1737: 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; Part X. 1738: 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; Part XI. 1739: 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 artificial; 43. Investigation of curves which produce evolutes that are similar to themselves; 44. Considerations about certain series; Part XII. 1740: 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: gymnastics with radical signs; Part XIII. 1741: 47. Proof of the sum of this series 1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/ 36 + etc; 48. Several analytic observations on combinations; 49. On the utility of higher mathematics; Topically related articles; Index; About the author.