The Early Mathematics of Leonhard Euler (Spectrum S.)
By: C. Edward Sandifer (author)Hardback
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Describes Euler's early mathematical works - the 50 mathematical articles he wrote before he left St. Petersburg in 1741 to join the Academy of Frederick the Great in Berlin. These works contain some of Euler's greatest mathematics: the Konigsburg bridge problem, his solution to the Basel problem, his first proof of the Euler-Fermat theorem. Also presented are important results that we seldom realize are due to Euler: that mixed partial derivatives are equal, our f(x) notation, and the integrating factor in differential equations. The book is a portrait of the world's most exciting mathematics between 1725 and 1741, rich in technical detail, woven with connections within Euler's work and with the work of other mathematicians in other times and places, laced with historical context.
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.
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