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LEONHARDI EULERI OPERA OMNIA [欧拉全集][VOL. X ET XI SERIEI SECUNDAE ]
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CONTENTS Pmewo rd . . . . . . .. ...................... 11 PNfato ry Note concerning the presumed technological origin of the science of elasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13 PROLOGUE 1. Remarks of the ancients on vibration and elasticity . . . . . . . . . . .. 15 2. Western researches before 1600 . . . . . . . . . . . . . . . . . . . .. 18 3. BEECKMAN on the suspension bridge (1614-1615), on vibration (1614-1618), and on elasticity (1620-1630) . . . . . . . . . . .. ... 24 4. MERSENNE on vibratiol1 and rupture (1623-1636). . . . . . . . . . . 28 PART I. THE EARLIEST SPECIAL PROBLEMS, 1638-1730 5. The vibrating string, the breaking of a beam, and the catenary in GALILEO'S Di800rsi (1638) . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34 6. The unpublished work of HUYGENS on the suspension bridge (1646), the breaking of a beam (1662), the vibrating string (1673), and the vibrating rod (1688) 44 7. P.A.RDIES' essays on the catenary and on elastic beams (1673) . . . . . . .. 50 8. HOOKE'S law of spring (1675, 1678) and researches on the arch (1675), on ropes (1669), and on sound (1675-1681) ................... 53 9. MARIOTTE and LEIBNIZ on elastic beams (1684) . . . . . . . . . . . . .. 59 10. The contest to find the catenary (1690) .... . . . . . . . . . . . .. 64 II. JAMES BERNOULLI'S researches on the general theory of flexible lines (1691-1704), and later work to 1717 . . . . . . . . . . . . . . . . .. 75 12. JAMES BERNOULLI'S first researches on the elastica (1691-1694) . . . . .. 88 13. JAMES BERNOULLI'S attempts toward a theory of the neutral fibre (1695-1705) 96 14. PARENT'S researches on the neutral fibre (1704-1713) . . . . . . . . . . . 109 15. Researches on theoretical and experimental elasticity by JAMES RICCATI (1720-1723) and others . . . . . . . . . . . . . . . . . . . . . . . . 114 16. Experiments on the nodes of vibrating bodies by NOBLE and PIGOT (c. 1674), SAUVEUR (1696-1701), DE LA HIRE (1709), and ZENDRINI (1715-1716) . 118 17. TAYLOR'S analysis of the continuous vibrating string (1713). . . . . 129 18. JOHN BERNOULLI'S analysis of the loaded vibrating string (1727) . . . . 132 19. Summary: EULER'S heritage . . . . . . . . . . . . . . . . . . . · · · 136 8 CONTENTS PART II. THE BEGINNINGS OF THE GENERAL THEORIES, 1727-1748 20. EULER'S derivation of JAMES BERNOULLI'S law of bending from HOOKE'S law of extension; introduction of the modulus of extension (1727) . . . . . . . .. 142 21. EULER'S unification of the catenary and the elastica (1728) . . . . . . . . . 146 22. MUSSCHENBROEK'S experimental discovery of the law of buckling in com- pression (1729) . . . . . . . . . . . . . . . . . . . . . . . . . . .. 150 23. DANIEL BERNOULLI'S discovery of the simple modes and proper frequencies of vibrating systems (1733), and related work of EULER. . . . . . . . . . . 154 24. DANIEL BERNOULLI'S and EULER'S first calculations of simple modes and proper frequencies for the transverse oscillations of bars (1734-1735) . . . . . . . 165 25. Further researches on the elastica; EULER'S general solution for linear vibration problems (1735-1739). . . . . . . . . . . . . . . . . . . . . . . . . 170 26. The first differential equations of motion: JOHN BERNOULLI'S and D'ALEMBERT'S treatments of the hanging cord (1742-1743) . . . . . . . . . . . . . . . 179 27. DANIEL BERNOULLI'S definitive work on the transverse vibrations of bars (1740-1742) . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 28. EULER'S treatise on elastic curves (1743). I. Static deflection . . . . . . . . 199 29. EULER'S treatise on elastic curves (1743). II. Vibrations . . . . . . . . . . 219 30. EULER'S equations of finite motion for linked systems and for the continuous string (1 744) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 31. EULER'S general solution for longitudinal vibration of the loaded elastic cord (1748) . . . . . . .. .......... ........ 229 32. Summary to 1748 . . . . . . . . . . . . . . . . . . . 234 PART III. THE CONTROVERSY OVER SMALL PLANE VIBRATIONS OF A STRING OF UNIFORM THICKNESS, 1746-1788 33. D'ALEMBERT'S first memoir (1746): The partial differential equation and its I t . b " t ." so U Ion y an equa Ion ....................... 34. EULER'S first memoir (1748): The solution by arbitrary functions . . · · 35. EULER'S "first principles of mechanics" (1750). . . . . . . · · . · · · 36. DANIEL BERNOULLI'S memoirs on the composition of simple modes (1753) 37. EULER'S second memoir (1754): The central importance of the partial differential 237 244 250 254 equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . · 259 38. LAGRANGE'S first memoir (1759): The explicit solution for the loaded string, and a faulty passage to the limit. . . . .. ............... 263 39. Miscellaneous polemics (1760-1767). . . . . . . . . . . . . . . . . . . 273 CONTENTS 9 40. EULER'S researches on the propagation and reflection of waves (1764-1765) . 281 41. Miscellaneous polemics to 1788 . . . . . . . . . .. ...... 286 42. Summary of the theory of the vibrating string to 1788. . . . . . . . . . . 295 PART IV. RESEARCHES SUBSEQUENT TO EULER'S "FIRST PRINCIPLES OF MECHANICS", 1752-1788 IV A. The non-uniform vibrating string 43. The earliest researches on non-uniform strings: Simple modes for special cases, and EULER'S general inverse method (1752-1765) . . . . . . 301 44. Polemics, errors, and further special cases (1770-1788) . . . . . . . . 311 IVB. Plane vibrations of a heavy cord hung from one end 45. Plane vibrations of a heavy cord hung from one end. .......... . 316 IVC. Plane vibrations of straight or curved rods 46, EULER'S faulty theories of curved rods (1760, 1774) . . . . . . . . . . . . 320 47. EULER'S definitive work on the six kinds of transverse vibration of straight rods (1772-1774), and its experimental verification by JORDAN RICCATI (1782) and by CHLADNI (1 787) . . . . . . . . . . . . . . . . . . . . . . . . . . 322 IVD. Vibrations of membranes and plates 48. EULER'B theory of the vibrating membrane (1759). . . . . . 49. CHLADNI'S experiments on the vibrations of free plates (1787). . IVE. Discrete models 50. Discrete models. . . . . . . 330 335 .......................... . 341 IVF. Plane static deflection and buckling of straight bars 51. EULER'S calculation of buckling loads for columns of non-uniform cross-seotion ( 1 757) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 52. The distinction of different kinds of buckling by DANIEL & JOHN III BERNOULLI (1766) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 53. LAQRANQE'S two memoirs (1770, 1773) ................. 349 54. EULER'S determination of the height at which a heavy vertical column buokleg (1776, 1778) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 IVG. The statics and dynamics of skew curves 55. LAGRANGE'S variational equations of motion for flexible lines (1761) . 56. EULER'S theory of the skew elastica (1774-1775) . · · . · · · · 371 374 10 CONTENTS lVR. The laws of elasticity and flexibility 57. Miscellaneous researches (1754-1768) . . . . . . . . . . . . . . . . . . 383 58. EULER'S introduction of shear force and derivation of the general eqtlations of equilibrium for a deformable line (1771, 1774) ......... ... 390 59. COULOMB'S introduction of interior shearing stress (1773) . . . . 396 60. EULER'S formal definition of tIle modulus of extension and final scaling laws for tIle frequencies and buckling loads of straight rods (1774, 1776) . . . . . 401 61. COULOMB'S experiments on torsion (1777, 1784) . . . . . . . . . . 405 PART V. EVALUATION 62. An evaluation from the eighteenth century: LAGRANGE'S Mechanique Analitique (1788) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 63. Some later evaluations: CHLADNI (1802), T. YOUNG (1807), LOVE (1892) . 412 64. A modern evaluation. . . . . . . . . . . . . . . . . . . . . . . . 416 Acknowledgment. The research here presented was done in the years 1956-1958, during parts of which the work was generously supported by the U. S. National Science Foundation, the American-Swiss FnlUldation for Scientific Exchange, the :r ohn Simon Guggenheim Memorial Foun- dation, and the Euler Commission of the Swiss Society of Natural Sciences. Among the s veral persons who helped me in various ways I wish particularly to thank Professor SPEISER for his friendship and encouragement, Professor SPIESS for his searching out and putting at my disposal hundreds of relevant unpublished documents, Dr. STRAUB and Professor KUERTI for criticism of some parts of the manu- script and proofs, and my wife, CHARLOTTE. I am happy to record that the writing was begun in the Conde-Zimmer in the Engelhof on the Nadelberg, which belonged to the BERNOULLI family during the latter part of the period described. ![LEONHARDI EULERI OPERA OMNIA [欧拉全集][VOL. X ET XI SERIEI SECUNDAE ]](https://muchongimg.xmcimg.com/data/bcs/2015/1030/w72h3338525_1446195552_163.png#opennewwindow) |
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