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[资源] LEONHARDI EULERI OPERA OMNIA [欧拉全集][VOL. X ET XI SERIEI SECUNDAE ]

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 The unpublished work of HUYGENS on the suspension bridge (1646), the breaking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.

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