George Howard Darwin (1845-1912) and the Genesis of the Moon:
Evolution and Catastrophe in Victorian Mathematical Astronomy
Robinson M. Yost
In a literary tradition stretching back to the ancient Greeks, the Romantic poet Percy Bysshe Shelley personified the ever-changing face of the moon:
Art thou pale for weariness
Of climbing heaven and gazing on the earth,
Among the stars that have a different birth,
And ever changing, like a joyless eye
That finds no object worth its constancy?
Despite the imperfect rhyme, Shelley alluded to both stellar and lunar origins in his verse. The stars were of a "different birth" than the moon. Where had they come from? Indeed, how had the moon come to be? Although Shelley, no doubt, had little interest in answering these questions, countless other for several millenia had contemplated the origins of the stars, the moon and other heavenly bodies. With the development of "science," ancient mythical descriptions gave way to more naturalistic, mechanistic explanations independent of supernatural forces or deities.
In the past few centuries, several general hypotheses have emerged explaining the birth of earth's only natural satellite. The binary theory states that moon and earth formed independently, condensing from a single cloud of gas. Another idea, the capture theory, says that the earth's gravity snared a close-approaching planetoid which had formed elsewhere in the solar system. The rupture or fission hypothesis, states that the moon emerged from the earth. Since the Apollo moon missions, the giant impact theory has gained wide acceptance. In it, a Mars-sized object collided obliquely with the earth and debris from this catastrophic collision coalesced to form the moon.
It is beyond the scope and purpose of this paper to discuss the histories and relative scientific merits of all these theories of the moon's genesis. Instead, the focus will be on the historical origins and development of the fission hypothesis first elaborated by George Howard Darwin. Although many historians of astronomy have described Darwin's hypothesis, few have scrutinized its development over time. How did Darwin's theory come about and who influenced his work? How did his ideas change over the three decades he examined the problem? How were the ideas linked with wider cosmological concerns? Ultimately, the fission hypothesis was influenced by the work or others, evolved over time and was inextricably linked to Darwin's cosmology.
George Howard, the second son of Charles and Emma Darwin, was born on July 9, 1845, at Down House. He attended Trinity College, Cambridge, where, in 1868, he graduated as Second Wrangler and second Smith's Prizeman. He was elected a Trinity Fellow later that year. After studying law for six years, he abandoned a legal career primarily due to deteriorating health. By October, 1873 he had returned to Cambridge pursuing a career in academia. Made a Fellow of the Royal Society of London in 1879, Darwin won the Copley Medal in 1911 for his work on the tides and was bestowed with numerous academic honors throughout his career. He served as president for the Cambridge Philosophical Society (1890-92, 1911-12), the Royal Astronomical Society (1899-1900) and the British Association for the Advancement of Science (1904-05) and was appointed to the Plumian Chair of astronomy and experimental physics at Cambridge in 1883, a position he held until his death on December 12, 1912.
George Darwin, from early in life no stranger to evolutionary ideas, involved himself directly and indirectly with his father's work. He provided some illustrations for Charles Darwin's books (e.g., Climbing Plants and Fertilisation of Orchids); edited the second edition of Descent of Man (1874) and published several of his own articles dealing with biological evolution. For instance, he wrote "On beneficial restriction to liberty of marriage", a eugenic article later attacked by the vigorous opponent of Charles Darwin, St. George Jackson Mivart. In this article he supported Francis Galton's proposals for a family register and supported legal divorce on the grounds of hereditary defects such as insanity, criminality and vice. Perhaps for personal reasons, he published several articles on the effects of marriages between first cousins. Like his father, George had recurring bouts of illness throughout his life. In an early paper Charles warned George to omit anti-theological, inflammatory statements. Accordingly the best strategy was to keep silent on controversial matters and let the work speak for itself.
When George focused his energies on astronomy, evolution on a cosmic scale became a recurring theme in his work. Instead of applying evolution to plants or animals, Darwin examined the evolution of planets, satellites, binary stars and the entire solar system. His eminent father’s outlook undoubtedly affected his approach to astronomy, although George's brand of cosmic evolution cannot be solely attributed to his father's influence. An excerpt from a speech George gave later in life suggests elements of continuity between father and son:
The physicist, like the biologist... watches the effect of slowly varying external conditions; he sees the quality of persistence or stability gradually decaying until it vanishes...
Judging by analogy we should rather expect to find slight continuous changes occurring during a long period of time, followed by a somewhat sudden transformation into a new species, or by rapid extinction. However this may be, when the stability of a mode of motion vanishes, the physicist either finds that it is replaced by a new persistent type of motion adapted to the changed conditions, or perhaps that no such transformation is possible and that the mode of motion has become extinct.
In George Darwin's view both physicist and biologist solved problems of stability and instability. However, the conditions of the living world were "incomparably more intricate than those of the world of matter" so that whereas the biologist settled for qualitative conclusions, the physicist remained unsatisfied until "he obtains a quantitative estimate of various causes and effects on the systems of matter." Practicing what he preached, Darwin used mathematical models to quantify previously qualitative cosmological ideas. Quantification was not a simple process. Frequently there were differences between the mathematical models and the physical reality that were not easy to reconcile. Simplifications and best estimates were often, if not always, necessary.
To understand Darwin's work, it is necessary to examine the scientific roots from which the fission hypothesis and associated ideas grew. Who were his predecessors? What were the mathematical, geophysical and cosmological traditions within which he worked?
Since the seventeenth century mathematicians had addressed rotating fluids whose general shapes depended on gravity and rotational velocity. In the Principia (1687), Sir Isaac Newton made his famous prediction of the earth’s shape. Diurnal rotation caused a centrifugal force, making the earth bulge at the equator and flatten at the poles. Although French-led expeditions of the 1730s to Lapland and South America confirmed that the globe was an oblate Newtonian onion rather than an elongated Cartesian lemon, the generalized quantitative relationship between rotational velocity and deviation from sphericity proved a more complex problem. In 1742 Scottish mathematician Colin Maclaurin generalized Newton's result. Maclaurin demonstrated that a homogeneous liquid rotating uniformly about a fixed axis maintained equilibrium, if the shape was an ellipsoid of revolution. This family of shapes became known as Maclaurin spheroids. Other prominent eighteenth-century figures such as D'Alembert, Laplace and Lagrange studied the same problem, but it was not until 1834 that Carl Gustav Jacobi determined that ellipsoids with three unequal axes could also be figures of equilibrium. Some nineteenth-century scientists found this shocking and counterintuitive because Jacobi’s shapes lacked complete rotational symmetry. Still, there were those who readily accepted Jacobian ellipsoids and applied them to explain variable stars. Maclaurin's spheroids and Jacobi's ellipsoids assumed an incompressible, homogeneous fluid with no viscosity— an ideal situation not met within the earth or other heavenly bodies. It was apparently believed that if the problem was solved for a more realistic model, only minor qualitative corrections would arise.
At Cambridge, Darwin learned of rotating fluid bodies in a tradition going back to the second Lucasian professor of mathematics, the incomparable Sir Isaac. He was taught hydrodynamics by one of the experts in the field, George Gabriel Stokes, the Lucasian professor during Darwin's undergraduate days. The Cambridge Mathematical Tripos included questions dealing with the mathematics of rotating fluid masses as did the Smith's Prize questions written by luminaries including Stokes, James Challis, John Couch Adams and William Whewell. The following, for example, were problems posed for Smith's Prizes in 1865 and 1871 respectively:
A mass (M) of homogeneous fluid revolves, in relative equilibrium, about a fixed axis with a uniform angular velocity such that the ellipticity (e) of its surface is small. If the part mM of the mass were collected into an infinitely dense material point at the centre, and the density of the remaining part (1-m)M were diminished in the ratio of (1-m) to 1, find what would be the ellipticity of the new surface of equilibrium supposing the time of rotation to be the same as before.
Shew that the attraction of the Earth, assumed to be an ellipsoid of revolution of small ellipticity, on an external particle is the same as if its mass were condensed into a ring in the plane of the equator; and find the diameter of that ring.
The Tripos also asked questions dealing with hydrodynamics and the movements of the earth, moon and sun. Darwin must have had comparable questions when he took the exams in 1868. A Cambridge education prepared him in hydrodynamics, celestial mechanics and mathematical physics in general.
Terrestrial physics, later called geophysics, questioned the nature of the earth's interior, shape and formation. It also linked the idealized mathematical solutions of Maclaurin and Jacobi to physical geological problems. The belief that the earth was once a molten mass that had cooled, solidified and contracted dated from the seventeenth century or earlier. In the nineteenth century, debate persisted with respect to the nature and extent of these processes. Many geologists pointed to earthquakes, volcanoes and crustal movements as evidence that the earth still retained a molten liquid interior beneath a thin solid crust. A report from 1871 illustrates the connection between the geological and mathematical ideas with respect to this "shell" hypothesis:
our globe must in reality be a sphere of molten matter surrounded by an external shell or crust of solid matter... and the figure of the earth itself, which is an ellipsoid of revolution...is regarded by natural philosophers in general as all but conclusive evidence that the earth at an early period of its history must have been in a fluid condition.
Beginning in the 1830s, the thin-shell hypothesis and the earth's fluidity were increasingly challenged, primarily by non-geologists. André-Marie Ampère argued that lunar tidal forces on a liquid terrestrial interior would tear apart any thin crust that formed on the surface. Herbert Spencer, George Stokes and others rejected the idea that the earth's oblateness confirmed its fluid interior. Obliquity did not prove that the earth was or had been a fluid in the ordinary sense because any sufficiently large object would act as a fluid under the tremendous gravitational and tidal forces. Although the earth's shape was no longer acceptable evidence for terrestrial fluidity, many geologists persisted with the shell idea because it remained useful for explaining volcanic eruptions and other geological activities on the surface.
In 1839, William Hopkins, the famed Cambridge mathematics tutor, showed that models for the earth's interior could be tested by combining mathematical analysis with data from physical chemistry and astronomy. Using pressure and temperature variations from the surface to the center and the effects of pressure on melting temperatures, in addition to astronomical evidence via rates of nutation and precession, Hopkins concluded that the earth must be solid or nearly solid with a crust of about 800 to 1000 miles thick. Because Hopkins' results were subsequently supported and strengthened by William Thomson and other physicists, many geologists "felt themselves quite unable to answer the arguments of the astronomers and mathematicians. . . they [were] compelled to bow to the decision of such eminent authorities."
By the 1860s terms such as "fluidists" and "solidists" were being used to label opposing sides of the debate. Many fluidists were geologists, arguing for a crust less than 50 miles thick. The solidists, who tended to be physicists, followed the lead of William Thomson, arguing for a completely solid earth (with possible isolated liquid regions).Utilizing a theory of heat conduction, Thomson argued that the earth had solidfied from the center outwards, a process that was now essentially completed. If a solid crust ever began to form on a liquid globe, it would be broken up and the colder, denser rock would sink to the center via convection currents. Thomson also defended the astronomical arguments from attacks by French astronomer Charles-Eugène Delaunay. In 1868 Delaunay concluded that the earth would act as a solid mass even if it had a liquid interior because the fluid would follow the slow rotational motion of the crust (e.g., nutation or precession). Thomson, rejecting this assessment, stated that "the viscosity assumed by Delaunay, to produce the effect he attributes to it, must be more than ten million million million times the viscosity of water." In 1876, Thomson summarized his case: "The hypothesis of a perfectly rigid crust containing liquid violates physics by assuming preternaturally rigid matter, and violates dynamical astronomy in the solar semiannual and lunar fortnightly nutations..." Many, including George Darwin, agreed with these judgments against the shell hypothesis.
Sir William Thomson (Lord Kelvin after 1892) profoundly influenced the young mathematical astronomer. Darwin often praised Thomson and frequently cited him in his work. In 1878 Darwin wrote:
Sir W. Thomson's investigation of the bodily tides of an elastic sphere has gone far to overthrow the idea of a semi-fluid interior to the earth, yet geologists are so strongly impressed by the fact that enormous masses of rock have been poured out of volcanic vents in the earth's surface that the belief is not yet extinct that we live on a thin shell over a sea of molten lava.
Investigations of the earth's age, rigidity and interior composition often crossed the boundaries between physics, geology and astronomy. Thomson enthusiastically supported Darwin's first papers which addressed these interdisciplinary problems. Darwin later commented:
Early in my scientific career it was my good fortune to be brought into close personal relationship with Lord Kelvin. Many visits to Glasgow and to Largs have brought me to look up to him as my master, and I cannot find words to express how much I owe to his friendship and to his inspiration.
Lasting ties developed between Thomson and Darwin. In fact, Darwin regarded his work on tidal friction and cosmogony as the scientific outcome of a conversation with Thomson in 1877. He later dubbed Thomson "amongst the greatest of those who have tried to guess the riddle of the history of the universe."
Geophysical problems often became linked with questions about the history of the universe and its beginnings. Scientists had long pondered the origins of the sun, the planets and their satellites. Simon-Pierre Laplace at the conclusion of his Exposition du Système du Monde (1796) briefly hypothesized on the initial development of the solar system. Earlier thinkers such as René Descartes, Thomas Wright and Immanuel Kant had speculated on the system's origins, but it was Laplacian notions which became most widely accepted by the early nineteenth century.
Laplace began his developmental process with a hot spherical cloud of gas which contracted and flattened as it rotated more rapidly, the sun formed at the center of this cloud, while the outer layer cooled and shed discrete rings of nebulous matter. These successively-shed rings eventually collected into smaller planetary nebulae. In an analogous manner, the planetary nebulae formed planets and shed their own rings to form satellites. This gave a plausible process for the formation of all satellites, including the moon. Laplace's idea also conveniently explained why the planets rotated and revolved in the same direction in approximately the same plane. However, these brief suggestions remained primarily qualitative in Laplace’s work. They did not rely on mathematical models or quantitative evidence.
In Great Britain of the 1830s, Laplace's "nebular hypothesis" was popularized by Scottish astronomer, natural philosopher and political economist John Pringle Nichol. For reformers like Nichol, the nebular hypothesis exemplified natural progressive laws and complemented a progressive political-theological agenda. Along similar lines, Robert Chambers (1844) and Herbert Spencer (1857) adopted versions of the nebular cosmogony.William Thomson, who was taught by Nichol at the University of Glasgow, also accepted the nebular hypothesis. In light of the emerging thermodynamics, it was no longer necessary to assume a hot original nebula because mutual gravitational attraction of a cold cloud of gas would convert potential and kinetic energy into heat. In its basic outline, the nebular hypothesis had become well established in British cosmological thought by the time Darwin turned his attention to cosmic evolution. Further modifications were being made to an already altered nebular hypothesis.
From the study of rotating liquid spheroids, to the interior composition of the earth, to the origins and evolution of the solar system, such are the strands of inquiry which came together in Darwin's thought. He used elements of mathematical physics, geophysics and developmental cosmology. In the process he altered the nebular hypothesis and developed his theory of lunar genesis.
III. The Fission Hypothesis
In the mid 1870s Darwin began writing a series of geophysical papers. One dealt with whether or not the earth's axis of rotation altered due to geological changes. While the earth's obliquity to the ecliptic (i.e., the tilt of the axis) was found "sensibly constant" over geological history, Darwin concluded that the earth was simply not rigid enough to resist significant departures from equilibrium due to internal changes. These types of changes led to realignments of the axis of rotation, resulting in earthquakes as the globe readjusted to a new figure of equilibrium. Not surprisingly, the problem of determining terrestrial axial adjustments led to studies of the earth's interior composition.
A paper from May 1878 began with the debate over the earth's interior. Darwin sided with William Thomson's theory of an elastic sphere, yet altered Thomson's approach to consider both viscous and elasto-viscous spheroids. The equations of hydrodynamic flow for Darwin's viscous fluid were analogous to Thomson's for the strain in an elastic solid. Darwin calculated that the viscosity of the earth was greater than that of frozen pitch. Hence, he said, "no very considerable portion of the interior of the earth can even distantly approach the fluid condition."If the earth did behave as a fluid, Darwin argued that oceanic tides would be insignificant when compared with tidal movements on a rigid terrestrial nucleus. Nevertheless, he believed that the earth's enormous interior pressures could induce even the most solid substances to flow, and, thus, small bodily tides remained a possibility. An "elasto-viscous" spheroid, neither purely elastic nor purely viscous, was Darwin's intermediate solution. Despite this differing approach to the question, the paper strongly confirmed Thomson's view of the earth's great effective rigidity. The detailed study of viscous spheroids remained a continuous strand in Darwin's work of subsequent decades.
Lunar origins, however, were not the primary focus of Darwin's early papers. Instead, the fission hypothesis emerged from a mathematical investigation of the precession of viscous spheroids. He initially used the moon's influence on the earth as a case study in theoretical dynamics. Darwin hoped to explain why the planets' axes of rotation had different angles of tilt. Such diversity of planetary inclinations did not follow from the nebular hypothesis because, according to it, all planets should spin with axes nearly parallel to the sun's axis of rotation (i.e., perpendicular to the planes of their orbits). In 1878 Darwin wrote that the nebular hypothesis contained "no explanation of the great diversity which the telescope shows us is to be found in the inclinations of planets to their orbits." His investigation sought to explain this diversity.
Precession, a slow retrogression of the intersection of a planet's equator with the plane of its orbit, was caused by the attraction of the sun and the planet's satellites on the equatorial bulge. This depended on four causes: 1) the amount of equatorial protuberance, 2) the tilt of the axis of rotation, 3) the rate of rotation, and 4) the planet's orbit around the sun and orbits of the satellites. Darwin argued that as the planets condensed and contracted from their planetary nebulae they rotated more quickly, causing increased equatorial bulging. The increase of equatorial bulge enlarged the inclination to the orbit and thereby affected the rate of precession. He admitted that these were "very speculative views" which immediately led to "serious difficulties" when applied. For instance, Darwin's argument applied only to rigid bodies, yet the planets in the remote past were thought to be partially gaseous or fluid. Also, decreasing obliquity was expected the further a planet was from the sun; this did not agree with the observations. Many questions emerged from Darwin's attempt to quantify the nebular hypothesis dealing with the following: the stability or instability of rotating nebulae, the motions of the parts of the system, and the conditions under which a nebula would shed a satellite.
Darwin explored the relationships between the earth, sun and moon with these wider questions in mind. In July 1878 he wrote:
The following paper contains the investigation of the mass-motion of viscous and imperfectly elastic spheroids, as modified by a relative motion of their parts, produced in them by the attraction of external disturbing bodies...
The problem, as it was so broadly defined, was first limited by Darwin to the case of the earth, as disturbed by the sun and moon. Although, he did not believe the earth to be in reality a homogeneous viscous or elastico-viscous spheroid, it seemed probable that if it were once molten, changes had occurred closely analogous to those he had determined. It was Darwin's hope that these investigations would eventually lead to important conclusions regarding the history of the solar system.
Assuming a single tide-generating body, the moon, moving in circular orbit in the ecliptic about a viscous earth resulted in a set of thirteen tides whose heights and retardations depended on their speeds and on the terrestrial coefficient of viscosity. These thirteen tides were reduced to seven and the seven simplified into three groups: semi-diurnal, diurnal, and fortnightly. Considering the attraction of the moon on the terrestrial tidal protuberances, Darwin found couples about three axes in the earth which yielded information about the changes in precession, obliquity, and diurnal rotation. Such quantitative arguments, based on the effects of tidal friction, were of primary importance in Darwin's evolutionary scheme for the solar system.
After considering the sun and moon's combined effects on the earth, Darwin calculated the reactions which the terrestrial tides had upon the disturbing bodies. His analysis illustrated that, for various obliquities and viscosities, there were a great variety of positions of dynamical equilibrium (some stable, others unstable). However, in each case, the moon retarded the earth's rotation because the tides always acted as a breaking mechanism. In the very distant future, earth and moon would move as if fixed together although the moon would be further away than at present. As tidal friction retarded the earth's rotation, conservation of angular momentum required that the earth exercise a force on the moon causing an acceleration in its linear velocity. Tidal friction not only retarded the earth's rotation, but also caused the moon gradually to recede from the earth and decrease its orbital angular velocity. As the braking process continued, the earth would eventually rotate once every fifty-five of our present days with the moon orbiting at a greater distance also in fifty-five days.Such was the predicted final equilibrium state of the earth-moon system.
Next Darwin traced the state of the system into the past. A general analytical solution was not feasible because of the mathematical complexities, so Darwin used numerical approximations with a fixed, moderate degree of terrestrial viscosity. The particular value of viscosity was assigned because it made the rate of change of the obliquity a maximum. Darwin explained,
...although it is not that degree of viscosity which will make all the changes proceed with the greatest possible rapidity, yet it is sufficiently near that value to enable us to estimate very well the smallest time which can possibly have elapsed in the history of the earth.
Beginning with the present values of the year, day, month and obliquity, he traced the system back in time. In the regression, the length of the year remained relatively constant, but the day, month and obliquity diminished. The changes occurred at a rapidly increasing rate the further back he went. In the future the moon receded, but in the retrogression it approached the earth causing greater tidal forces. When Darwin stopped the calculation, the obliquity had decreased by 9°, the day was 6 hours 50 minutes, and the month was only 1 day 14 hours. Darwin noted that these changes might have taken place in 54 to 57 million years— well within the time limits physicists had set for the earth and moon's existence by other methods.
A formerly molten earth would cool and stiffen with passing time, so the viscosity must decrease as one looked backwards. Therefore, Darwin's second solution used variable viscosity. No definite law of the viscosity's dimunition was given and the new solution gave "no indication of the time which may have elapsed." Now Darwin found the earth would have rotated and the moon would have orbited in 5 hours 40 minutes, as if fixed together. Because of the conditions of dynamic stability, he went no further because the moon would fall into the earth. Darwin reported to the B. A. A. S.,
This periodic time of the moon of 5 hours 40 minutes corresponds to an interval of only 6000 miles between the moon's centre and the earth's surface... The conclusion, therefore, appears to me almost irresistible, that if the moon and earth were ever molten viscous bodies, then they once formed parts of a common mass.
Although this seemed a plausible mechanism for the moon's genesis, Darwin realized the preliminary nature of his findings. There were, admittedly, many instances where the hypothetical conditions differed from physical reality. For instance, in assuming homogeneity for the calculations, Darwin neglected the heterogeneity of density and viscosity in the earth. He wrote that these would not make a material difference in the solution. Also unknown were the influence of great pressures on semi-solids and the conditions of stability for rotating fluid masses. He remarked:
A whole series of problems, some of them of great difficulty, still await solution; and not until they are solved will it be possible either decisively to accept or reject the modified form of the nebular hypothesis, to which my results obviously point.
Darwin's results "obviously pointed" to the possibility that satellites, or at least the moon, may not have come from a Laplacian coalescing ring of matter. Could, in fact, a uniformly-shed ring condense into a satellite? Darwin found that the ring must be of heterogeneous density for successful conglomeration and so its distance from the planet could increase. The moon was traced to a point in the past of near contact where it always faced the same part of the earth. Perhaps the newborn moon was a "ring" with heterogeneous density, or maybe a different mechanism was needed.
At the conclusion of a paper from December 1878, Darwin proposed a new mechanism for the rupture. He noted that a rotation of the earth before the moon had formed in 5 hours 40 minutes seemed to slow to render it unstable. Darwin, however, suggested an explanation for the cause of the fission:
Sir William Thomson has shown that a fluid spheroid of the same mean density as the earth would perform a complete gravitational oscillation in 1 hour 34 minutes... It seems quite possible that two complete gravitational oscillations of the earth in its primitive state might occupy 4 or 5 hours. But if this were the case, then the solar semi-diurnal tide would have very nearly the same period as the free oscillation of the spheroid, and accordingly the solar tides would be of enormous height.
These concurrent forces "might rupture the body into two or more parts." A uniform ring would not detach as it did in Laplacian satellite formation. Despite speculations, the question of the moon separated from the earth remained a mystery. Over the next three decades, Darwin addressed numerous difficulties, refining and elaborating his results.
IV. Refinement and Elaboration
Darwin applied thermodynamic principles to the earth and moon as early as December 1878.In particular, he proposed that heat generated within the earth from internal tidal friction had direct bearing on Thomson's investigation of the secular cooling of the earth (and hence estimating the age of the earth). After calculations, however, the effects were found negligible. In future papers, Darwin continued to connect thermodynamics with the problem of the earth-moon system.
In May 1879 Darwin examined the degradation of energy in systems such as the earth and moon.The conservation of energy, or first law of thermodynamics, said that energy was neither created nor destroyed. Energy was merely converted into other forms, such as frictional heat generated inside the earth. Darwin's "degraded" energy referred to the total potential and kinetic energies of the earth-moon system. Assuming imperfectly elastic or viscous spheroids, tidal friction acted to "degrade" the system's potential and kinetic energy over time. Given this definition, the system would lose energy via friction and its "whole" energy would diminish (i.e., kinetic energy + potential energy = "whole" energy). Using a method suggested by William Thomson, Darwin graphically illustrated that the "two bodies [earth and moon] once formed parts of a single one, which broke up in consequence of some kind of instability." The method traced the orbital momentum, rotational momentum and energy of the system over time. Through tidal friction, the earth-moon system was gradually evolving to its lowest state as potential and kinetic energy dissipated into heat. With this analysis Darwin incorporated thermodynamical ideas into his fission hypothesis to give further quantatitive support.
The sequel to Darwin's three previous papers on the precession of viscous spheroids was delivered to the Royal Society in December 1879. Earlier work had "proved that frictional tides in the earth are causing, and must have caused, changes in the configuration of the system."As Darwin traced these changes to the near contact point of earth and moon, new calculations revealed the following:
The present lunar period of 27.3 days was originally 2 to 4 hours.
The inclination of the lunar orbit's proper plane to the ecliptic of 8" was originally about 11° 45'.
The earth's present rotation of 24 hours was initially 2 to 4 hours.
Inclination of the earth's proper plane to the ecliptic, 23° 28', was initially 11° 45'.
Eccentricity of the lunar orbit must have been smaller in the past, early it was very small or zero.
These statements illustrated that, in the remote past, day and month were approximately equal (i.e., 2-4 hours) and that the inclination of the earth and moon's proper plane to the ecliptic had coincided (i.e., 11° 45'). At this point in its evolution, the moon moved in a more or less circular orbit just above the terrestrial equator. Taken together, these results suggested that "the moon was produced by the rupture, in consequence of rapid rotation or other causes, of a primeval planet, whose mass was made up of the present earth and moon."
What had Darwin concluded at this juncture? In his view, the earth's shape was distorted by the tidal forces of the moon and sun. More recent changes were in the form of oceanic tides on a rigid terrestrial nucleus, while effects in the remote past relied on bodily tides of a liquid earth. Bodily tidal friction must have played a much greater role when the moon was closer and the earth was not yet solidified. Of lasting interest to Darwin was the rupturing process which had formed the moon in the distant past, at least some fifty million years ago. He wrote of the primeval earth:
The rapidity of the planet's rotation causes so great a compression of its figure that it cannot continue to exist in an ellipsoidal form with stability; or else it is so nearly unstable that complete instability is induced by solar tides.
The planet then separates into two masses, the larger being the earth and the smaller the moon. I do not attempt to define the mode of separation, or to say whether the moon was initially more or less annular.
After lunar formation, Darwin used his theory of tidal friction (lunar and solar) to trace the evolution of earth-moon system. Tidal friction was "A theory... which brings into quantitative correlation the lengths of the present day and month, the obliquity of the ecliptic, and the inclination and eccentricity of the lunar orbit, must, I think, have strong claims to acceptance."
Although these ideas were admittedly speculative, quantitative support bolstered Darwin's confidence. Surely, similar processes had taken place elsewhere in the solar system. In January 1881 Darwin extended the theory to consider planets with more than one satellite and the evolution of the entire solar system. He found that solar tidal friction could not have enlarged the planetary orbits any appreciable amount since their formation. Also the satellites of Mars, Jupiter and Saturn could not be traced back to near contact configuration. These results implied that the sun had not given birth to the planets by fission, and that the earth-moon system was unique in the solar system. Darwin noted, "It has been shown that the case of the earth and moon does actually differ widely from that of the other planets, and we may therefore reasonably suppose that the history has also differed considerably."
Why was the earth-moon system unique? With respect to these differences, Darwin speculated on the following: (1) a measure of the relative efficiency of solar tidal friction in reducing rotational momentum and the rotation of the planets; (2) the effects of the planetary contraction and solar tidal friction acting simultaneously; (3) how the separation of a satellite is likely to affect the course of evolution. Solar tidal friction acted as a retarding force in opposition to the acceleration of planetary contraction. Solar influence diminished with distance making solar tidal friction more important for closer planets, while planetary contraction played a larger role for more remote ones. From the nebular hypothesis, Darwin argued that a planetary nebula contracted and accelerated as it evolved. Rapid enough rotation would cause "its form to become unstable, or, perhaps a portion gradually detaches itself"; the contraction and increase of rotation would continue in this manner recurring in a "series of epochs of instability or of abnormal change."While solar tidal friction slowed the onset of instability for planets closer to the sun, this instability occurred more often for the remote planets. Darwin believed this demonstrated a cause for the distribution of satellites in the solar system: "For Mercury and Venus have no satellites, and there is a progressive increase in the number of satellites as we recede from the sun." It also explained the uniqueness of the earth-moon system and its evolution. Darwin wrote:
In the case of the contracting terrestrial mass we may suppose that there was for a long time nearly a balance between the retardation due to solar tidal friction and the acceleration due to contraction, and that it was not until the planetary mass had contracted to nearly its present dimensions that an epoch of instability could occur...
If the contraction of the planetary mass be almost completed before the genesis of the satellite, tidal friction, due jointly to the satellite and to the sun, will thereafter be the great cause of change in the system, and thus the hypothesis that it is the sole cause of change will give an approximately accurate explanation of the motion of the planet and satellite at any subsequent time.
That this condition is fulfilled in the case of the earth and moon, I have endeavoured to show in previous papers of this series.
Still, the mode of a satellite's separation remained a mysterious process. Darwin returned to this question in the mid-1880s.
In 1882 the geologist Samuel Haughton suggested a revised version of Darwin's hypothesis. Haughton doubted whether bodies of the solar system were ever in a fluid state and gave evidence supporting this belief. The earth and moon, instead of being molten or fluid, separated from the solar nebula "as a swarm of solid meteoric stones." Haughton concluded that the motions of this meteoric swarm could be reduced to a hydrodynamical problem. In similar fashion, Darwin blended aspects of the meteoric hypothesis with the nebular hypothesis in 1888. Using the kinetic theory of gases, he concluded that if the meteorites:
. . .possess a virtual elasticity, a swarm of meteorites provides a gas-like medium of fine enough structure to satisfy the demands of the nebular hypothesis.
The result. . . appeared to justify the opinion that the meteoric theory may be reconciled with Laplace's hypothesis, and that they may both be held to be true.
Yet again, Darwin accommodated his new ideas to fit within a generally Laplacian framework.
With quantitative evidence, Darwin incorporated the meteoric hypothesis and tidal friction into the nebular hypothesis. He concluded in 1881, "These investigations afford no grounds for the rejection of the nebular hypothesis, but while they present evidence in favour of the main outlines of that theory, they introduce modifications of considerable importance." Writing in 1887, the historian of astronomy Agnes Clerke echoed Darwin's sentiment:
The general outcome of Mr. Darwin's researches has been to leave Laplace's cosmogony untouched. He concludes nothing against it... In one form or the other, if we speculate at all on the development of the planetary system, our speculations are driven into conformity with the broad lines of the Nebular Hypothesis.
Others also viewed the evolution of the solar system within broad Laplacian guidelines. As problems arose, Darwin and others adapted the nebular hypothesis rather than scrapping it and starting anew.
V. The Pear-shaped Figure
Early in the 1880s Darwin admitted that the process by which the moon separated from the primeval earth was unknown to him. By mid-decade Darwin resumed his search for this process. His intention was:
. . .to investigate the forms which two masses of fluid assume when they revolve in close proximity about one another and. . . to obtain a representation of the single form of equilibrium which must exist when two masses approach so near to one another as just to coalesce into a single mass.
Darwin discovered that when the two masses were equal, they were shaped like "flattened eggs" with the two smaller ends facing each other. The small elongated ends, in one instance, actually overlapped and became "a single mass of fluid consisting of two bulbs joined by a neck." William Thomson remarked in 1882 on the gap between the unstable Jacobian ellipsoid and two equal detached portions: "The consideration of how to fill up this gap with intermediate figures is a most attractive question, towards answering which we at present offer no contribution."
When Darwin proposed the dumb-bell shaped figure of equilibrium in the mid-1880s, he was not alone in considering a figure of this kind. He wrote, "These figures were already drawn when a paper by M. Poincaré appeared, in which, amongst other things, a similar conclusion was arrived at." In 1885, Jules Henri Poincaré, famed French astronomer-mathematician who was studying the stability of mechanical systems and was known for his work in pure mathematics, published an important paper dealing with the figures of equilibrium for rotating fluid masses. Like Darwin, Poincaré had read Thomson's discussion of equilibrium figures in Thomson and Tait's Treatise on Natural Philosophy (1883). A goal of his paper was to demonstrate the existence of figures of equilibrium different from those already discussed in Thomson and Tait (i.e., annular and ellipsoidal).
In the course of his investigation Poincaré found that a Jacobian ellipsoid, at a point of exchanging stabilities, gradually formed a dumb-bell shape with unequal bulbs. This became known as the pear-shaped or pyriform figure. Darwin was greatly influenced by the 1885 work. It came as a revelation to Darwin who had approached the problem from the other end, i.e., the coalescence of two bodies into a single body. For a year, he held back his paper in an attempt to apply Poincaré's principles. He admitted that his article was "complementary to, but far less perfect than, that of M. Poincaré."
Treating the earth and moon as homogeneous fluids with equal densities, Darwin found the moon at the point of disintegration when it was 380 miles from the terrestrial surface (6500 miles between the centers of earth and moon). Two problems were that "the rigorous method of discussing the stability of the system fails... and it appears that there cannot be such a form, unless the smaller of the two masses exceeds about one-thirtieth of the larger." Unfortunately, the lunar mass was known to be less than one-thirtieth of the earth's mass (i.e., about 1/82nd of the earth's mass) thereby confounding the stability problem. Although Darwin hoped that the results might shed light on the nebular hypothesis, three primary difficulties persisted.
First, the mathematical problem of separating two masses differed from the physical reality. As Poincaré noted, Laplace had assumed a heterogeneous fluid (i.e., denser near the center of rotation) while his mathematical models used a homogeneous fluid. Heterogenous models were complex because they had to determine the extent of and changes in heterogeneity over time. In a paper from 1881, Darwin concluded that tidal friction would be about the same whether the planet were homogeneous or heterogeneous. He again dismissed the difficulty in 1887, stating that it was "hardly probable that the heterogeneity of the central body can make so great a difference in the result."
A second problem was Roche's limit. In 1850 Edouard Roche had concluded that when two bodies were within a critical distance from each other, tidal forces would tear the smaller body apart. Darwin discussed this problem in the pages of Nature with James Nolan, an astronomer in Victoria, Australia. Nolan, who agreed with Roche, criticized Darwin in a pamphlet: "Darwin's Theory of the Genesis of the Moon" (1885). Nolan’s had three basic criticisms: "(1) That the moon could not have existed bodily so near the earth. . . (2) That in any form possible there she could not have receded by the agency assigned— tidal friction. (3) That, if a modification be made by allowing her to have separated at a greater radius than that corresponding to a period between 2 and 4 hours, the moon would no longer be traceable to the earth's present surface." Darwin, discounted the second and third objections, yet answered the first. He agreed with Nolan that the moon had not existed close to the earth. This was so because the moon, in its initial stages, was a swarm of meteorites rather than a continuous mass. Nolan responded:
Nothing can be gained by this, for the flock of meteorites cannot come nearly so close to the earth as the moon in a single mass, without the constituent members being separated and each compelled to describe an independent orbit with its own period. In other words, the tidal force would separate the flock of meteorites at a greater distance than it would the single body.
Undeterred, Darwin asked Nolan for proof of the law that two bodies could not revolve about each other with their surfaces in near contact, unless one was smaller and denser than the other by certain values.
A final difficulty involved mathematics. The mathematical methods employed were often so complex that the stability of the pear-shaped figure remained indeterminate except through uncertain numerical approximations. How and when in its evolution did the shape divide and was the fission process ever stable? Darwin addressed these questions over the next two decades, often with unsatisfying results. He wrote in 1887, "It seems then at present necessary to suppose that after the birth of a satellite, if it takes place at all in this way, a series of changes occur which are still quite unknown." A decade later not much had changed: "There is nothing to tell us whether this theory affords the true explanation of the birth of the moon, and I say that it is only a wild speculation, incapable of verification."
As the president of the Royal Astronomical Society, Darwin presented Poincaré with the Society's gold medal in 1900. In his presentation, Darwin summarized the 1885 paper which had come as "a revelation" to him some fifteen years earlier. Poincaré's 1885 work marked:
an epoch not only in the study of the subject itself, but also in that of many others. . . the theory of the stability of systems in equilibrium or in steady motion has undoubtedly been crystallised and rendered transparent by his efforts. So fundamental are the new conceptions introduced that a new phraseology has become necessary.
Mechanical systems in equilibrium, identical in all respects save one, could be arranged according to the magnitude of a measurable quantity (e.g., the family of Maclaurin spheroids arranged by speed of rotation). These systems, however, might form other families of equilibrium with completely different configurations of their parts (e.g., the Jacobian ellipsoids). Darwin continued:
Now it is possible that there would occur in one family a certain member which would resemble the corresponding member of the other family in all respects. If this were the case, this particular member of either family would be described as a form of bifurcation, because it would belong to both... Now, M. Poincaré proves that if we follow each family towards the form of bifurcation, the equilibrium in one of the families would be stable, while that in the other would be unstable; the same would also hold good after the passage through the form of bifurcation, but the family which was stable before would be unstable afterwards, and vice versa. There is accordingly exchange of stabilities between the two families.
These ideas applied to a rotating planet of homogeneous fluid. A molten rotating mass globe formed increasingly flattened Maclaurin spheroids as it cooled, contracted, and accelerated. At the point of bifurcation it became one of Jacobi's ellipsoids and, at another bifurcation, the shape acquired an unsymmetrical furrow and became pyriform. Darwin supposed that "the mass will go on deepening its furrow more and more, and then it will at last divide itself into two separate bodies by the throttling of the middle part. It is clear that a process of this kind may have played its part in the evolution of celestial systems."Still uncertain was the pear-shape's stability.
In May 1901 Darwin read the paper, "Ellipsoidal Harmonic Analysis", to the Royal Society. He had hoped to obtain "exact numerical results with respect to M. Poincaré's pear-shaped figure," but by broadening the paper's scope, Darwin's discussion of this specific problem was deferred.On November 21, 1901, Darwin and Poincaré both addressed the stability question in papers presented at the meeting of the Royal Society. Establishing stability was critical to the success of the fission hypothesis because, if the pyriform figure were stable at some point, then part of the cooling primeval earth could gradually separate. However, if the shape were never stable, then it would undergo drastic deformations and oscillations, followed by a catastrophic disintegration. Using the method of ellipsoidal harmonic analysis developed in his previous paper, Darwin concluded: "The pear-shaped bodies are almost certainly stable, but a rigorous and conclusive proof is wanting... To do this further approximation is needed."
After corresponding with Poincaré, Darwin attempted to carry out such an approximation. The result was another paper devoted to pyriform stability. In June 1902, Darwin reaffirmed the stability of the pear shape to his satisfaction, although the analysis admittedly fell short of "absolute algebraic proof." The pear shape was also found more elongated than previously thought. Darwin concluded:
Notwithstanding the warning of M. Poincaré as to the danger of applying these results to heterogeneous masses and thence to cosmogony, I cannot restrain myself from joining him in seeing in this almost life-like process a counterpart to at least one form of the birth of double stars, planets, and satellites.
The nebular hypothesis and stability of the pyriform figure were to face critical challenges at the beginning of twentieth century. These surfaced in the collaboration of two Americans— a geologist and an astronomer, and in the work of a Russian mathematician.
VI. Criticisms of the Nebular and Fission Hypotheses
In 1902 Darwin believed the pear-shaped figure was stable. Furthermore, if his investigations had their "counterpart in the genesis of satellites and planets, it seems clear that the birth of a new body, although not cataclysmal, is rapid."Darwin incorporated non-cataclysmal changes of the fission hypothesis into his nebular hypothesis relying on evolution by tidal friction. Apparently, Poincaré, not totally convinced of the pyriform figure's stability, was even more wary of applying the idea to the formation of satellites. At the same time, other scientists were challenging the validity of the entire nebular hypothesis and seeking alternatives.
Since the mid-nineteenth century, growing numbers of scientists had found problems with Laplace's hypothesis. For example, Jacques Babinet in 1861 and Maurice Fouché in 1884 described the problem of angular momentum. If the sun at the center of the nebula rotated fast enough to shed rings of matter, then it should now be spinning much faster than measurements indicated. Instead, the planets had the majority of the angular momentum of the solar system. The discrepancy was difficult to explain. This was but one of many criticisms which persisted into the twentieth century.
In 1900 F. R. Moulton, an American astronomer, began to convince others that the nebular hypothesis was untenable on dynamical grounds. Collaborating with University of Chicago geology professor, T. C. Chamberlin, Moulton launched a successful attack against Laplace by persuasively presenting the past criticisms. He wrote:
It [the nebular hypothesis] has been accepted with a few slight modifications almost without question by the highest authorities, as Helmholtz, Kelvin, Newcomb, and Darwin, and it has been made the basis for the most sweeping conclusions. It has held such a dominant sway over the thoughts of investigators in the field of the grosser modes of inorganic evolution that doubtless in many instances facts have been warped and perverted, and questionable methods of reasoning employed, in order to bring experience into harmony with it.
In 1905 Chamberlin and Moulton abandoned Laplace's ideas, favoring what they called the planetesimal hypothesis. This idea, which many American and British astronomers supported, supposed a massive star had passed near the sun, pulling out exaggerated solar prominences. The ejected material, resembling a spiral nebula, condensed to form small bodies called planetesimals which eventually accreted forming the planets and satellites. James Hopwood Jeans and Harold Jeffreys later developed similar ideas relying on the tidal actions of a massive passing star. These theories, rather than relying on the development of a cloud of gas, depended on an encounter of our sun with a passing star. Given the large mutual distances between stars, stellar encounters were statistically very rare.In addition, a star being ripped apart was a cataclysmic event. More random and catastrophic than the evolutionary development from a Laplacian nebula, these theories competed with and gained acceptance over the nebular hypothesis because they overcame many of its difficulties.
Still there were those, like Darwin, who retained the nebular hypothesis. Although, in 1905, Laplace still held "dominant sway" in his mind, he did not ignore the criticisms. In his presidential address to the British Association for the Advancement of Science, he summarized some of the questions raised by the nebular hypothesis. For instance, why were discrete rings thrown off rather than a continuous flow of gas? How did an evenly distributed ring of matter aggregate to form a planet or satellite without merely falling back to the planet? Why were several satellites moving in retrograde orbits? Darwin conceded, "...enough has been said to show that the Nebular Hypothesis cannot be considered as a connected intelligible whole, however much truth it may contain." Nonetheless, he concluded, "...we still have to rely on such theories as that of Laplace for the explanation of the main outlines of the solar system." As late as 1911, Darwin maintained this position. Returning to his views on the meteoric hypothesis, he wrote that gaseous and meteoric nebulae could be treated similarly and concluded: "If this be so the Planetesimal Hypothesis of Chamberlin and Moulton is nearer akin to the Nebular Hypothesis than the authors of the former seem disposed to admit."
In addition to Chamberlin and Moulton's assault on the nebular hypothesis, another blow to Darwin's ideas came in 1905 when the Russian mathematician Aleksandr Lyapunov published a paper concluding that the pear-shaped figure was unstable. Darwin found it difficult to accept Lyapunov's finding:
I imagined that I had proved that the pear-shaped figure with incipient furrowing was also stable. But M. Liapounoff now stated that he is able to prove the pear-shaped figure to be unstable from the beginning. For the present at least I still think it is stable, and this belief receives powerful support from Mr. Jeans' researches.
Jeans, a pupil of Darwin's at Cambridge, had investigated the stability of infinite rotating cylinders of liquid. In this two-dimensional analogue to Darwin's three-dimensional situation (i.e. two instead of three axes needed to describe the motions) he found conclusively that a pyriform cross section was stable. In 1907, Darwin re-examined his work using improved computational methods and obtained similar results.
Despite the reassurance of Jeans' work, "dissent from so distinguished a mathematician as M. Liapounoff" was not to be taken lightly. The difficulty hinged on calculating the value of an infinite series. Darwin found this value to be negative, while Lyapunov concluded it was positive. Darwin wrote:
Having made my revision, and completed the computations...I feel a conviction that the source of our disagreement will be found in some matter of principle, and not the neglected residue of this series. I can now only express a hope that some one else will take up the question.
Darwin held on to his belief in the pear-shape's stability. If the shape were never stable, then the rotating spheroid would always disintegrate catastrophically and the fission hypothesis could not work, no exchange of stabilities could take place. In 1911 he wrote, "According to my calculations this series of figures is stable... notwithstanding M. Liapounoff's deservedly great authority, I venture to state the conclusions in accordance with my own work."
After Poincaré's and Darwin's deaths in 1912, doubts continued with respect to pyriform stability. Jeans re-examined the problem in 1916 remarking:
My results agreed with those previously given by Darwin up to a certain distance, and where they began to disagree I was able to discover a quite simple error which invalidated Darwin's discussion of the problem. ...I confirmed the conclusion already reached by Liapounoff, that the pear-shaped series was initially unstable...
There seems, then, to be little room for doubt that the series of pear-shaped configurations is initially unstable.
With the confirmation that the pear-shape was always unstable and mounting criticisms against the nebular hypothesis came a decline in the popularity of Darwin's fission hypothesis. Some scientists, like Moulton, Chamberlin and Jeans favored accretion models for the formation of satellites. Others, such as American astronomer T. J. J. See, favored capture models for the origin of all satellites, including the moon. Some, like William H. Pickering and Harold Jeffreys, remained proponents of the fission hypothesis. Pickering popularized an idea, suggested in 1882 by geologist Osmond Fisher, that the Pacific Ocean basin was the point of moon's rupture from the earth.This idea did not gain a large following among professional geologists or astronomers.
Darwin's fission hypothesis, linked to an increasingly questionable nebular hypothesis, was no longer widely accepted in the 1930s. For instance, in 1924 Harold Jeffreys dedicated his book to "the memory of the late Sir George Howard Darwin the Founder of Modern Cosmogony and Geophysics" and devoted the third chapter to the origin of the moon stating that Darwin's theory was the "most plausible." By 1930, however, he had abandoned the fission hypothesis. Darwin's theory depended on finding the degree of friction in the rotating body, something that Darwin had been unable to do. Jeffreys estimated the friction and found it "to be sufficient to the invalidate the theory." His criticisms persuaded many other astronomers to give up the fission hypothesis.
When the Royal Society accepted Darwin's first paper in 1877, father Charles wrote:
All of us are delighted, for considering what a man Sir William Thomson is, it is most grand that you should have staggered him so quickly, and that he should speak of your 'discovery, etc.'...Hurrah for the bowels of earth and their viscosity and for the moon and for the Heavenly bodies and for my son George.
Charles Darwin's encouragement and evolutionary thinking had an impact on his second son, George. His Cambridge education was in scientific traditions influenced by Newton, Maclaurin, Laplace, Hopkins, Nichol, Stokes and many others. George Darwin, embedded within the interconnected disciplines of mathematical physics, terrestrial physics, evolutionary cosmology and other areas, developed original scientific ideas about cosmic evolution. Through these, in addition to personal and professional contacts with men such as Charles Darwin, William Thomson and Henri Poincaré, Darwin worked to quantify his version of the nebular hypothesis. From this, Darwin developed the theory of evolution of the solar system by tidal friction and the fission hypothesis for the genesis of the moon.
In the preface to volume three of his collected scientific papers, Darwin wrote, "The next four papers are devoted to an extension of M. Poincaré's results as to figures of equilibrium, and he must be regarded as the presiding genius— or shall I say my patron saint— in this volume, just as Lord Kelvin was for the two preceding ones." Both Thomson and Poincaré were great influences on Darwin's changing cosmological ideas.
Initially, Darwin worked on a theoretical problem that was typical for a Cambridge Wrangler. He eventually sought to quantify and mathematize problems that had remained, for the most part, qualitative in nature. Darwin proposed a heterogeneous Laplacian ring which which would condense into a satellite. Then switched from this idea to that of a single body rupturing into two distinct bodies. Investigation of the coalescence of two bodies led Darwin to seek stable forms of rotating fluid masses beyond those known. This led to an investigation of the dumb-bell shape and its stability.
Phases of stability and instability were important to Darwin beyond the problem of rotating fluids. They provided him with a basis for his evolutionary views of the heavens as well as biological and political realms. In his presidential address to the B. A. A. S. in 1900, Darwin said:
In the world of life the naturalist describes those forms which persist as species; similarly the physicist speaks of stable configurations or modes of motion of matter; and the politician speaks of States. The idea at the base of all these conceptions is that of stability, or the power of resisting disintegration. In other words, the degree of persistence or permanence of a species, or a configuration of matter, or of a State depends on the perfection of its adaptation to its surrounding conditions.
Darwin applied the above ideas for evolution on a wide scale (e.g., tidal friction on the solar system) and on a smaller scale (e.g., the earth-moon system). Poincaré's ideas on the stability of mechanical systems helped him prove, to his satisfaction, the stability of the pear-shape. Such a proof, backed by the mathematics, gave Darwin support for his fission hypothesis. In 1905 Lyapunov's proof of the instability of the pear-shaped figure and the Chamberlin-Moulton planetesimal hypothesis were difficult for Darwin to accept. In fact, Darwin's confidence was great enough for him to reject their ideas. The planetesimal hypothesis, relying on a chance encounter of the sun with another star, was rejected by Darwin in favor of a modified Laplacian nebular hypothesis— the more gradualistic alternative. He also retained his belief in the pear-shaped figure's stability. Although not completely averse to catastrophic transitions, George Darwin preferred to retain stable evolutionary change when possible. In this respect he should be viewed as one in a generation of nineteenth-century astronomers confronting the new scientific ideas emerging at the beginning of the twentieth century.