In a literary tradition stretching back to the ancient Greeks, the Romantic poet Percy Bysshe Shelley personified the ever-changing face of the moon:

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:

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:

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:

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."

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:

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:

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.

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.

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.

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 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.

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,

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.,

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:

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:

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.

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:

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."

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:

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:

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.

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:

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:

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:

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:

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:

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.

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:

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:

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:

After Poincaré's and Darwin's deaths in 1912, doubts continued with respect to pyriform stability. Jeans re-examined the problem in 1916 remarking:

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:

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:

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.