Transcriber’s Note
This book uses some unusual characters, such as those representing the constellation Aries (♈) and Libra (♎). These characters may fail to display correctly if the font you are using does not support them.
Some corrections have been made to the printed text. These are listed in a second transcriber’s note at the end of the text.
MARY SOMERVILLE
J. COOPER Sc.
ON
THE CONNEXION
OF
THE PHYSICAL SCIENCES.
By MARY SOMERVILLE,
AUTHORESS OF ‘MECHANISM OF THE HEAVENS,’ AND
‘PHYSICAL GEOGRAPHY.’
“No natural phenomenon can be adequately studied in itself alone—but, to be understood, it must be considered as it stands connected with all Nature.”—Bacon.
Ninth Edition, completely Revised.
LONDON:
JOHN MURRAY, ALBEMARLE STREET.
1858.
The right of Translation is reserved.
LONDON: PRINTED BY W. CLOWES AND SONS, DUKE STREET, STAMFORD STREET,
AND CHARING CROSS.
This Book is Dedicated
TO
HER DEAR CHILDREN,
BY THEIR AFFECTIONATE MOTHER,
MARY SOMERVILLE.
Florence, Nov. 1, 1858.
CONTENTS.
Introduction
Page [1]
SECTION I.
Attraction of a Sphere—Form of Celestial Bodies—Terrestrial Gravitation retains the Moon in her Orbit—The Heavenly Bodies move in Conic Sections—Gravitation Proportional to Mass—Gravitation of the Particles of Matter—Figure of the Planets—How it affects the Motions of their Satellites—Rotation and Translation impressed by the same Impulse—Motion of the Sun and Solar System
SECTION II.
Elliptical Motion—Mean and True Motion—Equinoctial—Ecliptic—Equinoxes—Mean and True Longitude—Equation of Centre—Inclination of the Orbits of Planets—Celestial Latitude—Nodes—Elements of an Orbit—Undisturbed or Elliptical Orbits—Great Inclination of the Orbits of the New Planets—Universal Gravitation the Cause of Perturbations in the Motions of the Heavenly Bodies—Problem of the Three Bodies—Stability of Solar System depends upon the Primitive Momentum of the Bodies
SECTION III.
Perturbations, Periodic and Secular—Disturbing Action equivalent to three Partial Forces—Tangential Force the cause of the Periodic Inequalities in Longitude, and Secular Inequalities in the Form and Position of the Orbit in its own Plane—Radial Force the cause of Variations in the Planet’s Distance from the Sun—It combines with the Tangential Force to produce the Secular Variations in the Form and Position of the Orbit in its own Plane—Perpendicular Force the cause of Periodic Perturbations in Latitude, and Secular Variations in the Position of the Orbit with regard to the Plane of the Ecliptic—Mean Motion and Major Axis Invariable—Stability of System—Effects of a Resisting Medium—Invariable Plane of the Solar System and of the Universe—Great Inequality of Jupiter and Saturn
SECTION IV.
Theory of Jupiter’s Satellites—Effects of the Figure of Jupiter upon his Satellites—Position of their Orbits—Singular Laws among the Motions of the first Three Satellites—Eclipses of the Satellites—Velocity of Light—Aberration—Ethereal Medium—Satellites of Saturn and Uranus
SECTION V.
Lunar Theory—Periodic Perturbations of the Moon—Equation of Centre—Evection—Variation—Annual Equation—Direct and Indirect Action of Planets—The Moon’s Action on the Earth disturbs her own Motion—Excentricity and Inclination of Lunar Orbit invariable—Acceleration—Secular Variation in Nodes and Perigee—Motion of Nodes and Perigee inseparably connected with the Acceleration—Nutation of Lunar Orbit—Form and Internal Structure of the Earth determined from it—Lunar, Solar, and Planetary Eclipses—Occultations and Lunar Distances—Mean Distance of the Sun from the Earth obtained from Lunar Theory—Absolute Distances of the Planets, how found
SECTION VI.
Form of the Earth and Planets—Figure of a Homogeneous Spheroid in Rotation—Figure of a Spheroid of variable Density—Figure of the Earth, supposing it to be an Ellipsoid of Revolution—Mensuration of a Degree of the Meridian—Compression and Size of the Earth from Degrees of Meridian—Figure of Earth from the Pendulum
SECTION VII.
Parallax—Lunar Parallax found from Direct Observation—Solar Parallax deduced from the Transit of Venus—Distance of the Sun from the Earth—Annual Parallax—Distance of the Fixed Stars
SECTION VIII.
Masses of Planets that have no Satellites determined from their Perturbations—Masses of the others obtained from the Motions of their Satellites—Masses of the Sun, the Earth, of Jupiter and of the Jovial System—Mass of the Moon—Real Diameters of Planets, how obtained—Size of Sun, Densities of the Heavenly Bodies—Formation of Astronomical Tables—Requisite Data and Means of obtaining them
SECTION IX.
Rotation of the Sun and Planets—Saturn’s Rings—Periods of the Rotation of the Moon and other Satellites equal to the Periods of their Revolutions—Form of Lunar Spheroid—Libration, Aspect, and Constitution of the Moon—Rotation of Jupiter’s Satellites
SECTION X.
Rotation of the Earth invariable—Decrease in the Earth’s mean Temperature—Earth originally in a state of Fusion—Length of Day constant—Decrease of Temperature ascribed by Sir John Herschel to the variation in the Excentricity of the Terrestrial Orbit—Difference in the Temperature of the two Hemispheres erroneously ascribed to the Excess in the Length of Spring and Summer in the Southern Hemisphere; attributed by Sir Charles Lyell to the Operation of existing Causes—Three principal Axes of Rotation—Position of the Axis of Rotation on the Surface of the Earth invariable—Ocean not sufficient to restore the Equilibrium of the Earth if deranged—Its Density and mean Depth—Internal Structure of the Earth
SECTION XI.
Precession and Nutation—Their Effects on the Apparent Places of the Fixed Stars
SECTION XII.
Mean and Apparent Sidereal Time—Mean and Apparent Solar Time—Equation of Time—English and French Subdivisions of Time—Leap Year—Christian Era—Equinoctial Time—Remarkable Eras depending upon the Position of the Solar Perigee—Inequality of the Lengths of the Seasons in the two Hemispheres—Application of Astronomy to Chronology—English and French Standards of Weights and Measures
SECTION XIII.
Tides—Forces that produce them—Origin and Course of Tidal Wave—Its Speed—Three kinds of Oscillations in the Ocean—The Semidiurnal Tides—Equinoctial Tides—Effects of the Declination of the Sun and Moon—Theory insufficient without Observation—Direction of the Tidal Wave—Height of Tides—Mass of Moon obtained from her Action on the Tides—Interference of Undulations—Impossibility of a Universal Inundation—Currents
SECTION XIV.
Molecular Forces—Permanency of the ultimate Particles of Matter—Interstices—Mossotti’s Theory—Rankin’s Theory of Molecular Vortices—Gases reduced to Liquids by Pressure—Gravitation of Particles—Cohesion—Crystallization—Cleavage—Isomorphism—Minuteness of the Particles—Height of Atmosphere—Chemical Affinity—Definite Proportions and Relative Weights of Atoms—Faraday’s Discovery with regard to Affinity—Capillary Attraction
SECTION XV.
Analysis of the Atmosphere—Its pressure—Law of Decrease in Density—Law of Decrease in Temperature—Measurement of Heights by the Barometer—Extent of the Atmosphere—Barometrical Variations—Oscillations—Trade-Winds—Cloud-Ring—Monsoons—Rotation of Winds—Laws of Hurricanes
SECTION XVI.
Sound—Propagation of Sound illustrated by a Field of Standing Corn—Nature of Waves—Propagation of Sound through the Atmosphere—Intensity—Noises—A Musical Sound—Quality—Pitch—Extent of Human Hearing—Velocity of Sound in Air, Water, and Solids—Causes of the Obstruction of Sound—Law of its Intensity—Reflection of Sound—Echoes—Thunder—Refraction of Sound—Interference of Sounds
SECTION XVII.
Vibration of Musical Strings—Harmonic Sounds—Nodes—Vibration of Air in Wind-Instruments—Vibration of Solids—Vibrating Plates—Bells—Harmony—Sounding Boards—Forced Vibrations—Resonance—Speaking Machines
SECTION XVIII.
Refraction—Astronomical Refraction and its Laws—Formation of Tables of Refraction—Terrestrial Refraction—Its Quantity—Instances of Extraordinary Refraction—Reflection—Instances of Extraordinary Reflection—Loss of Light by the Absorbing Power of the Atmosphere—Apparent Magnitude of Sun and Moon in the Horizon
SECTION XIX.
Constitution of Light according to Sir Isaac Newton—Absorption of Light—Colours of Bodies—Constitution of Light according to Sir David Brewster—New Colours—Fraunhofer’s Dark Lines—Dispersion of Light—The Achromatic Telescope—Homogeneous Light—Accidental and Complementary Colours—M. Plateau’s Experiments and Theory of Accidental Colours
SECTION XX.
Interference of Light—Undulatory Theory of Light—Propagation of Light—Newton’s Rings—Measurement of the Length of the Waves of Light, and of the Frequency of the Vibrations of Ether for each Colour—Newton’s Scale of Colours—Diffraction of Light—Sir John Herschel’s Theory of the Absorption of Light—Refraction and Reflection of Light
SECTION XXI.
Polarization of Light—Defined—Polarization by Refraction—Properties of the Tourmaline—Double Refraction—All doubly Refracted Light is Polarized—Properties of Iceland Spar—Tourmaline absorbs one of the two Refracted Rays—Undulations of Natural Light—Undulations of Polarized Light—The Optic Axes of Crystals—M. Fresnel’s Discoveries on the Rays passing along the Optic Axis—Polarization by Reflection
SECTION XXII.
Phenomena exhibited by the Passage of Polarized Light through Mica and Sulphate of Lime—The Coloured Images produced by Polarized Light passing through Crystals having one and two Optic Axes—Circular Polarization—Elliptical Polarization—Discoveries of MM. Biot, Fresnel, and Professor Airy—Coloured Images produced by the Interference of Polarized Rays—Fluorescence
SECTION XXIII.
Objections to the Undulatory Theory, from a difference in the Action of Sound and Light under the same circumstances, removed—The Dispersion of Light according to the Undulatory Theory—Arago’s final proof that the Undulatory Theory is the Law of Nature
SECTION XXIV.
Chemical or Photographic Rays of Solar Spectrum—Scheele, Ritter, and Wollaston’s Discoveries—Wedgwood’s and Sir Humphry Davy’s Photographic Pictures—The Calotype—The Daguerreotype—The Chromatype—The Cyanotype—Collodion—Sir John Herschel’s Discoveries in the Chemical Spectrum—M. Becquerel’s Discoveries of Inactive Lines in ditto—Thermic Spectrum—Phosphoric Spectrum—Electrical Properties—Parathermic Rays—Moser and Hunt’s Experiments—General Structure and antagonist Properties of Solar Spectrum—Defracted Spectrum
SECTION XXV.
Size and Constitution of the Sun—The Solar Spots—Intensity of the Sun’s Light and Heat—The Sun’s Atmosphere—His influence on the Planets—Atmospheres of the Planets—The Moon has none—Lunar heat—The Differential Telescope—Temperature of Space—Internal Heat of the Earth—Zone of constant Temperature—Increase of Heat with the Depth—Central Heat—Volcanic Action—Quantity of Heat received from the Sun—Isogeothermal Lines—Line of Perpetual Congelation—Climate—Isothermal Lines—Same quantity of Heat annually received and radiated by the Earth
SECTION XXVI.
Influence of Temperature on Vegetation—Vegetation varies with the Latitude and Height above the Sea—Geographical Distribution of Land Plants—Distribution of Marine Plants—Corallines, Shell-fish, Reptiles, Insects, Birds, and Quadrupeds—Varieties of Mankind, yet identity of Species
SECTION XXVII.
Terrestrial Heat—Radiation—Transmission—Melloni’s experiments—Heat in Solar Spectrum—Polarization of Heat—Nature of Heat—Absorptions—Dew—Rain—Combustion—Expansion—Compensation Pendulum—Transmission through Crystals—Propagation—Dynamic Theory of Heat—Mechanical equivalent of Heat—Latent Heat is the Force of Expansion—Steam—Work performed by Heat—Conservation of Force—Mechanical Power in the Tides—Dynamical Power of Light—Analogy between Light, Heat, and Sound
SECTION XXVIII.
Common or Static Electricity, or Electricity of Tension—A Dual Power—Methods of exciting it—Attraction and Repulsion—Conduction—Electrics and Non-electrics—Induction—Dielectrics—Tension—Law of the Electric Force—Distribution—Laws of Distribution—Heat of Electricity—Electrical Light and its Spectrum—Velocity—Atmospheric Electricity—Its cause—Electric Clouds—Violent effects of Lightning—Back Stroke—Electric Glow—Phosphorescence
SECTION XXIX.
Voltaic Electricity—The Voltaic Battery—Intensity—Quantity—Static Electricity, and Electricity in Motion—Luminous Effects—Mr. Grove on the Electric Arc and Light—Decomposition of Water—Formation of Crystals by Voltaic Electricity—Photo-galvanic Engraving—Conduction—Heat of Voltaic Electricity—Electric Fish
SECTION XXX.
Discovery of Electro-magnetism—Deflection of the Magnetic Needle by a Current of Electricity—Direction of the Force—Rotatory Motion by Electricity—Rotation of a Wire and a Magnet—Rotation of a Magnet about its Axis—Of Mercury and Water—Electro-Magnetic Cylinder or Helix—Suspension of a Needle in a Helix—Electro-Magnetic Induction—Temporary Magnets—The Galvanometer
SECTION XXXI.
Electro-Dynamics—Reciprocal Action of Electric Currents—Identity of Electro-Dynamic Cylinders and Magnets—Differences between the Action of Voltaic Electricity and Electricity of Tension—Effects of a Voltaic Current—Ampère’s Theory—Dr. Faraday’s Experiment of Electrifying and Magnetising a Ray of Light
SECTION XXXII.
Magneto-Electricity—Volta-Electric Induction—Magneto-Electric Induction—Identity in the Action of Electricity and Magnetism—Description of a Magneto-Electric Apparatus and its Effects—Identity of Magnetism and Electricity—The Submarine Telegraph
SECTION XXXIII.
Electricity produced by Rotation—Direction of the Currents—Electricity from the Rotation of a Magnet—M. Arago’s Experiment explained—Rotation of a Plate of Iron between the Poles of a Magnet—Relation of Substances to Magnets of three Kinds—Thermo-Electricity
SECTION XXXIV.
Magnetism a Dual Power—Antithetic Character of Paramagnetism and Diamagnetism—The Earth Paramagnetic—Properties of Paramagnetic Bodies—Polarity—Induction—Lines of Magnetic Force—Currents of Electricity induced by them—Proved to be Closed Curves—Analogy and Identity of Electricity and Magnetism—Terrestrial Magnetism—Mean Values of the Three Magnetic Elements—Their Variations in Double Progression proved to consist of Two Superposed Variations—Discovery of the Periodicity of the Magnetic Storms—The Decennial Period of the Magnetic Elements the same with that of the Solar Spots—Magnetism of the Atmosphere—Diamagnetism—Action of Electro-Magnetism on Paramagnetic, Diamagnetic Bodies, and on Copper, very different—Proof of Diamagnetic Polarity and Induction—Magnecrystallic Action—Effects of Compression, Heat, and Cleavage on Magnetic Bodies—Mutual Dependence of Light, Heat, Electricity, &c. &c.—The Conservation of Force and the Permanency of Matter Primary Laws of Nature—Definition of Gravity not according to that Law—Gravity only the Residual Force of a Universal Power—Magnetism of the Ethereal Medium
SECTION XXXV.
Ethereal Medium—Comets—Do not disturb the Solar System—Their Orbits and Disturbances—M. Faye’s Comet probably the same with Lexel’s—Periods of other three known—Acceleration in the mean Motions of Encke’s and Biela’s Comets—The Shock of a Comet—Disturbing Action of the Earth and Planets on Encke’s and Biela’s Comets—Velocity of Comets—The Comet of 1264—The great Comet of 1343—Physical Constitution—Shine by borrowed Light—Estimation of their Number
SECTION XXXVI.
The Fixed Stars—Their Number—The Milky Way—Double Stars—Binary Systems—Their Orbits and Periodic Times—Colours of the Stars—Stars that have vanished—Variable Stars—Variation in Sun’s Light—Parallax and Distances of the Fixed Stars—Masses of the Stars—Comparative Light of the Stars—Proper Motions of the Stars—Apparent Motions of the Stars—Motion and Velocity of the Sun and Solar System—The Nebulæ—Their Number—Catalogue of them—Consist of Two Classes—Diffuse Nebulæ—Definitely formed Nebulæ—Globular Clusters—Splendour of Milky Way—Distribution of the Nebulæ—The Magellanic Clouds—Nebulæ round η Argûs—Constitution of Nebulæ, and the Forces that maintain them—Meteorites and Shooting Stars
SECTION XXXVII.
Diffusion of Matter through Space—Gravitation—Its Velocity—Simplicity of its Laws—Gravitation independent of the Magnitude and Distances of the Bodies—Not impeded by the intervention of any Substance—Its Intensity invariable—General Laws—Recapitulation and Conclusion
Notes
Index
THE CONNECTION
OF
THE PHYSICAL SCIENCES.
INTRODUCTION.
Science, regarded as the pursuit of truth, must ever afford occupation of consummate interest, and subject of elevated meditation. The contemplation of the works of creation elevates the mind to the admiration of whatever is great and noble; accomplishing the object of all study, which, in the eloquent language of Sir James Mackintosh, “is to inspire the love of truth, of wisdom, of beauty—especially of goodness, the highest beauty—and of that supreme and eternal Mind, which contains all truth and wisdom, all beauty and goodness. By the love or delightful contemplation and pursuit of these transcendent aims, for their own sake only, the mind of man is raised from low and perishable objects, and prepared for those high destinies which are appointed for all those who are capable of them.”
Astronomy affords the most extensive example of the connection of the physical sciences. In it are combined the sciences of number and quantity, of rest and motion. In it we perceive the operation of a force which is mixed up with everything that exists in the heavens or on earth; which pervades every atom, rules the motions of animate and inanimate beings, and is as sensible in the descent of a rain-drop as in the falls of Niagara; in the weight of the air, as in the periods of the moon. Gravitation not only binds satellites to their planet, and planets to the sun, but it connects sun with sun throughout the wide extent of creation, and is the cause of the disturbances, as well as of the order of nature; since every tremor it excites in any one planet is immediately transmitted to the farthest limits of the system, in oscillations which correspond in their periods with the cause producing them, like sympathetic notes in music, or vibrations from the deep tones of an organ.
The heavens afford the most sublime subject of study which can be derived from science. The magnitude and splendour of the objects, the inconceivable rapidity with which they move, and the enormous distances between them, impress the mind with some notion of the energy that maintains them in their motions, with a durability to which we can see no limit. Equally conspicuous is the goodness of the great First Cause, in having endowed man with faculties, by which he can not only appreciate the magnificence of His works, but trace, with precision, the operation of His laws, use the globe he inhabits as a base wherewith to measure the magnitude and distance of the sun and planets, and make the diameter ([Note 1]) of the earth’s orbit the first step of a scale by which he may ascend to the starry firmament. Such pursuits, while they ennoble the mind, at the same time inculcate humility, by showing that there is a barrier which no energy, mental or physical, can ever enable us to pass: that, however profoundly we may penetrate the depths of space, there still remain innumerable systems, compared with which, those apparently so vast must dwindle into insignificance, or even become invisible; and that not only man, but the globe he inhabits—nay, the whole system of which it forms so small a part—might be annihilated, and its extinction be unperceived in the immensity of creation.
A complete acquaintance with physical astronomy can be attained by those only who are well versed in the higher branches of mathematical and mechanical science ([N. 2]), and they alone can appreciate the extreme beauty of the results, and of the means by which these results are obtained. It is nevertheless true, that a sufficient skill in analysis ([N. 3]) to follow the general outline—to see the mutual dependence of the different parts of the system, and to comprehend by what means the most extraordinary conclusions have been arrived at,—is within the reach of many who shrink from the task, appalled by difficulties, not more formidable than those incident to the study of the elements of every branch of knowledge. There is a wide distinction between the degree of mathematical acquirement necessary for making discoveries, and that which is requisite for understanding what others have done.
Our knowledge of external objects is founded upon experience, which furnishes facts; the comparison of these facts establishes relations, from which the belief that like causes will produce like effects leads to general laws. Thus, experience teaches that bodies fall at the surface of the earth with an accelerated velocity, and with a force proportional to their masses. By comparison, Newton proved that the force which occasions the fall of bodies at the earth’s surface is identical with that which retains the moon in her orbit; and he concluded, that, as the moon is kept in her orbit by the attraction of the earth, so the planets might be retained in their orbits by the attraction of the sun. By such steps he was led to the discovery of one of those powers with which the Creator has ordained that matter should reciprocally act upon matter.
Physical astronomy is the science which compares and identifies the laws of motion observed on earth with the motions that take place in the heavens: and which traces, by an uninterrupted chain of deduction from the great principle that governs the universe, the revolutions and rotations of the planets, and the oscillations ([N. 4]) of the fluids at their surfaces; and which estimates the changes the system has hitherto undergone, or may hereafter experience—changes which require millions of years for their accomplishment.
The accumulated efforts of astronomers, from the earliest dawn of civilization, have been necessary to establish the mechanical theory of astronomy. The courses of the planets have been observed for ages, with a degree of perseverance that is astonishing, if we consider the imperfection and even the want of instruments. The real motions of the earth have been separated from the apparent motions of the planets; the laws of the planetary revolutions have been discovered; and the discovery of these laws has led to the knowledge of the gravitation ([N. 5]) of matter. On the other hand, descending from the principle of gravitation, every motion in the solar system has been so completely explained, that the laws of any astronomical phenomena that may hereafter occur are already determined.
SECTION I.
Attraction of a Sphere—Form of Celestial Bodies—Terrestrial Gravitation retains the Moon in her Orbit—The Heavenly Bodies move in Conic Sections—Gravitation Proportional to Mass—Gravitation of the Particles of Matter—Figure of the Planets—How it affects the Motions of their Satellites—Rotation and Translation impressed by the same Impulse—Motion of the Sun and Solar System.
It has been proved by Newton, that a particle of matter ([N. 6]) placed without the surface of a hollow sphere ([N. 7]) is attracted by it in the same manner as if the mass of the hollow sphere, or the whole matter it contains, were collected into one dense particle in its centre. The same is therefore true of a solid sphere, which may be supposed to consist of an infinite number of concentric hollow spheres ([N. 8]). This, however, is not the case with a spheroid ([N. 9]); but the celestial bodies are so nearly spherical, and at such remote distances from one another, that they attract and are attracted as if each were condensed into a single particle situate in its centre of gravity ([N. 10])—a circumstance which greatly facilitates the investigation of their motions.
Newton has shown that the force which retains the moon in her orbit is the same with that which causes heavy substances to fall at the surface of the earth. If the earth were a sphere, and at rest, a body would be equally attracted, that is, it would have the same weight at every point of its surface, because the surface of a sphere is everywhere equally distant from its centre. But, as our planet is flattened at the poles ([N. 11]), and bulges at the equator, the weight of the same body gradually decreases from the poles, where it is greatest, to the equator, where it is least. There is, however, a certain mean ([N. 12]) latitude ([N. 13]), or part of the earth intermediate between the pole and the equator, where the attraction of the earth on bodies at its surface is the same as if it were a sphere; and experience shows that bodies there fall through 16·0697 feet in a second. The mean distance ([N. 14]) of the moon from the earth is about sixty times the mean radius ([N. 15]) of the earth. When the number 16·0697 is diminished in the ratio ([N. 16]) of 1 to 3600, which is the square of the moon’s distance ([N. 17]) from the earth’s centre, estimated in terrestrial radii, it is found to be exactly the space the moon would fall through in the first second of her descent to the earth, were she not prevented by the centrifugal force ([N. 18]) arising from the velocity with which she moves in her orbit. The moon is thus retained in her orbit by a force having the same origin, and regulated by the same law, with that which causes a stone to fall at the earth’s surface. The earth may, therefore, be regarded as the centre of a force which extends to the moon; and, as experience shows that the action and reaction of matter are equal and contrary ([N. 19]), the moon must attract the earth with an equal and contrary force.
Newton also ascertained that a body projected ([N. 20]) in space ([N. 21]) will move in a conic section ([N. 22]), if attracted by a force proceeding from a fixed point, with an intensity inversely as the square of the distance ([N. 23]); but that any deviation from that law will cause it to move in a curve of a different nature. Kepler found, by direct observation, that the planets describe ellipses ([N. 24]), or oval paths, round the sun. Later observations show that comets also move in conic sections. It consequently follows that the sun attracts all the planets and comets inversely as the square of their distances from its centre; the sun, therefore, is the centre of a force extending indefinitely in space, and including all the bodies of the system in its action.
Kepler also deduced from observation that the squares of the periodic times ([N. 25]) of the planets, or the times of their revolutions round the sun, are proportional to the cubes of their mean distances from its centre ([N. 26]). Hence the intensity of gravitation of all the bodies towards the sun is the same at equal distances. Consequently, gravitation is proportional to the masses ([N. 27]); for, if the planets and comets were at equal distances from the sun, and left to the effects of gravity, they would arrive at his surface at the same time ([N. 28]). The satellites also gravitate to their primaries ([N. 29]) according to the same law that their primaries do to the sun. Thus, by the law of action and reaction, each body is itself the centre of an attractive force extending indefinitely in space, causing all the mutual disturbances which render the celestial motions so complicated, and their investigation so difficult.
The gravitation of matter directed to a centre, and attracting directly as the mass and inversely as the square of the distance, does not belong to it when considered in mass only; particle acts on particle according to the same law when at sensible distances from each other. If the sun acted on the centre of the earth, without attracting each of its particles, the tides would be very much greater than they now are, and would also, in other respects, be very different. The gravitation of the earth to the sun results from the gravitation of all its particles, which, in their turn, attract the sun in the ratio of their respective masses. There is a reciprocal action likewise between the earth and every particle at its surface. The earth and a feather mutually attract each other in the proportion of the mass of the earth to the mass of the feather. Were this not the case, and were any portion of the earth, however small, to attract another portion, and not be itself attracted, the centre of gravity of the earth would be moved in space by this action, which is impossible.
The forms of the planets result from the reciprocal attraction of their component particles. A detached fluid mass, if at rest, would assume the form of a sphere, from the reciprocal attraction of its particles. But if the mass revolve about an axis, it becomes flattened at the poles and bulges at the equator ([N. 11]), in consequence of the centrifugal force arising from the velocity of rotation ([N. 30]); for the centrifugal force diminishes the gravity of the particles at the equator, and equilibrium can only exist where these two forces are balanced by an increase of gravity. Therefore, as the attractive force is the same on all particles at equal distances from the centre of a sphere, the equatorial particles would recede from the centre, till their increase in number balance the centrifugal force by their attraction. Consequently, the sphere would become an oblate or flattened spheroid, and a fluid, partially or entirely covering a solid, as the ocean and atmosphere cover the earth, must assume that form in order to remain in equilibrio. The surface of the sea is, therefore, spheroidal, and the surface of the earth only deviates from that figure where it rises above or sinks below the level of the sea. But the deviation is so small, that it is unimportant when compared with the magnitude of the earth; for the mighty chain of the Andes, and the yet more lofty Himalaya, bear about the same proportion to the earth that a grain of sand does to a globe three feet in diameter. Such is the form of the earth and planets. The compression ([N. 31]) or flattening at their poles is, however, so small, that even Jupiter, whose rotation is the most rapid, and therefore the most elliptical of the planets, may, from his great distance, be regarded as spherical. Although the planets attract each other as if they were spheres, on account of their distances, yet the satellites ([N. 32]) are near enough to be sensibly affected in their motions by the forms of their primaries. The moon, for example, is so near the earth, that the reciprocal attraction between each of her particles, and each of the particles in the prominent mass at the terrestrial equator, occasions considerable disturbances in the motions of both bodies; for the action of the moon on the matter at the earth’s equator produces a nutation ([N. 33]) in the axis ([N. 34]) of rotation, and the reaction of that matter on the moon is the cause of a corresponding nutation in the lunar orbit ([N. 35]).
If a sphere at rest in space receive an impulse passing through its centre of gravity, all its parts will move with an equal velocity in a straight line; but, if the impulse does not pass through the centre of gravity, its particles, having unequal velocities, will have a rotatory or revolving motion, at the same time that it is translated ([N. 36]) in space. These motions are independent of one another; so that a contrary impulse, passing through its centre of gravity, will impede its progress, without interfering with its rotation. The sun rotates about an axis, and modern observations show that an impulse in a contrary direction has not been given to his centre of gravity, for he moves in space, accompanied by all those bodies which compose the solar system—a circumstance which in no way interferes with their relative motions; for, in consequence of the principle that force is proportional to velocity ([N. 37]), the reciprocal attractions of a system remain the same whether its centre of gravity be at rest, or moving uniformly in space. It is computed that, had the earth received its motion from a single impulse, that impulse must have passed through a point about twenty-five miles from its centre.
Since the motions of rotation and translation of the planets are independent of each other, though probably communicated by the same impulse, they form separate subjects of investigation.
SECTION II.
Elliptical Motion—Mean and True Motion—Equinoctial—Ecliptic—Equinoxes—Mean and True Longitude—Equation of Centre—Inclination of the Orbits of Planets—Celestial Latitude—Nodes—Elements of an Orbit—Undisturbed or Elliptical Orbits—Great Inclination of the Orbits of the New Planets—Universal Gravitation the Cause of Perturbations in the Motions of the Heavenly Bodies—Problem of the Three Bodies—Stability of Solar System depends upon the Primitive Momentum of the Bodies.
A planet moves in its elliptical orbit with a velocity varying every instant, in consequence of two forces, one tending to the centre of the sun, and the other in the direction of a tangent ([N. 38]) to its orbit, arising from the primitive impulse given at the time when it was launched into space. Should the force in the tangent cease, the planet would fall to the sun by its gravity. Were the sun not to attract it, the planet would fly off in the tangent. Thus, when the planet is at the point of its orbit farthest from the sun, his action overcomes the planet’s velocity, and brings it towards him with such an accelerated motion, that at last it overcomes the sun’s attraction, and, shooting past him, gradually decreases in velocity until it arrives at the most distant point, where the sun’s attraction again prevails ([N. 39]). In this motion the radii vectores ([N. 40]), or imaginary lines joining the centres of the sun and the planets, pass over equal areas or spaces in equal times ([N. 41]).
The mean distance of a planet from the sun is equal to half the major axis ([N. 42]) of its orbit: if, therefore, the planet described a circle ([N. 43]) round the sun at its mean distance, the motion would be uniform, and the periodic time unaltered, because the planet would arrive at the extremities of the major axis at the same instant, and would have the same velocity, whether it moved in the circular or elliptical orbit, since the curves coincide in these points. But in every other part the elliptical, or true motion ([N. 44]), would either be faster or slower than the circular or mean motion ([N. 45]). As it is necessary to have some fixed point in the heavens from whence to estimate these motions, the vernal equinox ([N. 46]) at a given epoch has been chosen. The equinoctial, which is a great circle traced in the starry heavens by the imaginary extension of the plane of the terrestrial equator, is intersected by the ecliptic, or apparent path of the sun, in two points diametrically opposite to one another, called the vernal and autumnal equinoxes. The vernal equinox is the point through which the sun passes in going from the southern to the northern hemisphere; and the autumnal, that in which he crosses from the northern to the southern. The mean or circular motion of a body, estimated from the vernal equinox, is its mean longitude; and its elliptical, or true motion, reckoned from that point, is its true longitude ([N. 47]): both being estimated from west to east, the direction in which the bodies move. The difference between the two is called the equation of the centre ([N. 48]); which consequently vanishes at the apsides ([N. 49]), or extremities of the major axis, and is at its maximum ninety degrees ([N. 50]) distant from these points, or in quadratures ([N. 51]), where it measures the excentricity ([N. 52]) of the orbit; so that the place of the planet in its elliptical orbit is obtained by adding or subtracting the equation of the centre to or from its mean longitude.
The orbits of the principal planets have a very small obliquity or inclination ([N. 53]) to the plane of the ecliptic in which the earth moves; and, on that account, astronomers refer their motions to this plane at a given epoch as a known and fixed position. The angular distance of a planet from the plane of the ecliptic is its latitude ([N. 54]), which is south or north according as the planet is south or north of that plane. When the planet is in the plane of the ecliptic, its latitude is zero; it is then said to be in its nodes ([N. 55]). The ascending node is that point in the ecliptic through which the planet passes in going from the southern to the northern hemisphere. The descending node is a corresponding point in the plane of the ecliptic diametrically opposite to the other, through which the planet descends in going from the northern to the southern hemisphere. The longitude and latitude of a planet cannot be obtained by direct observation, but are deduced from observations made at the surface of the earth by a very simple computation. These two quantities, however, will not give the place of a planet in space. Its distance from the sun ([N. 56]) must also be known; and, for the complete determination of its elliptical motion, the nature and position of its orbit must be ascertained by observation. This depends upon seven quantities, called the elements of the orbit ([N. 57]). These are, the length of the major axis, and the excentricity, which determine the form of the orbit; the longitude of the planet when at its least distance from the sun, called the longitude of the perihelion; the inclination of the orbit to the plane of the ecliptic, and the longitude of its ascending node: these give the position of the orbit in space; but the periodic time, and the longitude of the planet at a given instant, called the longitude of the epoch, are necessary for finding the place of the body in its orbit at all times. A perfect knowledge of these seven elements is requisite for ascertaining all the circumstances of undisturbed elliptical motion. By such means it is found that the paths of the planets, when their mutual disturbances are omitted, are ellipses nearly approaching to circles, whose planes, slightly inclined to the ecliptic, cut it in straight lines, passing through the centre of the sun ([N. 58]). The orbits of the recently-discovered planets deviate more from the ecliptic than those of the ancient planets: that of Pallas, for instance, has an inclination of 34° 42ʹ 29·8ʺ to it; on which account it is more difficult to determine their motions.
Were the planets attracted by the sun only, they would always move in ellipses, invariable in form and position; and because his action is proportional to his mass, which is much larger than that of all the planets put together, the elliptical is the nearest approximation to their true motions. The true motions of the planets are extremely complicated, in consequence of their mutual attraction, so that they do not move in any known or symmetrical curve, but in paths now approaching to, now receding from, the elliptical form; and their radii vectores do not describe areas or spaces exactly proportional to the time, so that the areas become a test of disturbing forces.
To determine the motion of each body, when disturbed by all the rest, is beyond the power of analysis. It is therefore necessary to estimate the disturbing action of one planet at a time, whence the celebrated problem of the three bodies, originally applied to the moon, the earth, and the sun—namely, the masses being given of three bodies projected from three given points, with velocities given both in quantity and direction; and supposing the bodies to gravitate to one another with forces that are directly as their masses, and inversely as the squares of the distances, to find the lines described by these bodies, and their positions at any given instant; or, in other words, to determine the path of a celestial body when attracted by a second body, and disturbed in its motion round the second body by a third—a problem equally applicable to planets, satellites, and comets.
By this problem the motions of translation of the celestial bodies are determined. It is an extremely difficult one, and would be infinitely more so if the disturbing action were not very small when compared with the central force; that is, if the action of the planets on one another were not very small when compared with that of the sun. As the disturbing influence of each body may be found separately, it is assumed that the action of the whole system, in disturbing any one planet, is equal to the sum of all the particular disturbances it experiences, on the general mechanical principle, that the sum of any number of small oscillations is nearly equal to their simultaneous and joint effect.
On account of the reciprocal action of matter, the stability of the system depends upon the intensity of the primitive momentum ([N. 59]) of the planets, and the ratio of their masses to that of the sun; for the nature of the conic sections in which the celestial bodies move depends upon the velocity with which they were first propelled in space. Had that velocity been such as to make the planets move in orbits of unstable equilibrium ([N. 60]), their mutual attractions might have changed them into parabolas, or even hyperbolas ([N. 22]); so that the earth and planets might, ages ago, have been sweeping far from our sun through the abyss of space. But as the orbits differ very little from circles, the momentum of the planets, when projected, must have been exactly sufficient to ensure the permanency and stability of the system. Besides, the mass of the sun is vastly greater than that of any planet; and as their inequalities bear the same ratio to their elliptical motions that their masses do to that of the sun, their mutual disturbances only increase or diminish the excentricities of their orbits by very minute quantities; consequently the magnitude of the sun’s mass is the principal cause of the stability of the system. There is not in the physical world a more splendid example of the adaptation of means to the accomplishment of an end than is exhibited in the nice adjustment of these forces, at once the cause of the variety and of the order of Nature.
SECTION III.
Perturbations, Periodic and Secular—Disturbing Action equivalent to three Partial Forces—Tangential Force the cause of the Periodic Inequalities in Longitude, and Secular Inequalities in the Form and Position of the Orbit in its own Plane—Radial Force the cause of Variations in the Planet’s Distance from the Sun—It combines with the Tangential Force to produce the Secular Variations in the Form and Position of the Orbit in its own Plane—Perpendicular Force the cause of Periodic Perturbations in Latitude, and Secular Variations in the Position of the Orbit with regard to the Plane of the Ecliptic—Mean Motion and Major Axis Invariable—Stability of System—Effects of a Resisting Medium—Invariable Plane of the Solar System and of the Universe—Great Inequality of Jupiter and Saturn.
The planets are subject to disturbances of two kinds, both resulting from the constant operation of their reciprocal attraction: one kind, depending upon their positions with regard to each other, begins from zero, increases to a maximum, decreases, and becomes zero again, when the planets return to the same relative positions. In consequence of these, the disturbed planet is sometimes drawn away from the sun, sometimes brought nearer to him: sometimes it is accelerated in its motion, and sometimes retarded. At one time it is drawn above the plane of its orbit, at another time below it, according to the position of the disturbing body. All such changes, being accomplished in short periods, some in a few months, others in years, or in hundreds of years, are denominated periodic inequalities. The inequalities of the other kind, though occasioned likewise by the disturbing energy of the planets, are entirely independent of their relative positions. They depend upon the relative positions of the orbits alone, whose forms and places in space are altered by very minute quantities, in immense periods of time, and are therefore called secular inequalities.
The periodical perturbations are compensated when the bodies return to the same relative positions with regard to one another and to the sun: the secular inequalities are compensated when the orbits return to the same positions relatively to one another and to the plane of the ecliptic.
Planetary motion, including both these kinds of disturbance, may be represented by a body revolving in an ellipse, and making small and transient deviations, now on one side of its path, and now on the other, whilst the ellipse itself is slowly, but perpetually, changing both in form and position.
The periodic inequalities are merely transient deviations of a planet from its path, the most remarkable of which only lasts about 918 years; but, in consequence of the secular disturbances, the apsides, or extremities of the major axes of all the orbits, have a direct but variable motion in space, excepting those of the orbit of Venus, which are retrograde ([N. 61]), and the lines of the nodes move with a variable velocity in a contrary direction. Besides these, the inclination and excentricity of every orbit are in a state of perpetual but slow change. These effects result from the disturbing action of all the planets on each. But, as it is only necessary to estimate the disturbing influence of one body at a time, what follows may convey some idea of the manner in which one planet disturbs the elliptical motion of another.
Suppose two planets moving in ellipses round the sun; if one of them attracted the other and the sun with equal intensity, and in parallel directions ([N. 62]), it would have no effect in disturbing the elliptical motion. The inequality of this attraction is the sole cause of perturbation, and the difference between the disturbing planet’s action on the sun and on the disturbed planet constitutes the disturbing force, which consequently varies in intensity and direction with every change in the relative positions of the three bodies. Although both the sun and planet are under the influence of the disturbing force, the motion of the disturbed planet is referred to the centre of the sun as a fixed point, for convenience. The whole force ([N. 63]) which disturbs a planet is equivalent to three partial forces. One of these acts on the disturbed planet, in the direction of a tangent to its orbit, and is called the tangential force: it occasions secular inequalities in the form and position of the orbit in its own plane, and is the sole cause of the periodical perturbations in the planet’s longitude. Another acts upon the same body in the direction of its radius vector, that is, in the line joining the centres of the sun and planet, and is called the radial force: it produces periodical changes in the distance of the planet from the sun, and affects the form and position of the orbit in its own plane. The third, which may be called the perpendicular force, acts at right angles to the plane of the orbit, occasions the periodic inequalities in the planet’s latitude, and affects the position of the orbit with regard to the plane of the ecliptic.
It has been observed, that the radius vector of a planet, moving in a perfectly elliptical orbit, passes over equal spaces or areas in equal times; a circumstance which is independent of the law of the force, and would be the same whether it varied inversely as the square of the distance, or not, provided only that it be directed to the centre of the sun. Hence the tangential force, not being directed to the centre, occasions an unequable description of areas, or, what is the same thing, it disturbs the motion of the planet in longitude. The tangential force sometimes accelerates the planet’s motion, sometimes retards it, and occasionally has no effect at all. Were the orbits of both planets circular, a complete compensation would take place at each revolution of the two planets, because the arcs in which the accelerations and retardations take place would be symmetrical on each side of the disturbing force. For it is clear, that if the motion be accelerated through a certain space, and then retarded through as much, the motion at the end of the time will be the same as if no change had taken place. But, as the orbits of the planets are ellipses, this symmetry does not hold: for, as the planet moves unequably in its orbit, it is in some positions more directly, and for a longer time, under the influence of the disturbing force than in others. And, although multitudes of variations do compensate each other in short periods, there are others, depending on peculiar relations among the periodic times of the planets, which do not compensate each other till after one, or even till after many revolutions of both bodies. A periodical inequality of this kind in the motions of Jupiter and Saturn has a period of no less than 918 years.
The radial force, or that part of the disturbing force which acts in the direction of the line joining the centres of the sun and disturbed planet, has no effect on the areas, but is the cause of periodical changes of small extent in the distance of the planet from the sun. It has already been shown, that the force producing perfectly elliptical motion varies inversely as the square of the distance, and that a force following any other law would cause the body to move in a curve of a very different kind. Now, the radial disturbing force varies directly as the distance; and, as it sometimes combines with and increases the intensity of the sun’s attraction for the disturbed body, and at other times opposes and consequently diminishes it, in both cases it causes the sun’s attraction to deviate from the exact law of gravity, and the whole action of this compound central force on the disturbed body is either greater or less than what is requisite for perfectly elliptical motion. When greater, the curvature of the disturbed planet’s path, on leaving its perihelion ([N. 64]), or point nearest the sun, is greater than it would be in the ellipse, which brings the planet to its aphelion ([N. 65]), or point farthest from the sun, before it has passed through 180°, as it would do if undisturbed. So that in this case the apsides, or extremities of the major axis, advance in space. When the central force is less than the law of gravity requires, the curvature of the planet’s path is less than the curvature of the ellipse. So that the planet, on leaving its perihelion, would pass through more than 180° before arriving at its aphelion, which causes the apsides to recede in space ([N. 66]). Cases both of advance and recess occur during a revolution of the two planets; but those in which the apsides advance preponderate. This, however, is not the full amount of the motion of the apsides; part arises also from the tangential force ([N. 63]), which alternately accelerates and retards the velocity of the disturbed planet. An increase in the planet’s tangential velocity diminishes the curvature of its orbit, and is equivalent to a decrease of central force. On the contrary, a decrease of the tangential velocity, which increases the curvature of the orbit, is equivalent to an increase of central force. These fluctuations, owing to the tangential force, occasion an alternate recess and advance of the apsides, after the manner already explained ([N. 66]). An uncompensated portion of the direct motion, arising from this cause, conspires with that already impressed by the radial force, and in some cases even nearly doubles the direct motion of these points. The motion of the apsides may be represented by supposing a planet to move in an ellipse, while the ellipse itself is slowly revolving about the sun in the same plane ([N. 67]). This motion of the major axis, which is direct in all the orbits except that of the planet Venus, is irregular, and so slow that it requires more than 109,830 years for the major axis of the earth’s orbit to accomplish a sidereal revolution ([N. 68]), that is, to return to the same stars; and 20,984 years to complete its tropical revolution ([N. 69]), or to return to the same equinox. The difference between these two periods arises from a retrograde motion in the equinoctial point, which meets the advancing axis before it has completed its revolution with regard to the stars. The major axis of Jupiter’s orbit requires no less than 200,610 years to perform its sidereal revolution, and 22,748 years to accomplish its tropical revolution from the disturbing action of Saturn alone.
A variation in the excentricity of the disturbed planet’s orbit is an immediate consequence of the deviation from elliptical curvature, caused by the action of the disturbing force. When the path of the body, in proceeding from its perihelion to its aphelion, is more curved than it ought to be from the effect of the disturbing forces, it falls within the elliptical orbit, the excentricity is diminished, and the orbit becomes more nearly circular; when that curvature is less than it ought to be, the path of the planet falls without its elliptical orbit ([N. 66]), and the excentricity is increased; during these changes, the length of the major axis is not altered, the orbit only bulges out, or becomes more flat ([N. 70]). Thus the variation in the excentricity arises from the same cause that occasions the motion of the apsides ([N. 67]). There is an inseparable connection between these two elements: they vary simultaneously, and have the same period; so that, whilst the major axis revolves in an immense period of time, the excentricity increases and decreases by very small quantities, and at length returns to its original magnitude at each revolution of the apsides. The terrestrial excentricity is decreasing at the rate of about 40 miles annually; and, if it were to decrease equably, it would be 39,861 years before the earth’s orbit became a circle. The mutual action of Jupiter and Saturn occasions variations in the excentricity of both orbits, the greatest excentricity of Jupiter’s orbit corresponding to the least of Saturn’s. The period in which these vicissitudes are accomplished is 70,414 years, estimating the action of these two planets alone; but, if the action of all the planets were estimated, the cycle would extend to millions of years.
That part of the disturbing force is now to be considered which acts perpendicularly to the plane of the orbit, causing periodic perturbations in latitude, secular variations in the inclination of the orbit, and a retrograde motion to its nodes on the true plane of the ecliptic ([N. 71]). This force tends to pull the disturbed body above, or push ([N. 72]) it below, the plane of its orbit, according to the relative positions of the two planets with regard to the sun, considered to be fixed. By this action, it sometimes makes the plane of the orbit of the disturbed body tend to coincide with the plane of the ecliptic, and sometimes increases its inclination to that plane. In consequence of which, its nodes alternately recede or advance on the ecliptic ([N. 73]). When the disturbing planet is in the line of the disturbed planet’s nodes ([N. 74]), it neither affects these points, the latitude, nor the inclination, because both planets are then in the same plane. When it is at right angles to the line of the nodes, and the orbit symmetrical on each side of the disturbing force, the average motion of these points, after a revolution of the disturbed body, is retrograde, and comparatively rapid: but, when the disturbing planet is so situated that the orbit of the disturbed planet is not symmetrical on each side of the disturbing force, which is most frequently the case, every possible variety of action takes place. Consequently, the nodes are perpetually advancing or receding with unequal velocity; but, as a compensation is not effected, their motion is, on the whole, retrograde.
With regard to the variations in the inclination, it is clear, that, when the orbit is symmetrical on each side of the disturbing force, all its variations are compensated after a revolution of the disturbed body, and are merely periodical perturbations in the planet’s latitude; and no secular change is induced in the inclination of the orbit. When, on the contrary, that orbit is not symmetrical on each side of the disturbing force, although many of the variations in latitude are transient or periodical, still, after a complete revolution of the disturbed body, a portion remains uncompensated, which forms a secular change in the inclination of the orbit to the plane of the ecliptic. It is true, part of this secular change in the inclination is compensated by the revolution of the disturbing body, whose motion has not hitherto been taken into the account, so that perturbation compensates perturbation; but still a comparatively permanent change is effected in the inclination, which is not compensated till the nodes have accomplished a complete revolution.
The changes in the inclination are extremely minute ([N. 75]), compared with the motion of the nodes, and there is the same kind of inseparable connection between their secular changes that there is between the variation of the excentricity and the motion of the major axis. The nodes and inclinations vary simultaneously; their periods are the same, and very great. The nodes of Jupiter’s orbit, from the action of Saturn alone, require 36,261 years to accomplish even a tropical revolution. In what precedes, the influence of only one disturbing body has been considered; but, when the action and reaction of the whole system are taken into account, every planet is acted upon, and does itself act, in this manner, on all the others; and the joint effect keeps the inclinations and excentricities in a state of perpetual variation. It makes the major axes of all the orbits continually revolve, and causes, on an average, a retrograde motion of the nodes of each orbit upon every other. The ecliptic ([N. 71]) itself is in motion from the mutual action of the earth and planets, so that the whole is a compound phenomenon of great complexity, extending through unknown ages. At the present time the inclinations of all the orbits are decreasing, but so slowly, that the inclination of Jupiter’s orbit is only about six minutes less than it was in the age of Ptolemy.
But, in the midst of all these vicissitudes, the length of the major axes and the mean motions of the planets remain permanently independent of secular changes. They are so connected by Kepler’s law, of the squares of the periodic times being proportional to the cubes of the mean distances of the planets from the sun, that one cannot vary without affecting the other. And it is proved, that any variations which do take place are transient, and depend only on the relative positions of the bodies.
It is true that, according to theory, the radial disturbing force should permanently alter the dimensions of all the orbits, and the periodic times of all the planets, to a certain degree. For example, the masses of all the planets revolving within the orbit of any one, such as Mars, by adding to the interior mass, increase the attracting force of the sun, which, therefore, must contract the dimensions of the orbit of that planet, and diminish its periodic time; whilst the planets exterior to Mars’s orbit must have the contrary effect. But the mass of the whole of the planets and satellites taken together is so small, when compared with that of the sun, that these effects are quite insensible, and could only have been discovered by theory. And, as it is certain that the length of the major axes and the mean motions are not permanently changed by any other power whatever, it may be concluded that they are invariable.
With the exception of these two elements, it appears that all the bodies are in motion, and every orbit in a state of perpetual change. Minute as these changes are, they might be supposed to accumulate in the course of ages, sufficiently to derange the whole order of nature, to alter the relative positions of the planets, to put an end to the vicissitudes of the seasons, and to bring about collisions which would involve our whole system, now so harmonious, in chaotic confusion. It is natural to inquire, what proof exists that nature will be preserved from such a catastrophe? Nothing can be known from observation, since the existence of the human race has occupied comparatively but a point in duration, while these vicissitudes embrace myriads of ages. The proof is simple and conclusive. All the variations of the solar system, secular as well as periodic, are expressed analytically by the sines and cosines of circular arcs ([N. 76]), which increase with the time; and, as a sine or cosine can never exceed the radius, but must oscillate between zero and unity, however much the time may increase, it follows that when the variations have accumulated to a maximum by slow changes, in however long a time, they decrease, by the same slow degrees, till they arrive at their smallest value, again to begin a new course; thus for ever oscillating about a mean value. This circumstance, however, would be insufficient, were it not for the small excentricities of the planetary orbits, their minute inclinations to the plane of the ecliptic, and the revolutions of all the bodies, as well planets as satellites, in the same direction. These secure the perpetual stability of the solar system ([N. 77]). However, at the time that the stability was proved by La Grange and La Place, the telescopic planets between Mars and Jupiter had not been discovered; but La Grange, having investigated the subject under a very general point of view, showed that, if a planetary system be composed of very unequal masses, the whole of the larger would maintain an unalterable stability with regard to the form and position of their orbits, while the orbits of the lesser might undergo unlimited changes. M. Le Verrier has applied this to the solar system, and has found that the orbits of all the larger planets will for ever maintain an unalterable stability in form and position; for, though liable to mutations of very long periods, they return again exactly to what they originally were, oscillating between very narrow limits; but he found a zone of instability between the orbit of Mars, and twice the mean distance of the earth from the sun,[[1]] or between 1·5 and 2·00; therefore the position and form of the orbits of such of the telescopic planets as revolve within that zone will be subject to unlimited variations. But the orbits of those more remote from the sun than Flora, or beyond 2·20, will be stable, so that their excentricities and inclinations must always have been, and will always remain, very great, since they must have depended upon the primitive conditions that prevailed when these planetary atoms were launched into space. The 51st of these small bodies, which was discovered, and the elements of its orbit determined, by M. Valz, at Nimes, has a mean distance of 1·88; so it revolves within the zone of instability. It has a shorter periodic time than any of those previously discovered, and a greater excentricity, with the exception of Nysa. Its orbit cuts that of Mars, and comes nearer to the earth than the orbits of either Mars or Venus, a circumstance which would be favourable for correcting the parallax of the sun, or confirming its accuracy. The telescopic planets, numerous as they are, have no influence on the motions of the larger planets, for Jupiter has a diameter of 90,734 miles, while that of Pallas, his nearest neighbour, is only 97 miles, little more than the distance from London to Bath. The diameter of Mars, on the other side of the small planets, is 4546 miles, and that of the earth 79251⁄2 miles, so that the telescopic group are too minute to disturb the others. M. Le Verrier found another zone of instability between Venus and the sun, on the border of which Mercury is revolving, the inclination of whose orbit to the plane of the ecliptic is about 7°, which is more than that of any of the large planets. Neptune’s orbit is, no doubt, as stable as that of any other of the large planets, as the inclination is very small, but he will have periodical variations of very long duration from the reciprocal attraction between him and Uranus, one especially of an enormous duration, similar to those of Jupiter and Saturn, and, like them, depending on the time of his revolution round the sun, being nearly twice as long as that of Saturn. Mr. Adams has computed that Neptune produces a periodical perturbation in the motion of Uranus, whose duration is about 6800 years.
The equilibrium of the system, however, would be deranged if the planets moved in a resisting medium ([N. 78]) sufficiently dense to diminish their tangential velocity, for then both the excentricities and the major axes of the orbits would vary with the time, so that the stability of the system would be ultimately destroyed. The existence of an ethereal medium is now proved; and, although it is so extremely rare that hitherto its effects on the motions of the planets have been altogether insensible, there can be no doubt that, in the immensity of time, it will modify the forms of the planetary orbits, and may at last even cause the destruction of our system, which in itself contains no principle of decay, unless a rotatory motion from west to east has been given to this medium by the bodies of the solar system, which have all been revolving about the sun in that direction for unknown ages. This rotation, which seems to be highly probable, may even have been coeval with its creation. Such a vortex would have no effect on bodies moving with it, but it would influence the motions of those revolving in a contrary direction. It is possible that the disturbances experienced by comets, which have already revealed the existence of this medium, may also, in time, disclose its rotatory motion.
The form and position of the planetary orbits, and the motion of the bodies in the same direction, together with the periodicity of the terms in which the inequalities are expressed, assure us that the variations of the system are confined within very narrow limits, and that, although we do not know the extent of the limits, nor the period of that grand cycle which probably embraces millions of years, yet they never will exceed what is requisite for the stability and harmony of the whole; for the preservation of which every circumstance is so beautifully and wonderfully adapted.
The plane of the ecliptic itself, though assumed to be fixed at a given epoch for the convenience of astronomical computation, is subject to a minute secular variation of 45ʺ·7, occasioned by the reciprocal action of the planets. But, as this is also periodical, and cannot exceed 2° 42ʹ, the terrestrial equator, which is inclined to it at an angle[[2]] of 23° 27ʹ 28ʺ·29, will never coincide with the plane of the ecliptic: so there never can be perpetual spring ([N. 79]). The rotation of the earth is uniform; therefore day and night, summer and winter, will continue their vicissitudes while the system endures, or is undisturbed by foreign causes.
Yonder starry sphere
Of planets and of fix’d, in all her wheels,
Resembles nearest mazes intricate,
Eccentric, intervolved, yet regular,
Then most, when most irregular they seem.
The stability of our system was established by La Grange: “a discovery,” says Professor Playfair, “that must render the name for ever memorable in science, and revered by those who delight in the contemplation of whatever is excellent and sublime.” After Newton’s discovery of the mechanical laws of the elliptical orbits of the planets, that of their periodical inequalities, by La Grange, is, without doubt, the noblest truth in the mechanism of the heavens; and, in respect of the doctrine of final causes, it may be regarded as the greatest of all.
Notwithstanding the permanency of our system, the secular variations in the planetary orbits would have been extremely embarrassing to astronomers when it became necessary to compare observations separated by long periods. The difficulty was in part obviated, and the principle for accomplishing it established, by La Place, and has since been extended by M. Poinsot. It appears that there exists an invariable plane ([N. 80]), passing through the centre of gravity of the system, about which the whole oscillates within very narrow limits, and that this plane will always remain parallel to itself, whatever changes time may induce in the orbits of the planets, in the plane of the ecliptic, or even in the law of gravitation; provided only that our system remains unconnected with any other. The position of the plane is determined by this property—that, if each particle in the system be multiplied by the area described upon this plane in a given time, by the projection of its radius vector about the common centre of gravity of the whole, the sum of all these products will be a maximum ([N. 81]). La Place found that the plane in question is inclined to the ecliptic at an angle of nearly 1° 34ʹ 15ʺ, and that, in passing through the sun, and about midway between the orbits of Jupiter and Saturn, it may be regarded as the equator of the solar system, dividing it into two parts, which balance one another in all their motions. This plane of greatest inertia, by no means peculiar to the solar system, but existing in every system of bodies submitted to their mutual attractions only, always maintains a fixed position, whence the oscillations of the system may be estimated through unlimited time. Future astronomers will know, from its immutability or variation, whether the sun and his attendants are connected or not with the other systems of the universe. Should there be no link between them, it may be inferred, from the rotation of the sun, that the centre of gravity ([N. 82]) of the system situate within his mass describes a straight line in this invariable plane or great equator of the solar system, which, unaffected by the changes of time, will maintain its stability through endless ages. But, if the fixed stars, comets, or any unknown and unseen bodies, affect our sun and planets, the nodes of this plane will slowly recede on the plane of that immense orbit which the sun may describe about some most distant centre, in a period which it transcends the power of man to determine. There is every reason to believe that this is the case; for it is more than probable that, remote as the fixed stars are, they in some degree influence our system, and that even the invariability of this plane is relative, only appearing fixed to creatures incapable of estimating its minute and slow changes during the small extent of time and space granted to the human race. “The development of such changes,” as M. Poinsot justly observes, “is similar to an enormous curve, of which we see so small an arc that we imagine it to be a straight line.” If we raise our views to the whole extent of the universe, and consider the stars, together with the sun, to be wandering bodies, revolving about the common centre of creation, we may then recognise in the equatorial plane passing through the centre of gravity of the universe the only instance of absolute and eternal repose.
All the periodic and secular inequalities deduced from the law of gravitation are so perfectly confirmed by observation, that analysis has become one of the most certain means of discovering the planetary irregularities, either when they are too small, or too long in their periods, to be detected by other methods. Jupiter and Saturn, however, exhibit inequalities which for a long time seemed discordant with that law. All observations, from those of the Chinese and Arabs down to the present day, prove that for ages the mean motions of Jupiter and Saturn have been affected by a great inequality of a very long period, forming an apparent anomaly in the theory of the planets. It was long known by observation that five times the mean motion of Saturn is nearly equal to twice that of Jupiter; a relation which the sagacity of La Place perceived to be the cause of a periodic irregularity in the mean motion of each of these planets, which completes its period in nearly 918 years, the one being retarded while the other is accelerated; but both the magnitude and period of these quantities vary, in consequence of the secular variations in the elements of the orbits. Suppose the two planets to be on the same side of the sun, and all three in the same straight line, they are then said to be in conjunction ([N. 83]). Now, if they begin to move at the same time, one making exactly five revolutions in its orbit while the other only accomplishes two, it is clear that Saturn, the slow-moving body, will only have got through a part of its orbit during the time that Jupiter has made one whole revolution and part of another, before they be again in conjunction. It is found that during this time their mutual action is such as to produce a great many perturbations which compensate each other, but that there still remains a portion outstanding, owing to the length of time during which the forces act in the same manner; and, if the conjunction always happened in the same point of the orbit, this uncompensated inequality in the mean motion would go on increasing till the periodic times and forms of the orbits were completely and permanently changed: a case that would actually take place if Jupiter accomplished exactly five revolutions in the time Saturn performed two. These revolutions are, however, not exactly commensurable; the points in which the conjunctions take place are in advance each time as much as 8°·37; so that the conjunctions do not happen exactly in the same points of the orbits till after a period of 850 years; and, in consequence of this small advance, the planets are brought into such relative positions, that the inequality, which seemed to threaten the stability of the system, is completely compensated, and the bodies, having returned to the same relative positions with regard to one another and the sun, begin a new course. The secular variations in the elements of the orbit increase the period of the inequality to 918 years ([N. 84]). As any perturbation which affects the mean motion affects also the major axis, the disturbing forces tend to diminish the major axis of Jupiter’s orbit, and increase that of Saturn’s, during one half of the period, and the contrary during the other half. This inequality is strictly periodical, since it depends upon the configuration ([N. 85]) of the two planets; and theory is confirmed by observation, which shows that, in the course of twenty centuries, Jupiter’s mean motion has been accelerated by about 3° 23ʹ, and Saturn’s retarded by 5° 13ʹ. Several instances of perturbations of this kind occur in the solar system. One, in the mean motions of the Earth and Venus, only amounting to a few seconds, has been recently worked out with immense labour by Professor Airy. It accomplishes its changes in 240 years, and arises from the circumstance of thirteen times the periodic time of Venus being nearly equal to eight times that of the Earth. Small as it is, it is sensible in the motions of the Earth.
It might be imagined that the reciprocal action of such planets as have satellites would be different from the influence of those that have none. But the distances of the satellites from their primaries are incomparably less than the distances of the planets from the sun, and from one another. So that the system of a planet and its satellites moves nearly as if all these bodies were united in their common centre of gravity. The action of the sun, however, in some degree disturbs the motion of the satellites about their primary.
SECTION IV.
Theory of Jupiter’s Satellites—Effects of the Figure of Jupiter upon his Satellites—Position of their Orbits—Singular Laws among the Motions of the first Three Satellites—Eclipses of the Satellites—Velocity of Light—Aberration—Ethereal Medium—Satellites of Saturn and Uranus.
The changes which take place in the planetary system are exhibited on a smaller scale by Jupiter and his satellites; and, as the period requisite for the development of the inequalities of these moons only extends to a few centuries, it may be regarded as an epitome of that grand cycle which will not be accomplished by the planets in myriads of ages. The revolutions of the satellites about Jupiter are precisely similar to those of the planets about the sun; it is true they are disturbed by the sun, but his distance is so great, that their motions are nearly the same as if they were not under his influence. The satellites, like the planets, were probably projected in elliptical orbits: but, as the masses of the satellites are nearly 100,000 times less than that of Jupiter; and as the compression of Jupiter’s spheroid is so great, in consequence of his rapid rotation, that his equatorial diameter exceeds his polar diameter by no less than 6000 miles; the immense quantity of prominent matter at his equator must soon have given the circular form observed in the orbits of the first and second satellites, which its superior attraction will always maintain. The third and fourth satellites, being farther removed from its influence, revolve in orbits with a very small excentricity. And, although the first two sensibly move in circles, their orbits acquire a small ellipticity, from the disturbances they experience ([N. 86]).
It has been stated, that the attraction of a sphere on an exterior body is the same as if its mass were united in one particle in its centre of gravity, and therefore inversely as the square of the distance. In a spheroid, however, there is an additional force arising from the bulging mass at its equator, which, not following the exact law of gravity, acts as a disturbing force. One effect of this disturbing force in the spheroid of Jupiter is to occasion a direct motion in the greater axes of the orbits of all his satellites, which is more rapid the nearer the satellite is to the planet, and very much greater than that part of their motion which arises from the disturbing action of the sun. The same cause occasions the orbits of the satellites to remain nearly in the plane of Jupiter’s equator ([N. 87]), on account of which the satellites are always seen nearly in the same line ([N. 88]); and the powerful action of that quantity of prominent matter is the reason why the motions of the nodes of these small bodies are so much more rapid than those of the planet. The nodes of the fourth satellite accomplish a tropical revolution in 531 years, while those of Jupiter’s orbit require no less than 36,261 years;—a proof of the reciprocal attraction between each particle of Jupiter’s equator and of the satellites. In fact, if the satellites moved exactly in the plane of Jupiter’s equator, they would not be pulled out of that plane, because his attraction would be equal on both sides of it. But, as their orbits have a small inclination to the plane of the planet’s equator, there is a want of symmetry, and the action of the protuberant matter tends to make the nodes regress by pulling the satellites above or below the planes of their orbits; an action which is so great on the interior satellites, that the motions of their nodes are nearly the same as if no other disturbing force existed.
The orbits of the satellites do not retain a permanent inclination, either to the plane of Jupiter’s equator, or to that of his orbit, but to certain planes passing between the two, and through their intersection. These have a greater inclination to his equator the farther the satellite is removed, owing to the influence of Jupiter’s compression; and they have a slow motion corresponding to secular variations in the planes of Jupiter’s orbit and equator.
The satellites are not only subject to periodic and secular inequalities from their mutual attraction, similar to those which affect the motions and orbits of the planets, but also to others peculiar to themselves. Of the periodic inequalities arising from their mutual attraction the most remarkable take place in the angular motions ([N. 89]) of the three nearest to Jupiter, the second of which receives from the first a perturbation similar to that which it produces in the third; and it experiences from the third a perturbation similar to that which it communicates to the first. In the eclipses these two inequalities are combined into one, whose period is 437·659 days. The variations peculiar to the satellites arise from the secular inequalities occasioned by the action of the planets in the form and position of Jupiter’s orbit, and from the displacement of his equator. It is obvious that whatever alters the relative positions of the sun, Jupiter, and his satellites, must occasion a change in the directions and intensities of the forces, which will affect the motions and orbits of the satellites. For this reason the secular variations in the excentricity of Jupiter’s orbit occasion secular inequalities in the mean motions of the satellites, and in the motions of the nodes and apsides of their orbits. The displacement of the orbit of Jupiter, and the variation in the position of his equator, also affect these small bodies ([N. 90]). The plane of Jupiter’s equator is inclined to the plane of his orbit at an angle of 3° 5ʹ 30ʺ, so that the action of the sun and of the satellites themselves produces a nutation and precession ([N. 91]) in his equator, precisely similar to that which takes place in the rotation of the earth, from the action of the sun and moon. Hence the protuberant matter at Jupiter’s equator is continually changing its position with regard to the satellites, and produces corresponding mutations in their motions. And, as the cause must be proportional to the effect, these inequalities afford the means, not only of ascertaining the compression of Jupiter’s spheroid, but they prove that his mass is not homogeneous. Although the apparent diameters of the satellites are too small to be measured, yet their perturbations give the values of their masses with considerable accuracy—a striking proof of the power of analysis.
A singular law obtains among the mean motions and mean longitudes of the first three satellites. It appears from observation that the mean motion of the first satellite, plus twice that of the third, is equal to three times that of the second; and that the mean longitude of the first satellite, minus three times that of the second, plus twice that of the third, is always equal to two right angles. It is proved by theory, that, if these relations had only been approximate when the satellites were first launched into space, their mutual attractions would have established and maintained them, notwithstanding the secular inequalities to which they are liable. They extend to the synodic motions ([N. 92]) of the satellites; consequently they affect their eclipses, and have a very great influence on their whole theory. The satellites move so nearly in the plane of Jupiter’s equator, which has a very small inclination to his orbit, that the first three are eclipsed at each revolution by the shadow of the planet, which is much larger than the shadow of the moon: the fourth satellite is not eclipsed so frequently as the others. The eclipses take place close to the disc of Jupiter when he is near opposition ([N. 93]); but at times his shadow is so projected with regard to the earth, that the third and fourth satellites vanish and reappear on the same side of the disc ([N. 94]). These eclipses are in all respects similar to those of the moon: but, occasionally, the satellites eclipse Jupiter, sometimes passing like obscure spots across his surface, resembling annular eclipses of the sun, and sometimes like a bright spot traversing one of his dark belts. Before opposition, the shadow of the satellite, like a round black spot, precedes its passage over the disc of the planet; and, after opposition, the shadow follows the satellite.
In consequence of the relations already mentioned in the mean motions and mean longitudes of the first three satellites, they never can be all eclipsed at the same time: for, when the second and third are in one direction, the first is in the opposite direction; consequently, when the first is eclipsed, the other two must be between the sun and Jupiter. The instant of the beginning or end of an eclipse of a satellite marks the same instant of absolute time to all the inhabitants of the earth; therefore, the time of these eclipses observed by a traveller, when compared with the time of the eclipse computed for Greenwich, or any other fixed meridian ([N. 95]), gives the difference of the meridians in time, and, consequently, the longitude of the place of observation. The longitude is determined with extreme precision whenever it is possible to convey the time instantaneously by means of electricity from one place to another, since it obviates the errors of clocks and chronometers. The eclipses of Jupiter’s satellites have been the means of a discovery which, though not so immediately applicable to the wants of man, unfolds one of the properties of light—that medium without whose cheering influence all the beauties of the creation would have been to us a blank. It is observed, that those eclipses of the first satellite which happen when Jupiter is near conjunction ([N. 96]), are later by 16ʹ 26ʺ·6 than those which take place when the planet is in opposition. As Jupiter is nearer to us when in opposition by the whole breadth of the earth’s orbit than when in conjunction, this circumstance is to be attributed to the time employed by the rays of light in crossing the earth’s orbit, a distance of about 190,000,000 of miles; whence it is estimated that light travels at the rate of 192,000 miles in one second. Such is its velocity, that the earth, moving at the rate of nineteen miles in a second, would take two months to pass through a distance which a ray of light would dart over in eight minutes. The subsequent discovery of the aberration of light has fully confirmed this astonishing result.
Objects appear to be situate in the direction of the rays which proceed from them. Were light propagated instantaneously, every object, whether at rest or in motion, would appear in the direction of these rays; but, as light takes some time to travel, we see Jupiter in conjunction, by means of rays that left him 16m 26s·6 before; but, during that time, we have changed our position, in consequence of the motion of the earth in its orbit: we therefore refer Jupiter to a place in which he is not. His true position is in the diagonal ([N. 97]) of the parallelogram, whose sides are in the ratio of the velocity of light to the velocity of the earth in its orbit, which is as 192,000 to 19, or nearly as 10,000 to 1. In consequence of the aberration of light, the heavenly bodies seem to be in places in which they are not. In fact, if the earth were at rest, rays from a star would pass along the axis of a telescope directed to it; but, if the earth were to begin to move in its orbit with its usual velocity, these rays would strike against the side of the tube; it would, therefore, be necessary to incline the telescope a little, in order to see the star. The angle contained between the axis of the telescope and a line drawn to the true place of the star is its aberration, which varies in quantity and direction in different parts of the earth’s orbit; but, as it is only 20ʺ·481, it is insensible in ordinary cases ([N. 98]).
The velocity of light deduced from the observed aberration of the fixed stars perfectly corresponds with that given by the eclipses of the first satellite. The same result, obtained from sources so different, leaves not a doubt of its truth. Many such beautiful coincidences, derived from circumstances apparently the most unpromising and dissimilar, occur in physical astronomy, and prove connections which we might otherwise be unable to trace. The identity of the velocity of light, at the distance of Jupiter, and on the earth’s surface, shows that its velocity is uniform; and as light consists in the vibrations of an elastic medium or ether filling space, the uniformity of its velocity shows that the density of the medium throughout the whole extent of the solar system must be proportional to its elasticity ([N. 99]). Among the fortunate conjectures which have been confirmed by subsequent experience, that of Bacon is not the least remarkable, “It produces in me,” says the restorer of true philosophy, “a doubt whether the face of the serene and starry heavens be seen at the instant it really exists, or not till some time later: and whether there be not, with respect to the heavenly bodies, a true time and an apparent time, no less than a true place and an apparent place, as astronomers say, on account of parallax. For it seems incredible that the species or rays of the celestial bodies can pass through the immense interval between them and us in an instant, or that they do not even require some considerable portion of time.”
Great discoveries generally lead to a variety of conclusions: the aberration of light affords a direct proof of the motion of the earth in its orbit; and its rotation is proved by the theory of falling bodies, since the centrifugal force it induces retards the oscillations of the pendulum ([N. 100]) in going from the pole to the equator. Thus a high degree of scientific knowledge has been requisite to dispel the errors of the senses ([N. 237]).
The little that is known of the theories of the satellites of Saturn and Uranus is, in all respects, similar to that of Jupiter. Saturn is accompanied by eight satellites. The seventh is about the size of Mars, and the eighth was simultaneously discovered by Mr. Bond in America, and that distinguished astronomer Mr. Lassell, of Liverpool. The orbits of the two last have a sensible inclination to the plane of the ring; but the great compression of Saturn occasions the other satellites to move nearly in the plane of his equator. So many circumstances must concur to render the two interior satellites visible, that they have very rarely been seen. They move exactly at the edge of the ring, and their orbits never deviate from its plane. In 1789 Sir William Herschel saw them like beads, threading the slender line of light which the ring is reduced to when seen edgewise from the earth. And for a short time he perceived them advancing off it at each end, when turning round in their orbits. The eclipses of the exterior satellites only take place when the ring is in this position. Mr. Lassell, with a powerful telescope, made by himself, has seen Iapetus, the nearest of the two, on several occasions, even when the opening of the ring was very wide, which made it extremely difficult to see so minute an object. Of the situation of the equator of Uranus we know nothing, nor of his compression; but the orbits of his satellites are nearly perpendicular to the plane of the ecliptic; and, by analogy, they ought to be in the plane of his equator. Uranus is so remote that he has more the appearance of a planetary nebula than a planet, which renders it extremely difficult to distinguish the satellites at all; and quite hopeless without such a telescope as is rarely to be met with even in observatories. Sir William Herschel discovered the two that are farthest from the planet, and ascertained their approximate periods, which his son afterwards determined to be 13d 11h 7m 12s·6 and 8d 16h 56m 28s·6 respectively. The orbits of both seem to have an inclination of about 101°·2 to the plane of the ecliptic. The two interior satellites are so faint and small, and so near the edge of the planet, that they can with difficulty be seen even under the most favourable circumstances: however, Mr. Lassell has ascertained that the more distant of the two revolves about Uranus in 4 days, and that nearest to the planet in 21⁄2 days, and from a long and minute examination he is convinced that the system only consists of four satellites. Soon after Neptune was seen Mr. Lassell discovered the only satellite known to belong to that planet. The satellites of Uranus and Neptune, the two planets on the remotest verge of the solar system, offer the singular and only instance of a revolution from east to west, while all the planets and all the other satellites revolve from west to east. Retrograde motion is occasionally met with in the comets and double stars.
SECTION V.
Lunar Theory—Periodic Perturbations of the Moon—Equation of Centre—Evection—Variation—Annual Equation—Direct and Indirect Action of Planets—The Moon’s Action on the Earth disturbs her own Motion—Excentricity and Inclination of Lunar Orbit invariable—Acceleration—Secular Variation in Nodes and Perigee—Motion of Nodes and Perigee inseparably connected with the Acceleration—Nutation of Lunar Orbit—Form and Internal Structure of the Earth determined from it—Lunar, Solar, and Planetary Eclipses—Occultations and Lunar Distances—Mean Distance of the Sun from the Earth obtained from Lunar Theory—Absolute Distances of the Planets, how found.
Our constant companion, the moon, next claims our attention. Several circumstances concur to render her motions the most interesting, and at the same time the most difficult to investigate, of all the bodies of our system. In the solar system, planet troubles planet; but, in the lunar theory, the sun is the great disturbing cause, his vast distance being compensated by his enormous magnitude, so that the motions of the moon are more irregular than those of the planets; and, on account of the great ellipticity of her orbit, and the size of the sun, the approximations to her motions are tedious and difficult, beyond what those unaccustomed to such investigations could imagine. The average distance of the moon from the centre of the earth is only 238,793 miles, so that her motion among the stars is perceptible in a few hours. She completes a circuit of the heavens in 27d 7h 43m 11s·5, moving in an orbit whose excentricity is about 12,985 miles. The moon is about four hundred times nearer to the earth than the sun. The proximity of the moon to the earth keeps them together. For so great is the attraction of the sun, that, if the moon were farther from the earth, she would leave it altogether, and would revolve as an independent planet about the sun.
The disturbing action ([N. 101]) of the sun on the moon is equivalent to three forces. The first, acting in the direction of the line joining the moon and earth, increases or diminishes her gravity to the earth. The second, acting in the direction of a tangent to her orbit, disturbs her motion in longitude. And the third, acting perpendicularly to the plane of her orbit, disturbs her motion in latitude; that is, it brings her nearer to, or removes her farther from, the plane of the ecliptic than she would otherwise be. The periodic perturbations in the moon, arising from these forces, are perfectly similar to the periodic perturbations of the planets. But they are much greater and more numerous; because the sun is so large, that many inequalities which are quite insensible in the motions of the planets, are of great magnitude in those of the moon. Among the innumerable periodic inequalities to which the moon’s motion in longitude is liable, the most remarkable are, the Equation of the Centre, which is the difference between the moon’s mean and true longitude, the Evection, the Variation, and the Annual Equation. The disturbing force which acts in the line joining the moon and earth produces the Evection: it diminishes the excentricity of the lunar orbit in conjunction and opposition, thereby making it more circular, and augments it in quadrature, which consequently renders it more elliptical. The period of this inequality is less than thirty-two days. Were the increase and diminution always the same, the Evection would only depend upon the distance of the moon from the sun; but its absolute value also varies with her distance from the perigee ([N. 102]) of her orbit. Ancient astronomers, who observed the moon solely with a view to the prediction of eclipses, which can only happen in conjunction and opposition, where the excentricity is diminished by the Evection, assigned too small a value to the ellipticity of her orbit ([N. 103]). The Evection was discovered by Ptolemy from observation, about A.D. 140. The Variation produced by the tangential disturbing force, which is at its maximum when the moon is 45° distant from the sun, vanishes when that distance amounts to a quadrant, and also when the moon is in conjunction and opposition; consequently, that inequality never could have been discovered from the eclipses: its period is half a lunar month ([N. 104]). The Annual Equation depends upon the sun’s distance from the earth: it arises from the moon’s motion being accelerated when that of the earth is retarded, and vice versâ—for, when the earth is in its perihelion, the lunar orbit is enlarged by the action of the sun; therefore, the moon requires more time to perform her revolution. But, as the earth approaches its aphelion, the moon’s orbit contracts, and less time is necessary to accomplish her motion—its period, consequently, depends upon the time of the year. In the eclipses the Annual Equation combines with the Equation of the Centre of the terrestrial orbit, so that ancient astronomers imagined the earth’s orbit to have a greater excentricity than modern astronomers assign to it.
The planets disturb the motion of the moon both directly and indirectly; their action on the earth alters its relative position with regard to the sun and moon, and occasions inequalities in the moon’s motion, which are more considerable than those arising from their direct action; for the same reason the moon, by disturbing the earth, indirectly disturbs her own motion. Neither the excentricity of the lunar orbit, nor its mean inclination to the plane of the ecliptic, have experienced any changes from secular inequalities; for, although the mean action of the sun on the moon depends upon the inclination of the lunar orbit to the ecliptic, and the position of the ecliptic is subject to a secular inequality, yet analysis shows that it does not occasion a secular variation in the inclination of the lunar orbit, because the action of the sun constantly brings the moon’s orbit to the same inclination to the ecliptic. The mean motion, the nodes, and the perigee, however, are subject to very remarkable variations.
From the eclipse observed at Babylon, on the 19th of March, seven hundred and twenty-one years before the Christian era, the place of the moon is known from that of the sun at the instant of opposition ([N. 83]), whence her mean longitude may be found. But the comparison of this mean longitude with another mean longitude, computed back for the instant of the eclipse from modern observations, shows that the moon performs her revolution round the earth more rapidly and in a shorter time now than she did formerly, and that the acceleration in her mean motion has been increasing from age to age as the square of the time ([N. 105]). All ancient and intermediate eclipses confirm this result. As the mean motions of the planets have no secular inequalities, this seemed to be an unaccountable anomaly. It was at one time attributed to the resistance of an ethereal medium pervading space, and at another to the successive transmission of the gravitating force. But, as La Place proved that neither of these causes, even if they exist, have any influence on the motions of the lunar perigee ([N. 102]) or nodes, they could not affect the mean motion; a variation in the mean motion from such causes being inseparably connected with variations in the motions of the perigee and nodes. That great mathematician, in studying the theory of Jupiter’s satellites, perceived that the secular variation in the elements of Jupiter’s orbit, from the action of the planets, occasions corresponding changes in the motions of the satellites, which led him to suspect that the acceleration in the mean motion of the moon might be connected with the secular variation in the excentricity of the terrestrial orbit. Analysis has shown that he assigned the true cause of the acceleration.
It is proved that the greater the excentricity of the terrestrial orbit, the greater is the disturbing action of the sun on the moon. Now, as the excentricity has been decreasing for ages, the effect of the sun in disturbing the moon has been diminishing during that time. Consequently the attraction of the earth has had a more and more powerful effect on the moon, and has been continually diminishing the size of the lunar orbit. So that the moon’s velocity has been gradually augmenting for many centuries to balance the increase of the earth’s attraction. This secular increase in the moon’s velocity is called the Acceleration, a name peculiarly appropriate at present, and which will continue to be so for a vast number of ages; because, as long as the earth’s excentricity diminishes, the moon’s mean motion will be accelerated; but when the excentricity has passed its minimum, and begins to increase, the mean motion will be retarded from age to age. The secular acceleration is now about 11ʺ·9, but its effect on the moon’s place increases as the square of the time ([N. 106]). It is remarkable that the action of the planets, thus reflected by the sun to the moon, is much more sensible than their direct action either on the earth or moon. The secular diminution in the excentricity, which has not altered the equation of the centre of the sun by eight minutes since the earliest recorded eclipses, has produced a variation of about 1° 48ʹ in the moon’s longitude, and of 7° 12ʹ in her mean anomaly ([N. 107]).
The action of the sun occasions a rapid but variable motion in the nodes and perigee of the lunar orbit. Though the nodes recede during the greater part of the moon’s revolution, and advance during the smaller, they perform their sidereal revolution in 6793d 9h 23m 9s·3, or about 186⁄10 years; and the perigee accomplishes a revolution, called of the moon’s apsides, in 3232d 13h 48m 29s·6, or a little more than nine years, notwithstanding its motion is sometimes retrograde and sometimes direct: but such is the difference between the disturbing energy of the sun and that of all the planets put together, that it requires no less than 109,830 years for the greater axis of the terrestrial orbit to do the same, moving at the rate of 11ʺ·8 annually. The form of the earth has no sensible effect either on the lunar nodes or apsides. It is evident that the same secular variation which changes the sun’s distance from the earth, and occasions the acceleration in the moon’s mean motion, must affect the nodes and perigee. It consequently appears, from theory as well as observation, that both these elements are subject to a secular inequality, arising from the variation in the excentricity of the earth’s orbit, which connects them with the Acceleration, so that both are retarded when the mean motion is anticipated. The secular variations in these three elements are in the ratio of the numbers 3, 0·735, and 1; whence the three motions of the moon, with regard to the sun, to her perigee, and to her nodes, are continually accelerated, and their secular equations are as the numbers 1, 4·702, and 0·612. A comparison of ancient eclipses observed by the Arabs, Greeks, and Chaldeans, imperfect as they are, with modern observations, confirms these results of analysis. Future ages will develop these great inequalities, which at some most distant period will amount to many circumferences ([N. 108]). They are, indeed, periodic; but who shall tell their period? Millions of years must elapse before that great cycle is accomplished.
The moon is so near, that the excess of matter at the earth’s equator occasions periodic variations in her longitude, and also that remarkable inequality in her latitude, already mentioned as a nutation in the lunar orbit, which diminishes its inclination to the ecliptic when the moon’s ascending node coincides with the equinox of spring, and augments it when that node coincides with the equinox of autumn. As the cause must be proportional to the effect, a comparison of these inequalities, computed from theory, with the same given by observation, shows that the compression of the terrestrial spheroid, or the ratio of the difference between the polar and the equatorial diameters, to the diameter of the equator, is 1⁄305·05. It is proved analytically, that, if a fluid mass of homogeneous matter, whose particles attract each other inversely as the squares of the distance, were to revolve about an axis as the earth does, it would assume the form of a spheroid whose compression is 1⁄230. Since that is not the case, the earth cannot be homogeneous, but must decrease in density from its centre to its circumference. Thus the moon’s eclipses show the earth to be round; and her inequalities not only determine the form, but even the internal structure of our planet; results of analysis which could not have been anticipated. Similar inequalities in the motions of Jupiter’s satellites prove that his mass is not homogeneous, and that his compression is 1⁄13·8. His equatorial diameter exceeds his polar diameter by about 6000 miles.
The phases ([N. 109]) of the moon, which vary from a slender silvery crescent soon after conjunction, to a complete circular disc of light in opposition, decrease by the same degrees till the moon is again enveloped in the morning beams of the sun. These changes regulate the returns of the eclipses. Those of the sun can only happen in conjunction, when the moon, coming between the earth and the sun, intercepts his light. Those of the moon are occasioned by the earth intervening between the sun and moon when in opposition. As the earth is opaque and nearly spherical, it throws a conical shadow on the side of the moon opposite to the sun, the axis of which passes through the centres of the sun and earth ([N. 110]). The length of the shadow terminates at the point where the apparent diameters ([N. 111]) of the sun and earth would be the same. When the moon is in opposition, and at her mean distance, the diameter of the sun would be seen from her centre under an angle of 1918ʺ·1. That of the earth would appear under an angle of 6908ʺ·3. So that the length of the shadow is at least three times and a half greater than the distance of the moon from the earth, and the breadth of the shadow, where it is traversed by the moon, is about eight-thirds of the lunar diameter. Hence the moon would be eclipsed every time she is in opposition, were it not for the inclination of her orbit to the plane of the ecliptic, in consequence of which the moon, when in opposition, is either above or below the cone of the earth’s shadow, except when in or near her nodes. Her position with regard to them occasions all the varieties in the lunar eclipses. Every point of the moon’s surface successively loses the light of different parts of the sun’s disc before being eclipsed. Her brightness therefore gradually diminishes before she plunges into the earth’s shadow. The breadth of the space occupied by the penumbra ([N. 112]) is equal to the apparent diameter of the sun, as seen from the centre of the moon. The mean duration of a revolution of the sun, with regard to the node of the lunar orbit, is to the duration of a synodic revolution ([N. 113]) of the moon as 223 to 19. So that, after a period of 223 lunar months, the sun and moon would return to the same relative position with regard to the node of the moon’s orbit, and therefore the eclipses would recur in the same order were not the periods altered by irregularities in the motions of the sun and moon. In lunar eclipses, our atmosphere bends the sun’s rays which pass through it all round into the cone of the earth’s shadow. And as the horizontal refraction ([N. 114]) or bending of the rays surpasses half the sum of the semidiameters of the sun and moon, divided by their mutual distance, the centre of the lunar disc, supposed to be in the axis of the shadow, would receive the rays from the same point of the sun, round all sides of the earth; so that it would be more illuminated than in full moon, if the greater portion of the light were not stopped or absorbed by the atmosphere. Instances are recorded where this feeble light has been entirely absorbed, so that the moon has altogether disappeared in her eclipses.
The sun is eclipsed when the moon intercepts his rays ([N. 115]). The moon, though incomparably smaller than the sun, is so much nearer the earth, that her apparent diameter differs but little from his, but both are liable to such variations that they alternately surpass one another. Were the eye of a spectator in the same straight line with the centres of the sun and moon, he would see the sun eclipsed. If the apparent diameter of the moon surpassed that of the sun, the eclipse would be total. If it were less, the observer would see a ring of light round the disc of the moon, and the eclipse would be annular, as it was on the 17th of May, 1836, and on the 15th of March, 1858. If the centre of the moon should not be in the straight line joining the centres of the sun and the eye of the observer, the moon might only eclipse a part of the sun. The variation, therefore, in the distances of the sun and moon from the centre of the earth, and of the moon from her node at the instant of conjunction, occasions great varieties in the solar eclipses. Besides, the height of the moon above the horizon changes her apparent diameter, and may augment or diminish the apparent distances of the centres of the sun and moon, so that an eclipse of the sun may occur to the inhabitants of one country, and not to those of another. In this respect the solar eclipses differ from the lunar, which are the same for every part of the earth where the moon is above the horizon. In solar eclipses, the light reflected by the atmosphere diminishes the obscurity they produce. Even in total eclipses the higher part of the atmosphere is enlightened by a part of the sun’s disc, and reflects its rays to the earth. The whole disc of the new moon is frequently visible from atmospheric reflection. During the eclipse of the 19th of March, 1849, the spots on the lunar disc were distinctly visible, and during that of 1856 the moon was like a beautiful rose-coloured ball floating in the ether: the colour is owing to the refraction of the sun’s light passing through the earth’s atmosphere.
In total solar eclipses the slender luminous arc that is visible for a few seconds before the sun vanishes and also before he reappears, resembles a string of pearls surrounding the dark edge of the moon; it is occasioned by the sun’s rays passing between the tops of the lunar mountains: it occurs likewise in annular eclipses.
A phenomenon altogether unprecedented was seen during the total eclipse of the sun which happened on the 8th of July, 1842. The moon was like a black patch on the sky surrounded by a faint whitish light or corona about the eighth of the moon’s diameter in breadth, which is supposed to be the solar atmosphere rendered visible by the intervention of the moon. In this whitish corona there appeared three rose-coloured flames like the teeth of a saw. Similar flames were also seen in the white corona of the total eclipse which took place in 1851, and a long rose-coloured chain of what appeared to be jagged mountains or sierras united at the base by a red band seemed to be raised into the corona by mirage; but there is no doubt that the corona and red phenomena belong to the sun. This red chain was so bright that Mr. Airy saw it illuminate the northern horizon through an azimuth of 90° with red light. M. Faye attributes the rose-coloured protuberances to the constitution of the sun, which, like Sir William Herschel, he conceives to be an incandescent globe, consisting of two concentric parts of very unequal density, the internal part being a dark spherical mass, the external a very extensive atmosphere, at a certain height in which there is a stratum of luminous clouds which constitutes the photosphere of the sun; above this rises his real atmosphere, so rare as to be only visible as a white aureola or corona during total and annular eclipses. M. Faye conceives that from the central mass gaseous eruptions issue, which form the spots by dissipating and partly extinguishing the luminous clouds, and then rising into the rare atmosphere above that they appear as rose-coloured protuberances during annular eclipses. He estimates that the volume of these vapours sometimes surpasses that of the earth a thousand or even two thousand times. Sir William Herschel attributed the spots to occasional openings in the luminous coating, which seems to be always in motion; but whatever the cause of the spots may be, it is certainly periodical. The white corona and beads were seen during the eclipse of the 15th March, 1858, but there were no rose-coloured appearances, in England at least; but the sky was clouded, so that the eclipse was only visible at intervals.
Planets sometimes eclipse one another. On the 17th of May, 1737, Mercury was eclipsed by Venus near their inferior conjunction; Mars passed over Jupiter on the 9th of January, 1591; and on the 30th of October, 1825, the moon eclipsed Saturn. These phenomena, however, happen very seldom, because all the planets, or even a part of them, are very rarely seen in conjunction at once; that is, in the same part of the heavens at the same time. More than 2500 years before our era the five great planets were in conjunction. On the 15th of September, 1186, a similar assemblage took place between the constellations of Virgo and Libra; and in 1801 the Moon, Jupiter, Saturn, and Venus were united in the heart of the Lion. These conjunctions are so rare, that Lalande has computed that more than seventeen millions of millions of years separate the epochs of the contemporaneous conjunctions of the six great planets.
The motions of the moon have now become of more importance to the navigator and geographer than those of any other heavenly body, from the precision with which terrestrial longitude is determined by occultations of stars, and by lunar distances. In consequence of the retrograde motion of the nodes of the lunar orbit, at the rate of 3ʹ 10ʺ·64 daily, these points make a tour of the heavens in a little more than eighteen years and a half. This causes the moon to move round the earth in a kind of spiral, so that her disc at different times passes over every point in a zone of the heavens extending rather more than 5° 9ʹ on each side of the ecliptic. It is therefore evident that at one time or other she must eclipse every star and planet she meets with in this space. Therefore the occultation of a star by the moon is a phenomenon of frequent occurrence. The moon seems to pass over the star, which almost instantaneously vanishes at one side of her disc, and after a short time as suddenly reappears on the other. A lunar distance is the observed distance of the moon from the sun, or from a particular star or planet, at any instant. The lunar theory is brought to such perfection, that the times of these phenomena, observed under any meridian, when compared with those computed for that of Greenwich, and given in the Nautical Almanac, furnish the longitude of the observer within a few miles ([N. 95.])
From the lunar theory, the mean distance of the sun from the earth, and thence the whole dimensions of the solar system, are known; for the forces which retain the earth and moon in their orbits are respectively proportional to the radii vectores of the earth and moon, each being divided by the square of its periodic time. And, as the lunar theory gives the ratio of the forces, the ratio of the distances of the sun and moon from the earth is obtained. Hence it appears that the sun’s mean distance from the earth is 399·7 or nearly 400 times greater than that of the moon. The method of finding the absolute distances of the celestial bodies, in miles, is in fact the same with that employed in measuring the distances of terrestrial objects. From the extremities of a known base ([N. 116]), the angles which the visual rays from the object form with it are measured; their sum subtracted from two right angles gives the angle opposite the base; therefore, by trigonometry, all the angles and sides of the triangle may be computed—consequently the distance of the object is found. The angle under which the base of the triangle is seen from the object is the parallax of that object. It evidently increases and decreases with the distance. Therefore the base must be very great indeed to be visible from the celestial bodies. The globe itself, whose dimensions are obtained by actual admeasurement, furnishes a standard of measures with which we compare the distances, masses, densities, and volumes of the sun and planets.
SECTION VI.
Form of the Earth and Planets—Figure of a Homogeneous Spheroid in Rotation—Figure of a Spheroid of variable Density—Figure of the Earth, supposing it to be an Ellipsoid of Revolution—Mensuration of a Degree of the Meridian—Compression and Size of the Earth from Degrees of Meridian—Figure of Earth from the Pendulum.
The theoretical investigation of the figure of the earth and planets is so complicated, that neither the geometry of Newton, nor the refined analysis of La Place, has attained more than an approximation. The solution of that difficult problem has been accomplished by our distinguished countryman Mr. Ivory. The investigation has been conducted by successive steps, beginning with a simple case, and then proceeding to the more difficult. But, in all, the forces which occasion the revolutions of the earth and planets are omitted, because, by acting equally upon all the particles, they do not disturb their mutual relations. A fluid mass of uniform density, whose particles mutually gravitate to each other, will assume the form of a sphere when at rest. But, if the sphere begins to revolve, every particle will describe a circle ([N. 117]), having its centre in the axis of revolution. The planes of all these circles will be parallel to one another and perpendicular to the axis, and the particles will have a tendency to fly from that axis in consequence of the centrifugal force arising from the velocity of rotation. The force of gravity is everywhere perpendicular to the surface ([N. 118]), and tends to the interior of the fluid mass; whereas the centrifugal force acts perpendicularly to the axis of rotation, and is directed to the exterior. And, as its intensity diminishes with the distance from the axis of rotation, it decreases from the equator to the poles, where it ceases. Now it is clear that these two forces are in direct opposition to each other in the equator alone, and that gravity is there diminished by the whole effect of the centrifugal force, whereas, in every other part of the fluid, the centrifugal force is resolved into two parts, one of which, being perpendicular to the surface, diminishes the force of gravity; but the other, being at a tangent to the surface, urges the particles towards the equator, where they accumulate till their numbers compensate the diminution of gravity, which makes the mass bulge at the equator, and become flattened at the poles. It appears, then, that the influence of the centrifugal force is most powerful at the equator, not only because it is actually greater there than elsewhere, but because its whole effect is employed in diminishing gravity, whereas, in every other point of the fluid mass, it is only a part that is so employed. For both these reasons, it gradually decreases towards the poles, where it ceases. On the contrary, gravity is least at the equator, because the particles are farther from the centre of the mass, and increases towards the poles, where it is greatest. It is evident, therefore, that, as the centrifugal force is much less than the force of gravity—gravitation, which is the difference between the two, is least at the equator, and continually increases towards the poles, where it is a maximum. On these principles Sir Isaac Newton proved that a homogeneous fluid ([N. 119]) mass in rotation assumes the form of an ellipsoid of revolution ([N. 120]), whose compression is 1⁄230. Such, however, cannot be the form of the earth, because the strata increase in density towards the centre. The lunar inequalities also prove the earth to be so constructed; it was requisite, therefore, to consider the fluid mass to be of variable density. Including this condition, it has been found that the mass, when in rotation, would still assume the form of an ellipsoid of revolution ([N. 120]); that the particles of equal density would arrange themselves in concentric elliptical strata ([N. 121]), the most dense being in the centre; but that the compression or flattening would be less than in the case of the homogeneous fluid. The compression is still less when the mass is considered to be, as it actually is, a solid nucleus, decreasing regularly in density from the centre to the surface, and partially covered by the ocean, because the solid parts, by their cohesion, nearly destroy that part of the centrifugal force which gives the particles a tendency to accumulate at the equator, though not altogether; otherwise the sea, by the superior mobility of its particles, would flow towards the equator and leave the poles dry. Besides, it is well known that the continents at the equator are more elevated than they are in higher latitudes. It is also necessary for the equilibrium of the ocean that its density should be less than the mean density of the earth, otherwise the continents would be perpetually liable to inundations from storms and other causes. On the whole, it appears from theory, that a horizontal line passing round the earth through both poles must be nearly an ellipse, having its major axis in the plane of the equator, and its minor axis coincident with the axis of the earth’s rotation ([N. 122]). It is easy to show, in a spheroid whose strata are elliptical, that the increase in the length of the radii ([N. 123]), the decrease of gravitation, and the increase in the length of the arcs of the meridian, corresponding to angles of one degree, from the poles to the equator, are all proportional to the square of the cosine of the latitude ([N. 124]). These quantities are so connected with the ellipticity of the spheroid, that the total increase in the length of the radii is equal to the compression or flattening, and the total diminution in the length of the arcs is equal to the compression, multiplied by three times the length of an arc of one degree at the equator. Hence, by measuring the meridian curvature of the earth, the compression, and consequently its figure, become known. This, indeed, is assuming the earth to be an ellipsoid of revolution; but the actual measurement of the globe will show how far it corresponds with that solid in figure and constitution.
The courses of the great rivers, which are in general navigable to a considerable extent, prove that the curvature of the land differs but little from that of the ocean; and, as the heights of the mountains and continents are inconsiderable when compared with the magnitude of the earth, its figure is understood to be determined by a surface at every point perpendicular to the direction of gravitation, or of the plumb-line, and is the same which the sea would have if it were continued all round the earth beneath the continents. Such is the figure that has been measured in the following manner:—
A terrestrial meridian is a line passing through both poles, all the points of which have their noon contemporaneously. Were the lengths and curvatures of different meridians known, the figure of the earth might be determined. But the length of one degree is sufficient to give the figure of the earth, if it be measured on different meridians, and in a variety of latitudes. For, if the earth were a sphere, all degrees would be of the same length; but, if not, the lengths of the degrees would be greater, exactly in proportion as the curvature is less. A comparison of the length of a degree in different parts of the earth’s surface will therefore determine its size and form.
An arc of the meridian may be measured by determining the latitude of its extreme points by astronomical observations ([N. 125]), and then measuring the distance between them in feet or fathoms. The distance thus determined on the surface of the earth, divided by the degrees and parts of a degree contained in the difference of the latitudes, will give the exact length of one degree, the difference of the latitudes being the angle contained between the verticals at the extremities of the arc. This would be easily accomplished were the distance unobstructed and on a level with the sea. But, on account of the innumerable obstacles on the surface of the earth, it is necessary to connect the extreme points of the arc by a series of triangles ([N. 126]), the sides and angles of which are either measured or computed, so that the length of the arc is ascertained with much laborious calculation. In consequence of the irregularities of the surface each triangle is in a different plane. They must therefore be reduced by computation to what they would have been had they been measured on the surface of the sea. And, as the earth may in this case be esteemed spherical, they require a correction to reduce them to spherical triangles. The officers who conducted the trigonometrical survey, in measuring 500 feet of a base in Ireland twice over, found that the difference in the two measurements did not amount to the 800th part of an inch; and in the General Survey of Great Britain, five bases were measured from 5 to 7 miles long, and some of them 400 miles apart, yet, when connected by series of triangles, the measured and computed lengths did not differ by more than 3 inches, an unparalleled degree of accuracy; but such is the accuracy with which these operations are conducted.
Arcs of the meridian have been measured in a variety of latitudes in both hemispheres, as well as arcs perpendicular to the meridian. From these measurements it appears that the length of the degrees increases from the equator to the poles, nearly in proportion to the square of the sine of the latitude ([N. 127]). Consequently, the convexity of the earth diminishes from the equator to the poles.
Were the earth an ellipsoid of revolution, the meridians would be ellipses whose lesser axes would coincide with the axis of rotation, and all the degrees measured between the pole and the equator would give the same compression when combined two and two. That, however, is far from being the case. Scarcely any of the measurements give exactly the same results, chiefly on account of local attractions, which cause the plumb-line to deviate from the vertical. The vicinity of mountains produces that effect. One of the most remarkable anomalies of this kind has been observed in certain localities of northern Italy, where the action of some dense subterraneous matter causes the plumb-line to deviate seven or eight times more than it did from the attraction of Chimborazo, in the observations of Bouguer, while measuring a degree of the meridian at the equator. In consequence of this local attraction, the degrees of the meridian in that part of Italy seem to increase towards the equator through a small space, instead of decreasing, as if the earth was drawn out at the poles, instead of being flattened.
Many other discrepancies occur, but from the mean of the five principal measurements of arcs in Peru, India, France, England, and Lapland, Mr. Ivory has deduced that the figure which most nearly follows this law is an ellipsoid of revolution whose equatorial radius is 3962·824 miles, and the polar radius 3949·585 miles. The difference, or 13·239 miles, divided by the equatorial radius, is 1⁄299 nearly[[3]] ([N. 128]). This fraction is called the compression of the earth, and does not differ much from that given by the lunar inequalities. Since the preceding quantities were determined, arcs of the meridian have been measured in various parts of the globe, of which the most extensive are the Russian arc of 25° 20ʹ between the Glacial Sea and the Danube, conducted under the superintendence of M. Struve, and the Indian arc extended to 21° 21ʹ, by Colonel Everest. The compression deduced by Bessel from the sum of ten arcs is 2983⁄4, the equatorial radius 3962·802, and the polar 3949·554 miles, whilst Mr. Airy arrives at an almost identical result (3962·824, 3949·585, and 29883⁄100) from a consideration of all the arcs, measured up to 1831, including the great Indian and Russian ones. If we assume the earth to be a sphere, the length of a degree of the meridian is 6914⁄100 English miles. Therefore 360 degrees, or the whole equatorial circumference of the globe, is 24,899 English miles. Eratosthenes, who died 194 years before the Christian era, was the first to give an approximate value of the earth’s circumference, by the measurement of an arc between Alexandria and Syene.
There is another method of finding the figure of the earth, totally different from the preceding, solely depending upon the increase of gravitation from the equator to the poles. The force of gravitation at any place is measured by the descent of a heavy body during the first second of its fall. And the intensity of the centrifugal force is measured by the deflection of any point from the tangent in a second. For, since the centrifugal force balances the attraction of the earth, it is an exact measure of the gravitating force. Were the attraction to cease, a body on the surface of the earth would fly off in the tangent by the centrifugal force, instead of bending round in the circle of rotation. Therefore, the deflection of the circle from the tangent in a second measures the intensity of the earth’s attraction, and is equal to the versed sine of the arc described during that time, a quantity easily determined from the known velocity of the earth’s rotation. Whence it has been found that at the equator the centrifugal force is equal to the 289th part of gravity. Now, it is proved by analysis that, whatever the constitution of the earth and planets may be, if the intensity of gravitation at the equator be taken equal to unity, the sum of the compression of the ellipsoid, and the whole increase of gravitation from the equator to the pole, is equal to five halves of the ratio of the centrifugal force to gravitation at the equator. This quantity with regard to the earth is 5⁄2 of 1⁄289 or 1⁄115·2. Consequently, the compression of the earth is equal to 1⁄115·2 diminished by the whole increase of gravitation. So that its form will be known, if the whole increase of gravitation from the equator to the pole can be determined by experiment. This has been accomplished by a method founded upon the following considerations:—If the earth were a homogeneous sphere without rotation, its attraction on bodies at its surface would be everywhere the same. If it be elliptical and of variable density, the force of gravity, theoretically, ought to increase from the equator to the pole, as unity plus a constant quantity multiplied into the square of the sine of the latitude ([N. 127]). But for a spheroid in rotation the centrifugal force varies, by the laws of mechanics, as the square of the sine of the latitude, from the equator, where it is greatest, to the pole, where it vanishes. And, as it tends to make bodies fly off the surface, it diminishes the force of gravity by a small quantity. Hence, by gravitation, which is the difference of these two forces, the fall of bodies ought to be accelerated from the equator to the poles proportionably to the square of the sine of the latitude; and the weight of the same body ought to increase in that ratio. This is directly proved by the oscillations of the pendulum ([N. 129]), which, in fact, is a falling body; for, if the fall of bodies be accelerated, the oscillations will be more rapid: in order, therefore, that they may always be performed in the same time, the length of the pendulum must be altered. By numerous and careful experiments it is proved that a pendulum, which oscillates 86,400 times in a mean day at the equator, will do the same at every point of the earth’s surface, if its length be increased progressively to the pole, as the square of the sine of the latitude.
From the mean of these it appears that the whole decrease of gravitation from the poles to the equator is 0·0051449, which, subtracted from 1⁄115·2, shows that the compression of the terrestrial spheroid is about 1⁄285·26. This value has been deduced by the late Mr. Baily, president of the Astronomical Society, who devoted much attention to this subject; at the same time, it may be observed that no two sets of pendulum experiments give the same result, probably from local attractions. The compression obtained by this method does not differ much from that given by the lunar inequalities, nor from the arcs in the direction of the meridian, and those perpendicular to it. The near coincidence of these three values, deduced by methods so entirely independent of each other, shows that the mutual tendencies of the centres of the celestial bodies to one another, and the attraction of the earth for bodies at its surface, result from the reciprocal attraction of all their particles. Another proof may be added. The nutation of the earth’s axis and the precession of the equinoxes ([N. 146]) are occasioned by the action of the sun and moon on the protuberant matter at the earth’s equator. And, although these inequalities do not give the absolute value of the terrestrial compression, they show that the fraction expressing it is comprised between the limits 1⁄279 and 1⁄573.
It might be expected that the same compression should result from each, if the different methods of observation could be made without error. This, however, is not the case; for after allowance has been made for every cause of error, such discrepancies are found, both in the degrees of the meridian and in the length of the pendulum, as show that the figure of the earth is very complicated. But they are so small, when compared with the general results, that they may be disregarded. The compression deduced from the mean of the whole appears not to differ much from 1⁄300; that given by the lunar theory has the advantage of being independent of the irregularities of the earth’s surface and of local attractions. The regularity with which the observed variation in the length of the pendulum follows the law of the square of the sine of the latitude proves the strata to be elliptical, and symmetrically disposed round the centre of gravity of the earth, which affords a strong presumption in favour of its original fluidity. It is remarkable how little influence the sea has on the variation of the lengths of the arcs of the meridian, or on gravitation; neither does it much affect the lunar inequalities, from its density being only about a fifth of the mean density of the earth. For, if the earth were to become fluid, after being stripped of the ocean, it would assume the form of an ellipsoid of revolution whose compression is 1⁄304·8, which differs very little from that determined by observation, and proves, not only that the density of the ocean is inconsiderable, but that its mean depth is very small. There are profound cavities in the bottom of the sea, but its mean depth probably does not much exceed the mean height of the continents and islands above its level. On this account, immense tracts of land may be deserted or overwhelmed by the ocean, as appears really to have been the case, without any great change in the form of the terrestrial spheroid. The variation in the length of the pendulum was first remarked by Richter in 1672, while observing transits of the fixed stars across the meridian at Cayenne, about five degrees north of the equator. He found that his clock lost at the rate of 2m 28s daily, which induced him to determine the length of a pendulum beating seconds in that latitude; and, repeating the experiments on his return to Europe, he found the seconds’ pendulum at Paris to be more than the twelfth of an inch longer than that at Cayenne. The form and size of the earth being determined, a standard of measure is furnished with which the dimensions of the solar system may be compared.
SECTION VII.
Parallax—Lunar Parallax found from Direct Observation—Solar Parallax deduced from the Transit of Venus—Distance of the Sun from the Earth—Annual Parallax—Distance of the Fixed Stars.
The parallax of a celestial body is the angle under which the radius of the earth would be seen if viewed from the centre of that body; it affords the means of ascertaining the distances of the sun, moon, and planets ([N. 130]). When the moon is in the horizon at the instant of rising or setting, suppose lines to be drawn from her centre to the spectator and to the centre of the earth: these would form a right-angled triangle with the terrestrial radius, which is of a known length; and, as the parallax or angle at the moon can be measured, all the angles and one side are given; whence the distance of the moon from the centre of the earth may be computed. The parallax of an object may be found, if two observers under the same meridian, but at a very great distance from one another, observe its zenith distances on the same day at the time of its passage over the meridian. By such contemporaneous observations at the Cape of Good Hope and at Berlin, the mean horizontal parallax of the moon was found to be 3459ʺ, whence the mean distance of the moon is about sixty times the greatest terrestrial radius, or 237,608 miles nearly.[[4]] Since the parallax is equal to the radius of the earth divided by the distance of the moon, it varies with the distance of the moon from the earth under the same parallel of latitude, and proves the ellipticity of the lunar orbit. When the moon is at her mean distance, it varies with the terrestrial radii, thus showing that the earth is not a sphere ([N. 131]).
Although the method described is sufficiently accurate for finding the parallax of an object as near as the moon, it will not answer for the sun, which is so remote that the smallest error in observation would lead to a false result. But that difficulty is obviated by the transits of Venus. When that planet is in her nodes ([N. 132]), or within 11⁄4° of them, that is, in, or nearly in, the plane of the ecliptic, she is occasionally seen to pass over the sun like a black spot. If we could imagine that the sun and Venus had no parallax, the line described by the planet on his disc, and the duration of the transit, would be the same to all the inhabitants of the earth. But, as the semi-diameter of the earth has a sensible magnitude when viewed from the centre of the sun, the line described by the planet in its passage over his disc appears to be nearer to his centre, or farther from it, according to the position of the observer; so that the duration of the transit varies with the different points of the earth’s surface at which it is observed ([N. 133]). This difference of time, being entirely the effect of parallax, furnishes the means of computing it from the known motions of the earth and Venus, by the same method as for the eclipses of the sun. In fact, the ratio of the distances of Venus and the sun from the earth at the time of the transit is known from the theory of their elliptical motion. Consequently the ratio of the parallaxes of these two bodies, being inversely as their distances, is given; and as the transit gives the difference of the parallaxes, that of the sun is obtained. In 1769 the parallax of the sun was determined by observations of a transit of Venus made at Wardhus in Lapland, and at Tahiti in the South Sea. The latter observation was the object of Cook’s first voyage. The transit lasted about six hours at Tahiti, and the difference in duration at these two stations was eight minutes; whence the sun’s horizontal parallax was found to be 8ʺ·72. But by other considerations it has been reduced by Professor Encke to 8ʺ·5776; from which the mean distance of the sun appears to be about ninety-five millions of miles. This is confirmed by an inequality in the motion of the moon, which depends upon the parallax of the sun, and which, when compared with observation, gives 8ʺ·6 for the sun’s parallax. The transits of Venus in 1874 and 1882 will be unfavourable for ascertaining the accuracy of the solar parallax, and no other transit of that planet will take place till the twenty-first century; but in the mean time recourse may be had to the oppositions of Mars.
The parallax of Venus is determined by her transits; that of Mars by direct observation, and it is found to be nearly double that of the sun, when the planet is in opposition. The distance of these two planets from the earth is therefore known in terrestrial radii, consequently their mean distances from the sun may be computed; and as the ratios of the distances of the planets from the sun are known by Kepler’s law, of the squares of the periodic times of any two planets being as the cubes of their mean distances from the sun, their absolute distances in miles are easily found ([N. 134]). This law is very remarkable, in thus uniting all the bodies of the system, and extending to the satellites as well as the planets.
Far as the earth seems to be from the sun, Uranus is no less than nineteen, and Neptune thirty times farther. Situate on the verge of the system, the sun must appear from Uranus not much larger than Venus does to us, and from Neptune as a star of the fifth magnitude. The earth cannot even be visible as a telescopic object to a body so remote as either Uranus or Neptune. Yet man, the inhabitant of the earth, soars beyond the vast dimensions of the system to which his planet belongs, and assumes the diameter of its orbit as the base of a triangle whose apex extends to the stars.
Sublime as the idea is, this assumption proves ineffectual, except in a very few cases; for the apparent places of the fixed stars are not sensibly changed by the earth’s annual revolution. With the aid derived from the refinements of modern astronomy, and of the most perfect instruments, a sensible parallax has been detected only in a very few of these remote suns. α Centauri has a parallax of one second of space, therefore it is the nearest known star, and yet it is more than two hundred thousand times farther from us than the sun is. At such a distance not only the terrestrial orbit shrinks to a point, but the whole solar system, seen in the focus of the most powerful telescope, might be eclipsed by the thickness of a spider’s thread. Light, flying at the rate of 190,000 miles in a second, would take more than three years to travel over that space. One of the nearest stars may therefore have been kindled or extinguished more than three years before we could have been aware of so mighty an event. But this distance must be small when compared with that of the most remote of the bodies which are visible in the heavens. The fixed stars are undoubtedly luminous like the sun: it is therefore probable that they are not nearer to one another than the sun is to the nearest of them. In the milky way and the other starry nebulæ, some of the stars that seem to us to be close to others may be far behind them in the boundless depth of space; nay, may be rationally supposed to be situate many thousand times farther off. Light would therefore require thousands of years to come to the earth from those myriads of suns of which our own is but “the remote companion.”
SECTION VIII.
Masses of Planets that have no Satellites determined from their Perturbations—Masses of the others obtained from the Motions of their Satellites—Masses of the Sun, the Earth, of Jupiter and of the Jovial System—Mass of the Moon—Real Diameters of Planets, how obtained—Size of Sun, Densities of the Heavenly Bodies—Formation of Astronomical Tables—Requisite Data and Means of obtaining them.
The masses of such planets as have no satellites are known by comparing the inequalities they produce in the motions of the earth and of each other, determined theoretically, with the same inequalities given by observation; for the disturbing cause must necessarily be proportional to the effect it produces. The masses of the satellites themselves may also be compared with that of the sun by their perturbations. Thus, it is found, from the comparison of a vast number of observations with La Place’s theory of Jupiter’s satellites, that the mass of the sun is no less than 65,000,000 times greater than the least of these moons. But, as the quantities of matter in any two primary planets are directly as the cubes of the mean distances at which their satellites revolve, and inversely as the squares of their periodic times ([N. 135]), the mass of the sun and of any planets which have satellites may be compared with the mass of the earth. In this manner it is computed that the mass of the sun is 354,936 times that of the earth; whence the great perturbations of the moon, and the rapid motion of the perigee and nodes of her orbit ([N. 136]). Even Jupiter, the largest of the planets, has been found by Professor Airy to be 1047·871 times less than the sun; and, indeed, the mass of the whole Jovial system is not more than the 1054·4th part of that of the sun. So that the mass of the satellites bears a very small proportion to that of their primary. The mass of the moon is determined from several sources—from her action on the terrestrial equator, which occasions the nutation in the axis of rotation; from her horizontal parallax; from an inequality she produces in the sun’s longitude; and from her action on the tides. The three first quantities, computed from theory and compared with their observed values, give her mass respectively equal to the 1⁄71, 1⁄74·2, and 1⁄69·2, part of that of the earth, which do not differ much from each other. Dr. Brinkley has found it to be 1⁄80 from the constant of lunar nutation: but, from the moon’s action in raising the tides, her mass appears to be about the 1⁄75 part of that of the earth—a value that cannot differ much from the truth.
The apparent diameters of the sun, moon, and planets are determined by measurement; therefore their real diameters may be compared with that of the earth; for the real diameter of a planet is to the real diameter of the earth, or 7926 miles, as the apparent diameter of the planet to the apparent diameter of the earth as seen from the planet, that is, to twice the parallax of the planet. According to Bessel, the mean apparent diameter of the sun is 1923ʺ·64, and with the solar parallax 8ʺ·5776, it will be found that the diameter of the sun is about 886,877 miles. Therefore, if the centre of the sun were to coincide with the centre of the earth, his volume would not only include the orbit of the moon, but would extend nearly as far again; for the moon’s mean distance from the earth is about sixty times the earth’s equatorial radius, or 238,793 miles: so that twice the distance of the moon is 477,586 miles, which differs but little from the solar radius; his equatorial radius is probably not much less than the major axis of the lunar orbit. The diameter of the moon is only 2160 miles; and Jupiter’s diameter of 88,200 miles is very much less than that of the sun; the diameter of Pallas does not much exceed 79 miles, so that an inhabitant of that planet, in one of our steam carriages, might go round his world in a few hours. The diameters of Lutetia and Atalanta are only 8 and 4 miles respectively; but the whole of the 55 telescopic planets are so small, that their united mass is probably not more than the fifth or sixth part of that of the moon.
The densities of bodies are proportional to their masses, divided by their volumes. Hence, if the sun and planets be assumed to be spheres, their volumes will be as the cubes of their diameters. Now, the apparent diameters of the sun and earth, at their mean distance, are 1923ʺ·6 and 17ʺ·1552, and the mass of the earth is the 354,936th part of that of the sun taken as the unit. It follows, therefore, that the earth is four times as dense as the sun. But the sun is so large that his attractive force would cause bodies to fall through about 334·65 feet in a second. Consequently, if he were habitable by human beings, they would be unable to move, since their weight would be thirty times as great as it is here. A man of moderate size would weigh about two tons at the surface of the sun; whereas at the surface of some of the new planets he would be so light that it would be impossible to stand steady, since he would only weigh a few pounds. The mean density of the earth has been determined by the following method. Since a comparison of the action of two planets upon a third gives the ratio of the masses of these two planets, it is clear that, if we can compare the effect of the whole earth with the effect of any part of it, a comparison may be instituted between the mass of the whole earth and the mass of that part of it. Now a leaden ball was weighed against the earth by comparing the effects of each upon a pendulum; the nearness of the smaller mass making it produce a sensible effect as compared with that of the larger: for by the laws of attraction the whole earth must be considered as collected in its centre. By this method it has been found that the mean density of the earth is 5·660 times greater than that of water at the temperature of 62° of Fahrenheit’s thermometer. The late Mr. Baily, whose accuracy as an experimental philosopher is acknowledged, was unremittingly occupied nearly four years in accomplishing this very important object. In order to ascertain the mean density of the earth still more perfectly, Mr. Airy made a series of experiments to compare the simultaneous oscillations of two pendulums, one at the bottom of the Harton coal-pit, 1260 feet deep, in Northumberland, and the other on the surface of the earth immediately above it. The oscillations of the pendulums were compared with an astronomical clock at each station, and the time was instantaneously transmitted from one to the other by a telegraphic wire. The oscillations were observed for more than 100 hours continuously, when it was found that the lower pendulum made 21⁄2 oscillations more in 24 hours than the upper one. The experiment was repeated for the same length of time with the same result; but on this occasion the upper pendulum was taken to the bottom of the mine and the lower brought to the surface. From the difference between the oscillations at the two stations it appears that gravitation at the bottom of the mine exceeds that at the surface by the 1⁄19190 part, and that the mean density of the earth is 6·565, which is greater than that obtained by Mr. Baily by ·89. While employed on the trigonometrical survey of Scotland, Colonel James determined the mean density of the earth to be 5·316, from a deviation of the plumb-line amounting to 2ʺ, caused by the attraction of Arthur’s Seat and the heights east of Edinburgh: it agrees more nearly with the density found by Mr. Baily than with that deduced from Mr. Airy’s experiments. All the planets and satellites appear to be of less density than the earth. The motions of Jupiter’s satellites show that his density increases towards his centre. Were his mass homogeneous, his equatorial and polar axes would be in the ratio of 41 to 36, whereas they are observed to be only as 41 to 38. The singular irregularities in the form of Saturn, and the great compression of Mars, prove the internal structure of these two planets to be very far from uniform.
Before entering on the theory of rotation, it may not be foreign to the subject to give some idea of the methods of computing the places of the planets, and of forming astronomical tables. Astronomy is now divided into the three distinct departments of theory, observation, and computation. Since the problem of the three bodies can only be solved by approximation, the analytical astronomer determines the position of a planet in space by a series of corrections. Its place in its circular orbit is first found, then the addition or subtraction of the equation of the centre ([N. 48]) to or from its mean place gives its position in the ellipse. This again is corrected by the application of the principal periodic inequalities. But, as these are determined for some particular position of the three bodies, they require to be corrected to suit other relative positions. This process is continued till the corrections become less than the errors of observation, when it is obviously unnecessary to carry the approximation further. The true latitude and distance of the planet from the sun are obtained by methods similar to those employed for the longitude.
As the earth revolves equably about its axis in 24 hours, at the rate of 15° in an hour, time becomes a measure of angular motion, and the principal element in astronomy, where the object is to determine the exact state of the heavens and the successive changes it undergoes in all ages, past, present, and to come. Now, the longitude, latitude, and distance of a planet from the sun are given in terms of the time, by general analytical formulæ. These formulæ will consequently give the exact place of the body in the heavens, for any time assumed at pleasure, provided they can be reduced to numbers. But before the calculator begins his task the observer must furnish the necessary data, which are, obviously, the forms of the orbits, and their positions with regard to the plane of the ecliptic ([N. 57]). It is therefore necessary to determine by observation, for each planet, the length of the major axis of its orbit, the excentricity, the inclination of the orbit to the plane of the ecliptic, the longitudes of its perihelion and ascending node at a given time, the periodic time of the planet, and its longitude at any instant arbitrarily assumed, as an origin from whence all its subsequent and antecedent longitudes are estimated. Each of these quantities is determined from that position of the planet on which it has most influence. For example, the sum of the greatest and least distances of the planet from the sun is equal to the major axis of the orbit, and their difference is equal to twice the excentricity. The longitude of the planet, when at its least distance from the sun, is the same with the longitude of the perihelion; the greatest latitude of the planet is equal to the inclination of the orbit: the longitude of the planet, when in the plane of the ecliptic in passing towards the north, is the longitude of the ascending node, and the periodic time is the interval between two consecutive passages of the planet through the same node, a small correction being made for the precession of the node during the revolution of the planet ([N. 137]). Notwithstanding the excellence of instruments and the accuracy of modern observers, unavoidable errors of observation can only be compensated by finding the value of each element from the mean of a thousand, or even many thousands of observations. For as it is probable that the errors are not all in one direction, but that some are in excess and others in defect, they will compensate each other when combined.
However, the values of the elements determined separately can only be regarded as approximate, because they are so connected that the estimation of any one independently will induce errors in the others. The excentricity depends upon the longitude of the perihelion, the mean motion depends upon the major axis, the longitude of the node upon the inclination of the orbit, and vice versâ. Consequently, the place of a planet computed with the approximate data will differ from its observed place. Then the difficulty is to ascertain what elements are most in fault, since the difference in question is the error of all; that is obviated by finding the errors of some thousands of observations, and combining them, so as to correct the elements simultaneously, and to make the sum of the squares of the errors a minimum with regard to each element ([N. 138]). The method of accomplishing this depends upon the Theory of Probabilities; a subject fertile in most important results in the various departments of science and of civil life, and quite indispensable in the determination of astronomical data. A series of observations continued for some years will give approximate values of the secular and periodic inequalities, which must be corrected from time to time, till theory and observation agree. And these again will give values of the masses of the bodies forming the solar system, which are important data in computing their motions. The periodic inequalities derived from a great number of observations are employed for the determination of the values of the masses till such time as the secular inequalities shall be perfectly known, which will then give them with all the necessary precision. When all these quantities are determined in numbers, the longitude, latitude, and distance of the planet from the sun are computed for stated intervals, and formed into tables, arranged according to the time estimated from a given epoch, so that the place of the body may be determined from them by inspection alone, at any instant for perhaps a thousand years before and after that epoch. By this tedious process, tables have been computed for all the great planets, and several of the small, besides the moon and the satellites of Jupiter. In the present state of astronomy the masses and elements of the orbits are pretty well known, so that the tables only require to be corrected from time to time as observations become more accurate. Those containing the motions of Jupiter, Saturn, and Uranus have already been twice constructed within the last thirty years, and the tables of Jupiter and Saturn agree almost perfectly with modern observation. The following prediction will be found in the sixth edition of this book, published in the year 1842: “Those of Uranus, however, are already defective, probably because the discovery of that planet in 1781 is too recent to admit of much precision in the determination of its motions, or that possibly it may be subject to disturbances from some unseen planet revolving about the sun beyond the present boundaries of our system. If, after a lapse of years, the tables formed from a combination of numerous observations should be still inadequate to represent the motions of Uranus, the discrepancies may reveal the existence, nay, even the mass and orbit, of a body placed for ever beyond the sphere of vision.”[[5]]
That prediction has been fulfilled since the seventh edition of this book was published. Not only the existence of Neptune, revolving at the distance of three thousand millions of miles from the sun, has been discovered from his disturbing action on Uranus, but his mass, the form and position of his orbit in space, and his periodic time had been determined before the planet had been seen, and the planet itself was discovered in the very point of the heavens which had been assigned to it. It had been noticed for years that the perturbation of Uranus had increased in an unaccountable manner ([N. 139]). After the disturbing action of all the known planets had been determined, it was found that, between the years 1833 and 1837, the observed and computed distance of Uranus from the sun differed by 240,000 miles, which is about the mean distance of the moon from the earth, while, in 1841, the error in the geocentric longitude of the planet amounted to 96ʺ. These discrepancies were therefore attributed to the attraction of some unseen and unknown planet, consequently they gave rise to a case altogether unprecedented in the history of astronomy. Heretofore it was required to determine the disturbing action of one known planet upon another. Whereas the inverse problem had now to be solved, in which it was required to find the place of an unknown body in the heavens, at a given time, together with its mass, and the form and position of its orbit, from the disturbance it produced on the motions of another. The difficulty was extreme, because all the elements of the orbit of Uranus were erroneous from the action of Neptune, and those of Neptune’s orbit were unknown. In this dilemma it was necessary to form some hypothesis with regard to the unknown planet; it was therefore assumed, according to Bode’s empirical law on the mean distances of the planets, that it was revolving at twice the distance of Uranus from the sun. In fact, the periodic time of Uranus is about 84 years, and, as the discrepancies in his motions increased slowly and regularly, it was evident that it would require a planet with a much longer periodic time to produce them—moreover, it was clear that the new planet must be exterior to Uranus, otherwise it would have disturbed the motions of Saturn.
Another circumstance tended to lessen the difficulty; the latitude of Uranus was not much affected, therefore it was concluded that the inclination of the orbit of the unknown body must be very small, and, as that of the orbit of Uranus is only 46ʹ 28ʺ·4, both planets were assumed to be moving in the plane of the ecliptic, and thus the elements of the orbit of the unknown planet were reduced from six to four. Having thus assumed that the unknown body was revolving in a circle in the plane of the ecliptic, the analytical expression of its action on the motion of Uranus, when in numerous points of its orbit, was compared with the observed longitude of Uranus, through a regular series of years, by means of which the faulty elements of the orbit of Uranus were eliminated, or got rid of, and there only remained a relation between the mass of the new planet and three of the elements of its orbit; and it then was necessary to assume such a value for two of them as would suit the rest. That was accomplished so dexterously, that the perturbations of Uranus were perfectly conformable to the motions of Neptune, moving in the orbit thus found, and the place of the new planet exactly agreed with observation. Subsequently its orbit and motions have been determined more accurately.
The honour of this admirable effort of genius is shared by Mr. Adams and M. Le Verrier, who, independently of each other, arrived at these wonderful results. Mr. Adams had determined the mass and apparent diameter of Neptune, with all the circumstances of its motion, eight months before M. Le Verrier had terminated his results, and had also pointed out the exact spot where the planet would be found; but the English observers neglected to look for it till M. Le Verrier made known his researches, and communicated its position to Dr. Galle, at Berlin, who found it the very first night he looked for it, and then it was evident that it would have been seen in the place Mr. Adams had assigned to it eight months before had it been looked for. So closely did the results of these two great mathematicians agree.
Neptune has a diameter of 39,793 miles, consequently he is nearly 200 times larger than the earth, and may be seen with a telescope of moderate power. His motion is retrograde at present, and six times slower than that of the earth. At so great a distance from the sun it can only have the 1⁄1300th part of the light and heat the earth receives; but having a satellite, the deficiency of light may in some measure be supplied.
The prediction may now be transferred from Uranus to Neptune, whose perturbations may reveal the existence of a planet still further removed, which may for ever remain beyond the reach of telescopic vision—yet its mass, the form and position of its orbit, and all the circumstances of its motion may become known, and the limits of the solar system may still be extended hundreds of millions of miles.
The mean distance of Neptune from the sun has subsequently proved to be only 2893 millions of miles, and the period of his revolution 166 years, so that Baron Bode’s law, of the interval between the orbits of any two planets being twice as great as the inferior interval and half of the superior, fails in the case of Neptune, though it was useful on the first approximation to his motions; and since Bode’s time it has led to the discovery of fifty-five telescopic planets revolving between the orbits of Mars and Jupiter, some by chance, others by a systematic search on the faith that these minute planets are fragments of a larger body that has exploded, because their distances from the sun are nearly the same; the lines of the nodes of some of their orbits terminate in the same points of the heavens, and the inclinations of their orbits are such as might have taken place from their mutual disturbances at the time of the explosion, and while yet they were near enough for their forms to affect their motions. The orbits of the more recently discovered asteroids show that this hypothesis is untenable.
The tables of Mars, Venus, and even those of the sun, have been greatly improved, and still engage the attention of our Astronomer Royal, Mr. Airy, and other eminent astronomers. We are chiefly indebted to the German astronomers for tables of the four older telescopic planets, Vesta, Juno, Ceres, and Pallas; the others have only been discovered since the year 1845.
The determination of the path of a planet when disturbed by all the others, a problem which has employed the talents of the greatest astronomers, from Newton to the present day, is only successfully accomplished with regard to the older planets, which revolve in nearly circular orbits, but little inclined to the plane of the ecliptic. When the excentricity and inclination of the orbits are great, their analysis fails, because the series expressing the co-ordinates of the bodies become extremely complicated, and do not converge when applied to comets and the telescopic planets. This difficulty has been overcome by Sir John Lubbock, and other mathematicians, who have the honour of having completed the theory of planetary motion, which becomes every day of more importance, from the new planets that have been discovered, and also with regard to comets, many of which return to the sun at regular intervals, and from whose perturbations the masses of the planets will be more accurately determined, and the retarding influence of the ethereal medium better known.
SECTION IX.
Rotation of the Sun and Planets—Saturn’s Rings—Periods of the Rotation of the Moon and other Satellites equal to the Periods of their Revolutions—Form of Lunar Spheroid—Libration, Aspect, and Constitution of the Moon—Rotation of Jupiter’s Satellites.
The oblate form of several of the planets indicates rotatory motion. This has been confirmed in most cases by tracing spots on their surface, by which their poles and times of rotation have been determined. The rotation of Mercury is unknown, on account of his proximity to the sun; that of the new planets has not yet been ascertained. The sun revolves in twenty-five days and ten hours about an axis which is directed towards a point half-way between the pole-star and α of Lyra, the plane of rotation being inclined by 7° 30ʹ, or a little more than seven degrees, to the plane of the ecliptic: it may therefore be concluded that the sun’s mass is a spheroid, flattened at the poles. From the rotation of the sun, there was every reason to believe that he has a progressive motion in space, a circumstance which is confirmed by observation. But, in consequence of the reaction of the planets, he describes a small irregular orbit about the centre of gravity of the system, never deviating from his position by more than twice his own diameter, or a little more than seven times the distance of the moon from the earth. The sun and all his attendants rotate from west to east, on axes that remain nearly parallel to themselves ([N. 140]) in every point of their orbit, and with angular velocities that are sensibly uniform ([N. 141]). Although the uniformity in the direction of their rotation is a circumstance hitherto unaccounted for in the economy of nature, yet, from the design and adaptation of every other part to the perfection of the whole, a coincidence so remarkable cannot be accidental. And, as the revolutions of the planets and satellites are also from west to east, it is evident that both must have arisen from the primitive cause which determined the planetary motions.[[6]] Indeed, La Place has computed the probability to be as four millions to one that all the motions of the planets, both of rotation and revolution, were at once imparted by an original common cause, but of which we know neither the nature nor the epoch.
The larger planets rotate in shorter periods than the smaller planets and the earth. Their compression is consequently greater, and the action of the sun and of their satellites occasions a nutation in their axes and a precession of their equinoxes ([N. 147]) similar to that which obtains in the terrestrial spheroid, from the attraction of the sun and moon on the prominent matter at the equator. Jupiter revolves in less than ten hours round an axis at right angles to certain dark belts or bands, which always cross his equator. (See [Plate 1].) This rapid rotation occasions a very great compression in his form. His equatorial axis exceeds his polar axis by 6000 miles, whereas the difference in the axes of the earth is only about twenty-six and a half. It is an evident consequence of Kepler’s law of the squares of the periodic times of the planets being as the cubes of the major axes of their orbits, that the heavenly bodies move slower the farther they are from the sun. In comparing the periods of the revolutions of Jupiter and Saturn with the times of their rotation, it appears that a year of Jupiter contains nearly ten thousand of his days, and that of Saturn about thirty thousand Saturnian days.
The appearance of Saturn is unparalleled in the system of the world. He is a spheroid nearly 1000 times larger than the earth, surrounded by a ring even brighter than himself, which always remains suspended in the plane of his equator: and, viewed with a very good telescope, it is found to consist of two concentric rings, divided by a dark band. The exterior ring, as seen through Mr. Lassell’s great equatorial at Malta, has a dark-striped band through the centre, and is altogether less bright than the interior ring, one half of which is extremely brilliant; while the interior half is shaded in rings like the seats in an amphitheatre. Mr. Lassell made the remarkable discovery of a dark transparent ring, whose edge coincides with the inner edge of the interior ring, and which occupies about half the space between it and Saturn. He compares it to a band of dark-coloured crape drawn across a portion of the disc of the planet, and the part projected upon the blue sky is also transparent. At the time these observations were made at Malta, Captain Jacob discovered the transparent ring at Madras. It is conjectured to be fluid; even the luminous rings cannot be very dense, since the density of Saturn himself is known to be less than the eighth part of that of the earth. A transit of the ring across a star might reveal something concerning this wonderful object. The ball of Saturn is striped by belts of different colours. At the time of these observations the part above the ring was bright white; at his equator there was a ruddy belt divided in two, above which were belts of a bluish green alternately dark and light, while at the pole there was a circular space of a pale colour. (See [Plate 2].) The mean distance of the interior part of the double ring from the surface of the planet is about 22,240 miles, it is no less than 33,360 miles broad, but, by the estimation of Sir John Herschel, its thickness does not much exceed 100 miles, so that it appears like a plane. By the laws of mechanics, it is impossible that this body can retain its position by the adhesion of its particles alone. It must necessarily revolve with a velocity that will generate a centrifugal force sufficient to balance the attraction of Saturn. Observation confirms the truth of these principles, showing that the rings rotate from west to east about the planet in ten hours and a half, which is nearly the time a satellite would take to revolve about Saturn at the same distance. Their plane is inclined to the ecliptic, at an angle of 28° 10ʹ 44ʺ·5; in consequence of this obliquity of position, they always appear elliptical to us, but with an excentricity so variable as even to be occasionally like a straight line drawn across the planet. In the beginning of October, 1832, the plane of the rings passed through the centre of the earth; in that position they are only visible with very superior instruments, and appear like a fine line across the disc of Saturn. About the middle of December, in the same year, the rings became invisible, with ordinary instruments, on account of their plane passing through the sun. In the end of April, 1833, the rings vanished a second time, and reappeared in June of that year. Similar phenomena will occur as often as Saturn has the same longitude with either node of his rings. Each side of these rings has alternately fifteen years of sunshine and fifteen years of darkness.
It is a singular result of theory, that the rings could not maintain their stability of rotation if they were everywhere of uniform thickness; for the smallest disturbance would destroy the equilibrium, which would become more and more deranged, till, at last, they would be precipitated on the surface of the planet. The rings of Saturn must therefore be irregular solids, of unequal breadth in different parts of the circumference, so that their centres of gravity do not coincide with the centres of their figures. Professor Struve has also discovered that the centre of the rings is not concentric with the centre of Saturn. The interval between the outer edge of the globe of the planet and the outer edge of the rings on one side is 11ʺ·272, and, on the other side, the interval is 11ʺ·390, consequently there is an excentricity of the globe in the rings of 0ʺ·215. If the rings obeyed different forces, they would not remain in the same plane, but the powerful attraction of Saturn always maintains them and his satellites in the plane of his equator. The rings, by their mutual action, and that of the sun and satellites, must oscillate about the centre of Saturn, and produce phenomena of light and shadow whose periods extend to many years. According to M. Bessel the mass of Saturn’s ring is equal to the 1⁄118 part of that of the planet.
The periods of rotation of the moon and the other satellites are equal to the times of their revolutions, consequently these bodies always turn the same face to their primaries. However, as the mean motion of the moon is subject to a secular inequality, which will ultimately amount to many circumferences ([N. 108]), if the rotation of the moon were perfectly uniform and not affected by the same inequalities, it would cease exactly to counterbalance the motion of revolution; and the moon, in the course of ages, would successively and gradually discover every point of her surface to the earth. But theory proves that this never can happen; for the rotation of the moon, though it does not partake of the periodic inequalities of her revolution, is affected by the same secular variations, so that her motions of rotation and revolution round the earth will always balance each other, and remain equal. This circumstance arises from the form of the lunar spheroid, which has three principal axes of different lengths at right angles to each other.
The moon is flattened at her poles from her centrifugal force, therefore her polar axis is the least. The other two are in the plane of her equator, but that directed towards the earth is the greatest ([N. 142]). The attraction of the earth, as if it had drawn out that part of the moon’s equator, constantly brings the greatest axis, and consequently the same hemisphere, towards us, which makes her rotation participate in the secular variations of her mean motion of revolution. Even if the angular velocities of rotation and revolution had not been nicely balanced in the beginning of the moon’s motion, the attraction of the earth would have recalled the greatest axis to the direction of the line joining the centres of the moon and earth; so that it would have vibrated on each side of that line in the same manner as a pendulum oscillates on each side of the vertical from the influence of gravitation. No such libration is perceptible; and, as the smallest disturbance would make it evident, it is clear that, if the moon has ever been touched by a comet, the mass of the latter must have been extremely small. If it had been only the hundred thousandth part of that of the earth, it would have rendered the libration sensible. According to analysis, a similar libration exists in the motions of Jupiter’s satellites, which still remains insensible to observation, and yet the comet of 1770 passed twice through the midst of them.
The moon, it is true, is liable to librations depending upon the position of the spectator. At her rising, part of the western edge of her disc is visible, which is invisible at her setting, and the contrary takes place with regard to her eastern edge. There are also librations arising from the relative positions of the earth and moon in their respective orbits; but, as they are only optical appearances, one hemisphere will be eternally concealed from the earth. For the same reason the earth, which must be so splendid an object to one lunar hemisphere, will be for ever veiled from the other. On account of these circumstances, the remoter hemisphere of the moon has its day a fortnight long, and a night of the same duration, not even enlightened by a moon, while the favoured side is illuminated by the reflection of the earth during its long night. A planet exhibiting a surface thirteen times larger than that of the moon, with all the varieties of clouds, land, and water, coming successively into view, must be a splendid object to a lunar traveller in a journey to his antipodes. The great height of the lunar mountains probably has a considerable influence on the phenomena of her motion, the more so as her compression is small, and her mass considerable. In the curve passing through the poles, and that diameter of the moon which always points to the earth, nature has furnished a permanent meridian, to which the different spots on her surface have been referred, and their positions are determined with as much accuracy as those of many of the most remarkable places on the surface of our globe. According to the observations of Professor Secchi at Rome, the mountains of the moon are mostly volcanic and of three kinds. The first and oldest have their borders obliterated, so that they look like deep wells; the second, which are of an intermediate class, have elevated, and, for the most part, regular unbroken edges, with the ground around them raised to a prodigious extent in proportion to the size of the volcano, with generally an insulated rock in the centre of the crater. The third, and most recent class, are very small, and seem to be the last effort of the expiring volcanic force, which is probably now extinct.
The distance and minuteness of Jupiter’s satellites render it extremely difficult to ascertain their rotation. It was, however, accomplished by Sir William Herschel from their relative brightness. He observed that they alternately exceed each other in brilliancy, and, by comparing the maxima and minima of their illumination with their positions relatively to the sun and to their primary, he found that, like the moon, the time of their rotation is equal to the period of their revolution about Jupiter. Miraldi was led to the same conclusion with regard to the fourth satellite, from the motion of a spot on its surface.
SECTION X.
Rotation of the Earth invariable—Decrease in the Earth’s mean Temperature—Earth originally in a state of Fusion—Length of Day constant—Decrease of Temperature ascribed by Sir John Herschel to the variation in the Excentricity of the Terrestrial Orbit—Difference in the Temperature of the two Hemispheres erroneously ascribed to the Excess in the Length of Spring and Summer in the Southern Hemisphere; attributed by Sir Charles Lyell to the Operation of existing Causes—Three principal Axes of Rotation—Position of the Axis of Rotation on the Surface of the Earth invariable—Ocean not sufficient to restore the Equilibrium of the Earth if deranged—Its Density and mean Depth—Internal Structure of the Earth.
The rotation of the earth, which determines the length of the day, may be regarded as one of the most important elements in the system of the world. It serves as a measure of time, and forms the standard of comparison for the revolutions of the celestial bodies, which, by their proportional increase or decrease, would soon disclose any changes it might sustain. Theory and observation concur in proving that, among the innumerable vicissitudes which prevail throughout creation, the period of the earth’s diurnal rotation is immutable. The water of rivers, falling from a higher to a lower level, carries with it the velocity due to its revolution with the earth at a greater distance from the centre; it will therefore accelerate, although to an almost infinitesimal extent, the earth’s daily rotation. The sum of all these increments of velocity, arising from the descent of all the rivers on the earth’s surface, would in time become perceptible, did not nature, by the process of evaporation, raise the waters back to their sources, and thus, by again removing matter to a greater distance from the centre, destroy the velocity generated by its previous approach; so that the descent of rivers does not affect the earth’s rotation. Enormous masses projected by volcanoes from the equator to the poles, and the contrary, would indeed affect it, but there is no evidence of such convulsions. The disturbing action of the moon and planets, which has so powerful an effect on the revolution of the earth, in no way influences its rotation. The constant friction of the trade winds on the mountains and continents between the tropics does not impede its velocity, which theory even proves to be the same as if the sea, together with the earth, formed one solid mass. But, although these circumstances be insufficient, a variation in the mean temperature would certainly occasion a corresponding change in the velocity of rotation. In the science of dynamics it is a principle in a system of bodies or of particles revolving about a fixed centre, that the momentum or sum of the products of the mass of each into its angular velocity and distance from the centre is a constant quantity, if the system be not deranged by a foreign cause. Now, since the number of particles in the system is the same whatever its temperature may be, when their distances from the centre are diminished, their angular velocity must be increased, in order that the preceding quantity may still remain constant. It follows, then, that, as the primitive momentum of rotation with which the earth was projected into space must necessarily remain the same, the smallest decrease in heat, by contracting the terrestrial spheroid, would accelerate its rotation, and consequently diminish the length of the day. Notwithstanding the constant accession of heat from the sun’s rays, geologists have been induced to believe, from the fossil remains, that the mean temperature of the globe is decreasing.
The high temperature of mines, hot springs, and above all the internal fires which have produced, and do still occasion, such devastation on our planet, indicate an augmentation of heat towards its centre. The increase of density corresponding to the depth and the form of the spheroid, being what theory assigns to a fluid mass in rotation, concurs to induce the idea that the temperature of the earth was originally so high as to reduce all the substances of which it is composed to a state of fusion or of vapour, and that in the course of ages it has cooled down to its present state; that it is still becoming colder; and that it will continue to do so till the whole mass arrives at the temperature of the medium in which it is placed, or rather at a state of equilibrium between this temperature, the cooling power of its own radiation, and the heating effect of the sun’s rays.
Previous to the formation of ice at the poles, the ancient lands of northern latitudes might, no doubt, have been capable of producing those tropical plants preserved in the coal-measures, if indeed such plants could flourish without the intense light of a tropical sun. But, even if the decreasing temperature of the earth be sufficient to produce the observed effects, it must be extremely slow in its operation; for, in consequence of the rotation of the earth being a measure of the periods of the celestial motions, it has been proved that, if the length of the day had decreased by the three-thousandth part of a second since the observations of Hipparchus two thousand years ago, it would have diminished the secular equation of the moon by 44ʺ·4. It is, therefore, beyond a doubt that the mean temperature of the earth cannot have sensibly varied during that time. If, then, the appearances exhibited by the strata are really owing to a decrease of internal temperature, it either shows the immense periods requisite to produce geological changes, to which two thousand years are as nothing, or that the mean temperature of the earth had arrived at a state of equilibrium before these observations.
However strong the indications of the primitive fluidity of the earth, as there is no direct proof of it, the hypothesis can only be regarded as very probable. But one of the most profound philosophers and elegant writers of modern times has found in the secular variation of the excentricity of the terrestrial orbit an evident cause of decreasing temperature. That accomplished author, in pointing out the mutual dependencies of phenomena, says, “It is evident that the mean temperature of the whole surface of the globe, in so far as it is maintained by the action of the sun at a higher degree than it would have were the sun extinguished, must depend on the mean quantity of the sun’s rays which it receives, or—which comes to the same thing—on the total quantity received in a given invariable time; and, the length of the year being unchangeable in all the fluctuations of the planetary system, it follows that the total amount of solar radiation will determine, cæteris paribus, the general climate of the earth. Now, it is not difficult to show that this amount is inversely proportional to the minor axis of the ellipse described by the earth about the sun ([N. 143]), regarded as slowly variable; and that, therefore, the major axis remaining, as we know it to be, constant, and the orbit being actually in a state of approach to a circle, and consequently the minor axis being on the increase, the mean annual amount of solar radiation received by the whole earth must be actually on the decrease. We have, therefore, an evident real cause to account for the phenomenon.” The limits of the variation in the excentricity of the earth’s orbit are unknown. But, if its ellipticity has ever been as great as that of the orbit of Mercury or Pallas, the mean temperature of the earth must have been sensibly higher than it is at present. Whether it was great enough to render our northern climates fit for the production of tropical plants, and for the residence of the elephant and other animals now inhabitants of the torrid zone, it is impossible to say.
Of the decrease in temperature of the northern hemisphere there is abundant evidence in the fossil plants discovered in very high latitudes, which could only have existed in a tropical climate, and which must have grown near the spot where they are found, from the delicacy of their structure and the perfect state of their preservation. This change of temperature has been erroneously ascribed to an excess in the duration of spring and summer in the northern hemisphere, in consequence of the excentricity of the solar ellipse. The length of the seasons varies with the position of the perihelion ([N. 64]) of the earth’s orbit for two reasons. On account of the excentricity, small as it is, any line passing through the centre of the sun divides the terrestrial ellipse into two unequal parts, and by the laws of elliptical motion the earth moves through these two portions with unequal velocities. The perihelion always lies in the smaller portion, and there the earth’s motion is the most rapid. In the present position of the perihelion, spring and summer north of the equator exceed by about eight days the duration of the same seasons south of it. And 10,492 years ago the southern hemisphere enjoyed the advantage we now possess from the secular variation of the perihelion. Yet Sir John Herschel has shown that by this alternation neither hemisphere acquires any excess of light or heat above the other; for, although the earth is nearer to the sun while moving through that part of its orbit in which the perihelion lies than in the other part, and consequently receives a greater quantity of light and heat, yet as it moves faster it is exposed to the heat for a shorter time. In the other part of the orbit, on the contrary, the earth, being farther from the sun, receives fewer of his rays; but because its motion is slower, it is exposed to them for a longer time; and, as in both cases the quantity of heat and the angular velocity vary exactly in the same proportion, a perfect compensation takes place ([N. 144]). So that the excentricity of the earth’s orbit has little or no effect on the temperature corresponding to the difference of the seasons.
Sir Charles Lyell, in his excellent works on Geology, refers the increased cold of the northern hemisphere to the operation of existing causes with more probability than most theories that have been advanced in solution of this difficult subject. The loftiest mountains would be represented by a grain of sand on a globe six feet in diameter, and the depth of the ocean by a scratch on its surface. Consequently the gradual elevation of a continent or chain of mountains above the surface of the ocean, or their depression below it, is no very great event compared with the magnitude of the earth, and the energy of its subterranean fires, if the same periods of time be admitted in the progress of geological as in astronomical phenomena, which the successive and various races of extinct beings show to have been immense. Climate is always more intense in the interior of continents than in islands or sea-coasts. An increase of land within the tropics would therefore augment the general heat, and an increase in the temperate and frigid zones would render the cold more severe. Now it appears that most of the European, North Asiatic, and North American continents and islands were raised from the deep after the coal-measures were formed in which the fossil tropical plants are found; and a variety of geological facts indicate the existence of an ancient and extensive archipelago throughout the greater part of the northern hemisphere. Sir Charles Lyell is therefore of opinion that the climate of these islands must have been sufficiently mild, in consequence of the surrounding ocean, to clothe them with tropical plants, and render them a fit abode for the huge animals whose fossil remains are so often found; that the arborescent ferns and the palms of these regions, carried by streams to the bottom of the ocean, were imbedded in the strata which were by degrees heaved up by the subterranean fires during a long succession of ages, till the greater part of the northern hemisphere became dry land as it now is, and that the consequence has been a continual decrease of temperature.
It is evident, from the marine shells found on the tops of the highest mountains and in almost every part of the globe, that immense continents have been elevated above the ocean which must have engulfed others. Such a catastrophe would be occasioned by a variation in the position of the axis of rotation on the surface of the earth; for the seas tending to a new equator would leave some portions of the globe and overwhelm others. Now, it is found by the laws of mechanics that in every body, be its form or density what it may, there are at least three axes at right angles to each other, round any one of which, if the solid begins to rotate, it will continue to revolve for ever, provided it be not disturbed by a foreign cause, but that the rotation about any other axis will only be for an instant, and consequently the poles or extremities of the instantaneous axis of rotation would perpetually change their position on the surface of the body. In an ellipsoid of revolution the polar diameter and every diameter in the plane of the equator are the only permanent axes of rotation ([N. 145]). Hence, if the ellipsoid were to begin to revolve about any diameter between the pole and the equator, the motion would be so unstable that the axis of rotation and the position of the poles would change every instant. Therefore, as the earth does not differ much from this figure, if it did not turn round one of its principal axes, the position of the poles would change daily; the equator, which is 90° distant, would undergo corresponding variations; and the geographical latitudes of all places, being estimated from the equator, assumed to be fixed, would be perpetually changing. A displacement in the position of the poles of only two hundred miles would be sufficient to produce these effects, and would immediately be detected. But, as the latitudes are found to be invariable, it may be concluded that the terrestrial spheroid must have revolved about the same axis for ages. The earth and planets differ so little from ellipsoids of revolution, that in all probability any libration from one axis to another, produced by the primitive impulse which put them in motion, must have ceased soon after their creation from the friction of the fluids at their surface.
Theory also proves that neither nutation, precession, nor any of the disturbing forces that affect the system, have the smallest influence on the axis of rotation, which maintains a permanent position on the surface, if the earth be not disturbed in its rotation by a foreign cause, as the collision of a comet, which might have happened in the immensity of time. But, had that been the case, its effects would still have been perceptible in the variations of the geographical latitudes. If we suppose that such an event had taken place, and that the disturbance had been very great, equilibrium could then only have been restored with regard to a new axis of rotation by the rushing of the seas to the new equator, which they must have continued to do till the surface was everywhere perpendicular to the direction of gravity. But it is probable that such an accumulation of the waters would not be sufficient to restore equilibrium if the derangement had been great, for the mean density of the sea is only about a fifth part of the mean density of the earth, and the mean depth of the Pacific Ocean is supposed not to be more than four or five miles, whereas the equatorial diameter of the earth exceeds the polar diameter by about 261⁄2 miles. Consequently the influence of the sea on the direction of gravity is very small. And, as it thus appears that a great change in the position of the axis is incompatible with the law of equilibrium, the geological phenomena in question must be ascribed to an internal cause. Indeed it is now demonstrated that the strata containing marine diluvia, which are in lofty situations, must have been formed at the bottom of the ocean, and afterwards upheaved by the action of subterraneous fires. Besides, it is clear, from the mensuration of the arcs of the meridian and the length of the seconds’ pendulum, as well as from the lunar theory, that the internal strata and also the external outline of the globe are elliptical, their centres being coincident and their axes identical with that of the surface—a state of things which, according to the distinguished author lately quoted, is incompatible with a subsequent accommodation of the surface to a new and different state of rotation from that which determined the original distribution of the component matter. Thus, amidst the mighty revolutions which have swept innumerable races of organized beings from the earth, which have elevated plains and buried mountains in the ocean, the rotation of the earth and the position of the axes on its surface have undergone but slight variations.
The strata of the terrestrial spheroid are not only concentric and elliptical, but the lunar inequalities show that they increase in density from the surface of the earth to its centre. This would certainly have happened if the earth had originally been fluid, for the denser parts must have subsided towards the centre as it approached a state of equilibrium. But the enormous pressure of the superincumbent mass is a sufficient cause for the phenomenon. Professor Leslie observes that air compressed into the fiftieth part of its volume has its elasticity fifty times augmented. If it continues to contract at that rate, it would, from its own incumbent weight, acquire the density of water at the depth of thirty-four miles. But water itself would have its density doubled at the depth of ninety-three miles, and would even attain the density of quicksilver at a depth of 362 miles. Descending therefore towards the centre through nearly 4000 miles, the condensation of ordinary substances would surpass the utmost powers of conception. Dr. Young says that steel would be compressed into one-fourth and stone into one-eighth of its bulk at the earth’s centre. However, we are yet ignorant of the laws of compression of solid bodies beyond a certain limit; from the experiments of Mr. Perkins they appear to be capable of a greater degree of compression than has generally been imagined.
But a density so extreme is not borne out by astronomical observation. It might seem to follow therefore that our planet must have a widely cavernous structure, and that we tread on a crust or shell whose thickness bears a very small proportion to the diameter of its sphere. Possibly, too, this great condensation at the central regions may be counterbalanced by the increased elasticity due to a very elevated temperature.
SECTION XI.
Precession and Nutation—Their Effects on the Apparent Places of the Fixed Stars.
It has been shown that the axis of rotation is invariable on the surface of the earth; and observation as well as theory prove that, were it not for the action of the sun and moon on the matter at the equator, it would remain exactly parallel to itself in every point of its orbit.
The attraction of an external body not only draws a spheroid towards it, but, as the force varies inversely as the square of the distance, it gives it a motion about its centre of gravity, unless when the attracting body is situated in the prolongation of one of the axes of the spheroid. The plane of the equator is inclined to the plane of the ecliptic at an angle of 23° 27ʹ 28ʺ·29; and the inclination of the lunar orbit to the same is 5° 8ʹ 47ʺ·9. Consequently, from the oblate figure of the earth, the sun and moon, acting obliquely and unequally on the different parts of the terrestrial spheroid, urge the plane of the equator from its direction, and force it to move from east to west, so that the equinoctial points have a slow retrograde motion on the plane of the ecliptic of 50ʺ·41 annually. The direct tendency of this action is to make the planes of the equator and ecliptic coincide, but it is balanced by the tendency of the earth to return to stable rotation about the polar diameter, which is one of its principal axes of rotation. Therefore the inclination of the two planes remains constant, as a top spinning preserves the same inclination to the plane of the horizon. Were the earth spherical, this effect would not be produced, and the equinoxes would always correspond with the same points of the ecliptic, at least as far as this kind of motion is concerned. But another and totally different cause which operates on this motion has already been mentioned. The action of the planets on one another and on the sun occasions a very slow variation in the position of the plane of the ecliptic, which affects its inclination to the plane of the equator, and gives the equinoctial points a slow but direct motion on the ecliptic of 0ʺ·31 annually, which is entirely independent of the figure of the earth, and would be the same if it were a sphere. Thus the sun and moon by moving the plane of the equator cause the equinoctial points to retrograde on the ecliptic: and the planets by moving the plane of the ecliptic give them a direct motion, though much less than the former. Consequently the difference of the two is the mean precession, which is proved both by theory and observation to be about 50ʺ·1 annually ([N. 146]).
As the longitudes of all the fixed stars are increased by this quantity, the effects of precession are soon detected. It was accordingly discovered by Hipparchus in the year 128 before Christ, from a comparison of his own observations with those of Timocharis 155 years before. In the time of Hipparchus the entrance of the sun into the constellation Aries was the beginning of spring; but since that time the equinoctial points have receded 30°, so that the constellations called the signs of the zodiac are now at a considerable distance from those divisions of the ecliptic which bear their names. Moving at the rate of 50ʺ·1 annually, the equinoctial points will accomplish a revolution in 25,868 years. But, as the precession varies in different centuries, the extent of this period will be slightly modified. Since the motion of the sun is direct, and that of the equinoctial points retrograde, he takes a shorter time to return to the equator than to arrive at the same stars; so that the tropical year of 365d 5h 48m 49s·7 must be increased by the time he takes to move through an arc of 50ʺ·1, in order to have the length of the sidereal year. The time required is 20m 19s·6, so that the sidereal year contains 365d 6h 9m 9s·6 mean solar days.
The mean annual precession is subject to a secular variation; for, although the change in the plane of the ecliptic in which the orbit of the sun lies be independent of the form of the earth, yet, by bringing the sun, moon, and earth into different relative positions from age to age, it alters the direct action of the two first on the prominent matter at the equator: on this account the motion of the equinox is greater by 0ʺ·455 now than it was in the time of Hipparchus. Consequently the actual length of the tropical year is about 4s·21 shorter than it was at that time. The utmost change that it can experience from this cause amounts to 43 seconds.
Such is the secular motion of the equinoxes. But it is sometimes increased and sometimes diminished by periodic variations, whose periods depend upon the relative positions of the sun and moon with regard to the earth, and which are occasioned by the direct action of these bodies on the equator. Dr. Bradley discovered that by this action the moon causes the pole of the equator to describe a small ellipse in the heavens, the axes of which are 18ʺ·5 and 13ʺ·674, the longer being directed towards the pole of the ecliptic. The period of this inequality is about 19 years, the time employed by the nodes of the lunar orbit to accomplish a revolution. The sun causes a small variation in the description of this ellipse; it runs through its period in half a year. Since the whole earth obeys these motions, they affect the position of its axis of rotation with regard to the starry heavens, though not with regard to the surface of the earth; for in consequence of precession alone the pole of the equator moves in a circle round the pole of the ecliptic in 25,868 years, and by nutation alone it describes a small ellipse in the heavens every 19 years, on each side of which it deviates every half-year from the action of the sun. The real curve traced in the starry heavens by the imaginary prolongation of the earth’s axis is compounded of these three motions ([N. 147]). This nutation in the earth’s axis affects both the precession and obliquity with small periodic variations. But in consequence of the secular variation in the position of the terrestrial orbit, which is chiefly owing to the disturbing energy of Jupiter on the earth, the obliquity of the ecliptic is annually diminished, according to M. Bessel, by 0ʺ·457. This variation in the course of ages may amount to 10 or 11 degrees; but the obliquity of the ecliptic to the equator can never vary more than 2° 42ʹ or 3°, since the equator will follow in some measure the motion of the ecliptic.
It is evident that the places of all the celestial bodies are affected by precession and nutation. Their longitudes estimated from the equinox are augmented by precession; but, as it affects all the bodies equally, it makes no change in their relative positions. Both the celestial latitudes and longitudes are altered to a small degree by nutation; hence all observations must be corrected for these inequalities. In consequence of this real motion in the earth’s axis, the pole-star, forming part of the constellation of the Little Bear, which was formerly 12° from the celestial pole, is now within 1° 24ʹ of it, and will continue to approach it till it is within 1⁄2°, after which it will retreat from the pole for ages; and 12,934 years hence the star α Lyræ will come within 5° of the celestial pole, and become the polar star of the northern hemisphere.
SECTION XII.
Mean and Apparent Sidereal Time—Mean and Apparent Solar Time—Equation of Time—English and French Subdivisions of Time—Leap Year—Christian Era—Equinoctial Time—Remarkable Eras depending upon the Position of the Solar Perigee—Inequality of the Lengths of the Seasons in the two Hemispheres—Application of Astronomy to Chronology—English and French Standards of Weights and Measures.
Astronomy has been of immediate and essential use in affording invariable standards for measuring duration, distance, magnitude, and velocity. The mean sidereal day measured by the time elapsed between two consecutive transits of any star at the same meridian ([N. 148]), and the mean sidereal year which is the time included between two consecutive returns of the sun to the same star, are immutable units with which all great periods of time are compared; the oscillations of the isochronous pendulum measure its smaller portions. By these invariable standards alone we can judge of the slow changes that other elements of the system may have undergone. Apparent sidereal time, which is measured by the transit of the equinoctial point at the meridian of any place, is a variable quantity, from the effects of precession and nutation. Clocks showing apparent sidereal time are employed for observation, and are so regulated that they indicate 0h 0m 0s at the instant the equinoctial point passes the meridian of the observatory. And as time is a measure of angular motion, the clock gives the distances of the heavenly bodies from the equinox by observing the instant at which each passes the meridian, and converting the interval into arcs at the rate of 15° to an hour.
The returns of the sun to the meridian and to the same equinox or solstice have been universally adopted as the measure of our civil days and years. The solar or astronomical day is the time that elapses between two consecutive noons or midnights. It is consequently longer than the sidereal day, on account of the proper motion of the sun during a revolution of the celestial sphere. But, as the sun moves with greater rapidity at the winter than at the summer solstice, the astronomical day is more nearly equal to the sidereal day in summer than in winter. The obliquity of the ecliptic also affects its duration; for near the equinoxes the arc of the equator is less than the corresponding arc of the ecliptic, and in the solstices it is greater ([N. 149]). The astronomical day is therefore diminished in the first case, and increased in the second. If the sun moved uniformly in the equator at the rate of 59ʹ 8ʺ·33 every day, the solar days would be all equal. The time therefore which is reckoned by the arrival of an imaginary sun at the meridian, or of one which is supposed to move uniformly in the equator, is denominated mean solar time, and is given by clocks and watches in common life. When it is reckoned by the arrival of the real sun at the meridian, it is true or apparent time, and is given by dials. The difference between the time shown by a clock and a dial is the equation of time given in the Nautical Almanac, sometimes amounting to as much as sixteen minutes. The apparent and mean time coincide four times in the year; when the sun’s daily motion in right ascension is equal to 59ʹ 8ʺ·33 in a mean solar day, which happens about the 16th of April, the 16th of June, the 1st of September, and the 25th of December.
The astronomical day begins at noon, but in common reckoning the day begins at midnight. In England it is divided into twenty-four hours, which are counted by twelve and twelve; but in France astronomers, adopting the decimal division, divide the day into ten hours, the hour into one hundred minutes, and the minute into a hundred seconds, because of the facility in computation, and in conformity with their decimal system of weights and measures. This subdivision is not now used in common life, nor has it been adopted in any other country; and although some scientific writers in France still employ that division of time, the custom is beginning to wear out. At one period during the French Revolution, the clock in the gardens of the Tuileries was regulated to show decimal time. The mean length of the day, though accurately determined, is not sufficient for the purposes either of astronomy or civil life. The tropical or civil year of 365d 5h 48m 49s·7, which is the time elapsed between the consecutive returns of the sun to the mean equinoxes or solstices, including all the changes of the seasons, is a natural cycle peculiarly suited for a measure of duration. It is estimated from the winter solstice, the middle of the long annual night under the north pole. But although the length of the civil year is pointed out by nature as a measure of long periods, the incommensurability that exists between the length of the day and the revolution of the sun renders it difficult to adjust the estimation of both in whole numbers. If the revolution of the sun were accomplished in 365 days, all the years would be of precisely the same number of days, and would begin and end with the sun at the same point of the ecliptic. But as the sun’s revolution includes the fraction of a day, a civil year and a revolution of the sun have not the same duration. Since the fraction is nearly the fourth of a day, in four years it is nearly equal to a revolution of the sun, so that the addition of a supernumerary day every fourth year nearly compensates the difference. But in process of time further correction will be necessary, because the fraction is less than the fourth of a day. In fact, if a bissextile be suppressed at the end of three out of four centuries, the year so determined will only exceed the true year by an extremely small fraction of a day; and if in addition to this a bissextile be suppressed every 4000 years, the length of the year will be nearly equal to that given by observation. Were the fraction neglected, the beginning of the year would precede that of the tropical year, so that it would retrograde through the different seasons in a period of about 1507 years. The Egyptian year began with the heliacal rising of Sirius ([N. 150]), and contained only 365 days, by which they lost one year in every 1461 years, their Sothaic period, or that cycle in which the heliacal rising of Sirius passes through the whole year and takes place again on the same day. The division of the year into months is very old and almost universal. But the period of seven days, by far the most permanent division of time, and the most ancient monument of astronomical knowledge, was used by the Brahmins in India with the same denominations employed by us, and was alike found in the calendars of the Jews, Egyptians, Arabs, and Assyrians. It has survived the fall of empires, and has existed among all successive generations, a proof of their common origin.
The day of the new moon immediately following the winter solstice in the 707th year of Rome was made the 1st of January of the first year of Julius Cæsar. The 25th of December of his forty-fifth year is considered as the date of Christ’s nativity; and the forty-sixth year of the Julian Calendar is assumed to be the first of our era. The preceding year is called the first year before Christ by chronologists, but by astronomers it is called the year 0. The astronomical year begins on the 31st of December at noon; and the date of an observation expresses the days and hours which have actually elapsed since that time.