  During the period of the intellectual and aesthetic revival, at the beginning of the sixteenth century, the “spirit of the age” was fostered by the invention of printing, by the downfall of the Byzantine Empire, and the scattering of Greek fugitives, carrying the treasures of literature through Western Europe, by the works of Raphael and Michael Angelo, by the Reformation, and by the extension of the known world through the voyages of Spaniards and Portuguese. During that period there came to the front the founder of accurate observational astronomy. Tycho Brahe, a Dane, born in 1546 of noble parents, was the most distinguished, diligent, and accurate observer of the heavens since the days of Hipparchus, 1,700 years before. Tycho was devoted entirely to his science from childhood, and the opposition of his parents only stimulated him in his efforts to overcome difficulties. He soon grasped the hopelessness of the old deductive methods of reasoning, and decided that no theories ought to be indulged in until preparations had been made by the accumulation of accurate observations. We may claim for him the title of founder of the inductive method. For a complete life of this great man the reader is referred to Dreyer’s Tycho Brahe, Edinburgh, 1890, containing a complete bibliography. The present notice must be limited to noting the work done, and the qualities of character which enabled him to attain his scientific aims, and which have been conspicuous in many of his successors. He studied in Germany, but King Frederick of Denmark, appreciating his great talents, invited him to carry out his life’s work in that country. He granted to him the island of Hveen, gave him a pension, and made him a canon of the Cathedral of Roskilde. On that island Tycho Brahe built the splendid observatory which he called Uraniborg, and, later, a second one for his assistants and students, called Stjerneborg. These he fitted up with the most perfect instruments, and never lost a chance of adding to his stock of careful observations.[1] The account of all these instruments and observations, printed at his own press on the island, was published by Tycho Brahe himself, and the admirable and numerous engravings bear witness to the excellence of design and the stability of his instruments. His mechanical skill was very great, and in his workmanship he was satisfied with nothing but the best. He recognised the importance of rigidity in the instruments, and, whereas these had generally been made of wood, he designed them in metal. His instruments included armillae like those which had been used in Alexandria, and other armillae designed by himself—sextants, mural quadrants, large celestial globes and various instruments for special purposes. He lived before the days of telescopes and accurate clocks. He invented the method of sub-dividing the degrees on the arc of an instrument by transversals somewhat in the way that Pedro Nunez had proposed. He originated the true system of observation and reduction of observations, recognising the fact that the best instrument in the world is not perfect; and with each of his instruments he set to work to find out the errors of graduation and the errors of mounting, the necessary correction being applied to each observation. When he wanted to point his instrument exactly to a star he was confronted with precisely the same difficulty as is met in gunnery and rifle-shooting. The sights and the object aimed at cannot be in focus together, and a great deal depends on the form of sight. Tycho Brahe invented, and applied to the pointers of his instruments, an aperture-sight of variable area, like the iris diaphragm used now in photography. This enabled him to get the best result with stars of different brightness. The telescope not having been invented, he could not use a telescopic-sight as we now do in gunnery. This not only removes the difficulty of focussing, but makes the minimum visible angle smaller. Helmholtz has defined the minimum angle measurable with the naked eye as being one minute of arc. In view of this it is simply marvellous that, when the positions of Tycho’s standard stars are compared with the best modern catalogues, his probable error in right ascension is only ± 24”, 1, and in declination only ± 25”, 9. Clocks of a sort had been made, but Tycho Brahe found them so unreliable that he seldom used them, and many of his position-measurements were made by measuring the angular distances from known stars. Taking into consideration the absence of either a telescope or a clock, and reading his account of the labour he bestowed upon each observation, we must all agree that Kepler, who inherited these observations in MS., was justified, under the conditions then existing, in declaring that there was no hope of anyone ever improving upon them. In the year 1572, on November 11th, Tycho discovered in Cassiopeia a new star of great brilliance, and continued to observe it until the end of January, 1573. So incredible to him was such an event that he refused to believe his own eyes until he got others to confirm what he saw. He made accurate observations of its distance from the nine principal stars in Casseiopeia, and proved that it had no measurable parallax. Later he employed the same method with the comets of 1577, 1580, 1582, 1585, 1590, 1593, and 1596, and proved that they too had no measurable parallax and must be very distant. The startling discovery that stars are not necessarily permanent, that new stars may appear, and possibly that old ones may disappear, had upon him exactly the same effect that a similar occurrence had upon Hipparchus 1,700 years before. He felt it his duty to catalogue all the principal stars, so that there should be no mistake in the future. During the construction of his catalogue of 1,000 stars he prepared and used accurate tables of refraction deduced from his own observations. Thus he eliminated (so far as naked eye observations required) the effect of atmospheric refraction which makes the altitude of a star seem greater than it really is. Tycho Brahe was able to correct the lunar theory by his observations. Copernicus had introduced two epicycles on the lunar orbit in the hope of obtaining a better accordance between theory and observation; and he was not too ambitious, as his desire was to get the tables accurate to ten minutes. Tycho Brahe found that the tables of Copernicus were in error as much as two degrees. He re-discovered the inequality called “variation” by observing the moon in all phases—a thing which had not been attended to. [It is remarkable that in the nineteenth century Sir George Airy established an altazimuth at Greenwich Observatory with this special object, to get observations of the moon in all phases.] He also discovered other lunar equalities, and wanted to add another epicycle to the moon’s orbit, but he feared that these would soon become unmanageable if further observations showed more new inequalities. But, as it turned out, the most fruitful work of Tycho Brahe was on the motions of the planets, and especially of the planet Mars, for it was by an examination of these results that Kepler was led to the discovery of his immortal laws. After the death of King Frederick the observatories of Tycho Brahe were not supported. The gigantic power and industry displayed by this determined man were accompanied, as often happens, by an overbearing manner, intolerant of obstacles. This led to friction, and eventually the observatories were dismantled, and Tycho Brahe was received by the Emperor Rudolph II., who placed a house in Prague at his disposal. Here he worked for a few years, with Kepler as one of his assistants, and he died in the year 1601. It is an interesting fact that Tycho Brahe had a firm conviction that mundane events could be predicted by astrology, and that this belief was supported by his own predictions. It has already been stated that Tycho Brahe maintained that observation must precede theory. He did not accept the Copernican theory that the earth moves, but for a working hypothesis he used a modification of an old Egyptian theory, mathematically identical with that of Copernicus, but not involving a stellar parallax. He says (De Mundi, etc.) that the Ptolemean system was too complicated, and the new one which that great man Copernicus had proposed, following in the footsteps of Aristarchus of Samos, though there was nothing in it contrary to mathematical principles, was in opposition to those of physics, as the heavy and sluggish earth is unfit to move, and the system is even opposed to the authority of Scripture. The absence of annual parallax further involves an incredible distance between the outermost planet and the fixed stars. We are bound to admit that in the circumstances of the case, so long as there was no question of dynamical forces connecting the members of the solar system, his reasoning, as we should expect from such a man, is practical and sound. It is not surprising, then, that astronomers generally did not readily accept the views of Copernicus, that Luther (Luther’s Tischreden, pp. 22, 60) derided him in his usual pithy manner, that Melancthon (Initia doctrinae physicae) said that Scripture, and also science, are against the earth’s motion; and that the men of science whose opinion was asked for by the cardinals (who wished to know whether Galileo was right or wrong) looked upon Copernicus as a weaver of fanciful theories. Johann Kepler is the name of the man whose place, as is generally agreed, would have been the most difficult to fill among all those who have contributed to the advance of astronomical knowledge. He was born at Wiel, in the Duchy of Wurtemberg, in 1571. He held an appointment at Gratz, in Styria, and went to join Tycho Brahe in Prague, and to assist in reducing his observations. These came into his possession when Tycho Brahe died, the Emperor Rudolph entrusting to him the preparation of new tables (called the Rudolphine tables) founded on the new and accurate observations. He had the most profound respect for the knowledge, skill, determination, and perseverance of the man who had reaped such a harvest of most accurate data; and though Tycho hardly recognised the transcendent genius of the man who was working as his assistant, and although there were disagreements between them, Kepler held to his post, sustained by the conviction that, with these observations to test any theory, he would be in a position to settle for ever the problem of the solar system. PORTRAIT OF JOHANNES KEPLER. It has seemed to many that Plato’s demand for uniform circular motion (linear or angular) was responsible for a loss to astronomy of good work during fifteen hundred years, for a hundred ill-considered speculative cosmogonies, for dissatisfaction, amounting to disgust, with these À priori guesses, and for the relegation of the science to less intellectual races than Greeks and other Europeans. Nobody seemed to dare to depart from this fetish of uniform angular motion and circular orbits until the insight, boldness, and independence of Johann Kepler opened up a new world of thought and of intellectual delight. While at work on the Rudolphine tables he used the old epicycles and deferents and excentrics, but he could not make theory agree with observation. His instincts told him that these apologists for uniform motion were a fraud; and he proved it to himself by trying every possible variation of the elements and finding them fail. The number of hypotheses which he examined and rejected was almost incredible (for example, that the planets turn round centres at a little distance from the sun, that the epicycles have centres at a little distance from the deferent, and so on). He says that, after using all these devices to make theory agree with Tycho’s observations, he still found errors amounting to eight minutes of a degree. Then he said boldly that it was impossible that so good an observer as Tycho could have made a mistake of eight minutes, and added: “Out of these eight minutes we will construct a new theory that will explain the motions of all the planets.” And he did it, with elliptic orbits having the sun in a focus of each.[2] It is often difficult to define the boundaries between fancies, imagination, hypothesis, and sound theory. This extraordinary genius was a master in all these modes of attacking a problem. His analogy between the spaces occupied by the five regular solids and the distances of the planets from the sun, which filled him with so much delight, was a display of pure fancy. His demonstration of the three fundamental laws of planetary motion was the most strict and complete theory that had ever been attempted. It has been often suggested that the revival by Copernicus of the notion of a moving earth was a help to Kepler. No one who reads Kepler’s great book could hold such an opinion for a moment. In fact, the excellence of Copernicus’s book helped to prolong the life of the epicyclical theories in opposition to Kepler’s teaching. All of the best theories were compared by him with observation. These were the Ptolemaic, the Copernican, and the Tychonic. The two latter placed all of the planetary orbits concentric with one another, the sun being placed a little away from their common centre, and having no apparent relation to them, and being actually outside the planes in which they move. Kepler’s first great discovery was that the planes of all the orbits pass through the sun; his second was that the line of apses of each planet passes through the sun; both were contradictory to the Copernican theory. He proceeds cautiously with his propositions until he arrives at his great laws, and he concludes his book by comparing observations of Mars, of all dates, with his theory. His first law states that the planets describe ellipses with the sun at a focus of each ellipse. His second law (a far more difficult one to prove) states that a line drawn from a planet to the sun sweeps over equal areas in equal times. These two laws were published in his great work, Astronomia Nova, sen. Physica Coelestis tradita commentariis de Motibus Stelloe; Martis, Prague, 1609. It took him nine years more[3] to discover his third law, that the squares of the periodic times are proportional to the cubes of the mean distances from the sun. These three laws contain implicitly the law of universal gravitation. They are simply an alternative way of expressing that law in dealing with planets, not particles. Only, the power of the greatest human intellect is so utterly feeble that the meaning of the words in Kepler’s three laws could not be understood until expounded by the logic of Newton’s dynamics. The joy with which Kepler contemplated the final demonstration of these laws, the evolution of which had occupied twenty years, can hardly be imagined by us. He has given some idea of it in a passage in his work on Harmonics, which is not now quoted, only lest someone might say it was egotistical—a term which is simply grotesque when applied to such a man with such a life’s work accomplished. The whole book, Astronomia Nova, is a pleasure to read; the mass of observations that are used, and the ingenuity of the propositions, contrast strongly with the loose and imperfectly supported explanations of all his predecessors; and the indulgent reader will excuse the devotion of a few lines to an example of the ingenuity and beauty of his methods. It may seem a hopeless task to find out the true paths of Mars and the earth (at that time when their shape even was not known) from the observations giving only the relative direction from night to night. Now, Kepler had twenty years of observations of Mars to deal with. This enabled him to use a new method, to find the earth’s orbit. Observe the date at any time when Mars is in opposition. The earth’s position E at that date gives the longitude of Mars M. His period is 687 days. Now choose dates before and after the principal date at intervals of 687 days and its multiples. Mars is in each case in the same position. Now for any date when Mars is at M and the earth at E3 the date of the year gives the angle E3SM. And the observation of Tycho gives the direction of Mars compared with the sun, SE3M. So all the angles of the triangle SEM in any of these positions of E are known, and also the ratios of SE1, SE2, SE3, SE4 to SM and to each other. For the orbit of Mars observations were chosen at intervals of a year, when the earth was always in the same place. But Kepler saw much farther than the geometrical facts. He realised that the orbits are followed owing to a force directed to the sun; and he guessed that this is the same force as the gravity that makes a stone fall. He saw the difficulty of gravitation acting through the void space. He compared universal gravitation to magnetism, and speaks of the work of Gilbert of Colchester. (Gilbert’s book, De Mundo Nostro Sublunari, Philosophia Nova, Amstelodami, 1651, containing similar views, was published forty-eight years after Gilbert’s death, and forty-two years after Kepler’s book and reference. His book De Magnete was published in 1600.) A few of Kepler’s views on gravitation, extracted from the Introduction to his Astronomia Nova, may now be mentioned:— 1. Every body at rest remains at rest if outside the attractive power of other bodies. 2. Gravity is a property of masses mutually attracting in such manner that the earth attracts a stone much more than a stone attracts the earth. 3. Bodies are attracted to the earth’s centre, not because it is the centre of the universe, but because it is the centre of the attracting particles of the earth. 4. If the earth be not round (but spheroidal?), then bodies at different latitudes will not be attracted to its centre, but to different points in the neighbourhood of that centre. 5. If the earth and moon were not retained in their orbits by vital force (aut alia aligua aequipollenti), the earth and moon would come together. 6. If the earth were to cease to attract its waters, the oceans would all rise and flow to the moon. 7. He attributes the tides to lunar attraction. Kepler had been appointed Imperial Astronomer with a handsome salary (on paper), a fraction of which was doled out to him very irregularly. He was led to miserable makeshifts to earn enough to keep his family from starvation; and proceeded to Ratisbon in 1630 to represent his claims to the Diet. He arrived worn out and debilitated; he failed in his appeal, and died from fever, contracted under, and fed upon, disappointment and exhaustion. Those were not the days when men could adopt as a profession the “research of endowment.” Before taking leave of Kepler, who was by no means a man of one idea, it ought to be here recorded that he was the first to suggest that a telescope made with both lenses convex (not a Galilean telescope) can have cross wires in the focus, for use as a pointer to fix accurately the positions of stars. An Englishman, Gascoigne, was the first to use this in practice. From the all too brief epitome here given of Kepler’s greatest book, it must be obvious that he had at that time some inkling of the meaning of his laws—universal gravitation. From that moment the idea of universal gravitation was in the air, and hints and guesses were thrown out by many; and in time the law of gravitation would doubtless have been discovered, though probably not by the work of one man, even if Newton had not lived. But, if Kepler had not lived, who else could have discovered his laws? FOOTNOTES: It is now necessary to leave the subject of dynamical astronomy for a short time in order to give some account of work in a different direction originated by a contemporary of Kepler’s, his senior in fact by seven years. Galileo Galilei was born at Pisa in 1564. The most scientific part of his work dealt with terrestrial dynamics; but one of those fortunate chances which happen only to really great men put him in the way of originating a new branch of astronomy. The laws of motion had not been correctly defined. The only man of Galileo’s time who seems to have worked successfully in the same direction as himself was that Admirable Crichton of the Italians, Leonardo da Vinci. Galileo cleared the ground. It had always been noticed that things tend to come to rest; a ball rolled on the ground, a boat moved on the water, a shot fired in the air. Galileo realised that in all of these cases a resisting force acts to stop the motion, and he was the first to arrive at the not very obvious law that the motion of a body will never stop, nor vary its speed, nor change its direction, except by the action of some force. It is not very obvious that a light body and a heavy one fall at the same speed (except for the resistance of the air). Galileo proved this on paper, but to convince the world he had to experiment from the leaning tower of Pisa. At an early age he discovered the principle of isochronism of the pendulum, which, in the hands of Huyghens in the middle of the seventeenth century, led to the invention of the pendulum clock, perhaps the most valuable astronomical instrument ever produced. These and other discoveries in dynamics may seem very obvious now; but it is often the most every-day matters which have been found to elude the inquiries of ordinary minds, and it required a high order of intellect to unravel the truth and discard the stupid maxims scattered through the works of Aristotle and accepted on his authority. A blind worship of scientific authorities has often delayed the progress of human knowledge, just as too much “instruction” of a youth often ruins his “education.” Grant, in his history of Physical Astronomy, has well said that “the sagacity and skill which Galileo displays in resolving the phenomena of motion into their constituent elements, and hence deriving the original principles involved in them, will ever assure to him a distinguished place among those who have extended the domains of science.” But it was work of a different kind that established Galileo’s popular reputation. In 1609 Galileo heard that a Dutch spectacle-maker had combined a pair of lenses so as to magnify distant objects. Working on this hint, he solved the same problem, first on paper and then in practice. So he came to make one of the first telescopes ever used in astronomy. No sooner had he turned it on the heavenly bodies than he was rewarded by such a shower of startling discoveries as forthwith made his name the best known in Europe. He found curious irregular black spots on the sun, revolving round it in twenty-seven days; hills and valleys on the moon; the planets showing discs of sensible size, not points like the fixed stars; Venus showing phases according to her position in relation to the sun; Jupiter accompanied by four moons; Saturn with appendages that he could not explain, but unlike the other planets; the Milky Way composed of a multitude of separate stars. His fame flew over Europe like magic, and his discoveries were much discussed—and there were many who refused to believe. Cosmo de Medici induced him to migrate to Florence to carry on his observations. He was received by Paul V., the Pope, at Rome, to whom he explained his discoveries. He thought that these discoveries proved the truth of the Copernican theory of the Earth’s motion; and he urged this view on friends and foes alike. Although in frequent correspondence with Kepler, he never alluded to the New Astronomy, and wrote to him extolling the virtue of epicycles. He loved to argue, never shirked an encounter with any number of disputants, and laughed as he broke down their arguments. Through some strange course of events, not easy to follow, the Copernican theory, whose birth was welcomed by the Church, had now been taken up by certain anti-clerical agitators, and was opposed by the cardinals as well as by the dignitaries of the Reformed Church. Galileo—a good Catholic—got mixed up in these discussions, although on excellent terms with the Pope and his entourage. At last it came about that Galileo was summoned to appear at Rome, where he was charged with holding and teaching heretical opinions about the movement of the earth; and he then solemnly abjured these opinions. There has been much exaggeration and misstatement about his trial and punishment, and for a long time there was a great deal of bitterness shown on both sides. But the general verdict of the present day seems to be that, although Galileo himself was treated with consideration, the hostility of the Church to the views of Copernicus placed it in opposition also to the true Keplerian system, and this led to unprofitable controversies. From the time of Galileo onwards, for some time, opponents of religion included the theory of the Earth’s motion in their disputations, not so much for the love, or knowledge, of astronomy, as for the pleasure of putting the Church in the wrong. This created a great deal of bitterness and intolerance on both sides. Among the sufferers was Giordano Bruno, a learned speculative philosopher, who was condemned to be burnt at the stake. Galileo died on Christmas Day, 1642—the day of Newton’s birth. The further consideration of the grand field of discovery opened out by Galileo with his telescopes must be now postponed, to avoid discontinuity in the history of the intellectual development of this period, which lay in the direction of dynamical, or physical, astronomy. Until the time of Kepler no one seems to have conceived the idea of universal physical forces controlling terrestrial phenomena, and equally applicable to the heavenly bodies. The grand discovery by Kepler of the true relationship of the Sun to the Planets, and the telescopic discoveries of Galileo and of those who followed him, spread a spirit of inquiry and philosophic thought throughout Europe, and once more did astronomy rise in estimation; and the irresistible logic of its mathematical process of reasoning soon placed it in the position it has ever since occupied as the foremost of the exact sciences. The practical application of this process of reasoning was enormously facilitated by the invention of logarithms by Napier. He was born at Merchistoun, near Edinburgh, in 1550, and died in 1617. By this system the tedious arithmetical operations necessary in astronomical calculations, especially those dealing with the trigonometrical functions of angles, were so much simplified that Laplace declared that by this invention the life-work of an astronomer was doubled. Jeremiah Horrocks (born 1619, died 1641) was an ardent admirer of Tycho Brahe and Kepler, and was able to improve the Rudolphine tables so much that he foretold a transit of Venus, in 1639, which these tables failed to indicate, and was the only observer of it. His life was short, but he accomplished a great deal, and rightly ascribed the lunar inequality called evection to variations in the value of the eccentricity and in the direction of the line of apses, at the same time correctly assigning the disturbing force of the Sun as the cause. He discovered the errors in Jupiter’s calculated place, due to what we now know as the long inequality of Jupiter and Saturn, and measured with considerable accuracy the acceleration at that date of Jupiter’s mean motion, and indicated the retardation of Saturn’s mean motion. Horrocks’ investigations, so far as they could be collected, were published posthumously in 1672, and seldom, if ever, has a man who lived only twenty-two years originated so much scientific knowledge. At this period British science received a lasting impetus by the wise initiation of a much-abused man, Charles II., who founded the Royal Society of London, and also the Royal Observatory of Greeenwich, where he established Flamsteed as first Astronomer Royal, especially for lunar and stellar observations likely to be useful for navigation. At the same time the French Academy and the Paris Observatory were founded. All this within fourteen years, 1662-1675. Meanwhile gravitation in general terms was being discussed by Hooke, Wren, Halley, and many others. All of these men felt a repugnance to accept the idea of a force acting across the empty void of space. Descartes (1596-1650) proposed an ethereal medium whirling round the sun with the planets, and having local whirls revolving with the satellites. As Delambre and Grant have said, this fiction only retarded the progress of pure science. It had no sort of relation to the more modern, but equally misleading, “nebular hypothesis.” While many were talking and guessing, a giant mind was needed at this stage to make things clear. We now reach the period which is the culminating point of interest in the history of dynamical astronomy. Isaac Newton was born in 1642. Pemberton states that Newton, having quitted Cambridge to avoid the plague, was residing at Wolsthorpe, in Lincolnshire, where he had been born; that he was sitting one day in the garden, reflecting upon the force which prevents a planet from flying off at a tangent and which draws it to the sun, and upon the force which draws the moon to the earth; and that he saw in the case of the planets that the sun’s force must clearly be unequal at different distances, for the pull out of the tangential line in a minute is less for Jupiter than for Mars. He then saw that the pull of the earth on the moon would be less than for a nearer object. It is said that while thus meditating he saw an apple fall from a tree to the ground, and that this fact suggested the questions: Is the force that pulled that apple from the tree the same as the force which draws the moon to the earth? Does the attraction for both of them follow the same law as to distance as is given by the planetary motions round the sun? It has been stated that in this way the first conception of universal gravitation arose.[1] Quite the most important event in the whole history of physical astronomy was the publication, in 1687, of Newton’s Principia (Philosophiae Naturalis Principia Mathematica). In this great work Newton started from the beginning of things, the laws of motion, and carried his argument, step by step, into every branch of physical astronomy; giving the physical meaning of Kepler’s three laws, and explaining, or indicating the explanation of, all the known heavenly motions and their irregularities; showing that all of these were included in his simple statement about the law of universal gravitation; and proceeding to deduce from that law new irregularities in the motions of the moon which had never been noticed, and to discover the oblate figure of the earth and the cause of the tides. These investigations occupied the best part of his life; but he wrote the whole of his great book in fifteen months. Having developed and enunciated the true laws of motion, he was able to show that Kepler’s second law (that equal areas are described by the line from the planet to the sun in equal times) was only another way of saying that the centripetal force on a planet is always directed to the sun. Also that Kepler’s first law (elliptic orbits with the sun in one focus) was only another way of saying that the force urging a planet to the sun varies inversely as the square of the distance. Also (if these two be granted) it follows that Kepler’s third law is only another way of saying that the sun’s force on different planets (besides depending as above on distance) is proportional to their masses. Having further proved the, for that day, wonderful proposition that, with the law of inverse squares, the attraction by the separate particles of a sphere of uniform density (or one composed of concentric spherical shells, each of uniform density) acts as if the whole mass were collected at the centre, he was able to express the meaning of Kepler’s laws in propositions which have been summarised as follows:— The law of universal gravitation.—Every particle of matter in the universe attracts every other particle with a force varying inversely as the square of the distance between them, and directly as the product of the masses of the two particles.[2] But Newton did not commit himself to the law until he had answered that question about the apple; and the above proposition now enabled him to deal with the Moon and the apple. Gravity makes a stone fall 16.1 feet in a second. The moon is 60 times farther from the earth’s centre than the stone, so it ought to be drawn out of a straight course through 16.1 feet in a minute. Newton found the distance through which she is actually drawn as a fraction of the earth’s diameter. But when he first examined this matter he proceeded to use a wrong diameter for the earth, and he found a serious discrepancy. This, for a time, seemed to condemn his theory, and regretfully he laid that part of his work aside. Fortunately, before Newton wrote the Principia the French astronomer Picard made a new and correct measure of an arc of the meridian, from which he obtained an accurate value of the earth’s diameter. Newton applied this value, and found, to his great joy, that when the distance of the moon is 60 times the radius of the earth she is attracted out of the straight course 16.1 feet per minute, and that the force acting on a stone or an apple follows the same law as the force acting upon the heavenly bodies.[3] The universality claimed for the law—if not by Newton, at least by his commentators—was bold, and warranted only by the large number of cases in which Newton had found it to apply. Its universality has been under test ever since, and so far it has stood the test. There has often been a suspicion of a doubt, when some inequality of motion in the heavenly bodies has, for a time, foiled the astronomers in their attempts to explain it. But improved mathematical methods have always succeeded in the end, and so the seeming doubt has been converted into a surer conviction of the universality of the law. Having once established the law, Newton proceeded to trace some of its consequences. He saw that the figure of the earth depends partly on the mutual gravitation of its parts, and partly on the centrifugal tendency due to the earth’s rotation, and that these should cause a flattening of the poles. He invented a mathematical method which he used for computing the ratio of the polar to the equatorial diameter. He then noticed that the consequent bulging of matter at the equator would be attracted by the moon unequally, the nearest parts being most attracted; and so the moon would tend to tilt the earth when in some parts of her orbit; and the sun would do this to a less extent, because of its great distance. Then he proved that the effect ought to be a rotation of the earth’s axis over a conical surface in space, exactly as the axis of a top describes a cone, if the top has a sharp point, and is set spinning and displaced from the vertical. He actually calculated the amount; and so he explained the cause of the precession of the equinoxes discovered by Hipparchus about 150 B.C. One of his grandest discoveries was a method of weighing the heavenly bodies by their action on each other. By means of this principle he was able to compare the mass of the sun with the masses of those planets that have moons, and also to compare the mass of our moon with the mass of the earth. Thus Newton, after having established his great principle, devoted his splendid intellect to the calculation of its consequences. He proved that if a body be projected with any velocity in free space, subject only to a central force, varying inversely as the square of the distance, the body must revolve in a curve which may be any one of the sections of a cone—a circle, ellipse, parabola, or hyperbola; and he found that those comets of which he had observations move in parabolae round the Sun, and are thus subject to the universal law. Newton realised that, while planets and satellites are chiefly controlled by the central body about which they revolve, the new law must involve irregularities, due to their mutual action—such, in fact, as Horrocks had indicated. He determined to put this to a test in the case of the moon, and to calculate the sun’s effect, from its mass compared with that of the earth, and from its distance. He proved that the average effect upon the plane of the orbit would be to cause the line in which it cuts the plane of the ecliptic (i.e., the line of nodes) to revolve in the ecliptic once in about nineteen years. This had been a known fact from the earliest ages. He also concluded that the line of apses would revolve in the plane of the lunar orbit also in about nineteen years; but the observed period is only ten years. For a long time this was the one weak point in the Newtonian theory. It was not till 1747 that Clairaut reconciled this with the theory, and showed why Newton’s calculation was not exact. Newton proceeded to explain the other inequalities recognised by Tycho Brahe and older observers, and to calculate their maximum amounts as indicated by his theory. He further discovered from his calculations two new inequalities, one of the apogee, the other of the nodes, and assigned the maximum value. Grant has shown the values of some of these as given by observation in the tables of Meyer and more modern tables, and has compared them with the values assigned by Newton from his theory; and the comparison is very remarkable. Newton. Modern Tables. ° ’ " ° ’ " Mean monthly motion of Apses 1.31.28 3.4.0 Mean annual motion of nodes 19.18.1,23 19.21.22,50 Mean value of “variation” 36.10 35.47 Annual equation 11.51 11.14 Inequality of mean motion of apogee 19.43 22.17 Inequality of mean motion of nodes 9.24 9.0 The only serious discrepancy is the first, which has been already mentioned. Considering that some of these perturbations had never been discovered, that the cause of none of them had ever been known, and that he exhibited his results, if he did not also make the discoveries, by the synthetic methods of geometry, it is simply marvellous that he reached to such a degree of accuracy. He invented the infinitesimal calculus which is more suited for such calculations, but had he expressed his results in that language he would have been unintelligible to many. Newton’s method of calculating the precession of the equinoxes, already referred to, is as beautiful as anything in the Principia. He had already proved the regression of the nodes of a satellite moving in an orbit inclined to the ecliptic. He now said that the nodes of a ring of satellites revolving round the earth’s equator would consequently all regress. And if joined into a solid ring its node would regress; and it would do so, only more slowly, if encumbered by the spherical part of the earth’s mass. Therefore the axis of the equatorial belt of the earth must revolve round the pole of the ecliptic. Then he set to work and found the amount due to the moon and that due to the sun, and so he solved the mystery of 2,000 years. When Newton applied his law of gravitation to an explanation of the tides he started a new field for the application of mathematics to physical problems; and there can be little doubt that, if he could have been furnished with complete tidal observations from different parts of the world, his extraordinary powers of analysis would have enabled him to reach a satisfactory theory. He certainly opened up many mines full of intellectual gems; and his successors have never ceased in their explorations. This has led to improved mathematical methods, which, combined with the greater accuracy of observation, have rendered physical astronomy of to-day the most exact of the sciences. Laplace only expressed the universal opinion of posterity when he said that to the Principia is assured “a pre-eminence above all the other productions of the human intellect.” The name of Flamsteed, First Astronomer Royal, must here be mentioned as having supplied Newton with the accurate data required for completing the theory. The name of Edmund Halley, Second Astronomer Royal, must ever be held in repute, not only for his own discoveries, but for the part he played in urging Newton to commit to writing, and present to the Royal Society, the results of his investigations. But for his friendly insistence it is possible that the Principia would never have been written; and but for his generosity in supplying the means the Royal Society could not have published the book. DEATH MASK OF SIR ISAAC NEWTON. Sir Isaac Newton died in 1727, at the age of eighty-five. His body lay in state in the Jerusalem Chamber, and was buried in Westminster Abbey. FOOTNOTES:
Edmund Halley succeeded Flamsteed as Second Astronomer Royal in 1721. Although he did not contribute directly to the mathematical proofs of Newton’s theory, yet his name is closely associated with some of its greatest successes. He was the first to detect the acceleration of the moon’s mean motion. Hipparchus, having compared his own observations with those of more ancient astronomers, supplied an accurate value of the moon’s mean motion in his time. Halley similarly deduced a value for modern times, and found it sensibly greater. He announced this in 1693, but it was not until 1749 that Dunthorne used modern lunar tables to compute a lunar eclipse observed in Babylon 721 B.C., another at Alexandria 201 B.C., a solar eclipse observed by Theon 360 A.D., and two later ones up to the tenth century. He found that to explain these eclipses Halley’s suggestion must be adopted, the acceleration being 10” in one century. In 1757 Lalande again fixed it at 10.” The Paris Academy, in 1770, offered their prize for an investigation to see if this could be explained by the theory of gravitation. Euler won the prize, but failed to explain the effect, and said: “It appears to be established by indisputable evidence that the secular inequality of the moon’s mean motion cannot be produced by the forces of gravitation.” The same subject was again proposed for a prize which was shared by Lagrange[1] and Euler, neither finding a solution, while the latter asserted the existence of a resisting medium in space. Again, in 1774, the Academy submitted the same subject, a third time, for the prize; and again Lagrange failed to detect a cause in gravitation. Laplace[2] now took the matter in hand. He tried the effect of a non-instantaneous action of gravity, to no purpose. But in 1787 he gave the true explanation. The principal effect of the sun on the moon’s orbit is to diminish the earth’s influence, thus lengthening the period to a new value generally taken as constant. But Laplace’s calculations showed the new value to depend upon the excentricity of the earth’s orbit, which, according; to theory, has a periodical variation of enormous period, and has been continually diminishing for thousands of years. Thus the solar influence has been diminishing, and the moon’s mean motion increased. Laplace computed the amount at 10” in one century, agreeing with observation. (Later on Adams showed that Laplace’s calculation was wrong, and that the value he found was too large; so, part of the acceleration is now attributed by some astronomers to a lengthening of the day by tidal friction.) Another contribution by Halley to the verification of Newton’s law was made when he went to St. Helena to catalogue the southern stars. He measured the change in length of the second’s pendulum in different latitudes due to the changes in gravity foretold by Newton. Furthermore, he discovered the long inequality of Jupiter and Saturn, whose period is 929 years. For an investigation of this also the Academy of Sciences offered their prize. This led Euler to write a valuable essay disclosing a new method of computing perturbations, called the instantaneous ellipse with variable elements. The method was much developed by Lagrange. But again it was Laplace who solved the problem of the inequalities of Jupiter and Saturn by the theory of gravitation, reducing the errors of the tables from 20’ down to 12”, thus abolishing the use of empirical corrections to the planetary tables, and providing another glorious triumph for the law of gravitation. As Laplace justly said: “These inequalities appeared formerly to be inexplicable by the law of gravitation—they now form one of its most striking proofs.” Let us take one more discovery of Halley, furnishing directly a new triumph for the theory. He noticed that Newton ascribed parabolic orbits to the comets which he studied, so that they come from infinity, sweep round the sun, and go off to infinity for ever, after having been visible a few weeks or months. He collected all the reliable observations of comets he could find, to the number of twenty-four, and computed their parabolic orbits by the rules laid down by Newton. His object was to find out if any of them really travelled in elongated ellipses, practically undistinguishable, in the visible part of their paths, from parabolÆ, in which case they would be seen more than once. He found two old comets whose orbits, in shape and position, resembled the orbit of a comet observed by himself in 1682. Apian observed one in 1531; Kepler the other in 1607. The intervals between these appearances is seventy-five or seventy-six years. He then examined and found old records of similar appearance in 1456, 1380, and 1305. It is true, he noticed, that the intervals varied by a year and a-half, and the inclination of the orbit to the ecliptic diminished with successive apparitions. But he knew from previous calculations that this might easily be due to planetary perturbations. Finally, he arrived at the conclusion that all of these comets were identical, travelling in an ellipse so elongated that the part where the comet was seen seemed to be part of a parabolic orbit. He then predicted its return at the end of 1758 or beginning of 1759, when he should be dead; but, as he said, “if it should return, according to our prediction, about the year 1758, impartial posterity will not refuse to acknowledge that this was first discovered by an Englishman.”[3] [Synopsis Astronomiae Cometicae, 1749.] Once again Halley’s suggestion became an inspiration for the mathematical astronomer. Clairaut, assisted by Lalande, found that Saturn would retard the comet 100 days, Jupiter 518 days, and predicted its return to perihelion on April 13th, 1759. In his communication to the French Academy, he said that a comet travelling into such distant regions might be exposed to the influence of forces totally unknown, and “even of some planet too far removed from the sun to be ever perceived.” The excitement of astronomers towards the end of 1758 became intense; and the honour of first catching sight of the traveller fell to an amateur in Saxony, George Palitsch, on Christmas Day, 1758. It reached perihelion on March 13th, 1759. This fact was a startling confirmation of the Newtonian theory, because it was a new kind of calculation of perturbations, and also it added a new member to the solar system, and gave a prospect of adding many more. When Halley’s comet reappeared in 1835, Pontecoulant’s computations for the date of perihelion passage were very exact, and afterwards he showed that, with more exact values of the masses of Jupiter and Saturn, his prediction was correct within two days, after an invisible voyage of seventy-five years! Hind afterwards searched out many old appearances of this comet, going back to 11 B.C., and most of these have been identified as being really Halley’s comet by the calculations of Cowell and Cromellin[4] (of Greenwich Observatory), who have also predicted its next perihelion passage for April 8th to 16th, 1910, and have traced back its history still farther, to 240 B.C. Already, in November, 1907, the Astronomer Royal was trying to catch it by the aid of photography. FOOTNOTES: It would be very interesting, but quite impossible in these pages, to discuss all the exquisite researches of the mathematical astronomers, and to inspire a reverence for the names connected with these researches, which for two hundred years have been establishing the universality of Newton’s law. The lunar and planetary theories, the beautiful theory of Jupiter’s satellites, the figure of the earth, and the tides, were mathematically treated by Maclaurin, D’Alembert, Legendre, Clairaut, Euler, Lagrange, Laplace, Walmsley, Bailly, Lalande, Delambre, Mayer, Hansen, Burchardt, Binet, Damoiseau, Plana, Poisson, Gauss, Bessel, Bouvard, Airy, Ivory, Delaunay, Le Verrier, Adams, and others of later date. By passing over these important developments it is possible to trace some of the steps in the crowning triumph of the Newtonian theory, by which the planet Neptune was added to the known members of the solar system by the independent researches of Professor J.C. Adams and of M. Le Verrier, in 1846. It will be best to introduce this subject by relating how the eighteenth century increased the number of known planets, which was then only six, including the earth. On March 13th, 1781, Sir William Herschel was, as usual, engaged on examining some small stars, and, noticing that one of them appeared to be larger than the fixed stars, suspected that it might be a comet. To test this he increased his magnifying power from 227 to 460 and 932, finding that, unlike the fixed stars near it, its definition was impaired and its size increased. This convinced him that the object was a comet, and he was not surprised to find on succeeding nights that the position was changed, the motion being in the ecliptic. He gave the observations of five weeks to the Royal Society without a suspicion that the object was a new planet. For a long time people could not compute a satisfactory orbit for the supposed comet, because it seemed to be near the perihelion, and no comet had ever been observed with a perihelion distance from the sun greater than four times the earth’s distance. Lexell was the first to suspect that this was a new planet eighteen times as far from the sun as the earth is. In January, 1783, Laplace published the elliptic elements. The discoverer of a planet has a right to name it, so Herschel called it Georgium Sidus, after the king. But Lalande urged the adoption of the name Herschel. Bode suggested Uranus, and this was adopted. The new planet was found to rank in size next to Jupiter and Saturn, being 4.3 times the diameter of the earth. In 1787 Herschel discovered two satellites, both revolving in nearly the same plane, inclined 80° to the ecliptic, and the motion of both was retrograde. In 1772, before Herschel’s discovery, Bode[1] had discovered a curious arbitrary law of planetary distances. Opposite each planet’s name write the figure 4; and, in succession, add the numbers 0, 3, 6, 12, 24, 48, 96, etc., to the 4, always doubling the last numbers. You then get the planetary distances. Mercury, dist.-- 4 4 + 0 = 4 Venus " 7 4 + 3 = 7 Earth " 10 4 + 6 = 10 Mars " 15 4 + 12 = 16 -- 4 + 24 = 28 Jupiter dist. 52 4 + 48 = 52 Saturn " 95 4 + 96 = 100 (Uranus) " 192 4 + 192 = 196 -- 4 + 384 = 388 All the five planets, and the earth, fitted this rule, except that there was a blank between Mars and Jupiter. When Uranus was discovered, also fitting the rule, the conclusion was irresistible that there is probably a planet between Mars and Jupiter. An association of twenty-four astronomers was now formed in Germany to search for the planet. Almost immediately afterwards the planet was discovered, not by any member of the association, but by Piazzi, when engaged upon his great catalogue of stars. On January 1st, 1801, he observed a star which had changed its place the next night. Its motion was retrograde till January 11th, direct after the 13th. Piazzi fell ill before he had enough observations for computing the orbit with certainty, and the planet disappeared in the sun’s rays. Gauss published an approximate ephemeris of probable positions when the planet should emerge from the sun’s light. There was an exciting hunt, and on December 31st (the day before its birthday) De Zach captured the truant, and Piazzi christened it Ceres. The mean distance from the sun was found to be 2.767, agreeing with the 2.8 given by Bode’s law. Its orbit was found to be inclined over 10° to the ecliptic, and its diameter was only 161 miles. On March 28th, 1802, Olbers discovered a new seventh magnitude star, which turned out to be a planet resembling Ceres. It was called Pallas. Gauss found its orbit to be inclined 35° to the ecliptic, and to cut the orbit of Ceres; whence Olbers considered that these might be fragments of a broken-up planet. He then commenced a search for other fragments. In 1804 Harding discovered Juno, and in 1807 Olbers found Vesta. The next one was not discovered until 1845, from which date asteroids, or minor planets (as these small planets are called), have been found almost every year. They now number about 700. It is impossible to give any idea of the interest with which the first additions since prehistoric times to the planetary system were received. All of those who showered congratulations upon the discoverers regarded these discoveries in the light of rewards for patient and continuous labours, the very highest rewards that could be desired. And yet there remained still the most brilliant triumph of all, the addition of another planet like Uranus, before it had ever been seen, when the analysis of Adams and Le Verrier gave a final proof of the powers of Newton’s great law to explain any planetary irregularity. After Sir William Herschel discovered Uranus, in 1781, it was found that astronomers had observed it on many previous occasions, mistaking it for a fixed star of the sixth or seventh magnitude. Altogether, nineteen observations of Uranus’s position, from the time of Flamsteed, in 1690, had been recorded. In 1790 Delambre, using all these observations, prepared tables for computing its position. These worked well enough for a time, but at last the differences between the calculated and observed longitudes of the planet became serious. In 1821 Bouvard undertook a revision of the tables, but found it impossible to reconcile all the observations of 130 years (the period of revolution of Uranus is eighty-four years). So he deliberately rejected the old ones, expressing the opinion that the discrepancies might depend upon “some foreign and unperceived cause which may have been acting upon the planet.” In a few years the errors even of these tables became intolerable. In 1835 the error of longitude was 30”; in 1838, 50”; in 1841, 70”; and, by comparing the errors derived from observations made before and after opposition, a serious error of the distance (radius vector) became apparent. In 1843 John Couch Adams came out Senior Wrangler at Cambridge, and was free to undertake the research which as an undergraduate he had set himself—to see whether the disturbances of Uranus could be explained by assuming a certain orbit, and position in that orbit, of a hypothetical planet even more distant than Uranus. Such an explanation had been suggested, but until 1843 no one had the boldness to attack the problem. Bessel had intended to try, but a fatal illness overtook him. Adams first recalculated all known causes of disturbance, using the latest determinations of the planetary masses. Still the errors were nearly as great as ever. He could now, however, use these errors as being actually due to the perturbations produced by the unknown planet. In 1844, assuming a circular orbit, and a mean distance agreeing with Bode’s law, he obtained a first approximation to the position of the supposed planet. He then asked Professor Challis, of Cambridge, to procure the latest observations of Uranus from Greenwich, which Airy immediately supplied. Then the whole work was recalculated from the beginning, with more exactness, and assuming a smaller mean distance. In September, 1845, he handed to Challis the elements of the hypothetical planet, its mass, and its apparent position for September 30th, 1845. On September 22nd Challis wrote to Airy explaining the matter, and declaring his belief in Adams’s capabilities. When Adams called on him Airy was away from home, but at the end of October, 1845, he called again, and left a paper with full particulars of his results, which had, for the most part, reduced the discrepancies to about 1”. As a matter of fact, it has since been found that the heliocentric place of the new planet then given was correct within about 2°. Airy wrote expressing his interest, and asked for particulars about the radius vector. Adams did not then reply, as the answer to this question could be seen to be satisfactory by looking at the data already supplied. He was a most unassuming man, and would not push himself forward. He may have felt, after all the work he had done, that Airy’s very natural inquiry showed no proportionate desire to search for the planet. Anyway, the matter lay in embryo for nine months. Meanwhile, one of the ablest French astronomers, Le Verrier, experienced in computing perturbations, was independently at work, knowing nothing about Adams. He applied to his calculations every possible refinement, and, considering the novelty of the problem, his calculation was one of the most brilliant in the records of astronomy. In criticism it has been said that these were exhibitions of skill rather than helps to a solution of the particular problem, and that, in claiming to find the elements of the orbit within certain limits, he was claiming what was, under the circumstances, impossible, as the result proved. In June, 1846, Le Verrier announced, in the Comptes Rendus de l’Academie des Sciences, that the longitude of the disturbing planet, for January 1st, 1847, was 325, and that the probable error did not exceed 10°. This result agreed so well with Adams’s (within 1°) that Airy urged Challis to apply the splendid Northumberland equatoreal, at Cambridge, to the search. Challis, however, had already prepared an exhaustive plan of attack which must in time settle the point. His first work was to observe, and make a catalogue, or chart, of all stars near Adams’s position. On August 31st, 1846, Le Verrier published the concluding part of his labours. On September 18th, 1846, Le Verrier communicated his results to the Astronomers at Berlin, and asked them to assist in searching for the planet. By good luck Dr. Bremiker had just completed a star-chart of the very part of the heavens including Le Verrier’s position; thus eliminating all of Challis’s preliminary work. The letter was received in Berlin on September 23rd; and the same evening Galle found the new planet, of the eighth magnitude, the size of its disc agreeing with Le Verrier’s prediction, and the heliocentric longitude agreeing within 57’. By this time Challis had recorded, without reduction, the observations of 3,150 stars, as a commencement for his search. On reducing these, he found a star, observed on August 12th, which was not in the same place on July 30th. This was the planet, and he had also observed it on August 4th. The feeling of wonder, admiration, and enthusiasm aroused by this intellectual triumph was overwhelming. In the world of astronomy reminders are met every day of the terrible limitations of human reasoning powers; and every success that enables the mind’s eye to see a little more clearly the meaning of things has always been heartily welcomed by those who have themselves been engaged in like researches. But, since the publication of the Principia, in 1687, there is probably no analytical success which has raised among astronomers such a feeling of admiration and gratitude as when Adams and Le Verrier showed the inequalities in Uranus’s motion to mean that an unknown planet was in a certain place in the heavens, where it was found. At the time there was an unpleasant display of international jealousy. The British people thought that the earlier date of Adams’s work, and of the observation by Challis, entitled him to at least an equal share of credit with Le Verrier. The French, on the other hand, who, on the announcement of the discovery by Galle, glowed with pride in the new proof of the great powers of their astronomer, Le Verrier, whose life had a long record of successes in calculation, were incredulous on being told that it had all been already done by a young man whom they had never heard of. These displays of jealousy have long since passed away, and there is now universally an entente cordiale that to each of these great men belongs equally the merit of having so thoroughly calculated this inverse problem of perturbations as to lead to the immediate discovery of the unknown planet, since called Neptune. It was soon found that the planet had been observed, and its position recorded as a fixed star by Lalande, on May 8th and 10th, 1795. Mr. Lassel, in the same year, 1846, with his two-feet reflector, discovered a satellite, with retrograde motion, which gave the mass of the planet about a twentieth of that of Jupiter. FOOTNOTES: |
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