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Table of Contents | |
Section | Page |
Start of eBook | 1 |
CHAPTER I. | 1 |
CHAPTER II. | 5 |
CHAPTER III. | 9 |
CHAPTER IV. | 14 |
CHAPTER V. | 19 |
CHAPTER VI. | 30 |
APPENDIX I. | 33 |
APPENDIX II. | 33 |
GLOSSARY. | 33 |
Astronomy before Kepler.
In order to emphasise the importance of the reforms introduced into astronomy by Kepler, it will be well to sketch briefly the history of the theories which he had to overthrow. In very early times it must have been realised that the sun and moon were continually changing their places among the stars. The day, the month, and the year were obvious divisions of time, and longer periods were suggested by the tabulation of eclipses. We can imagine the respect accorded to the Chaldaean sages who first discovered that eclipses could be predicted, and how the philosophers of Mesopotamia must have sought eagerly for evidence of fresh periodic laws. Certain of the stars, which appeared to wander, and were hence called planets, provided an extended field for these speculations. Among the Chaldaeans and Babylonians the knowledge gradually acquired was probably confined to the priests and utilised mainly for astrological prediction or the fixing of religious observances. Such speculations as were current among them, and also among the Egyptians and others who came to share their knowledge, were almost entirely devoted to mythology, assigning fanciful terrestrial origins to constellations, with occasional controversies as to how the earth is supported in space. The Greeks, too, had an elaborate mythology largely adapted from their neighbours, but they were not satisfied with this, and made persistent attempts to reduce the apparent motions of celestial objects to geometrical laws. Some of the Pythagoreans, if not Pythagoras himself, held that the earth is a sphere, and that the apparent daily revolution of the sun and stars is really due to a motion of the earth, though at first this motion of the earth was not supposed to be one of rotation about an axis. These notions, and also that the planets on the whole move round from west to east with reference to the stars, were made known to a larger circle through the writings of Plato. To Plato moreover is attributed the challenge to astronomers to represent all the motions of the heavenly bodies by uniformly described circles, a challenge generally held responsible for a vast amount of wasted effort, and the postponement, for many centuries, of real progress. Eudoxus of Cnidus, endeavouring to account for the fact that the planets, during every apparent revolution round the earth, come to rest twice, and in the shorter interval between these “stationary points,” move in the opposite direction, found that he could represent the phenomena fairly well by a system of concentric spheres, each rotating with its own velocity, and carrying its own particular planet round its own equator, the outermost sphere carrying the fixed stars. It was necessary to assume that the axes about which the various spheres revolved should have circular motions also, and gradually an increased number of spheres was evolved, the total
Aristarchus of Samos seems to have been the first to suggest that the planets revolved not about the earth but about the sun, but the idea seemed so improbable that it was hardly noticed, especially as Aristarchus himself did not expand it into a treatise.
About this time the necessity for more accurate places of the sun and moon, and the liberality of the Ptolemys who ruled Egypt, combined to provide regular observations at Alexandria, so that, when Hipparchus came upon the scene, there was a considerable amount of material for him to use. His discoveries marked a great advance in the science of astronomy. He noted the irregular motion of the sun, and, to explain it, assumed that it revolved uniformly not exactly about the earth but about a point some distance away, called the “excentric".[1] The line joining the centre of the earth to the excentric passes through the apses of the sun’s orbit, where its distance from the earth is greatest and least. The same result he could obtain by assuming that the sun moved round a small circle, whose centre described a larger circle about the earth; this larger circle carrying the other was called the “deferent”: so that the actual motion of the sun was in an epicycle. Of the two methods of expression Hipparchus ultimately preferred the second. He applied the same process to the moon but found that he could depend upon its being right only at new and full moon. The irregularity at first and third quarters he left to be investigated by his successors. He also considered the planetary observations at his disposal insufficient and so gave up the attempt at a complete planetary theory. He made improved determinations of some of the elements of the motions of the sun and moon, and discovered the Precession of the Equinoxes, from the Alexandrian observations which showed that each year as the sun came to cross the equator at the vernal equinox it did so at a point about fifty seconds of arc earlier on the ecliptic, thus producing in 150 years an unmistakable change of a couple of degrees, or four times the sun’s diameter. He also invented trigonometry. His star catalogue was due to the appearance of a new star which caused him to search for possible previous similar phenomena, and also to prepare for checking future ones. No advance was made in theoretical
[Footnote 1: See Glossary for this and other technical terms.]
After Ptolemy’s time very little was heard for many centuries of any fresh planetary theory, though advances in some points of detail were made, notably by some of the Arab philosophers, who obtained improved values for some of the elements by using better instruments. From time to time various modifications of Ptolemy’s theory were suggested, but none of any real value. The Moors in Spain did their share of the work carried on by their Eastern co-religionists, and the first independent star catalogue since the time of Hipparchus was made by another Oriental, Tamerlane’s grandson, Ulugh Begh, who built a fine observatory at Samarcand in the fifteenth century. In Spain the work was not monopolised by the Moors, for in the thirteenth century Alphonso of Castile, with the assistance of Jewish and Christian computers, compiled the Alphonsine tables, completed in 1252, in which year he ascended the throne as Alphonso X. They were long circulated in Ms. and were first printed in 1483, not long before the end of the period of stagnation.
Copernicus was born in 1473 at Thorn in Polish Prussia. In the course of his studies at Cracow and at several Italian universities, he learnt all that was known of the Ptolemaic astronomy and determined to reform it. His maternal uncle, the Bishop of Ermland, having provided him with a lay canonry in the Cathedral of Frauenburg, he had leisure to devote himself to Science. Reviewing the suggestions of the ancient Greeks, he was struck by the simplification that would be introduced by reviving the idea that the annual motion should be attributed to the earth itself instead of having a separate annual epicycle for each planet
It is rather an exaggeration to call the present accepted system the Copernican system, as it is really due to Kepler, half a century after the death of Copernicus, but much credit is due to the latter for his successful attempt to provide a real alternative for the Ptolemaic system, instead of tinkering with it. The old geocentric system once shaken, the way was gradually smoothed for the heliocentric system, which Copernicus, still hampered by tradition, did not quite reach. He was hardly a practical astronomer in the observational sense. His first recorded observation, of an occultation of Aldebaran, was made in 1497, and he is not known to have made as many as fifty astronomical observations, while, of the few he did make and use, at least one was more than half a degree in error, which would have been intolerable to such an observer as Hipparchus. Copernicus in fact seems to have considered accurate observations unattainable with the instruments at hand. He refused to give any opinion on the projected reform of the calendar, on the ground that the motions of the sun and moon were not known with sufficient accuracy. It is possible that with better data he might have made much more progress. He was in no hurry to publish anything, perhaps on account of possible opposition. Certainly Luther, with his obstinate conviction of the verbal accuracy of the Scriptures, rejected as mere folly the idea of a moving earth, and Melanchthon thought such opinions should be prohibited, but Rheticus, a professor at the Protestant University of Wittenberg and an enthusiastic pupil of Copernicus, urged publication, and undertook to see the work through the press. This, however, he was unable to complete and another Lutheran, Osiander, to whom he entrusted it, wrote a preface, with the apparent intention of disarming opposition, in which he stated that the principles laid down were only abstract hypotheses convenient for purposes of calculation. This unauthorised interpolation may have had its share in postponing the prohibition of the book by the Church of Rome.
According to Copernicus the earth is only a planet like the others, and not even the biggest one, while the sun is the most important body in the system, and the stars probably too far away for any motion of the earth to affect their apparent places. The earth in fact is very small in comparison with the distance of the stars, as evidenced by the fact that an observer anywhere on the earth appears to be in the middle of the universe. He shows that the revolution of the earth will account for the seasons, and for the stationary points and retrograde motions of the planets. He corrects definitely the order of the planets outwards from the sun, a matter which had been in dispute. A notable defect is due to the idea that a body can only revolve about another body or a point, as if rigidly connected with it, so that, in order to keep the earth’s axis in a constant direction in space, he has to invent a third motion. His discussion of precession, which he rightly attributes to a slow motion of the earth’s axis, is marred by the idea that the precession is variable. With all its defects, partly due to reliance on bad observations, the work showed a great advance in the interpretation of the motions of the planets; and his determinations of the periods both in relation to the earth and to the stars were adopted by Reinhold, Professor of Astronomy at Wittenberg, for the new Prutenic or Prussian Tables, which were to supersede the obsolete Alphonsine Tables of the thirteenth century.
In comparison with the question of the motion of the earth, no other astronomical detail of the time seems to be of much consequence. Comets, such as from time to time appeared, bright enough for naked eye observation, were still regarded as atmospheric phenomena, and their principal interest, as well as that of eclipses and planetary conjunctions, was in relation to astrology. Reform, however, was obviously in the air. The doctrine of Copernicus was destined very soon to divide others besides the Lutheran leaders. The leaven of inquiry was working, and not long after the death of Copernicus real advances were to come, first in the accuracy of observations, and, as a necessary result of these, in the planetary theory itself.
Early life of Kepler.
On 21st December, 1571, at Weil in the Duchy of Wurtemberg, was born a weak and sickly seven-months’ child, to whom his parents Henry and Catherine Kepler gave the name of John. Henry Kepler was a petty officer in the service of the reigning Duke, and in 1576 joined the army serving in the Netherlands. His wife followed him, leaving her young son in his grandfather’s care at Leonberg, where he barely recovered from a severe attack of smallpox. It was from this place that John derived the Latinised name of Leonmontanus, in accordance with the common practice of the time, but he was not known by it to any great extent.
[Footnote 2: Since the sum of the plane angles at a corner of a regular solid must be less than four right angles, it is easily seen that few regular solids are possible. Hexagonal faces are clearly impossible, or any polygonal faces with more than five sides. The possible forms are the dodecahedron with twelve pentagonal faces, three meeting at each corner; the cube, six square faces, three meeting at each corner; and three figures with triangular faces, the tetrahedron of four faces, three meeting at each corner; the octahedron of eight faces, four meeting at each corner; and the icosahedron of twenty faces, five meeting at each corner.]
Tycho brahe.
The age following that of Copernicus produced three outstanding figures associated with the science of astronomy, then reaching the close of what Professor Forbes so aptly styles the geometrical period. These three Sir David Brewster has termed “Martyrs of Science”; Galileo, the great Italian philosopher, has his own place among the “Pioneers of Science”; and invaluable though Tycho Brahe’s work was, the latter can hardly be claimed as a pioneer in the same sense as the other two. Nevertheless, Kepler, the third member of the trio, could not have made his most valuable discoveries without Tycho’s observations.
Of noble family, born a twin on 14th December, 1546, at Knudstrup in Scania (the southernmost part of Sweden, then forming part of the kingdom of Denmark), Tycho was kidnapped a year later by a childless uncle. This uncle brought him up as his own son, provided him at the age of seven with a tutor, and sent him in 1559 to the University of Copenhagen, to study for a political career by taking courses in rhetoric and philosophy. On 21st August, 1560, however, a solar eclipse took place, total in Portugal, and therefore of small proportions in Denmark, and Tycho’s keen interest was awakened, not so much by the phenomenon, as by the fact that it had occurred according to prediction.
On 11th November, 1572, Tycho noticed an unfamiliar bright star in the constellation of Cassiopeia, and continued to observe it with a sextant. It was a very brilliant object, equal to Venus at its brightest for the rest of November, not falling below the first magnitude for another four months, and remaining visible for more than a year afterwards. Tycho wrote a little book on the new star, maintaining that it had practically no parallax, and therefore could not be, as some supposed, a comet. Deeming authorship beneath the dignity of a noble he was very reluctant to publish, but he was convinced of the importance of increasing the number and accuracy of observations, though he was by no means free from all the erroneous ideas of his time. The little book contained a certain amount of astrology, but Tycho evidently did not regard this as of very great importance. He adopted the view that the very rarity of the phenomenon of a new star must prevent the formulation and adoption of definite rules for determining its significance. We gather from lectures which he was persuaded to deliver at the University of Copenhagen that, though in agreement with the accepted canons of astrology as to the influence of planetary conjunctions and such phenomena on the course of human events, he did not consider the fate predicted by anyone’s horoscope to be unavoidable, but thought the great value of astrology lay in the warnings derived from such computations, which should enable the believer to avoid threatened calamities. In 1575 he left Denmark once more and made his way to Cassel, where he found a kindred spirit in the studious Landgrave, William IV. of Hesse, whose astronomical pursuits had been interrupted by his accession to the government of Hesse, in 1567. Tycho observed with him for some time, the two forming a firm friendship, and then visited successively Frankfort, Basle, and Venice, returning by way of Augsburg, Ratisbon, and Saalfeld to Wittenberg; on the way he acquired various astronomical manuscripts, made friends among practical astronomers, and examined new instruments. He seemed to have considered the advantages of the several places thus visited and decided on Basle, but on his return to Denmark to fetch his family with the object of transferring them to Basle, he found that his friend the Landgrave had written to King Frederick on his behalf, urging him to provide the means to enable Tycho to pursue his astronomical work, promising that not only should credit result for the king and for Denmark but that science itself would be greatly advanced. The ultimate result of this letter was that after refusing various offers, Tycho accepted from the king a grant of the small island of Hveen, in the Sound, with a guaranteed income, in addition to a large sum from the treasury for building an observatory on the island, far removed from the distractions of court life. Here Tycho built his celebrated observatory of Uraniborg and began observations in December, 1576, using the
King Frederick, in consideration of various grants to Tycho, relied upon his assistance in scientific matters, and especially in astrological calculations; such as the horoscope of the heir apparent, Prince Christian, born in 1577, which has been preserved among Tycho’s writings. There is, however, no known copy in existence of any of the series of annual almanacs with predictions which he prepared for the King. In November, 1577, appeared a bright comet, which Tycho carefully observed with his sextant, proving that it had no perceptible parallax, and must therefore be further off than the moon. He thus definitely overthrew the common belief in the atmospheric origin of comets, which he had himself hitherto shared. With increasing accuracy he observed several other comets, notably one in 1585, when he had a full equipment of instruments and a large staff of assistants. The year 1588, which saw the death of his royal benefactor, saw also the publication of a volume of Tycho’s great work “Introduction to the New Astronomy”. The first volume, devoted to the new star of 1572, was not ready, because the reduction of the observations involved so much research to correct the star places for refraction, precession, etc.; it was not completed in fact until Tycho’s death, but the second volume, dealing with the comet of 1577, was printed at Uraniborg and some copies were issued in 1588. Besides the comet observations it included an account of Tycho’s system of the world. He would not accept the Copernican system, as he considered the earth too heavy and sluggish to move, and also that the authority of Scripture was against such an
King Frederick’s death did not at first affect Tycho’s position, for the new king, Christian, was only eleven years old, and for some years the council of regents included two of his supporters. After their deaths, however, his emoluments began to be cut down on the plea of economy, and as he took very little trouble to carry out any other than scientific duties it was easy enough for his enemies to find fault. One after another source of income was cut off, but he persevered with his scientific work, including a catalogue of stars. He had obtained plenty of good observations of 777 stars, but thought his catalogue should contain 1000 stars, so he hastily observed as many more as he could up to the time of his leaving Hveen, though even then he had not completed his programme. About the time that King Christian reached the age of eighteen, Tycho began to look about for a new patron, and to consider the prospects offered by transferring himself with his instruments and activities to the patronage of the Emperor Rudolph II. In 1597, when even his pension from the Royal treasury was cut off, he hurriedly packed up his instruments and library, and after a few weeks’ sojourn at Copenhagen, proceeded to Rostock, in Mecklenburg, whence he sent an appeal to King Christian. It is possible that had he done this before leaving Hveen it might have had more effect, but it can be readily seen from the tone of the king’s unfavourable reply that his departure was regarded as an aggravation of previous shortcomings. Driven from Rostock by the plague, Tycho settled temporarily at Wandsbeck, in Holstein, but towards the end of 1598 set out to meet the Emperor at Prague. Once more plague intervened and he spent some time at Dresden, afterwards going to Wittenberg for the winter. He ultimately reached Prague in June, 1599. Rudolph granted him a salary of at least 3000 florins, promising also to settle on him the first hereditary estate that should lapse to the Crown. He offered, moreover, the choice between three castles outside Prague, of which Tycho chose Benatek. There he set about altering the buildings in readiness for his instruments, for which he sent
His magnificent Observatory of Uraniborg, the finest building for astronomical purposes that the world had hitherto seen, was allowed to fall into decay, and scarcely more than mere indications of the site may now be seen.
Kepler joins tycho.
The association of Kepler with Tycho was one of the most important landmarks in the history of astronomy. The younger man hoped, by the aid of Tycho’s planetary observations, to obtain better support for some of his fanciful speculative theories, while the latter, who had certainly not gained in prestige by leaving Denmark, was in great need of a competent staff of assistants. Of the two it would almost seem that Tycho thought himself the greater gainer, for in spite of his reputation for brusqueness and want of consideration, he not only made light of Kepler’s apology in the matter of Reymers, but treated him with
In 1604 the constellation of Cassiopeia was once more temporarily enriched by the appearance of a new star, said by some to be brighter than Tycho’s nova, and by others to be twice as bright as Jupiter. Kepler at once wrote a short account of it, from which may be gathered some idea of his attitude towards astrology. Contrasting the two novae, he says: “Yonder one chose for its appearance a time no way remarkable, and came into the world quite unexpectedly, like an enemy storming a town and breaking into the market-place before the citizens are aware of his approach; but ours has come exactly in the year of which astrologers have written so much about the fiery trigon that happens in it; just in the month in which (according to Cyprian), Mars comes up to a very perfect conjunction with the other two superior planets; just in the day when Mars has joined Jupiter, and just in the region where this conjunction has taken place. Therefore the apparition of this star is not like a secret hostile irruption, as was that one of 1572, but the spectacle of a public triumph, or the entry of a mighty potentate; when the couriers ride in some time before to prepare his lodgings, and the crowd of young urchins begin to think the time over long to wait, then roll in, one after another, the ammunition and money, and baggage waggons, and presently the trampling of horse and the rush of people from every side to the streets and windows; and when the crowd have gazed with their jaws all agape at the troops of knights; then at last the trumpeters and archers and lackeys so distinguish the person of the monarch, that there is no occasion to point him out, but every one cries of his own accord—’Here we have him’. What it may portend is hard to determine, and this much only is certain, that it comes to tell mankind either nothing at all or high and mighty news, quite beyond human sense and understanding. It will have an important influence on political and social relations; not indeed by its own nature, but as it were accidentally through the disposition of mankind. First, it portends to the booksellers great disturbances and tolerable gains; for almost every Theologus, Philosophicus, Medicus, and Mathematicus, or whoever else, having no laborious occupation entrusted to him, seeks his pleasure in studiis, will make particular remarks upon it, and will wish to bring these remarks to the light. Just so will others, learned and unlearned, wish to know its meaning, and they will buy the authors who profess to tell them. I mention these things merely by way of example, because although thus much can be easily predicted without great skill, yet may it happen just as easily, and in the same manner, that the vulgar, or whoever else is of easy faith, or, it may be, crazy, may wish to exalt himself into a great prophet; or it may even happen that some powerful lord, who has good foundation and beginning of great dignities, will be cheered on by this phenomenon to
Meanwhile the projected Rudolphine Tables were continually delayed by the want of money. Kepler’s nominal salary should have been ample for his expenses, increased though they were by his growing family, but in the depleted state of the treasury there were many who objected to any payment for such “unpractical” purposes. This particular attitude has not been confined to any special epoch or country, but the obvious result in Kepler’s case was to compel him to apply himself to less expensive matters than the Planetary Tables, and among these must be included not only the horoscopes or nativities, which owing to his reputation were always in demand, but also other writings which probably did not pay so well. In 1604 he published “A Supplement to Vitellion,” containing the earliest known reasonable theory of optics, and especially of dioptrics or vision through lenses. He compared the mechanism of the eye with that of Porta’s “Camera Obscura,” but made no attempt to explain how the image formed on the retina is understood by the brain. He went carefully into the question of refraction, the importance of which Tycho had been the first astronomer to recognise, though he only applied it at low altitudes, and had not arrived at a true theory or accurate values. Kepler wasted a good deal of time and ingenuity on trial theories. He would invariably start with some hypothesis, and work out the effect. He would then test it by experiment, and when it failed would at once recognise that his hypothesis was a priori bound to fail. He rarely seems to have noticed the fatal objections in time to save himself trouble. He would then at once start again on a new hypothesis, equally gratuitous and equally unfounded. It never seems to have occurred to him that there might be a better way of approaching a problem. Among the lines he followed in this particular investigation were, first, that refraction depends only on the angle of incidence, which, he says, cannot be correct as it would thus be the same for all refracting substances; next, that it depended also on the density of the medium. This was a good shot, but he unfortunately assumed that all rays passing into a denser medium would apparently penetrate it to a depth depending only on the medium, which means that there is a constant ratio between the tangents, instead of the sines, of the inclination of the incident and refracted rays to the normal. Experiment proved that this gave too high values for refraction near the vertical compared
In 1606 came a second treatise on the new star, discussing various theories to account for its appearance, and refusing to accept the notion that it was a “fortuitous concourse of atoms”. This was followed in 1607 by a treatise on comets, suggested by the comet appearing that year, known as Halley’s comet after its next return. He regarded comets as “planets” moving in straight lines, never having examined sufficient observations of any comet to convince himself that their paths are curved. If he had not assumed that they were external to the system and so could not be expected to return, he might have anticipated Halley’s discovery. Another suggestive remark of his was to the effect that the planets must be self-luminous, as otherwise Mercury and Venus, at any rate, ought to show phases. This was put to the test not long afterwards by means of Galileo’s telescope.
In 1607 Kepler rushed into print with an alleged observation of Mercury crossing the sun, but after Galileo’s discovery of sun-spots, Kepler at once cheerfully retracted his observation of “Mercury,” and so far was he from being annoyed or bigoted in his views, that he warmly adopted Galileo’s side, in contrast to most of those whose opinions were liable to be overthrown by the new discoveries. Maestlin and others of Kepler’s friends took the opposite view.
Kepler’s laws.
When Gilbert of Colchester, in his “New Philosophy,” founded on his researches in magnetism, was dealing with tides, he did not suggest that the moon attracted the water, but that “subterranean spirits and humours, rising in sympathy with the moon, cause the sea also to rise and flow to the shores and up rivers”. It appears that an idea, presented in some such way as this, was more readily received than a plain statement. This so-called philosophical method was, in fact, very generally applied, and Kepler, who shared Galileo’s admiration for Gilbert’s work, adopted it in his own attempt to extend the idea of magnetic attraction to the planets. The general idea of “gravity” opposed the hypothesis of the rotation of the earth on the ground that loose objects would fly off: moreover, the latest refinements of the old system of planetary motions necessitated their orbits being described about a mere empty point. Kepler very strongly combated these notions, pointing out the absurdity of the conclusions to which they tended, and proceeded in set terms to describe his own theory.
“Every corporeal substance, so far forth as it is corporeal, has a natural fitness for resting in every place where it may be situated by itself beyond the sphere of influence of a body cognate with it. Gravity is a mutual affection between cognate bodies towards union or conjunction (similar in kind to the magnetic virtue), so that the earth attracts a stone much rather than the stone seeks the earth. Heavy bodies (if we begin by assuming the earth to be in the centre of the world) are not carried to the centre of the world in its quality of centre of the world, but as to the centre of a cognate round body, namely, the earth; so that wheresoever the earth may be placed, or whithersoever it may be carried by its animal faculty, heavy bodies will always be carried towards it. If the earth were not round, heavy bodies would not tend from every side in a straight line towards the centre of the earth, but to different points from different sides. If two stones were placed in any part of the world near each other, and beyond the sphere of influence of a third cognate body, these stones, like two magnetic needles, would come together in the intermediate point, each approaching the other by a space proportional to the comparative mass of the other. If the moon and earth were not retained in their orbits by their animal force
The time taken from one opposition of Mars to the next is decidedly unequal at different parts of his orbit, so that many oppositions must be used to determine the mean motion. The ancients had noticed that what was called the “second inequality,” due as we now know to the orbital motion of the earth, only vanished when earth, sun, and planet were in line, i.e. at the planet’s opposition; therefore they used oppositions to determine the mean motion, but deemed it necessary to apply a correction to the true opposition to reduce to mean opposition, thus sacrificing part of the advantage of using oppositions. Tycho and Longomontanus had followed this method in their calculations from Tycho’s twenty years’ observations. Their aim was to find a position of the “equant,” such that these observations would show a constant angular motion about it; and that the computed positions would agree in latitude and longitude with the actual observed positions. When Kepler arrived he was told that their longitudes agreed within a couple of minutes of arc, but that something was wrong with the latitudes. He found, however, that even in longitude their positions showed discordances ten times as great as they admitted, and so, to clear the ground of assumptions as far as possible, he determined to use true oppositions. To this Tycho objected, and Kepler had great difficulty in convincing him that the new move would be any improvement, but undertook to prove to him by actual examples that a false position of the orbit could by adjusting the equant be made to fit the longitudes within five minutes of arc, while giving quite erroneous values of the latitudes and second inequalities. To avoid the possibility of further objection he carried out this demonstration separately for each of the systems of Ptolemy, Copernicus, and Tycho. For the new method he noticed that great accuracy was required in the reduction of the observed places of Mars to the ecliptic, and for this purpose the value obtained for the parallax by Tycho’s assistants fell far short of the requisite accuracy. Kepler therefore was obliged to recompute the parallax from the original observations, as also the position of the line of nodes and the inclination of the orbit. The last he found to be constant, thus corroborating his theory that the plane of the orbit passed through the sun. He repeated his calculations no fewer than seventy times (and that before the invention of logarithms), and at length adopted values for the mean longitude and longitude of aphelion. He found no discordance greater than two minutes of arc in Tycho’s observed longitudes in opposition, but the latitudes, and also longitudes in other parts of the orbit were much more discordant, and he found to his chagrin that four years’ work was practically wasted. Before making a fresh start he looked for some simplification of the labour; and determined to adopt Ptolemy’s assumption known as the principle of the bisection of the excentricity. Hitherto, since Ptolemy had given no reason for this assumption, Kepler had preferred not to make it, only taking for granted that the centre was at some point on the line called the excentricity (see Figs. 1, 2).
A marked improvement in residuals was the result of this step, proving, so far, the correctness of Ptolemy’s principle, but there still remained discordances amounting to eight minutes of arc. Copernicus, who had no idea of the accuracy obtainable in observations, would probably have regarded such an agreement as remarkably good; but Kepler refused to admit the possibility of an error of eight minutes in any of Tycho’s observations. He thereupon vowed to construct from these eight minutes a new planetary theory that should account for them all. His repeated failures had by this time convinced him that no uniformly described circle could possibly represent the motion of Mars. Either the orbit could not be circular, or else the angular velocity could not be constant about any point whatever. He determined to attack the “second inequality,” i.e. the optical illusion caused by the earth’s annual motion, but first revived an old idea of his own that for the sake of uniformity the sun, or as he preferred to regard it, the earth, should have an equant as well as the planets. From the irregularities of the solar motion he soon found that this was the case, and that the motion was uniform about a point on the line from the sun to the centre of the earth’s orbit, such that the centre bisected the distance from the sun to the “Equant”; this fully supported Ptolemy’s principle. Clearly then the earth’s linear velocity could not be constant, and Kepler was encouraged to revive another of his speculations as to a force which was weaker at greater distances. He found the velocity greater at the nearer apse, so that the time over an equal arc at either apse was proportional to the distance. He conjectured that this might prove to be true for arcs at all parts of the orbit, and to test this he divided the orbit into 360 equal parts, and calculated the distances to the points of division. Archimedes had obtained an approximation to the area of a circle by dividing it radially into a very large number of triangles, and Kepler had this device in mind. He found that the sums of successive distances from his 360 points were approximately proportional to the times from point to point, and was thus enabled to represent much more accurately the annual motion of the earth which produced the second inequality of Mars, to whose motion he now returned. Three points are sufficient to define a circle, so he took three observed positions of Mars and found a circle; he then took three other positions, but obtained a different circle, and a third set gave yet another. It thus began to appear that the orbit could not be a circle. He next tried to divide into 360 equal parts, as he had in the case of the earth, but the sums of distances failed to fit the times, and he realised that the sums of distances were not a good measure of the area of successive triangles. He noted, however, that the errors at the apses were now smaller than with a central circular orbit, and of the opposite
[Footnote 3: This is clearly a maximum at AMC in Fig. 2, when its tangent AC/cm = the eccentricity.]
It is often said that a coincidence like this only happens to somebody who “deserves his luck,” but this simply means that recognition is essential to the coincidence. In the same way the appearance of one of a large number of people mentioned is hailed as a case of the old adage “Talk of the devil, etc.,” ignoring all the people who failed to appear. No one, however, will consider Kepler unduly favoured. His genius, in his case certainly “an infinite capacity for taking pains,” enabled him out of his medley of hypotheses, mainly unsound, by dint of enormous labour and patience, to arrive thus at the first two of the laws which established his title of “Legislator of the Heavens”.
Figures explanatory of Kepler’s theory of the motion of Mars.
[Illustration: Fig. 1.]
_______ / \ / \ | | |___________| Q| E C A |P | | \ / \_______/
[Illustration: Fig. 2.]
___M___ /___|\__\ // N|\\ \\ |/ | \\ \| |_____|__\\_| Q| E C A |P |\ | /| \\___|___// \___|___/
[Transcriber’s Note: Approximate renditions of these figures are provided. Fig. 1 is a circle. Fig. 2 is a circle which contains an ellipse, tangent to the circle at Q and P. Line segments from M (on the circle) and N (on the ellipse) meet at point A.]
Fig. 1.—In Ptolemy’s excentric theory, A may be taken to represent the earth, C the centre of a planet’s orbit, and E the equant, P (perigee) and Q (apogee) being the apses of the orbit. Ptolemy’s idea was that uniform motion in a circle must be provided, and since the motion was not uniform about the earth, A could not coincide with C; and since the motion still failed to be uniform about A or C, some point E must be found about which the motion should be uniform.
Fig. 2.—This is not drawn to scale, but is intended to illustrate Kepler’s modification of Ptolemy’s excentric. Kepler found velocities at P and Q proportional not to AP and AQ but to AQ and AP, or to EP and EQ if EC = CA (bisection of the excentricity). The velocity at M was wrong, and am appeared too great. Kepler’s first ellipse had M moved too near C. The distance AC is much exaggerated in the figure, as also is MN. An = CP, the radius of the circle. MN should be .00429 of the radius, and Mc/NC should be 1.00429. The velocity at N appeared to be proportional to en ( = an). Kepler concluded that Mars moved round PNQ, so that the area described about A (the sun) was equal in equal times, A being the focus of the ellipse PNQ. The angular velocity is not quite constant about E, the equant or empty focus, but the difference could hardly have been detected in Kepler’s time.
Kepler’s improved determination of the earth’s orbit was obtained by plotting the different positions of the earth corresponding to successive rotations of Mars, i.e. intervals of 687 days. At each of these the date of the year would give the angle MSE (Mars-Sun-Earth), and Tycho’s observation the angle MES. So the triangle could be solved except for scale, and the ratio of Se to SM would give the distance of Mars from the sun in terms of that of the earth. Measuring from a fixed position of Mars (e.g. perihelion), this gave the variation of Se, showing the earth’s inequality. Measuring from a fixed position of the earth, it would give similarly a series of positions of Mars, which, though lying not far from the circle whose diameter was the axis of Mars’ orbit, joining perihelion and aphelion, always fell inside the circle except at those two points. It was a long time before it dawned upon Kepler that the simplest figure falling within the circle except at the two extremities of the diameter, was an ellipse, and it is not clear why his first attempt with an ellipse should have been just as much too narrow as the circle was too wide. The fact remains that he recognised suddenly that halving this error was tantamount to reducing the circle to the ellipse whose eccentricity was that of the old theory, i.e. that in which the sun would be in one focus and the equant in the other.
Having now fitted the ends of both major and minor axes of the ellipse, he leaped to the conclusion that the orbit would fit everywhere.
The practical effect of his clearing of the “second inequality” was to refer the orbit of Mars directly to the sun, and he found that the area between successive distances of Mars from the sun (instead of the sum of the distances) was strictly proportional to the time taken, in short, equal areas were described in equal times (2nd Law) when referred to the sun in the focus of the ellipse (1st Law).
He announced that (1) The planet describes an ellipse, the sun being in one focus; and (2) The straight line joining the planet to the sun sweeps out equal areas in any two equal intervals of time. These are Kepler’s first and second Laws though not discovered in that order, and it was at once clear that Ptolemy’s “bisection of the excentricity” simply amounted to the fact that the centre of an ellipse bisects the distance between the foci, the sun being in one focus and the angular velocity being uniform about the empty focus. For so many centuries had the fetish of circular motion postponed discovery. It was natural that Kepler should assume that his laws would apply equally to all the planets, but the proof of this, as well as the reason underlying the laws, was only given by Newton, who approached the subject from a totally different standpoint.
This commentary on Mars was published in 1609, the year of the invention of the telescope, and Kepler petitioned the Emperor for further funds to enable him to complete the study of the other planets, but once more there was delay; in 1612 Rudolph died, and his brother Matthias who succeeded him, cared very little for astronomy or even astrology, though Kepler was reappointed to his post of Imperial Mathematician. He left Prague to take up a permanent professorship at the University of Linz. His own account of the circumstances is gloomy enough. He says, “In the first place I could get no money from the Court, and my wife, who had for a long time been suffering from low spirits and despondency, was taken violently ill towards the end of 1610, with the Hungarian fever, epilepsy and phrenitis. She was scarcely convalescent when all my three children were at once attacked with smallpox. Leopold with his army occupied the town beyond the river just as I lost the dearest of my sons, him whose nativity you will find in my book on the new star. The town on this side of the river where I lived was harassed by the Bohemian troops, whose new levies were insubordinate and insolent; to complete the whole, the Austrian army brought the plague with them into the city. I went into Austria and endeavoured to procure the situation which I now hold. Returning in June, I found my wife in a decline from her grief at the death of her son, and on the eve of an infectious fever, and I lost her also within eleven days of my return. Then came fresh annoyance, of course, and her fortune was to be divided with my step-sisters. The Emperor Rudolph would not agree to my departure; vain hopes were given me of being paid from Saxony; my time and money were wasted together, till on the death of the Emperor in 1612, I was named again by his successor, and suffered to depart to Linz.”
Being thus left a widower with a ten-year-old daughter Susanna, and a boy Louis of half her age, he looked for a second wife to take charge of them. He has given an account of eleven ladies whose suitability he considered. The first, an intimate friend of his first wife, ultimately declined; one was too old, another an invalid, another too proud of her birth and quarterings, another could do nothing useful, and so on. Number eight kept him guessing for three months, until he tired of her constant indecision, and confided his disappointment to number nine, who was not impressed. Number ten, introduced by a friend, Kepler found exceedingly ugly and enormously fat, and number eleven apparently too young. Kepler then reconsidered one of the earlier ones, disregarding the advice of his friends who objected to her lowly station. She was the orphan daughter of a cabinetmaker, educated for twelve years by favour of the Lady of Stahrenburg, and Kepler writes of her: “Her person and manners are suitable to mine; no pride, no extravagance; she can bear to work; she has a tolerable knowledge of how to manage a family; middle-aged and of a disposition and capability to acquire what she still wants”.
Wine from the Austrian vineyards was plentiful and cheap at the time of the marriage, and Kepler bought a few casks for his household. When the seller came to ascertain the quantity, Kepler noticed that no proper allowance was made for the bulging parts, and the upshot of his objections was that he wrote a book on a new method of gauging—one of the earliest specimens of modern analysis, extending the properties of plane figures to segments of cones and cylinders as being “incorporated circles”. He was summoned before the Diet at Ratisbon to give his opinion on the Gregorian Reform of the Calendar, and soon afterwards was excommunicated, having fallen foul of the Roman Catholic party at Linz just as he had previously at Gratz, the reason apparently being that he desired to think for himself. Meanwhile his salary was not paid any more regularly than before, and he was forced to supplement it by publishing what he called a “vile prophesying almanac which is scarcely more respectable than begging unless it be because it saves the Emperor’s credit, who abandons me entirely, and with all his frequent and recent orders in council, would suffer me to perish with hunger”.
In 1617 he was invited to Italy to succeed Magini as Professor of Mathematics at Bologna. Galileo urged him to accept the post, but he excused himself on the ground that he was a German and brought up among Germans with such liberty of speech as he thought might get him into trouble in Italy. In 1619 Matthias died and was succeeded by Ferdinand III, who again retained Kepler in his post. In the same year Kepler reprinted his “Mysterium Cosmographicum,” and also published his “Harmonics” in five books dedicated to James I of England. “The first geometrical, on the origin and demonstration
In the same book Kepler enlarges again on his views in reference to the basis of astrology as concerned with nativities and the importance of planetary conjunctions. He gives particulars of his own nativity. “Jupiter nearest the nonagesimal had passed by four degrees the trine of Saturn; the Sun and Venus in conjunction were moving from the latter towards the former, nearly in sextiles with both: they were also removing from quadratures with Mars, to which Mercury was closely approaching: the moon drew near to the trine of the same planet, close to the Bull’s Eye even in latitude. The 25th degree of Gemini was rising, and the 22nd of Aquarius culminating. That there was this triple configuration on that day—namely the sextile of Saturn and the Sun, the sextile of Mars and Jupiter, and the quadrature of Mercury and Mars, is proved by the change of weather; for after a frost of some days, that very day became warmer, there was a thaw and a fall of rain.” This alleged “proof” is interesting as it relies on the same principle which was held to justify the correction of an uncertain birth-time, by reference to illnesses, etc., met with later. Kepler however goes on to say, “If I am to speak of the results of my studies, what, I pray, can I find in the sky, even remotely alluding to it? The learned confess that several not despicable branches of philosophy have been newly extricated or amended or brought to perfection by me: but here my constellations were, not Mercury from the East in the angle of the seventh, and in quadratures with Mars, but Copernicus, but Tycho Brahe, without whose books of observations everything now set by me in the clearest light must have remained buried in darkness; not Saturn predominating Mercury, but my lords the Emperors Rudolph and Matthias, not Capricorn the house of Saturn but Upper Austria, the house of the Emperor, and the ready and unexampled bounty of his nobles to my petition. Here is that corner, not the western one of the horoscope, but on the earth whither, by permission of my Imperial master, I have betaken myself from a too uneasy Court; and whence, during these years of my life, which now tends towards its setting, emanate these Harmonics and the other matters on which I am engaged.”
The fifth book contains a great deal of nonsense about the harmony of the spheres; the notes contributed by the several planets are gravely set down, that of Mercury having the greatest resemblance to a melody, though perhaps more reminiscent of a bugle-call. Yet the book is not all worthless for it includes Kepler’s Third Law, which he had diligently sought for years. In his own words, “The proportion existing between the periodic times of any two planets is exactly the sesquiplicate proportion of the mean distances of the orbits,” or as generally given, “the squares of the periodic times are proportional to the cubes of the mean distances.” Kepler was evidently transported with delight and
Closing years.
Soon after its publication Kepler’s “Epitome” was placed along with the book of Copernicus, on the list of books prohibited by the Congregation of the Index at Rome, and he feared that this might prevent the publication or sale of his books in Austria also, but was told that though Galileo’s violence was getting him into trouble, there would be no difficulty in obtaining permission for learned men to read any prohibited books, and that he (Kepler) need fear nothing so long as he remained quiet.
In his various works on Comets, he adhered to the opinion that they travelled in straight lines with varying velocity. He suggested that comets come from the remotest parts of ether, as whales and monsters from the depth of the sea, and that perhaps they are something of the nature of silkworms, and are wasted and consumed in spinning their own tails. Napier’s invention of logarithms at once attracted Kepler’s attention. He must have regretted that the discovery was not made early enough to save him a vast amount of labour in computations, but he managed to find time to compute some logarithm tables for himself, though he does not seem to have understood quite what Napier had done, and though with his usual honesty he gave full credit to the Scottish baron for his invention.
Though Eugenists may find a difficulty in reconciling Napier’s brilliancy with the extreme youth of his parents, they may at any rate attribute Kepler’s occasional fits of bad temper to heredity. His cantankerous mother, Catherine Kepler, had for some years been carrying on an action for slander against a woman who had accused her of administering a poisonous potion. Dame Kepler employed a young advocate who for reasons of his own “nursed” the case so long that after five years had elapsed without any conclusion being reached another judge was appointed, who had himself suffered from the caustic tongue of the prosecutrix, and so was already prejudiced against her. The defendant, knowing this, turned the tables on her opponent by bringing an accusation of witchcraft against her, and Catherine Kepler was imprisoned and condemned to the torture in July, 1620. Kepler, hearing of the sentence, hurried back from Linz, and succeeded in stopping the completion of the sentence, securing his mother’s release the following year, as it was made clear that the only support for the case against her was her own intemperate language. Kepler returned to Linz, and his mother at once brought another action for costs and damages against her late opponent, but died before the case could be tried.
A few months before this Sir Henry Wotton, English Ambassador to Venice, visited Kepler, and finding him as usual, almost penniless, urged him to go to England, promising him a warm welcome there. Kepler, however, would not at that time leave Germany, giving several reasons, one of which was that he dreaded the confinement of an island. Later on he expressed his willingness to go as soon as his Rudolphine Tables were published, and lecture on them, even in England, if he could not do it in Germany, and if a good enough salary were forthcoming.
In 1624 he went to Vienna, and managed to extract from the Treasury 6000 florins on account of expenses connected with the Tables, but, instead of a further grant, was given letters to the States of Swabia, which owed money to the Imperial treasury. Some of this he succeeded in collecting, but the Tables were still further delayed by the religious disturbances then becoming violent. The Jesuits contrived to have Kepler’s library sealed up, and, but for the Imperial protection, would have imprisoned him also; moreover the peasants revolted and blockaded Linz. In 1627, however, the long promised Tables, the first to discard the conventional circular motion, were at last published at Ulm in four parts. Two of these parts consisted of subsidiary Tables, of logarithms and other computing devices, another contained Tables of the elements of the sun, moon, and planets, and the fourth gave the places of a thousand stars as determined by Tycho, with Tycho’s refraction Tables, which had the peculiarity of using different values for the refraction of the sun, moon, and stars. From a map prefixed to some copies of the Tables, we may infer that Kepler was one of the first, if not actually the first, to suggest the method of determining differences of longitude by occultations of stars at the moon’s limb. In an Appendix, he showed how his Tables could be used by astrologers for their predictions, saying “Astronomy is the daughter of Astrology, and this modern Astrology again is the daughter of Astronomy, bearing something of the lineaments of her grandmother; and, as I have already said, this foolish daughter, Astrology, supports her wise but needy mother, Astronomy, from the profits of a profession not generally considered creditable”. There is no doubt that Kepler strongly resented having to depend so much for his income on such methods which he certainly did not consider creditable.
It was probably Galileo whose praise of the new Tables induced the Grand Duke of Tuscany to send Kepler a gold chain soon after their publication, and we may perhaps regard it as a mark of favour from the Emperor Ferdinand that he permitted Kepler to attach himself to the great Wallenstein, now Duke of Friedland, and a firm believer in Astrology. The Duke was a better paymaster than either of the three successive Emperors. He furnished Kepler with an assistant and a printing press; and obtained for him the Professorship of Astronomy at the University of Rostock in Mecklenburg. Apparently, however, the Emperor could not induce Wallenstein to take over the responsibility of the 8000 crowns, still owing from the Imperial treasury on account of the Rudolphine Tables. Kepler made a last attempt to secure payment at Ratisbon, but his journey thither brought disappointment and fatigue and left him in such a condition that he rapidly succumbed to an attack of fever, dying in November, 1630, in his fifty-ninth year. His body was buried at Ratisbon, but the tombstone was destroyed during the
Kepler’s fame does not rest upon his voluminous works. With his peculiar method of approaching problems there was bound to be an inordinate amount of chaff mixed with the grain, and he used no winnowing machine. His simplicity and transparent honesty induced him to include everything, in fact he seemed to glory in the number of false trails he laboriously followed. He was one who might be expected to find the proverbial “needle in a haystack,” but unfortunately the needle was not always there. Delambre says, “Ardent, restless, burning to distinguish himself by his discoveries he attempted everything, and having once obtained a glimpse of one, no labour was too hard for him in following or verifying it. All his attempts had not the same success, and in fact that was impossible. Those which have failed seem to us only fanciful; those which have been more fortunate appear sublime. When in search of that which really existed, he has sometimes found it; when he devoted himself to the pursuit of a chimera, he could not but fail, but even then he unfolded the same qualities, and that obstinate perseverance that must triumph over all difficulties but those which are insurmountable.” Berry, in his “Short History of Astronomy,” says “as one reads chapter after chapter without a lucid, still less a correct idea, it is impossible to refrain from regrets that the intelligence of Kepler should have been so wasted, and it is difficult not to suspect at times that some of the valuable results which lie embedded in this great mass of tedious speculation were arrived at by a mere accident. On the other hand it must not be forgotten that such accidents have a habit of happening only to great men, and that if
Professor Forbes is more enthusiastic. In his “History of Astronomy,” he refers to Kepler as “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,” and again a propos of Kepler’s great 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?”
List of Dates.
Johann Kepler, born 1571; school at Maulbronn, 1586; University of Tuebingen, 1589; M.A. of Tuebingen, 1591; Professor at Gratz, 1594; “Prodromus,” with “Mysterium Cosmographicum,” published 1596; first marriage, 1597; joins Tycho Brahe at Prague, 1600; death of Tycho, 1601; Kepler’s optics, 1603; Nova, 1604; on Comets, 1607; Commentary on Mars, including First and Second Laws, 1609; Professor at Linz, 1612; second marriage, 1613; Third Law discovered, 1618; Epitome of Copernican Astronomy, 1618-1621; Rudolphine Tables published, 1627; died, 1630.
Bibliography.
For a full account of the various systems of Kepler and his predecessors the reader cannot do better than consult the “History of the Planetary Systems, from Thales to Kepler,” by Dr. J.L.E. Dreyer (Cambridge Univ. Press, 1906). The same author’s “Tycho Brahe” gives a wealth of detail about that “Phoenix of Astronomers,” as Kepler styles him. A great proportion of the literature relating to Kepler is German, but he has his place in the histories of astronomy, from Delambre and the more modern R. Wolfs “Geschichte” to those of A. Berry, “History of Astronomy” (University Extension Manuals, Murray, 1898), and Professor G. Forbes, “History of Astronomy” (History of Science Series, Watts, 1909).
Apogee: The point in the orbit of a celestial
body when it is furthest
from the earth.
Apse: An extremity of the major axis of the orbit
of a body; a body is
at its greatest and least
distances from the body about which it
revolves, when at one or other
apse.
Conjunction: When a plane containing the earth’s
axis and passing
through the centre of the
sun also passes through that of the moon
or a planet, at the same side
of the earth, the moon or planet is in
conjunction, or if on opposite
sides of the earth, the moon or
planet is in opposition.
Mercury and Venus cannot be in opposition,
but are in inferior or superior
conjunction according as they are
nearer or further than the
sun.
Deferent: In the epicyclic theory, uneven motion
is represented by
motion round a circle whose
centre travels round another circle, the
latter is called the deferent.
Ecliptic: The plane of the earth’s orbital
motion about the sun, which
cuts the heavens in a great
circle. It is so called because
obviously eclipses can only
occur when the moon is also
approximately in this plane,
besides being in conjunction or
opposition with the sun.
Epicycle: A point moving on the circumference
of a circle whose centre
describes another circle,
traces an epicycle with reference to the
centre of the second circle.
Equant: In Ptolemy’s excentric theory,
when a planet is describing a
circle about a centre which
is not the earth, in order to satisfy
the convention that the motion
must be uniform, a point was found
about which the motion was
apparently uniform,[4] and this point was
called the equant.
[Footnote 4: I.e. the angular motion about the equant was uniform.]
Equinox: When the sun is in the plane of the
earth’s equator the lengths
of day and night are equal.
This happens twice a year, and the times
when the sun passes the equator
are called the vernal or spring
equinox and the autumnal equinox
respectively.
Evection: The second inequality of the moon,
which vanishes at new and
full moon and is a maximum
at first and last quarter.
Excentric: As an alternative to epicycles, planets
whose motion round
the earth was not uniform
could be represented as moving round a
point some distance from the
earth called the excentric.
Geocentric: Referred to the centre of the earth; e.g. Ptolemy’s theory.
Heliocentric: Referred to the centre of the sun;
e.g. the theory
commonly called Copernican.
Inequality: The difference between the actual
position of a planet and
its theoretical position on
the hypothesis of uniform circular
motion.
Node: The points where the orbit of the moon
or a planet intersect the
plane of the ecliptic.
The ascending node is the one when the planet
is moving northwards, and
the line of intersection of the orbital
plane with the ecliptic is
the line of nodes.
Occultation: Usually means when a planet or star
is hidden by the moon,
but it also includes “occultation”
of a star by a planet or of a
satellite by a planet or of
one planet by another.
Opposition v. Conjunction.
Parallax: The error introduced by observing from
some point other than
that required in theory, e.g.
in geocentric places because the
observations are made from
the surface of the earth instead of the
centre, or in heliocentric
places because observations are made from
the earth and not from the
sun.
Perigee: The point in the orbit of a celestial
body when it is nearest
to the earth.
Precession: Owing to the slow motion of the earth’s
pole around the pole
of the ecliptic, the equator
cuts the ecliptic a little earlier
every year, so that the equinox
each year slightly precedes, with
reference to the stars, that
of the previous year.