“But in this our age, one rare witte (seeing the continuall errors that from time to time more and more continually have been discovered, besides the infinite absurdities in their Theoricks, which they have been forced to admit that would not confesse any Mobilitie in the ball of the Earth) hath by long studye, paynfull practise, and rare invention delivered a new Theorick or Model of the world, shewing that the Earth resteth not in the Center of the whole world or globe of elements, which encircled and enclosed in the Moone’s orbit, and together with the whole globe of mortality is carried yearly round about the Sunne, which like a king in the middest of all, rayneth and giveth laws of motion to all the rest, sphaerically dispersing his glorious beames of light through all this sacred coelestiall Temple.”

Thomas Digges, 1590.

70. The growing interest in astronomy shewn by the work of such men as Regiomontanus was one of the early results in the region of science of the great movement of thought to different aspects of which are given the names of Revival of Learning, Renaissance, and Reformation, The movement may be regarded primarily as a general quickening of intelligence and of interest in matters of thought and knowledge. The invention of printing early in the 15th century, the stimulus to the study of the Greek authors, due in part to the scholars who were driven westwards after the capture of Constantinople by the Turks (1453), and the discovery of America by Columbus in 1492, all helped on a movement the beginning of which has to be looked for much earlier.

Every stimulus to the intelligence naturally brings with it a tendency towards inquiry into opinions received through tradition and based on some great authority. The effective discovery and the study of Greek philosophers other than Aristotle naturally did much to shake the supreme authority of that great philosopher, just as the Reformers shook the authority of the Church by pointing out what they considered to be inconsistencies between its doctrines and those of the Bible. At first there was little avowed opposition to the principle that truth was to be derived from some authority, rather than to be sought independently by the light of reason; the new scholars replaced the authority of Aristotle by that of Plato or of Greek and Roman antiquity in general, and the religious Reformers replaced the Church by the Bible. Naturally, however, the conflict between authorities produced in some minds scepticism as to the principle of authority itself; when freedom of judgment had to be exercised to the extent of deciding between authorities, it was but a step further—a step, it is true, that comparatively few took—to use the individual judgment on the matter at issue itself.

In astronomy the conflict between authorities had already arisen, partly in connection with certain divergencies between Ptolemy and Aristotle, partly in connection with the various astronomical tables which, though on substantially the same lines, differed in minor points. The time was therefore ripe for some fundamental criticism of the traditional astronomy, and for its reconstruction on a new basis.

Such a fundamental change was planned and worked out by the great astronomer whose work has next to be considered.71. Nicholas Coppernic or Coppernicus45 was born on February 19th, 1473, in a house still pointed out in the little trading town of Thorn on the Vistula. Thorn now lies just within the eastern frontier of the present kingdom of Prussia; in the time of Coppernicus it lay in a region over which the King of Poland had some sort of suzerainty, the precise nature of which was a continual subject of quarrel between him, the citizens, and the order of Teutonic knights, who claimed a good deal of the neighbouring country. The astronomer’s father (whose name was most commonly written Koppernigk) was a merchant who came to Thorn from Cracow, then the capital of Poland, in 1462. Whether Coppernicus should be counted as a Pole or as a German is an intricate question, over which his biographers have fought at great length and with some acrimony, but which is not worth further discussion here.

Nicholas, after the death of his father in 1483, was under the care of his uncle, Lucas Watzelrode, afterwards bishop of the neighbouring diocese of Ermland, and was destined by him from a very early date for an ecclesiastical career. He attended the school at Thorn, and at the age of 17 entered the University of Cracow. Here he seems to have first acquired (or shewn) a decided taste for astronomy and mathematics, subjects in which he probably received help from Albert Brudzewski, who had a great reputation as a learned and stimulating teacher; the lecture lists of the University show that the comparatively modern treatises of Purbach and Regiomontanus (chapter III., §68) were the standard textbooks used. Coppernicus had no intention of graduating at Cracow, and probably left after three years (1494). During the next year or two he lived partly at home, partly at his uncle’s palace at Heilsberg, and spent some of the time in an unsuccessful candidature for a canonry at Frauenburg, the cathedral city of his uncle’s diocese.

The next nine or ten years of his life (from 1496 to 1505 or 1506) were devoted to studying in Italy, his stay there being broken only by a short visit to Frauenburg in 1501. He worked chiefly at Bologna and Padua, but graduated at Ferrara, and also spent some time at Rome, where his astronomical knowledge evidently made a favourable impression. Although he was supposed to be in Italy primarily with a view to studying law and medicine, it is evident that much of his best work was being put into mathematics and astronomy, while he also paid a good deal of attention to Greek.

During his absence he was appointed (about 1497) to a canonry at Frauenburg, and at some uncertain date be also received a sinecure ecclesiastical appointment at Breslau.72. On returning to Frauenburg from Italy Coppernicus almost immediately obtained fresh leave of absence, and joined his uncle at Heilsberg, ostensibly as his medical adviser and really as his companion.

It was probably during the quiet years spent at Heilsberg that he first put into shape his new ideas about astronomy, and wrote the first draft of his book. He kept the manuscript by him, revising and rewriting from time to time, partly from a desire to make his work as perfect as possible, partly from complete indifference to reputation, coupled with dislike of the controversy to which the publication of his book would almost certainly give rise. In 1509 he published at Cracow his first book, a Latin translation of a set of Greek letters by Theophylactus, interesting as being probably the first translation from the Greek ever published in Poland or the adjacent districts. In 1512, on the death of his uncle, he finally settled in Frauenburg, in a set of rooms which he occupied, with short intervals, for the next 31 years. Once fairly in residence, he took his share in conducting the business of the Chapter: he acted, for example, more than once as their representative in various quarrels with the King of Poland and the Teutonic knights; in 1523 he was general administrator of the diocese for a few months after the death of the bishop; and for two periods, amounting altogether to six years (1516-1519 and 1520-1521), he lived at the castle of Allenstein, administering some of the outlying property of the Chapter. In 1521 he was commissioned to draw up a statement of the grievances of the Chapter against the Teutonic knights for presentation to the Prussian Estates, and in the following year wrote a memorandum on the debased and confused state of the coinage in the district, a paper which was also laid before the Estates, and was afterwards rewritten in Latin at the special request of the bishop. He also gave a certain amount of medical advice to his friends as well as to the poor of Frauenburg, though he never practised regularly as a physician; but notwithstanding these various occupations it is probable that a very large part of his time during the last 30 years of his life was devoted to astronomy.73. We are so accustomed to associate the revival of astronomy, as of other branches of natural science, with increased care in the collection of observed facts, and to think of Coppernicus as the chief agent in the revival, that it is worth while here to emphasise the fact that he was in no sense a great observer. His instruments, which were mostly of his own construction, were far inferior to those of Nassir Eddin and of Ulugh Begh (chapter III., §§62, 63), and not even as good as those which he could have procured if he had wished from the workshops of NÜrnberg; his observations were not at all numerous (only 27, which occur in his book, and a dozen or two besides being known), and he appears to have made no serious attempt to secure great accuracy. His determination of the position of one star, which was extensively used by him as a standard of reference and was therefore of special importance, was in error to the extent of nearly 40' (more than the apparent breadth of the sun or moon), an error which Hipparchus would have considered very serious. His pupil Rheticus (§74) reports an interesting discussion between his master and himself, in which the pupil urged the importance of making observations with all imaginable accuracy; Coppernicus answered that minute accuracy was not to be looked for at that time, and that a rough agreement between theory and observation was all that he could hope to attain. Coppernicus moreover points out in more than one place that the high latitude of Frauenburg and the thickness of the air were so detrimental to good observation that, for example, though he had occasionally been able to see the planet Mercury, he had never been able to observe it properly.

Although he published nothing of importance till towards the end of his life, his reputation as an astronomer and mathematician appears to have been established among experts from the date of his leaving Italy, and to have steadily increased as time went on.

In 1515 he was consulted by a committee appointed by the Lateran Council to consider the reform of the calendar, which had now fallen into some confusion (chapter II., §22), but he declined to give any advice on the ground that the motions of the sun and moon were as yet too imperfectly known for a satisfactory reform to be possible. A few years later (1524) he wrote an open letter, intended for publication, to one of his Cracow friends, in reply to a tract on precession, in which, after the manner of the time, he used strong language about the errors of his opponent.46

It was meanwhile gradually becoming known that he held the novel doctrine that the earth was in motion and the sun and stars at rest, a doctrine which was sufficiently startling to attract notice outside astronomical circles. About 1531 he had the distinction of being ridiculed on the stage at some popular performance in the neighbourhood; and it is interesting to note (especially in view of the famous persecution of Galilei at Rome a century later) that Luther in his Table Talk frankly described Coppernicus as a fool for holding such opinions, which were obviously contrary to the Bible, and that Melanchthon, perhaps the most learned of the Reformers, added to a somewhat similar criticism a broad hint that such opinions should not be tolerated. Coppernicus appears to have taken no notice of these or similar attacks, and still continued to publish nothing. No observation made later than 1529 occurs in his great book, which seems to have been nearly in its final form by that date; and to about this time belongs an extremely interesting paper, known as the Commentariolus, which contains a short account of his system of the world, with some of the evidence for it, but without any calculations. It was apparently written to be shewn or lent to friends, and was not published; the manuscript disappeared after the death of the author and was only rediscovered in 1878. The Commentariolus was probably the basis of a lecture on the ideas of Coppernicus given in 1533 by one of the Roman astronomers at the request of Pope Clement VII. Three years later Cardinal Schomberg wrote to ask Coppernicus for further information as to his views, the letter showing that the chief features were already pretty accurately known.74. Similar requests must have been made by others, but his final decision to publish his ideas seems to have been due to the arrival at Frauenburg in 1539 of the enthusiastic young astronomer generally known as Rheticus.47 Born in 1514, he studied astronomy under Schoner at NÜrnberg, and was appointed in 1536 to one of the chairs of mathematics created by the influence of Melanchthon at Wittenberg, at that time the chief Protestant University.

Having heard, probably through the Commentariolus, of Coppernicus and his doctrines, he was so much interested in them that he decided to visit the great astronomer at Frauenburg. Coppernicus received him with extreme kindness, and the visit, which was originally intended to last a few days or weeks, extended over nearly two years. Rheticus set to work to study Coppernicus’s manuscript, and wrote within a few weeks of his arrival an extremely interesting and valuable account of it, known as the First Narrative (Prima Narratio), in the form of an open letter to his old master Schoner, a letter which was printed in the following spring and was the first easily accessible account of the new doctrines.48

When Rheticus returned to Wittenberg, towards the end of 1541, he took with him a copy of a purely mathematical section of the great book, and had it printed as a textbook of the subject (Trigonometry); it had probably been already settled that he was to superintend the printing of the complete book itself. Coppernicus, who was now an old man and would naturally feel that his end was approaching, sent the manuscript to his friend Giese, Bishop of Kulm, to do what he pleased with. Giese sent it at once to Rheticus, who made arrangements for having it printed at NÜrnberg. Unfortunately Rheticus was not able to see it all through the press, and the work had to be entrusted to Osiander, a Lutheran preacher interested in astronomy. Osiander appears to have been much alarmed at the thought of the disturbance which the heretical ideas of Coppernicus would cause, and added a prefatory note of his own (which he omitted to sign), praising the book in a vulgar way, and declaring (what was quite contrary to the views of the author) that the fundamental principles laid down in it were merely abstract hypotheses convenient for purposes of calculation; he also gave the book the title De Revolutionibus Orbium Celestium (On the Revolutions of the Celestial Spheres), the last two words of which were probably his own addition. The printing was finished in the winter 1542-3, and the author received a copy of his book on the day of his death (May 24th, 1543), when his memory and mental vigour had already gone.75. The central idea with which the name of Coppernicus is associated, and which makes the De Revolutionibus one of the most important books in all astronomical literature, by the side of which perhaps only the Almagest and Newton’s Principia (chapter IX., §§177 seqq.) can be placed, is that the apparent motions of the celestial bodies are to a great extent not real motions, but are due to the motion of the earth carrying the observer with it. Coppernicus tells us that he had long been struck by the unsatisfactory nature of the current explanations of astronomical observations, and that, while searching in philosophical writings for some better explanation, he had found a reference of Cicero to the opinion of Hicetas that the earth turned round on its axis daily. He found similar views held by other Pythagoreans, while Philolaus and Aristarchus of Samos had also held that the earth not only rotates, but moves bodily round the sun or some other centre (cf. chapter II., §24). The opinion that the earth is not the sole centre of motion, but that Venus and Mercury revolve round the sun, he found to be an old Egyptian belief, supported also by Martianus Capella, who wrote a compendium of science and philosophy in the 5th or 6th century A.D. A more modern authority, Nicholas of Cusa (1401-1464), a mystic writer who refers to a possible motion of the earth, was ignored or not noticed by Coppernicus. None of the writers here named, with the possible exception of Aristarchus of Samos, to whom Coppernicus apparently paid little attention, presented the opinions quoted as more than vague speculations; none of them gave any substantial reasons for, much less a proof of, their views; and Coppernicus, though he may have been glad, after the fashion of the age, to have the support of recognised authorities, had practically to make a fresh start and elaborate his own evidence for his opinions.

It has sometimes been said that Coppernicus proved what earlier writers had guessed at or suggested; it would perhaps be truer to say that he took up certain floating ideas, which were extremely vague and had never been worked out scientifically, based on them certain definite fundamental principles, and from these principles developed mathematically an astronomical system which he shewed to be at least as capable of explaining the observed celestial motions as any existing variety of the traditional Ptolemaic system. The Coppernican system, as it left the hands of the author, was in fact decidedly superior to its rivals as an explanation of ordinary observations, an advantage which it owed quite as much to the mathematical skill with which it was developed as to its first principles; it was in many respects very much simpler; and it avoided certain fundamental difficulties of the older system. It was however liable to certain serious objections, which were only overcome by fresh evidence which was subsequently brought to light. For the predecessors of Coppernicus there was, apart from variations of minor importance, but one scientific system which made any serious attempt to account for known facts; for his immediate successors there were two, the newer of which would to an impartial mind appear on the whole the more satisfactory, and the further study of the two systems, with a view to the discovery of fresh arguments or fresh observations tending to support the one or the other, was immediately suggested as an inquiry of first-rate importance.76. The plan of the De Revolutionibus bears a general resemblance to that of the Almagest. In form at least the book is not primarily an argument in favour of the motion of the earth, and it is possible to read much of it without ever noticing the presence of this doctrine.

Coppernicus, like Ptolemy, begins with certain first principles or postulates, but on account of their novelty takes a little more trouble than his predecessor (cf. chapter II., §47) to make them at once appear probable. With these postulates as a basis he proceeds to develop, by means of elaborate and rather tedious mathematical reasoning, aided here and there by references to observations, detailed schemes of the various celestial motions; and it is by the agreement of these calculations with observations, far more than by the general reasoning given at the beginning, that the various postulates are in effect justified.

His first postulate, that the universe is spherical, is supported by vague and inconclusive reasons similar to those given by Ptolemy and others; for the spherical form of the earth he gives several of the usual valid arguments, one of his proofs for its curvature from east to west being the fact that eclipses visible at one place are not visible at another. A third postulate, that the motions of the celestial bodies are uniform circular motions or are compounded of such motions, is, as might be expected, supported only by reasons of the most unsatisfactory character. He argues, for example, that any want of uniformity in motion

“must arise either from irregularity in the moving power, whether this be within the body or foreign to it, or from some inequality of the body in revolution.... Both of which things the intellect shrinks from with horror, it being unworthy to hold such a view about bodies which are constituted in the most perfect order.”

77. The discussion of the possibility that the earth may move, and may even have more than one motion, then follows, and is more satisfactory though by no means conclusive. Coppernicus has a firm grasp of the principle, which Aristotle had also enunciated, sometimes known as that of relative motion, which he states somewhat as follows:—

“For all change in position which is seen is due to a motion either of the observer or of the thing looked at, or to changes in the position of both, provided that these are different. For when things are moved equally relatively to the same things, no motion is perceived, as between the object seen and the observer.”49

Coppernicus gives no proof of this principle, regarding it probably as sufficiently obvious, when once stated, to the mathematicians and astronomers for whom he was writing. It is, however, so fundamental that it may be worth while to discuss it a little more fully.

Let, for example, the observer be at A and an object at B, then whether the object move from B to B', the observer remaining at rest, or the observer move an equal distance in the opposite direction, from A to A', the object remaining at rest, the effect is to the eye exactly the same, since in either case the distance between the observer and object and the direction in which the object is seen, represented in the first case by A B' and in the second by A' B, are the same.

Thus if in the course of a year either the sun passes successively through the positions A, B, C, D (fig. 38), the earth remaining at rest at E, or if the sun is at rest and the earth passes successively through the positions a, b, c, d, at the corresponding times, the sun remaining at rest at S, exactly the same effect is produced on the eye, provided that the lines a S, b S, c S, d S are, as in the figure, equal in length and parallel in direction to E A, E B, E C, E D respectively. The same being true of intermediate points, exactly the same apparent effect is produced whether the sun describe the circle A B C D, or the earth describe at the same rate the equal circle a b c d. It will be noticed further that, although the corresponding motions in the two cases are at the same times in opposite directions (as at A and a), yet each circle as a whole is described, as indicated by the arrowheads, in the same direction (contrary to that of the motion of the hands of a clock, in the figures given). It follows in the same sort of way that an apparent motion (as of a planet) may be explained as due partially to the motion of the object, partially to that of the observer.

Coppernicus gives the familiar illustration of the passenger in a boat who sees the land apparently moving away from him, by quoting and explaining Virgil’s line:—

“Provehimur portu, terrÆque urbesque recedunt.”

78. The application of the same ideas to an apparent rotation round the observer, as in the case of the apparent daily motion of the celestial sphere, is a little more difficult. It must be remembered that the eye has no means of judging the direction of an object taken by itself; it can only judge the difference between the direction of the object and some other direction, whether that of another object or a direction fixed in some way by the body of the observer. Thus when after looking at a star twice at an interval of time we decide that it has moved, this means that its direction has changed relatively to, say, some tree or house which we had noticed nearly in its direction, or that its direction has changed relatively to the direction in which we are directing our eyes or holding our bodies. Such a change can evidently be interpreted as a change of direction, either of the star or of the line from the eye to the tree which we used as a line of reference. To apply this to the case of the celestial sphere, let us suppose that S represents a star on the celestial sphere, which (for simplicity) is overhead to an observer on the earth at A, this being determined by comparison with a line A B drawn upright on the earth. Next, earth and celestial sphere being supposed to have a common centre at O, let us suppose firstly that the celestial sphere turns round (in the direction of the hands of a clock) till S comes to S', and that the observer now sees the star on his horizon or in a direction at right angles to the original direction A B, the angle turned through by the celestial sphere being S O S'; and secondly that, the celestial sphere being unchanged, the earth turns round in the opposite direction, till A B comes to A' B', and the star is again seen by the observer on his horizon. Whichever of these motions has taken place, the observer sees exactly the same apparent motion in the sky; and the figure shews at once that the angle S O S' through which the celestial sphere was supposed to turn in the first case is equal to the angle A O A' through which the earth turns in the second case, but that the two rotations are in opposite directions. A similar explanation evidently applies to more complicated cases.

Hence the apparent daily rotation of the celestial sphere about an axis through the poles would be produced equally well, either by an actual rotation of this character, or by a rotation of the earth about an axis also passing through the poles, and at the same rate, but in the opposite direction, i.e. from west to east. This is the first motion which Coppernicus assigns to the earth.79. The apparent annual motion of the sun, in accordance with which it appears to revolve round the earth in a path which is nearly a circle, can be equally well explained by supposing the sun to be at rest, and the earth to describe an exactly equal path round the sun, the direction of the revolution being the same. This is virtually the second motion which Coppernicus gives to the earth, though, on account of a peculiarity in his geometrical method, he resolves this motion into two others, and combines with one of these a further small motion which is required for precession.5080. Coppernicus’s conception then is that the earth revolves round the sun in the plane of the ecliptic, while rotating daily on an axis which continually points to the poles of the celestial sphere, and therefore retains (save for precession) a fixed direction in space.

It should be noticed that the two motions thus assigned to the earth are perfectly distinct; each requires its own proof, and explains a different set of appearances. It was quite possible, with perfect consistency, to believe in one motion without believing in the other, as in fact a very few of the 16th-century astronomers did (chapter V., §105).

In giving his reasons for believing in the motion of the earth Coppernicus discusses the chief objections which had been urged by Ptolemy. To the objection that if the earth had a rapid motion of rotation about its axis, the earth would be in danger of flying to pieces, and the air, as well as loose objects on the surface, would be left behind, he replies that if such a motion were dangerous to the solid earth, it must be much more so to the celestial sphere, which, on account of its vastly greater size, would have to move enormously faster than the earth to complete its daily rotation; he enters also into an obscure discussion of difference between a “natural” and an “artificial” motion, of which the former might be expected not to disturb anything on the earth.

Coppernicus shews that the earth is very small compared to the sphere of the stars, because wherever the observer is on the earth the horizon appears to divide the celestial sphere into two equal parts and the observer appears always to be at the centre of the sphere, so that any distance through which the observer moves on the earth is imperceptible as compared with the distance of the stars.81. He goes on to argue that the chief irregularity in the motion of the planets, in virtue of which they move backwards at intervals (chapter I., §14, and chapter II., §51), can readily be explained in general by the motion of the earth and by a motion of each planet round the sun, in its own time and at its own distance. From the fact that Venus and Mercury were never seen very far from the sun, it could be inferred that their paths were nearer to the sun than that of the earth. Mercury being the nearer to the sun of the two, because never seen so far from it in the sky as Venus. The other three planets, being seen at times in a direction opposite to that of the sun, must necessarily evolve round the sun in orbits larger than that of the earth, a view confirmed by the fact that they were brightest when opposite the sun (in which positions they would be nearest to us). The order of their respective distances from the sun could be at once inferred from the disturbing effects produced on their apparent motions by the motion of the earth; Saturn being least affected must on the whole be farthest from the earth, Jupiter next, and Mars next. The earth thus became one of six planets revolving round the sun, the order of distance—Mercury, Venus, Earth, Mars, Jupiter, Saturn—being also in accordance with the rates of motion round the sun, Mercury performing its revolution most rapidly (in about 88 days51), Saturn most slowly (in about 30 years). On the Coppernican system the moon alone still revolved round the earth, being the only celestial body the status of which was substantially unchanged; and thus Coppernicus was able to give the accompanying diagram of the solar system (fig. 40), representing his view of its general arrangement (though not of the right proportions of the different parts) and of the various motions.

82. The effect of the motion of the earth round the sun on the length of the day and other seasonal effects is discussed in some detail, and illustrated by diagrams which are here reproduced.52

In fig. 41 A, B, C, D represent the centre of the earth in four positions, occupied by it about December 23rd, March 21st, June 22nd, and September 22nd respectively (i.e. at the beginnings of the four seasons, according to astronomical reckoning); the circle F G H I in each of its positions represents the equator of the earth, i.e. a great circle on the earth the plane of which is perpendicular to the axis of the earth and is consequently always parallel to the celestial equator. This circle is not in the plane of the ecliptic, but tilted up at an angle of 23-1/2°, so that F must always be supposed below and H above the plane of the paper (which represents the ecliptic); the equator cuts the ecliptic along G I. The diagram (in accordance with the common custom in astronomical diagrams) represents the various circles as seen from the north side of the equator and ecliptic. When the earth is at A, the north pole (as is shewn more clearly in fig. 42, in which P, P' denote the north pole and south pole respectively) is turned away from the sun, E, which is on the lower or south side of the plane of the equator, and consequently inhabitants of the northern hemisphere see the sun for less than half the day, while those on the southern hemisphere see the sun for more than half the day, and those beyond the line K L (in fig. 42) see the sun during the whole day. Three months later, when the earth’s centre is at B (fig. 41), the sun lies in the plane of the equator, the poles of the earth are turned neither towards nor away from the sun, but aside, and all over the earth daylight lasts for 12 hours and night for an equal time. Three months later still, when the earth’s centre is at C, the sun is above the plane of the equator, and the inhabitants of the northern hemisphere see the sun for more than half the day, those on the southern hemisphere for less than half, while those in parts of the earth farther north than the line M N (in fig. 42) see the sun for the whole 24 hours. Finally, when, at the autumn equinox, the earth has reached D (fig. 41), the sun is again in the plane of the equator, and the day is everywhere equal to the night.

83. Coppernicus devotes the first eleven chapters of the first book to this preliminary sketch of his system; the remainder of this book he fills with some mathematical propositions and tables, which, as previously mentioned (§74), had already been separately printed by Rheticus. The second book contains chiefly a number of the usual results relating to the celestial sphere and its apparent daily motion, treated much as by earlier writers, but with greater mathematical skill. Incidentally Coppernicus gives his measurement of the obliquity of the ecliptic, and infers from a comparison with earlier observations that the obliquity had decreased, which was in fact the case, though to a much less extent than his imperfect observations indicated. The book ends with a catalogue of stars, which is Ptolemy’s catalogue, occasionally corrected by fresh observations, and rearranged so as to avoid the effects of precession.53 When, as frequently happened, the Greek and Latin versions of the Almagest gave, owing to copyists’ or printers’ errors, different results, Coppernicus appears to have followed sometimes the Latin and sometimes the Greek version, without in general attempting to ascertain by fresh observations which was right.84. The third book begins with an elaborate discussion of the precession of the equinoxes (chapter II., §42). From a comparison of results obtained by Timocharis, by later Greek astronomers, and by Albategnius, Coppernicus infers that the amount of precession has varied, but that its average value is 50·2 annually (almost exactly the true value), and accepts accordingly Tabit ben Korra’s unhappy suggestion of the trepidation (chapter III., §58). An examination of the data used by Coppernicus shews that the erroneous or fraudulent observations of Ptolemy (chapter II., §50) are chiefly responsible for the perpetuation of this mistake.

Of much more interest than the detailed discussion of trepidation and of geometrical schemes for representing it is the interpretation of precession as the result of a motion of the earth’s axis. Precession was originally recognised by Hipparchus as a motion of the celestial equator, in which its inclination to the ecliptic was sensibly unchanged. Now the ideas of Coppernicus make the celestial equator dependent on the equator of the earth, and hence on its axis; it is in fact a great circle of the celestial sphere which is always perpendicular to the axis about which the earth rotates daily. Hence precession, on the theory of Coppernicus, arises from a slow motion of the axis of the earth, which moves so as always to remain inclined at the same angle to the ecliptic, and to return to its original position after a period of about 26,000 years (since a motion of 50·2 annually is equivalent to 360° or a complete circuit in that period); in other words, the earth’s axis has a slow conical motion, the central line (or axis) of the cone being at right angles to the plane of the ecliptic.85. Precession being dealt with, the greater part of the remainder of the third book is devoted to a discussion in detail of the apparent annual motion of the sun round the earth, corresponding to the real annual motion of the earth round the sun. The geometrical theory of the Almagest was capable of being immediately applied to the new system, and Coppernicus, like Ptolemy, uses an eccentric. He makes the calculations afresh, arrives at a smaller and more accurate value of the eccentricity (about 1/31 instead of 1/24), fixes the position of the apogee and perigee (chapter II., §39), or rather of the equivalent aphelion and perihelion (i.e. the points in the earth’s orbit where it is respectively farthest from and nearest to the sun), and thus verifies Albategnius’s discovery (chapter III., §59) of the motion of the line of apses. The theory of the earth’s motion is worked out in some detail, and tables are given whereby the apparent place of the sun at any time can be easily computed.

The fourth book deals with the theory of the moon. As has been already noticed, the moon was the only celestial body the position of which in the universe was substantially unchanged by Coppernicus, and it might hence have been expected that little alteration would have been required in the traditional theory. Actually, however, there is scarcely any part of the subject in which Coppernicus did more to diminish the discrepancies between theory and observation. He rejects Ptolemy’s equant (chapter II., §51), partly on the ground that it produces an irregular motion unsuitable for the heavenly bodies, partly on the more substantial ground that, as already pointed out (chapter II., §48), Ptolemy’s theory makes the apparent size of the moon at times twice as great as at others. By an arrangement of epicycles Coppernicus succeeded in representing the chief irregularities in the moon’s motion, including evection, but without Ptolemy’s prosneusis (chapter II., §48) or Abul Wafa’s inequality (chapter III., §60), while he made the changes in the moon’s distance, and consequently in its apparent size, not very much greater than those which actually take place, the difference being imperceptible by the rough methods of observation which he used.54

In discussing the distances and sizes of the sun and moon Coppernicus follows Ptolemy closely (chapter II., §49; cf. also fig. 20); he arrives at substantially the same estimate of the distance of the moon, but makes the sun’s distance 1,500 times the earth’s radius, thus improving to some extent on the traditional estimate, which was based on Ptolemy’s. He also develops in some detail the effect of parallax on the apparent place of the moon, and the variations in the apparent size, owing to the variations in distance: and the book ends with a discussion of eclipses.86. The last two books (V. and VI.) deal at length with the motion of the planets.

In the cases of Mercury and Venus, Ptolemy’s explanation of the motion could with little difficulty be rearranged so as to fit the ideas of Coppernicus. We have seen (chapter II., §51) that, minor irregularities being ignored, the motion of either of these planets could be represented by means of an epicycle moving on a deferent, the centre of the epicycle being always in the direction of the sun, the ratio of the sizes of the epicycle and deferent being fixed, but the actual dimensions being practically arbitrary. Ptolemy preferred on the whole to regard the epicycles of both these planets as lying between the earth and the sun. The idea of making the sun a centre of motion having once been accepted, it was an obvious simplification to make the centre of the epicycle not merely lie in the direction of the sun, but actually be the sun. In fact, if the planet in question revolved round the sun at the proper distance and at the proper rate, the same appearances would be produced as by Ptolemy’s epicycle and deferent, the path of the planet round the sun replacing the epicycle, and the apparent path of the sun round the earth (or the path of the earth round the sun) replacing the deferent.

In discussing the time of revolution of a planet a distinction has to be made, as in the case of the moon (chapter II., §40), between the synodic and sidereal periods of revolution. Venus, for example, is seen as an evening star at its greatest angular distance from the sun (as at V in fig. 43) at intervals of about 584 days. This is therefore the time which Venus takes to return to the same position relatively to the sun, as seen from the earth, or relatively to the earth, as seen from the sun; this time is called the synodic period. But as during this time the line E S has changed its direction, Venus is no longer in the same position relatively to the stars, as seen either from the sun or from the earth. If at first Venus and the earth are at V₁, E₁; respectively, after 584 days (or about a year and seven months) the earth will have performed rather more than a revolution and a half round the sun and will be at E₂; Venus being again at the greatest distance from the sun will therefore be at V₂, but will evidently be seen in quite a different part of the sky, and will not have performed an exact revolution round the sun. It is important to know how long the line S V₁ takes to return to the same position, i.e. how long Venus takes to return to the same position with respect to the stars, as seen from the sun, an interval of time known as the sidereal period. This can evidently be calculated by a simple rule-of-three sum from the data given. For Venus has in 584 days gained a complete revolution on the earth, or has gone as far as the earth would have gone in 584 + 365 or 949 days (fractions of days being omitted for simplicity); hence Venus goes in 584 × 365/949 days as far as the earth in 365 days, i.e. Venus completes a revolution in 584 × 365/949 or 225 days. This is therefore the sidereal period of Venus. The process used by Coppernicus was different, as he saw the advantage of using a long period of time, so as to diminish the error due to minor irregularities, and he therefore obtained two observations of Venus at a considerable interval of time, in which Venus occupied very nearly the same position both with respect to the sun and to the stars, so that the interval of time contained very nearly an exact number of sidereal periods as well as of synodic periods. By dividing therefore the observed interval of time by the number of sidereal periods (which being a whole number could readily be estimated), the sidereal period was easily obtained. A similar process shewed that the synodic period of Mercury was about 116 days, and the sidereal period about 88 days.

The comparative sizes of the orbits of Venus and Mercury as compared with that of the earth could easily be ascertained from observations of the position of either planet when most distant from the sun. Venus, for example, appears at its greatest distance from the sun when at a point V₁ (fig. 44) such that V₁ E₁ touches the circle in which Venus moves, and the angle E₁ V₁ S is then (by a known property of a circle) a right angle. The angle S E₁ V₁ being observed, the shape of the triangle S E₁ V₁ is known, and the ratio of its sides can be readily calculated. Thus Coppernicus found that the average distance of Venus from the sun was about 72 and that of Mercury about 36, the distance of the earth from the sun being taken to be 100; the corresponding modern figures are 72·3 and 38·7.

87. In the case of the superior planets. Mars, Jupiter, and Saturn, it was much more difficult to recognise that their motions could be explained by supposing them to revolve round the sun, since the centre of the epicycle did not always lie in the direction of the sun, but might be anywhere in the ecliptic. One peculiarity, however, in the motion of any of the superior planets might easily have suggested their motion round the sun, and was either completely overlooked by Ptolemy or not recognised by him as important. It is possible that it was one of the clues which led Coppernicus to his system. This peculiarity is that the radius of the epicycle of the planet, j J, is always parallel to the line E S joining the earth and sun, and consequently performs a complete revolution in a year. This connection between the motion of the planet and that of the sun received no explanation from Ptolemy’s theory. Now if we draw E J' parallel to j J and equal to it in length, it is easily seen55 that the line J' J is equal and parallel to E j, that consequently J describes a circle round J' just as j round E. Hence the motion of the planet can equally well be represented by supposing it to move in an epicycle (represented by the large dotted circle in the figure) of which J' is the centre and J' J the radius, while the centre of the epicycle, remaining always in the direction of the sun, describes a deferent (represented by the small circle round E) of which the earth is the centre. By this method of representation the motion of the superior planet is exactly like that of an inferior planet, except that its epicycle is larger than its deferent; the same reasoning as before shows that the motion can be represented simply by supposing the centre J' of the epicycle to be actually the sun. Ptolemy’s epicycle and deferent are therefore capable of being replaced, without affecting the position of the planet in the sky, by a motion of the planet in a circle round the sun, while the sun moves round the earth, or, more simply, the earth round the sun.

The synodic period of a superior planet could best be determined by observing when the planet was in opposition, i.e. when it was (nearly) opposite the sun, or, more accurately (since a planet does not move exactly in the ecliptic), when the longitudes of the planet and sun differed by 180° (or two right angles, chapter II., §43). The sidereal period could then be deduced nearly as in the case of an inferior planet, with this difference, that the superior planet moves more slowly than the earth, and therefore loses one complete revolution in each synodic period; or the sidereal period might be found as before by observing when oppositions occurred nearly in the same part of the sky.56 Coppernicus thus obtained very fairly accurate values for the synodic and sidereal periods, viz. 780 days and 687 days respectively for Mars, 399 days and about 12 years for Jupiter, 378 days and 30 years for Saturn (cf. fig. 40).

The calculation of the distance of a superior planet from the sun is a good deal more complicated than that of Venus or Mercury. If we ignore various details, the process followed by Coppernicus is to compute the position of the planet as seen from the sun, and then to notice when this position differs most from its position as seen from the earth, i.e. when the earth and sun are farthest apart as seen from the planet. This is clearly when (fig. 46) the line joining the planet (P) to the earth (E) touches the circle described by the earth, so that the angle S P E is then as great as possible. The angle P E S is a right angle, and the angle S P E is the difference between the observed place of the planet and its computed place as seen from the sun; these two angles being thus known, the shape of the triangle S P E is known, and therefore also the ratio of its sides. In this way Coppernicus found the average distances of Mars, Jupiter, and Saturn from the sun to be respectively about 1-1/2, 5, and 9 times that of the earth; the corresponding modern figures are 1·5, 5·2, 9·5.88. The explanation of the stationary points of the planets (chapter I., §14) is much simplified by the ideas of Coppernicus. If we take first an inferior planet, say Mercury (fig. 47), then when it lies between the earth and sun, as at M (or as on Sept. 5 in fig. 7), both the earth and Mercury are moving in the same direction, but a comparison of the sizes of the paths of Mercury and the earth, and of their respective times of performing complete circuits, shews that Mercury is moving faster than the earth. Consequently to the observer at E, Mercury appears to be moving from left to right (in the figure), or from east to west; but this is contrary to the general direction of motion of the planets, i.e. Mercury appears to be retrograding. On the other hand, when Mercury appears at the greatest distance from the sun, as at M₁ and M₂, its own motion is directly towards or away from the earth, and is therefore imperceptible; but the earth is moving towards the observer’s right, and therefore Mercury appears to be moving towards the left, or from west to east. Hence between M₁ and M its motion has changed from direct to retrograde, and therefore at some intermediate point, say m₁, (about Aug, 23 in fig. 7), Mercury appears for the moment to be stationary, and similarly it appears to be stationary again when at some point m₂ between M and M₂ (about Sept. 13 in fig. 7).

In the case of a superior planet, say Jupiter, the argument is nearly the same. When in opposition at J (as on Mar. 26 in fig. 6), Jupiter moves more slowly than the earth, and in the same direction, and therefore appears to be moving in the opposite direction to the earth, i.e. as seen from E (fig. 48), from left to right, or from east to west, that is in the retrograde direction. But when Jupiter is in either of the positions J₁ or J (in which the earth appears to the observer on Jupiter to be at its greatest distance from the sun), the motion of the earth itself being directly to or from Jupiter produces no effect on the apparent motion of Jupiter (since any displacement directly to or from the observer makes no difference in the object’s place on the celestial sphere); but Jupiter itself is actually moving towards the left, and therefore the motion of Jupiter appears to be also from right to left, or from west to east. Hence, as before, between J₁ and J and between J and J₂ there must be points j₁, j₂ (Jan. 24 and May 27, in fig. 6) at which Jupiter appears for the moment to be stationary.

The actual discussion of the stationary points given by Coppernicus is a good deal more elaborate and more technical than the outline given here, as he not only shews that the stationary points must exist, but shews how to calculate their exact positions.89. So far the theory of the planets has only been sketched very roughly, in order to bring into prominence the essential differences between the Coppernican and the Ptolemaic explanations of their motions, and no account has been taken of the minor irregularities for which Ptolemy devised his system of equants, eccentrics, etc., nor of the motion in latitude, i.e. to and from the ecliptic. Coppernicus, as already mentioned, rejected the equant, as being productive of an irregularity “unworthy” of the celestial bodies, and constructed for each planet a fairly complicated system of epicycles. For the motion in latitude discussed in Book VI. he supposed the orbit of each planet round the sun to be inclined to the ecliptic at a small angle, different for each planet, but found it necessary, in order that his theory should agree with observation, to introduce the wholly imaginary complication of a regular increase and decrease in the inclinations of the orbits of the planets to the ecliptic.

The actual details of the epicycles employed are of no great interest now, but it may be worth while to notice that for the motions of the moon, earth, and five other planets Coppernicus required altogether 34 circles, viz. four for the moon, three for the earth, seven for Mercury (the motion of which is peculiarly irregular), and five for each of the other planets; this number being a good deal less than that required in most versions of Ptolemy’s system: Fracastor (chapter III., §69), for example, writing in 1538, required 79 spheres, of which six were required for the fixed stars.90. The planetary theory of Coppernicus necessarily suffered from one of the essential defects of the system of epicycles. It is, in fact, always possible to choose a system of epicycles in such a way as to make either the direction of any body or its distance vary in any required manner, but not to satisfy both requirements at once. In the case of the motion of the moon round the earth, or of the earth round the sun, cases in which variations in distance could not readily be observed, epicycles might therefore be expected to give a satisfactory result, at any rate until methods of observation were sufficiently improved to measure with some accuracy the apparent sizes of the sun and moon, and so check the variations in their distances. But any variation in the distance of the earth from the sun would affect not merely the distance, but also the direction in which a planet would be seen; in the figure, for example, when the planet is at P and the sun at S, the apparent position of the planet, as seen from the earth, will be different according as the earth is at E or E'. Hence the epicycles and eccentrics of Coppernicus, which had to be adjusted in such a way that they necessarily involved incorrect values of the distances between the sun and earth, gave rise to corresponding errors in the observed places of the planets. The observations which Coppernicus used were hardly extensive or accurate enough to show this discrepancy clearly; but a crucial test was thus virtually suggested by means of which, when further observations of the planets had been made, a decision could be taken between an epicyclic representation of the motion of the planets and some other geometrical scheme.

91. The merits of Coppernicus are so great, and the part which he played in the overthrow of the Ptolemaic system is so conspicuous, that we are sometimes liable to forget that, so far from rejecting the epicycles and eccentrics of the Greeks, he used no other geometrical devices, and was even a more orthodox “epicyclist” than Ptolemy himself, as he rejected the equants of the latter.57 Milton’s famous description (Par. Lost, VIII. 82-5) of

applies therefore just as well to the astronomy of Coppernicus as to that of his predecessors; and it was Kepler (chapter VII.), writing more than half a century later, not Coppernicus, to whom the rejection of the epicycle and eccentric is due.92. One point which was of importance in later controversies deserves special mention here. The basis of the Coppernican system was that a motion of the earth carrying the observer with it produced an apparent motion of other bodies. The apparent motions of the sun and planets were thus shewn to be in great part explicable as the result of the motion of the earth round the sun. Similar reasoning ought apparently to lead to the conclusion that the fixed stars would also appear to have an annual motion. There would, in fact, be a displacement of the apparent position of a star due to the alteration of the earth’s position in its orbit, closely resembling the alteration in the apparent position of the moon due to the alteration of the observer’s position on the earth which had long been studied under the name of parallax (chapter II., §43). As such a displacement had never been observed, Coppernicus explained the apparent contradiction by supposing the fixed stars so far off that any motion due to this cause was too small to be noticed. If, for example, the earth moves in six months from E to E', the change in direction of a star at S' is the angle E' S' E, which is less than that of a nearer star at S; and by supposing the star S' sufficiently remote, the angle E' S' E can be made as small as may be required. For instance, if the distance of the star were 300 times the distance E E', i.e. 600 times as far from the earth as the sun is, the angle E S' E' would be less than 12', a quantity which the instruments of the time were barely capable of detecting.58 But more accurate observations of the fixed stars might be expected to throw further light on this problem.