Kepler Part 1

Kepler.

by Walter W. Bryant.

Chapter I.

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, a.s.signing fanciful terrestrial origins to constellations, with occasional controversies as to how the earth is supported in s.p.a.ce. 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 a.s.sume 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 number required by Aristotle reaching fifty-five. It may be regarded as counting in Aristotle's favour that he did consider the earth to be a sphere and not a flat disc, but he seems to have thought that the mathematical spheres of Eudoxus had a real solid existence, and that not only meteors, shooting stars and aurora, but also comets and the milky way belong to the atmosphere. His really great service to science in collating and criticising all that was known of natural science would have been greater if so much of the discussion had not been on the exact meaning of words used to describe phenomena, instead of on the facts and causes of the phenomena themselves.

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, a.s.sumed 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 pa.s.ses through the apses of the sun's...o...b..t, where its distance from the earth is greatest and least. The same result he could obtain by a.s.suming 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 astronomy for 260 years, the interval between Hipparchus and Ptolemy of Alexandria. Ptolemy accepted the spherical form of the earth but denied its rotation or any other movement. He made no advance on Hipparchus in regard to the sun, though the lapse of time had largely increased the errors of the elements adopted by the latter. In the case of the moon, however, Ptolemy traced the variable inequality noticed sometimes by Hipparchus at first and last quarter, which vanished when the moon was in apogee or perigee. This he called the evection, and introduced another epicycle to represent it. In his planetary theory he found that the places given by his adopted excentric did not fit, being one way at apogee and the other at perigee; so that the centre of distance must be nearer the earth. He found it best to a.s.sume the centre of distance half-way between the centre of the earth and the excentric, thus "bisecting the excentricity". Even this did not fit in the case of Mercury, and in general the agreement between theory and observation was spoilt by the necessity of making all the orbital planes pa.s.s through the centre of the earth, instead of the sun, thus making a good accordance practically impossible.

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 a.s.sistance 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 and for the sun. Of the seventy odd circles or epicycles required by the latest form of the Ptolemaic system, Copernicus succeeded in dispensing with rather more than half, but he still required thirty-four, which was the exact number a.s.sumed before the time of Aristotle. His considerations were almost entirely mathematical, his only invasion into physics being in defence of the "moving earth" against the stock objection that if the earth moved, loose objects would fly off, and towers fall. He did not break sufficiently away from the old tradition of uniform circular motion. Ptolemy's efforts at exactness were baulked, as we have seen, by the supposed necessity of all the orbit planes pa.s.sing through the earth, and if Copernicus had simply transferred this responsibility to the sun he would have done better. But he would not sacrifice the old fetish, and so, the orbit of the earth being clearly not circular with respect to the sun, he made all his planetary planes pa.s.s through the centre of the earth's...o...b..t, instead of through the sun, thus handicapping himself in the same way though not in the same degree as Ptolemy. His thirty-four circles or epicycles comprised four for the earth, three for the moon, seven for Mercury (on account of his highly eccentric orbit) and five each for the other planets.

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 s.p.a.ce, 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 princ.i.p.al 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.

Chapter II.

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 Leonmonta.n.u.s, in accordance with the common practice of the time, but he was not known by it to any great extent. He was sent to school in 1577, but in the following year his father returned to Germany, almost ruined by the absconding of an acquaintance for whom he had become surety. Henry Kepler was obliged to sell his house and most of his belongings, and to keep a tavern at Elmendingen, withdrawing his son from school to help him with the rough work. In 1583 young Kepler was sent to the school at Elmendingen, and in 1584 had another narrow escape from death by a violent illness. In 1586 he was sent, at the charges of the Duke, to the monastic school of Maulbronn; from whence, in accordance with the school regulations, he pa.s.sed at the end of his first year the examination for the bachelor's degree at Tubingen, returning for two more years as a "veteran" to Maulbronn before being admitted as a resident student at Tubingen. The three years thus spent at Maulbronn were marked by recurrences of several of the diseases from which he had suffered in childhood, and also by family troubles at his home. His father went away after a quarrel with his wife Catherine, and died abroad. Catherine herself, who seems to have been of a very unamiable disposition, next quarrelled with her own relatives. It is not surprising therefore that Kepler after taking his M.A. degree in August, 1591, coming out second in the examination lists, was ready to accept the first appointment offered him, even if it should involve leaving home. This happened to be the lectures.h.i.+p in astronomy at Gratz, the chief town in Styria. Kepler's knowledge of astronomy was limited to the compulsory school course, nor had he as yet any particular leaning towards the science; the post, moreover, was a meagre and unimportant one. On the other hand he had frequently expressed disgust at the way in which one after another of his companions had refused "foreign" appointments which had been arranged for them under the Duke's scheme of education. His tutors also strongly urged him to accept the lectures.h.i.+p, and he had not the usual reluctance to leave home. He therefore proceeded to Gratz, protesting that he did not thereby forfeit his claim to a more promising opening, when such should appear. His astronomical tutor, Maestlin, encouraged him to devote himself to his newly adopted science, and the first result of this advice appeared before very long in Kepler's "Mysterium Cosmographic.u.m". The bent of his mind was towards philosophical speculation, to which he had been attracted in his youthful studies of Scaliger's "Exoteric Exercises". He says he devoted much time "to the examination of the nature of heaven, of souls, of genii, of the elements, of the essence of fire, of the cause of fountains, the ebb and flow of the tides, the shape of the continents and inland seas, and things of this sort". Following his tutor in his admiration for the Copernican theory, he wrote an essay on the primary motion, attributing it to the rotation of the earth, and this not for the mathematical reasons brought forward by Copernicus, but, as he himself says, on physical or metaphysical grounds. In 1595, having more leisure from lectures, he turned his speculative mind to the number, size, and motion of the planetary orbits. He first tried simple numerical relations, but none of them appeared to be twice, thrice, or four times as great as another, although he felt convinced that there was some relation between the motions and the distances, seeing that when a gap appeared in one series, there was a corresponding gap in the other. These gaps he attempted to fill by hypothetical planets between Mars and Jupiter, and between Mercury and Venus, but this method also failed to provide the regular proportion which he sought, besides being open to the objection that on the same principle there might be many more equally invisible planets at either end of the series. He was nevertheless unwilling to adopt the opinion of Rheticus that the number six was sacred, maintaining that the "sacredness" of the number was of much more recent date than the creation of the worlds, and could not therefore account for it. He next tried an ingenious idea, comparing the perpendiculars from different points of a quadrant of a circle on a tangent at its extremity. The greatest of these, the tangent, not being cut by the quadrant, he called the line of the sun, and a.s.sociated with infinite force. The shortest, being the point at the other end of the quadrant, thus corresponded to the fixed stars or zero force; intermediate ones were to be found proportional to the "forces" of the six planets. After a great amount of unfinished trial calculations, which took nearly a whole summer, he convinced himself that success did not lie that way. In July, 1595, while lecturing on the great planetary conjunctions, he drew quasi-triangles in a circular zodiac showing the slow progression of these points of conjunction at intervals of just over 240 or eight signs. The successive chords marked out a smaller circle to which they were tangents, about half the diameter of the zodiacal circle as drawn, and Kepler at once saw a similarity to the orbits of Saturn and Jupiter, the radius of the inscribed circle of an equilateral triangle being half that of the circ.u.mscribed circle. His natural sequence of ideas impelled him to try a square, in the hope that the circ.u.mscribed and inscribed circles might give him a similar "a.n.a.logy" for the orbits of Jupiter and Mars. He next tried a pentagon and so on, but he soon noted that he would never reach the sun that way, nor would he find any such limitation as six, the number of "possibles" being obviously infinite. The actual planets moreover were not even six but only five, so far as he knew, so he next pondered the question of what sort of things these could be of which only five different figures were possible and suddenly thought of the five regular solids.[2] He immediately pounced upon this idea and ultimately evolved the following scheme. "The earth is the sphere, the measure of all; round it describe a dodecahedron; the sphere including this will be Mars. Round Mars describe a tetrahedron; the sphere including this will be Jupiter. Describe a cube round Jupiter; the sphere including this will be Saturn. Now, inscribe in the earth an icosahedron, the sphere inscribed in it will be Venus: inscribe an octahedron in Venus: the circle inscribed in it will be Mercury." With this result Kepler was inordinately pleased, and regretted not a moment of the time spent in obtaining it, though to us this "Mysterium Cosmographic.u.m" can only appear useless, even without the more recent additions to the known planets. He admitted that a certain thickness must be a.s.signed to the intervening spheres to cover the greatest and least distances of the several planets from the sun, but even then some of the numbers obtained are not a very close fit for the corresponding planetary orbits. Kepler's own suggested explanation of the discordances was that they must be due to erroneous measures of the planetary distances, and this, in those days of crude and infrequent observations, could not easily be disproved. He next thought of a variety of reasons why the five regular solids should occur in precisely the order given and in no other, diverging from this into a subtle and not very intelligible process of reasoning to account for the division of the zodiac into 360. The next subject was more important, and dealt with the relation between the distances of the planets and their times of revolution round the sun. It was obvious that the period was not simply proportional to the distance, as the outer planets were all too slow for this, and he concluded "either that the moving intelligences of the planets are weakest in those that are farthest from the sun, or that there is one moving intelligence in the sun, the common centre, forcing them all round, but those most violently which are nearest, and that it languishes in some sort and grows weaker at the most distant, because of the remoteness and the attenuation of the virtue". This is not so near a guess at the theory of gravitation as might be supposed, for Kepler imagined that a repulsive force was necessary to account for the planets being sometimes further from the sun, and so laid aside the idea of a constant attractive force. He made several other attempts to find a law connecting the distances and periods of the planets, but without success at that time, and only desisted when by unconsciously arguing in a circle he appeared to get the same result from two totally different hypotheses. He sent copies of his book to several leading astronomers, of whom Galileo praised his ingenuity and good faith, while Tycho Brahe was evidently much struck with the work and advised him to adapt something similar to the Tychonic system instead of the Copernican. He also intimated that his Uraniborg observations would provide more accurate determinations of the planetary orbits, and thus made Kepler eager to visit him, a project which as we shall see was more than fulfilled. Another copy of the book Kepler sent to Reymers the Imperial astronomer with a most fulsome letter, which Tycho, who a.s.serted that Reymers had simply plagiarised his work, very strongly resented, thus drawing from Kepler a long letter of apology. About the same time Kepler had married a lady already twice widowed, and become involved in difficulties with her relatives on financial grounds, and with the Styrian authorities in connection with the religious disputes then coming to a head. On account of these latter he thought it expedient, the year after his marriage, to withdraw to Hungary, from whence he sent short treatises to Tubingen, "On the magnet" (following the ideas of Gilbert of Colchester), "On the cause of the obliquity of the ecliptic" and "On the Divine wisdom as shown in the Creation". His next important step makes it desirable to devote a chapter to a short notice of Tycho Brahe.

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.

Chapter III.

Tycho Brahe.

The age following that of Copernicus produced three outstanding figures a.s.sociated 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 n.o.ble 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. Soon afterwards he purchased an edition of Ptolemy in order to read up the subject of astronomy, to which, and to mathematics, he devoted most of the remainder of his three years' course at Copenhagen. His uncle next sent him to Leipzig to study law, but he managed to continue his astronomical researches. He obtained the Alphonsine and the new Prutenic Tables, but soon found that the latter, though more accurate than the former, failed to represent the true positions of the planets, and grasped the fact that continuous observation was essential in order to determine the true motions. He began by observing a conjunction of Jupiter and Saturn in August, 1563, and found the Prutenic Tables several days in error, and the Alphonsine a whole month. He provided himself with a cross-staff for determining the angular distance between stars or other objects, and, finding the divisions of the scale inaccurate, constructed a table of corrections, an improvement that seems to have been a decided innovation, the previous practice having been to use the best available instrument and ignore its errors. About this time war broke out between Denmark and Sweden, and Tycho returned to his uncle, who was vice-admiral and attached to the king's suite. The uncle died in the following month, and early in the next year Tycho went abroad again, this time to Wittenberg. After five months, however, an outbreak of plague drove him away, and he matriculated at Rostock, where he found little astronomy but a good deal of astrology. While there he fought a duel in the dark and lost part of his nose, which he replaced by a composition of gold and silver. He carried on regular observations with his cross-staff and persevered with his astronomical studies in spite of the objections and want of sympathy of his fellow-countrymen. The King of Denmark, however, having a higher opinion of the value of science, promised Tycho the first canonry that should fall vacant in the cathedral chapter of Roskilde, so that he might be a.s.sured of an income while devoting himself to financially unproductive work. In 1568 Tycho left Rostock, and matriculated at Basle, but soon moved on to Augsburg, where he found more enthusiasm for astronomy, and induced one of his new friends to order the construction of a large 19-foot quadrant of heavy oak beams. This was the first of the series of great instruments a.s.sociated with Tycho's name, and it remained in use for five years, being destroyed by a great storm in 1574. Tycho meanwhile had left Augsburg in 1570 and returned to live with his father, now governor of Helsingborg Castle, until the latter's death in the following year. Tycho then joined his mother's brother, Steen Bille, the only one of his relatives who showed any sympathy with his desire for a scientific career.

On 11th November, 1572, Tycho noticed an unfamiliar bright star in the constellation of Ca.s.siopeia, and continued to observe it with a s.e.xtant. 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 authors.h.i.+p beneath the dignity of a n.o.ble 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 Ca.s.sel, 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 friends.h.i.+p, 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 ma.n.u.scripts, 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 large instruments then found necessary in order to attain the accuracy of observation which within the next half-century was to be so greatly facilitated by the invention of the telescope. Here also he built several smaller observing rooms, so that his pupils should be able to observe independently. For more than twenty years he continued his observations at Uraniborg, surrounded by his family, and attracting numerous pupils. His constant aim was to acc.u.mulate a large store of observations of a high order of accuracy, and thus to provide data for the complete reform of astronomy. As we have seen, few of the Danish n.o.bles had any sympathy with Tycho's pursuits, and most of them strongly resented the continual expense borne by the King's treasury. Tycho moreover was so absorbed in his scientific pursuits that he would not take the trouble to be a good landlord, nor to carry out all the duties laid upon him in return for certain of his grants of income. His buildings included a chemical laboratory, and he was in the habit of making up elixirs for various medical purposes; these were quite popular, particularly as he made no charge for them. He seems to have been something of a h.o.m.opathist, for he recommends sulphur to cure infectious diseases "brought on by the sulphurous vapours of the Aurora Borealis"!

King Frederick, in consideration of various grants to Tycho, relied upon his a.s.sistance 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 s.e.xtant, 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 a.s.sistants. 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 hypothesis. He therefore a.s.sumed that the other planets revolved about the sun, while the sun, moon, and stars revolved about the earth as a centre. Geometrically this is much the same as the Copernican system, but physically it involves the grotesque demand that the whole system of stars revolves round our insignificant little earth every twenty-four hours. Since his previous small book on the comet, Tycho had evidently considered more fully its possible astrological significance, for he foretold a religious war, giving the date of its commencement, and also the rising of a great Protestant champion. These predictions were apparently fulfilled almost to the letter by the great religious wars that broke out towards the end of the sixteenth century, and in the person of Gustavus Adolphus.

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 to Uraniborg. Before they reached him, after many vexatious delays, he had given up waiting for the funds promised for his building expenses, and removed from Benatek to Prague. It was during this interval that after considerable negotiation, Kepler, who had been in correspondence with Tycho, consented to join him as an a.s.sistant. Another a.s.sistant, Longomonta.n.u.s, who had been with Tycho at Uraniborg, was finding difficulty with the long series of Mars observations, and it was arranged that he should transfer his energies to the lunar observations, leaving those of Mars for Kepler. Before very much could be done with them, however, Tycho died at the end of October, 1601. He may have regretted the peaceful island of Hveen, considering the troubles in which Bohemia was rapidly becoming involved, but there is little doubt that had it not been for his self-imposed exile, his observations would not have come into Kepler's hands, and their great value might have been lost. In any case it was at Uraniborg that the ma.s.s of observations was produced upon which the fame of Tycho Brahe rests. His own discoveries, though in themselves the most important made in astronomy for many centuries, are far less valuable than those for which his observations furnished the material. He discovered the third and fourth inequalities of the moon in longitude, called respectively the variation and the annual equation, also the variability of the motion of the moon's nodes and the inclination of its...o...b..t to the ecliptic. He obtained an improved value of the constant of precession, and did good service by rejecting the idea that it was variable, an idea which, under the name of trepidation, had for many centuries been accepted. He discovered the effect of refraction, though only approximately its amount, and determined improved values of many other astronomical constants, but singularly enough made no determination of the distance of the sun, adopting instead the ancient and erroneous value given by Hipparchus.

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.

Chapter IV.

Kepler Joins Tycho.

The a.s.sociation 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 a.s.sistants. 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 uniform kindness in the face of great rudeness and ingrat.i.tude. He begged him to come "as a welcome friend," though Kepler, very touchy on the subject of his own astronomical powers, was afraid he might be regarded as simply a subordinate a.s.sistant. An arrangement had been suggested by which Kepler should obtain two years' leave of absence from Gratz on full pay, which, because of the higher cost of living in Prague, should be supplemented by the Emperor; but before this could be concluded, Kepler threw up his professors.h.i.+p, and thinking he had thereby also lost the chance of going to Prague, applied to Maestlin and others of his Tubingen friends to make interest for him with the Duke of Wurtemberg and secure the professors.h.i.+p of medicine. Tycho, however, still urged him to come to Prague, promising to do his utmost to secure for him a permanent appointment, or in any event to see that he was not the loser by coming. Kepler was delayed by illness on the way, but ultimately reached Prague, accompanied by his wife, and for some time lived entirely at Tycho's expense, writing by way of return essays against Reymers and another man, who had claimed the credit of the Tychonic system. This Kepler could do with a clear conscience, as it was only a question of priority and did not involve any support of the system, which he deemed far inferior to that of Copernicus. The following year saw friction between the two astronomers, and we learn from Kepler's abject letter of apology that he was entirely in the wrong. It was about money matters, which in one way or another embittered the rest of Kepler's life, and it arose during his absence from Prague. On his return in September, 1601, Tycho presented him to the Emperor, who gave him the t.i.tle of Imperial Mathematician, on condition of a.s.sisting Tycho in his calculations, the very thing Kepler was most anxious to be allowed to do: for nowhere else in the world was there such a collection of good observations sufficient for his purpose of reforming the whole theory of astronomy. The Emperor's interest was still mainly with astrology, but he liked to think that his name would be handed down to posterity in connection with the new Planetary Tables in the same way as that of Alphonso of Castile, and he made liberal promises to pay the expenses. Tycho's other princ.i.p.al a.s.sistant, Longomonta.n.u.s, did not stay long after giving up the Mars observations to Kepler, but instead of working at the new lunar theory, suddenly left to take up a professors.h.i.+p of astronomy in his native Denmark. Very shortly afterwards Tycho himself died of acute distemper; Kepler began to prepare the ma.s.s of ma.n.u.scripts for publication, but, as everything was claimed by the Brahe family, he was not allowed to finish the work. He succeeded to Tycho's post of princ.i.p.al mathematician to the Emperor, at a reduced official salary, which owing to the emptiness of the Imperial treasury was almost always in arrear. In order to meet his expenses he had recourse to the casting of nativities, for which he gained considerable reputation and received very good pay. He worked by the conventional rules of astrology, and was quite prepared to take fees for so doing, although he had very little faith in them, preferring his own fanciful ideas.

In 1604 the constellation of Ca.s.siopeia 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 att.i.tude 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 venture on some new scheme, just as if G.o.d had set up this star in the darkness merely to enlighten them." He made no secret of his views on conventional astrology, as to which he claimed to speak with the authority of one fully conversant with its principles, but he nevertheless expressed his sincere conviction that the conjunctions and aspects of the planets certainly did affect things on the earth, maintaining that he was driven to this belief against his will by "most unfailing experiences".

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 att.i.tude 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 alt.i.tudes, 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 a.s.sumed that all rays pa.s.sing 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 with those near the horizon, so Kepler "went off at a tangent" and tried a totally new set of ideas, which all reduced to the absurdity of a refraction which vanished at the horizon. These were followed by another set, involving either a constant amount of refraction or one becoming infinite. He then came to the conclusion that these geometrical methods must fail because the refracted image is not real, and determined to try by a.n.a.logy only, comparing the equally unreal image formed by a mirror with that formed by refraction in water. He noticed how the bottom of a vessel containing water appears to rise more and more away from the vertical, and at once jumped to the a.n.a.logy of a concave mirror, which magnifies the image, while a convex mirror was likened to a rarer medium. This line of attack also failed him, as did various attempts to find relations between his measurements of refraction and conic sections, and he broke off suddenly with a diatribe against Tycho's critics, whom he likened to blind men disputing about colours. Not many years later Snell discovered the true law of refraction, but Kepler's contribution to the subject, though he failed to discover the actual law, includes several of the adopted "by-laws". He noted that atmospheric refraction would alter with the height of the atmosphere and with temperature, and also recognised the fact that rainbow colours depend on the angle of refraction, whether seen in the rainbow itself, or in dew, gla.s.s, water, or any similar medium. He thus came near to antic.i.p.ating Newton. Before leaving the subject of Kepler's optics it will be well to recall that a few years later after hearing of Galileo's telescope, Kepler suggested that for astronomical purposes two convex lenses should be used, so that there should be a real image where measuring wires could be placed for reference. He did not carry out the idea himself, and it was left to the Englishman Gascoigne to produce the first instrument on this "Keplerian" principle, universally known as the Astronomical Telescope.

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 a.s.sumed that they were external to the system and so could not be expected to return, he might have antic.i.p.ated 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.

Chapter V.

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 sh.o.r.es 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 a.s.suming 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 s.p.a.ce proportional to the comparative ma.s.s of the other. If the moon and earth were not retained in their orbits by their animal force or some other equivalent, the earth would mount to the moon by a fifty-fourth part of their distance, and the moon fall towards the earth through the other fifty-three parts, and they would there meet, a.s.suming, however, that the substance of both is of the same density. If the earth should cease to attract its waters to itself all the waters of the sea would he raised and would flow to the body of the moon. The sphere of the attractive virtue which is in the moon extends as far as the earth, and entices up the waters; but as the moon flies rapidly across the zenith, and the waters cannot follow so quickly, a flow of the ocean is occasioned in the torrid zone towards the westward. If the attractive virtue of the moon extends as far as the earth, it follows with greater reason that the attractive virtue of the earth extends as far as the moon and much farther; and, in short, nothing which consists of earthly substance anyhow const.i.tuted although thrown up to any height, can ever escape the powerful operation of this attractive virtue. Nothing which consists of corporeal matter is absolutely light, but that is comparatively lighter which is rarer, either by its own nature, or by accidental heat. And it is not to be thought that light bodies are escaping to the surface of the universe while they are carried upwards, or that they are not attracted by the earth. They are attracted, but in a less degree, and so are driven outwards by the heavy bodies; which being done, they stop, and are kept by the earth in their own place. But although the attractive virtue of the earth extends upwards, as has been said, so very far, yet if any stone should be at a distance great enough to become sensible compared with the earth's diameter, it is true that on the motion of the earth such a stone would not follow altogether; its own force of resistance would be combined with the attractive force of the earth, and thus it would extricate itself in some degree from the motion of the earth." The above pa.s.sage from the Introduction to Kepler's "Commentaries on the Motion of Mars," always regarded as his most valuable work, must have been known to Newton, so that no such incident as the fall of an apple was required to provide a necessary and sufficient explanation of the genesis of his Theory of Universal Gravitation. Kepler's glimpse at such a theory could have been no more than a glimpse, for he went no further with it. This seems a pity, as it is far less fanciful than many of his ideas, though not free from the "virtues" and "animal faculties," that correspond to Gilbert's "spirits and humours". We must, however, proceed to the subject of Mars, which was, as before noted, the first important investigation entrusted to Kepler on his arrival at Prague.

The time taken from one opposition of Mars to the next is decidedly unequal at different parts of his...o...b..t, 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 Longomonta.n.u.s 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 lat.i.tude 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 lat.i.tudes. 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 a.s.sumptions 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 lat.i.tudes 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 a.s.sistants 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 pa.s.sed 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 lat.i.tudes, 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 a.s.sumption known as the principle of the bisection of the excentricity. Hitherto, since Ptolemy had given no reason for this a.s.sumption, 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...o...b..t, 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