,illustration3 ,,kepler ,pioneers of ,progress ,men of ,science ,edited by ,s4 ,chapman1 ,m4,a41 ,d4,sc41 ,f4,r4,s4 ,,kepler by ,,walter ,w4 ,,bryant of the ,royal ,observatory1 ,greenwich #aibj ,,contents4 ,i4 ,astronomy ,before ,kepler ,,ii4 ,early ,life of ,kepler ,,iii4 ,tycho ,brahe ,,iv4 ,kepler ,joins ,tycho ,v4 ,kepler's ,laws ,,vi4 ,closing ,years ,appendix ,i4--,list of ,dates ,appendix ,,ii4--,bibliography ,glossary ,,chapter ,i4 ,,astronomy ,,before ,,kepler4 ,in order to emphasise the importance of the reforms introduced into astronomy by ,kepler1 it will be well to sketch briefly the history of the theories which he had to overthrow4 ,in very early times it must have been realised that the sun and moon were continually changing their places among the stars4 ,the day1 the month1 and the year were obvious divisions of time1 and longer periods were suggested by the tabulation of eclipses4 ,we can imagine the respect accorded to the ,chaldaean sages who first discovered that eclipses could be predicted1 and how the philosophers of ,mesopotamia must have sought eagerly for evidence of fresh periodic laws4 ,certain of the stars1 which appeared to wander1 and were hence called planets1 provided an extended field for these speculations4 ,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 observances4 ,such speculations as were current among them1 and also among the ,egyptians and others who came to share their knowledge1 were almost entirely devoted to mythology1 assigning fanciful terrestrial origins to constellations1 with occasional controversies as to how the earth is supported in space4 ,the ,greeks1 too1 had an elaborate mythology largely adapted from their neighbours1 but they were not satisfied with this1 and made persistent attempts to reduce the apparent motions of celestial objects to geometrical laws4 ,some of the ,pythagoreans1 if not ,pythagoras himself1 held that the earth is a sphere1 and that the apparent daily revolution of the sun and stars is really due to a motion of the earth1 though at first this motion of the earth was not supposed to be one of rotation about an axis4 ,these notions1 and also that the planets on the whole move round from west to east with reference to the stars1 were made known to a larger circle through the writings of ,plato4 ,to ,plato moreover is attributed the challenge to astronomers to represent all the motions of the heavenly bodies by uniformly described circles1 a challenge generally held responsible for a vast amount of wasted effort1 and the postponement1 for many centuries1 of real progress4 ,eudoxus of ,cnidus1 endeavouring to account for the fact that the planets1 during every apparent revolution round the earth1 come to rest twice1 and in the shorter interval between these stationary points1 move in the opposite direction1 found that he could represent the phenomena fairly well by a system of concentric spheres1 each rotating with its own velocity1 and carrying its own particular planet round its own equator1 the outermost sphere carrying the fixed stars4 ,it was necessary to assume that the axes about which the various spheres revolved should have circular motions also1 and gradually an increased number of spheres was evolved1 the total number required by ,aristotle reaching fifty-five4 ,it may be regarded as counting in ,aristotle's favour that he did consider the earth to be a sphere and not a flat disc1 but he seems to have thought that the mathematical spheres of ,eudoxus had a real solid existence1 and that not only meteors1 shooting stars and aurora1 but also comets and the milky way belong to the atmosphere4 ,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 phenomena1 instead of on the facts and causes of the phenomena themselves4 ,aristarchus of ,samos seems to have been the first to suggest that the planets revolved not about the earth but about the sun1 but the idea seemed so improbable that it was hardly noticed1 especially as ,aristarchus himself did not expand it into a treatise4 ,about this time the necessity for more accurate places of the sun and moon1 and the liberality of the ,ptolemys who ruled ,egypt1 combined to provide regular observations at ,alexandria1 so that1 when ,hipparchus came upon the scene1 there was a considerable amount of material for him to use4 ,his discoveries marked a great advance in the science of astronomy4 ,he noted the irregular motion of the sun1 and1 to explain it1 assumed that it revolved uniformly not exactly about the earth but about a point some distance away1 called the excentric4#a ,the line joining the centre of the earth to the excentric passes through the apses of the sun's orbit1 where its distance from the earth is greatest and least4 ,the same result he could obtain by assuming that the sun moved round a small circle1 whose centre described a larger circle about the earth2 this larger circle carrying the other was called the deferent3 so that the actual motion of the sun was in an epicycle4 ,of the two methods of expression ,hipparchus ultimately preferred the second4 ,he applied the same process to the moon but found that he could depend upon its being right only at new and full moon4 ,the irregularity at first and third quarters he left to be investigated by his successors4 ,he also considered the planetary observations at his disposal insufficient and so gave up the attempt at a complete planetary theory4 ,he made improved determinations of some of the elements of the motions of the sun and moon1 and discovered the ,precession of the ,equinoxes1 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 ecliptic1 thus producing in #aej years an unmistakable change of a couple of degrees1 or four times the sun's diameter4 ,he also invented trigonometry4 ,his star catalogue was due to the appearance of a new star which caused him to search for possible previous similar phenomena1 and also to prepare for checking future ones4 ,no advance was made in theoretical astronomy for #bfj years1 the interval between ,hipparchus and ,ptolemy of ,alexandria4 ,ptolemy accepted the spherical form of the earth but denied its rotation or any other movement4 ,he made no advance on ,hipparchus in regard to the sun1 though the lapse of time had largely increased the errors of the elements adopted by the latter4 ,in the case of the moon1 however1 ,ptolemy traced the variable inequality noticed sometimes by ,hipparchus at first and last quarter1 which vanished when the moon was in apogee or perigee4 ,this he called the evection1 and introduced another epicycle to represent it4 ,in his planetary theory he found that the places given by his adopted excentric did not fit1 being one way at apogee and the other at perigee2 so that the centre of distance must be nearer the earth4 ,he found it best to assume the centre of distance half-way between the centre of the earth and the excentric1 thus bisecting the excentricity4 ,even this did not fit in the case of ,mercury1 and in general the agreement between theory and observation was spoilt by the necessity of making all the orbital planes pass through the centre of the earth1 instead of the sun1 thus making a good accordance practically impossible4 ,footnote #a3 ,see ,glossary for this and other technical terms4 ,after ,ptolemy's time very little was heard for many centuries of any fresh planetary theory1 though advances in some points of detail were made1 notably by some of the ,arab philosophers1 who obtained improved values for some of the elements by using better instruments4 ,from time to time various modifications of ,ptolemy's theory were suggested1 but none of any real value4 ,the ,moors in ,spain did their share of the work carried on by their ,eastern co-religionists1 and the first independent star catalogue since the time of ,hipparchus was made by another ,oriental1 ,tamerlane's grandson1 ,ulugh ,begh1 who built a fine observatory at ,samarcand in the fifteenth century4 ,in ,spain the work was not monopolised by the ,moors1 for in the thirteenth century ,alphonso of ,castile1 with the assistance of ,jewish and ,christian computers1 compiled the ,alphonsine tables1 completed in #abeb1 in which year he ascended the throne as ,alphonso ,x4 ,they were long circulated in ,,ms4 and were first printed in #adhc1 not long before the end of the period of stagnation4 ,copernicus was born in #adgc at ,thorn in ,polish ,prussia4 ,in the course of his studies at ,cracow and at several ,italian universities1 he learnt all that was known of the ,ptolemaic astronomy and determined to reform it4 ,his maternal uncle1 the ,bishop of ,ermland1 having provided him with a lay canonry in the ,cathedral of ,frauenburg1 he had leisure to devote himself to ,science4 ,reviewing the suggestions of the ancient ,greeks1 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 sun4 ,of the seventy odd circles or epicycles required by the latest form of the ,ptolemaic system1 ,copernicus succeeded in dispensing with rather more than half1 but he still required thirty-four1 which was the exact number assumed before the time of ,aristotle4 ,his considerations were almost entirely mathematical1 his only invasion into physics being in defence of the moving earth against the stock objection that if the earth moved1 loose objects would fly off1 and towers fall4 ,he did not break sufficiently away from the old tradition of uniform circular motion4 ,ptolemy's efforts at exactness were baulked1 as we have seen1 by the supposed necessity of all the orbit planes passing through the earth1 and if ,copernicus had simply transferred this responsibility to the sun he would have done better4 ,but he would not sacrifice the old fetish1 and so1 the orbit of the earth being clearly not circular with respect to the sun1 he made all his planetary planes pass through the centre of the earth's orbit1 instead of through the sun1 thus handicapping himself in the same way though not in the same degree as ,ptolemy4 ,his thirty-four circles or epicycles comprised four for the earth1 three for the moon1 seven for ,mercury 7on account of his highly eccentric orbit7 and five each for the other planets4 ,it is rather an exaggeration to call the present accepted system the ,copernican system1 as it is really due to ,kepler1 half a century after the death of ,copernicus1 but much credit is due to the latter for his successful attempt to provide a real alternative for the ,ptolemaic system1 instead of tinkering with it4 ,the old geocentric system once shaken1 the way was gradually smoothed for the heliocentric system1 which ,copernicus1 still hampered by tradition1 did not quite reach4 ,he was hardly a practical astronomer in the observational sense4 ,his first recorded observation1 of an occultation of ,aldebaran1 was made in #adig1 and he is not known to have made as many as fifty astronomical observations1 while1 of the few he did make and use1 at least one was more than half a degree in error1 which would have been intolerable to such an observer as ,hipparchus4 ,copernicus in fact seems to have considered accurate observations unattainable with the instruments at hand4 ,he refused to give any opinion on the projected reform of the calendar1 on the ground that the motions of the sun and moon were not known with sufficient accuracy4 ,it is possible that with better data he might have made much more progress4 ,he was in no hurry to publish anything1 perhaps on account of possible opposition4 ,certainly ,luther1 with his obstinate conviction of the verbal accuracy of the ,scriptures1 rejected as mere folly the idea of a moving earth1 and ,melanchthon thought such opinions should be prohibited1 but ,rheticus1 a professor at the ,protestant ,university of ,wittenberg and an enthusiastic pupil of ,copernicus1 urged publication1 and undertook to see the work through the press4 ,this1 however1 he was unable to complete and another ,lutheran1 ,osiander1 to whom he entrusted it1 wrote a preface1 with the apparent intention of disarming opposition1 in which he stated that the principles laid down were only abstract hypotheses convenient for purposes of calculation4 ,this unauthorised interpolation may have had its share in postponing the prohibition of the book by the ,church of ,rome4 ,according to ,copernicus the earth is only a planet like the others1 and not even the biggest one1 while the sun is the most important body in the system1 and the stars probably too far away for any motion of the earth to affect their apparent places4 ,the earth in fact is very small in comparison with the distance of the stars1 as evidenced by the fact that an observer anywhere on the earth appears to be in the middle of the universe4 ,he shows that the revolution of the earth will account for the seasons1 and for the stationary points and retrograde motions of the planets4 ,he corrects definitely the order of the planets outwards from the sun1 a matter which had been in dispute4 ,a notable defect is due to the idea that a body can only revolve about another body or a point1 as if rigidly connected with it1 so that1 in order to keep the earth's axis in a constant direction in space1 he has to invent a third motion4 ,his discussion of precession1 which he rightly attributes to a slow motion of the earth's axis1 is marred by the idea that the precession is variable4 ,with all its defects1 partly due to reliance on bad observations1 the work showed a great advance in the interpretation of the motions of the planets2 and his determinations of the periods both in relation to the earth and to the stars were adopted by ,reinhold1 ,professor of ,astronomy at ,wittenberg1 for the new ,prutenic or ,prussian ,tables1 which were to supersede the obsolete ,alphonsine ,tables of the thirteenth century4 ,in comparison with the question of the motion of the earth1 no other astronomical detail of the time seems to be of much consequence4 ,comets1 such as from time to time appeared1 bright enough for naked eye observation1 were still regarded as atmospheric phenomena1 and their principal interest1 as well as that of eclipses and planetary conjunctions1 was in relation to astrology4 ,reform1 however1 was obviously in the air4 ,the doctrine of ,copernicus was destined very soon to divide others besides the ,lutheran leaders4 ,the leaven of inquiry was working1 and not long after the death of ,copernicus real advances were to come1 first in the accuracy of observations1 and1 as a necessary result of these1 in the planetary theory itself4 ,,chapter ,,ii4 ,,early ,,life ,,of ,,kepler4 ,on #bast ,december1 #aega1 at ,weil in the ,duchy of ,wurtemberg1 was born a weak and sickly seven-months' child1 to whom his parents ,henry and ,catherine ,kepler gave the name of ,john4 ,henry ,kepler was a petty officer in the service of the reigning ,duke1 and in #aegf joined the army serving in the ,netherlands4 ,his wife followed him1 leaving her young son in his grandfather's care at ,leonberg1 where he barely recovered from a severe attack of smallpox4 ,it was from this place that ,john derived the ,latinised name of ,leonmontanus1 in accordance with the common practice of the time1 but he was not known by it to any great extent4 ,he was sent to school in #aegg1 but in the following year his father returned to ,germany1 almost ruined by the absconding of an acquaintance for whom he had become surety4 ,henry ,kepler was obliged to sell his house and most of his belongings1 and to keep a tavern at ,elmendingen1 withdrawing his son from school to help him with the rough work4 ,in #aehc young ,kepler was sent to the school at ,elmendingen1 and in #aehd had another narrow escape from death by a violent illness4 ,in #aehf he was sent1 at the charges of the ,duke1 to the monastic school of ,maulbronn2 from whence1 in accordance with the school regulations1 he passed at the end of his first year the examination for the bachelor's degree at ,tbingen1 returning for two more years as a veteran to ,maulbronn before being admitted as a resident student at ,tbingen4 ,the three years thus spent at ,maulbronn were marked by recurrences of several of the diseases from which he had suffered in childhood1 and also by family troubles at his home4 ,his father went away after a quarrel with his wife ,catherine1 and died abroad4 ,catherine herself1 who seems to have been of a very unamiable disposition1 next quarrelled with her own relatives4 ,it is not surprising therefore that ,kepler after taking his ,m4,a4 degree in ,august1 #aeia1 coming out second in the examination lists1 was ready to accept the first appointment offered him1 even if it should involve leaving home4 ,this happened to be the lectureship in astronomy at ,gratz1 the chief town in ,styria4 ,kepler's knowledge of astronomy was limited to the compulsory school course1 nor had he as yet any particular leaning towards the science2 the post1 moreover1 was a meagre and unimportant one4 ,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 education4 ,his tutors also strongly urged him to accept the lectureship1 and he had not the usual reluctance to leave home4 ,he therefore proceeded to ,gratz1 protesting that he did not thereby forfeit his claim to a more promising opening1 when such should appear4 ,his astronomical tutor1 ,maestlin1 encouraged him to devote himself to his newly adopted science1 and the first result of this advice appeared before very long in ,kepler's ,mysterium ,cosmographicum4 ,the bent of his mind was towards philosophical speculation1 to which he had been attracted in his youthful studies of ,scaliger's ,exoteric ,exercises4 ,he says he devoted much time to the examination of the nature of heaven1 of souls1 of genii1 of the elements1 of the essence of fire1 of the cause of fountains1 the ebb and flow of the tides1 the shape of the continents and inland seas1 and things of this sort4 ,following his tutor in his admiration for the ,copernican theory1 he wrote an essay on the primary motion1 attributing it to the rotation of the earth1 and this not for the mathematical reasons brought forward by ,copernicus1 but1 as he himself says1 on physical or metaphysical grounds4 ,in #aeie1 having more leisure from lectures1 he turned his speculative mind to the number1 size1 and motion of the planetary orbits4 ,he first tried simple numerical relations1 but none of them appeared to be twice1 thrice1 or four times as great as another1 although he felt convinced that there was some relation between the motions and the distances1 seeing that when a gap appeared in one series1 there was a corresponding gap in the other4 ,these gaps he attempted to fill by hypothetical planets between ,mars and ,jupiter1 and between ,mercury and ,venus1 but this method also failed to provide the regular proportion which he sought1 besides being open to the objection that on the same principle there might be many more equally invisible planets at either end of the series4 ,he was nevertheless unwilling to adopt the opinion of ,rheticus that the number six was sacred1 maintaining that the sacredness of the number was of much more recent date than the creation of the worlds1 and could not therefore account for it4 ,he next tried an ingenious idea1 comparing the perpendiculars from different points of a quadrant of a circle on a tangent at its extremity4 ,the greatest of these1 the tangent1 not being cut by the quadrant1 he called the line of the sun1 and associated with infinite force4 ,the shortest1 being the point at the other end of the quadrant1 thus corresponded to the fixed stars or zero force2 intermediate ones were to be found proportional to the forces of the six planets4 ,after a great amount of unfinished trial calculations1 which took nearly a whole summer1 he convinced himself that success did not lie that way4 ,in ,july1 #aeie1 while lecturing on the great planetary conjunctions1 he drew quasi-triangles in a circular zodiac showing the slow progression of these points of conjunction at intervals of just over #bdj or eight signs4 ,the successive chords marked out a smaller circle to which they were tangents1 about half the diameter of the zodiacal circle as drawn1 and ,kepler at once saw a similarity to the orbits of ,saturn and ,jupiter1 the radius of the inscribed circle of an equilateral triangle being half that of the circumscribed circle4 ,his natural sequence of ideas impelled him to try a square1 in the hope that the circumscribed and inscribed circles might give him a similar analogy for the orbits of ,jupiter and ,mars4 ,he next tried a pentagon and so on1 but he soon noted that he would never reach the sun that way1 nor would he find any such limitation as six1 the number of possibles being obviously infinite4 ,the actual planets moreover were not even six but only five1 so far as he knew1 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 solids4#b ,he immediately pounced upon this idea and ultimately evolved the following scheme4 ,the earth is the sphere1 the measure of all2 round it describe a dodecahedron2 the sphere including this will be ,mars4 ,round ,mars describe a tetrahedron2 the sphere including this will be ,jupiter4 ,describe a cube round ,jupiter2 the sphere including this will be ,saturn4 ,now1 inscribe in the earth an icosahedron1 the sphere inscribed in it will be ,venus3 inscribe an octahedron in ,venus3 the circle inscribed in it will be ,mercury4 ,with this result ,kepler was inordinately pleased1 and regretted not a moment of the time spent in obtaining it1 though to us this ,mysterium ,cosmographicum can only appear useless1 even without the more recent additions to the known planets4 ,he admitted that a certain thickness must be assigned to the intervening spheres to cover the greatest and least distances of the several planets from the sun1 but even then some of the numbers obtained are not a very close fit for the corresponding planetary orbits4 ,kepler's own suggested explanation of the discordances was that they must be due to erroneous measures of the planetary distances1 and this1 in those days of crude and infrequent observations1 could not easily be disproved4 ,he next thought of a variety of reasons why the five regular solids should occur in precisely the order given and in no other1 diverging from this into a subtle and not very intelligible process of reasoning to account for the division of the zodiac into #cfj4 ,the next subject was more important1 and dealt with the relation between the distances of the planets and their times of revolution round the sun4 ,it was obvious that the period was not simply proportional to the distance1 as the outer planets were all too slow for this1 and he concluded either that the moving intelligences of the planets are weakest in those that are farthest from the sun1 or that there is one moving intelligence in the sun1 the common centre1 forcing them all round1 but those most violently which are nearest1 and that it languishes in some sort and grows weaker at the most distant1 because of the remoteness and the attenuation of the virtue4 ,this is not so near a guess at the theory of gravitation as might be supposed1 for ,kepler imagined that a repulsive force was necessary to account for the planets being sometimes further from the sun1 and so laid aside the idea of a constant attractive force4 ,he made several other attempts to find a law connecting the distances and periods of the planets1 but without success at that time1 and only desisted when by unconsciously arguing in a circle he appeared to get the same result from two totally different hypotheses4 ,he sent copies of his book to several leading astronomers1 of whom ,galileo praised his ingenuity and good faith1 while ,tycho ,brahe was evidently much struck with the work and advised him to adapt something similar to the ,tychonic system instead of the ,copernican4 ,he also intimated that his ,uraniborg observations would provide more accurate determinations of the planetary orbits1 and thus made ,kepler eager to visit him1 a project which as we shall see was more than fulfilled4 ,another copy of the book ,kepler sent to ,reymers the ,imperial astronomer with a most fulsome letter1 which ,tycho1 who asserted that ,reymers had simply plagiarised his work1 very strongly resented1 thus drawing from ,kepler a long letter of apology4 ,about the same time ,kepler had married a lady already twice widowed1 and become involved in difficulties with her relatives on financial grounds1 and with the ,styrian authorities in connection with the religious disputes then coming to a head4 ,on account of these latter he thought it expedient1 the year after his marriage1 to withdraw to ,hungary1 from whence he sent short treatises to ,tbingen1 ,on the magnet 7following the ideas of ,gilbert of ,colchester71 ,on the cause of the obliquity of the ecliptic and ,on the ,divine wisdom as shown in the ,creation4 ,his next important step makes it desirable to devote a chapter to a short notice of ,tycho ,brahe4 ,footnote #b3 ,since the sum of the plane angles at a corner of a regular solid must be less than four right angles1 it is easily seen that few regular solids are possible4 ,hexagonal faces are clearly impossible1 or any polygonal faces with more than five sides4 ,the possible forms are the dodecahedron with twelve pentagonal faces1 three meeting at each corner2 the cube1 six square faces1 three meeting at each corner2 and three figures with triangular faces1 the tetrahedron of four faces1 three meeting at each corner2 the octahedron of eight faces1 four meeting at each corner2 and the icosahedron of twenty faces1 five meeting at each corner4 ,,chapter ,,iii4 ,,tycho ,,brahe4 ,the age following that of ,copernicus produced three outstanding figures associated with the science of astronomy1 then reaching the close of what ,professor ,forbes so aptly styles the geometrical period4 ,these three ,sir ,david ,brewster has termed ,martyrs of ,science2 ,galileo1 the great ,italian philosopher1 has his own place among the ,pioneers of ,science2 and invaluable though ,tycho ,brahe's work was1 the latter can hardly be claimed as a pioneer in the same sense as the other two4 ,nevertheless1 ,kepler1 the third member of the trio1 could not have made his most valuable discoveries without ,tycho's observations4 ,of noble family1 born a twin on #adth ,december1 #aedf1 at ,knudstrup in ,scania 7the southernmost part of ,sweden1 then forming part of the kingdom of ,denmark71 ,tycho was kidnapped a year later by a childless uncle4 ,this uncle brought him up as his own son1 provided him at the age of seven with a tutor1 and sent him in #aeei to the ,university of ,copenhagen1 to study for a political career by taking courses in rhetoric and philosophy4 ,on #bast ,august1 #aefj1 however1 a solar eclipse took place1 total in ,portugal1 and therefore of small proportions in ,denmark1 and ,tycho's keen interest was awakened1 not so much by the phenomenon1 as by the fact that it had occurred according to prediction4 ,soon afterwards he purchased an edition of ,ptolemy in order to read up the subject of astronomy1 to which1 and to mathematics1 he devoted most of the remainder of his three years' course at ,copenhagen4 ,his uncle next sent him to ,leipzig to study law1 but he managed to continue his astronomical researches4 ,he obtained the ,alphonsine and the new ,prutenic ,tables1 but soon found that the latter1 though more accurate than the former1 failed to represent the true positions of the planets1 and grasped the fact that continuous observation was essential in order to determine the true motions4 ,he began by observing a conjunction of ,jupiter and ,saturn in ,august1 #aefc1 and found the ,prutenic ,tables several days in error1 and the ,alphonsine a whole month4 ,he provided himself with a cross-staff for determining the angular distance between stars or other objects1 and1 finding the divisions of the scale inaccurate1 constructed a table of corrections1 an improvement that seems to have been a decided innovation1 the previous practice having been to use the best available instrument and ignore its errors4 ,about this time war broke out between ,denmark and ,sweden1 and ,tycho returned to his uncle1 who was vice-admiral and attached to the king's suite4 ,the uncle died in the following month1 and early in the next year ,tycho went abroad again1 this time to ,wittenberg4 ,after five months1 however1 an outbreak of plague drove him away1 and he matriculated at ,rostock1 where he found little astronomy but a good deal of astrology4 ,while there he fought a duel in the dark and lost part of his nose1 which he replaced by a composition of gold and silver4 ,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-countrymen4 ,the ,king of ,denmark1 however1 having a higher opinion of the value of science1 promised ,tycho the first canonry that should fall vacant in the cathedral chapter of ,roskilde1 so that he might be assured of an income while devoting himself to financially unproductive work4 ,in #aefh ,tycho left ,rostock1 and matriculated at ,basle1 but soon moved on to ,augsburg1 where he found more enthusiasm for astronomy1 and induced one of his new friends to order the construction of a large #ai-;foot quadrant of heavy oak beams4 ,this was the first of the series of great instruments associated with ,tycho's name1 and it remained in use for five years1 being destroyed by a great storm in #aegd4 ,tycho meanwhile had left ,augsburg in #aegj and returned to live with his father1 now governor of ,helsingborg ,castle1 until the latter's death in the following year4 ,tycho then joined his mother's brother1 ,steen ,bille1 the only one of his relatives who showed any sympathy with his desire for a scientific career4 ,on #aath ,november1 #aegb1 ,tycho noticed an unfamiliar bright star in the constellation of ,cassiopeia1 and continued to observe it with a sextant4 ,it was a very brilliant object1 equal to ,venus at its brightest for the rest of ,november1 not falling below the first magnitude for another four months1 and remaining visible for more than a year afterwards4 ,tycho wrote a little book on the new star1 maintaining that it had practically no parallax1 and therefore could not be1 as some supposed1 a comet4 ,deeming authorship beneath the dignity of a noble he was very reluctant to publish1 but he was convinced of the importance of increasing the number and accuracy of observations1 though he was by no means free from all the erroneous ideas of his time4 ,the little book contained a certain amount of astrology1 but ,tycho evidently did not regard this as of very great importance4 ,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 significance4 ,we gather from lectures which he was persuaded to deliver at the ,university of ,copenhagen that1 though in agreement with the accepted canons of astrology as to the influence of planetary conjunctions and such phenomena on the course of human events1 he did not consider the fate predicted by anyone's horoscope to be unavoidable1 but thought the great value of astrology lay in the warnings derived from such computations1 which should enable the believer to avoid threatened calamities4 ,in #aege he left ,denmark once more and made his way to ,cassel1 where he found a kindred spirit in the studious ,landgrave1 ,william ,,iv4 of ,hesse1 whose astronomical pursuits had been interrupted by his accession to the government of ,hesse1 in #aefg4 ,tycho observed with him for some time1 the two forming a firm friendship1 and then visited successively ,frankfort1 ,basle1 and ,venice1 returning by way of ,augsburg1 ,ratisbon1 and ,saalfeld to ,wittenberg2 on the way he acquired various astronomical manuscripts1 made friends among practical astronomers1 and examined new instruments4 ,he seemed to have considered the advantages of the several places thus visited and decided on ,basle1 but on his return to ,denmark to fetch his family with the object of transferring them to ,basle1 he found that his friend the ,landgrave had written to ,king ,frederick on his behalf1 urging him to provide the means to enable ,tycho to pursue his astronomical work1 promising that not only should credit result for the king and for ,denmark but that science itself would be greatly advanced4 ,the ultimate result of this letter was that after refusing various offers1 ,tycho accepted from the king a grant of the small island of ,hveen1 in the ,sound1 with a guaranteed income1 in addition to a large sum from the treasury for building an observatory on the island1 far removed from the distractions of court life4 ,here ,tycho built his celebrated observatory of ,uraniborg and began observations in ,december1 #aegf1 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 telescope4 ,here also he built several smaller observing rooms1 so that his pupils should be able to observe independently4 ,for more than twenty years he continued his observations at ,uraniborg1 surrounded by his family1 and attracting numerous pupils4 ,his constant aim was to accumulate a large store of observations of a high order of accuracy1 and thus to provide data for the complete reform of astronomy4 ,as we have seen1 few of the ,danish nobles had any sympathy with ,tycho's pursuits1 and most of them strongly resented the continual expense borne by the ,king's treasury4 ,tycho moreover was so absorbed in his scientific pursuits that he would not take the trouble to be a good landlord1 nor to carry out all the duties laid upon him in return for certain of his grants of income4 ,his buildings included a chemical laboratory1 and he was in the habit of making up elixirs for various medical purposes2 these were quite popular1 particularly as he made no charge for them4 ,he seems to have been something of a homoeopathist1 for he recommends sulphur to cure infectious diseases brought on by the sulphurous vapours of the ,aurora ,borealis6 ,king ,frederick1 in consideration of various grants to ,tycho1 relied upon his assistance in scientific matters1 and especially in astrological calculations2 such as the horoscope of the heir apparent1 ,prince ,christian1 born in #aegg1 which has been preserved among ,tycho's writings4 ,there is1 however1 no known copy in existence of any of the series of annual almanacs with predictions which he prepared for the ,king4 ,in ,november1 #aegg1 appeared a bright comet1 which ,tycho carefully observed with his sextant1 proving that it had no perceptible parallax1 and must therefore be further off than the moon4 ,he thus definitely overthrew the common belief in the atmospheric origin of comets1 which he had himself hitherto shared4 ,with increasing accuracy he observed several other comets1 notably one in #aehe1 when he had a full equipment of instruments and a large staff of assistants4 ,the year #aehh1 which saw the death of his royal benefactor1 saw also the publication of a volume of ,tycho's great work ,introduction to the ,new ,astronomy4 ,the first volume1 devoted to the new star of #aegb1 was not ready1 because the reduction of the observations involved so much research to correct the star places for refraction1 precession1 etc42 it was not completed in fact until ,tycho's death1 but the second volume1 dealing with the comet of #aegg1 was printed at ,uraniborg and some copies were issued in #aehh4 ,besides the comet observations it included an account of ,tycho's system of the world4 ,he would not accept the ,copernican system1 as he considered the earth too heavy and sluggish to move1 and also that the authority of ,scripture was against such an hypothesis4 ,he therefore assumed that the other planets revolved about the sun1 while the sun1 moon1 and stars revolved about the earth as a centre4 ,geometrically this is much the same as the ,copernican system1 but physically it involves the grotesque demand that the whole system of stars revolves round our insignificant little earth every twenty-four hours4 ,since his previous small book on the comet1 ,tycho had evidently considered more fully its possible astrological significance1 for he foretold a religious war1 giving the date of its commencement1 and also the rising of a great ,protestant champion4 ,these predictions were apparently fulfilled almost to the letter by the great religious wars that broke out towards the end of the sixteenth century1 and in the person of ,gustavus ,adolphus4 ,king ,frederick's death did not at first affect ,tycho's position1 for the new king1 ,christian1 was only eleven years old1 and for some years the council of regents included two of his supporters4 ,after their deaths1 however1 his emoluments began to be cut down on the plea of economy1 and as he took very little trouble to carry out any other than scientific duties it was easy enough for his enemies to find fault4 ,one after another source of income was cut off1 but he persevered with his scientific work1 including a catalogue of stars4 ,he had obtained plenty of good observations of #ggg stars1 but thought his catalogue should contain #ajjj stars1 so he hastily observed as many more as he could up to the time of his leaving ,hveen1 though even then he had not completed his programme4 ,about the time that ,king ,christian reached the age of eighteen1 ,tycho began to look about for a new patron1 and to consider the prospects offered by transferring himself with his instruments and activities to the patronage of the ,emperor ,rudolph ,,ii4 ,in #aeig1 when even his pension from the ,royal treasury was cut off1 he hurriedly packed up his instruments and library1 and after a few weeks' sojourn at ,copenhagen1 proceeded to ,rostock1 in ,mecklenburg1 whence he sent an appeal to ,king ,christian4 ,it is possible that had he done this before leaving ,hveen it might have had more effect1 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 shortcomings4 ,driven from ,rostock by the plague1 ,tycho settled temporarily at ,wandsbeck1 in ,holstein1 but towards the end of #aeih set out to meet the ,emperor at ,prague4 ,once more plague intervened and he spent some time at ,dresden1 afterwards going to ,wittenberg for the winter4 ,he ultimately reached ,prague in ,june1 #aeii4 ,rudolph granted him a salary of at least #cjjj florins1 promising also to settle on him the first hereditary estate that should lapse to the ,crown4 ,he offered1 moreover1 the choice between three castles outside ,prague1 of which ,tycho chose ,benatek4 ,there he set about altering the buildings in readiness for his instruments1 for which he sent to ,uraniborg4 ,before they reached him1 after many vexatious delays1 he had given up waiting for the funds promised for his building expenses1 and removed from ,benatek to ,prague4 ,it was during this interval that after considerable negotiation1 ,kepler1 who had been in correspondence with ,tycho1 consented to join him as an assistant4 ,another assistant1 ,longomontanus1 who had been with ,tycho at ,uraniborg1 was finding difficulty with the long series of ,mars observations1 and it was arranged that he should transfer his energies to the lunar observations1 leaving those of ,mars for ,kepler4 ,before very much could be done with them1 however1 ,tycho died at the end of ,october1 #afja4 ,he may have regretted the peaceful island of ,hveen1 considering the troubles in which ,bohemia was rapidly becoming involved1 but there is little doubt that had it not been for his self-imposed exile1 his observations would not have come into ,kepler's hands1 and their great value might have been lost4 ,in any case it was at ,uraniborg that the mass of observations was produced upon which the fame of ,tycho ,brahe rests4 ,his own discoveries1 though in themselves the most important made in astronomy for many centuries1 are far less valuable than those for which his observations furnished the material4 ,he discovered the third and fourth inequalities of the moon in longitude1 called respectively the variation and the annual equation1 also the variability of the motion of the moon's nodes and the inclination of its orbit to the ecliptic4 ,he obtained an improved value of the constant of precession1 and did good service by rejecting the idea that it was variable1 an idea which1 under the name of trepidation1 had for many centuries been accepted4 ,he discovered the effect of refraction1 though only approximately its amount1 and determined improved values of many other astronomical constants1 but singularly enough made no determination of the distance of the sun1 adopting instead the ancient and erroneous value given by ,hipparchus4 ,his magnificent ,observatory of ,uraniborg1 the finest building for astronomical purposes that the world had hitherto seen1 was allowed to fall into decay1 and scarcely more than mere indications of the site may now be seen4 ,,chapter ,,iv4 ,,kepler ,,joins ,,tycho4 ,the association of ,kepler with ,tycho was one of the most important landmarks in the history of astronomy4 ,the younger man hoped1 by the aid of ,tycho's planetary observations1 to obtain better support for some of his fanciful speculative theories1 while the latter1 who had certainly not gained in prestige by leaving ,denmark1 was in great need of a competent staff of assistants4 ,of the two it would almost seem that ,tycho thought himself the greater gainer1 for in spite of his reputation for brusqueness and want of consideration1 he not only made light of ,kepler's apology in the matter of ,reymers1 but treated him with uniform kindness in the face of great rudeness and ingratitude4 ,he begged him to come as a welcome friend1 though ,kepler1 very touchy on the subject of his own astronomical powers1 was afraid he might be regarded as simply a subordinate assistant4 ,an arrangement had been suggested by which ,kepler should obtain two years' leave of absence from ,gratz on full pay1 which1 because of the higher cost of living in ,prague1 should be supplemented by the ,emperor2 but before this could be concluded1 ,kepler threw up his professorship1 and thinking he had thereby also lost the chance of going to ,prague1 applied to ,maestlin and others of his ,tbingen friends to make interest for him with the ,duke of ,wurtemberg and secure the professorship of medicine4 ,tycho1 however1 still urged him to come to ,prague1 promising to do his utmost to secure for him a permanent appointment1 or in any event to see that he was not the loser by coming4 ,kepler was delayed by illness on the way1 but ultimately reached ,prague1 accompanied by his wife1 and for some time lived entirely at ,tycho's expense1 writing by way of return essays against ,reymers and another man1 who had claimed the credit of the ,tychonic system4 ,this ,kepler could do with a clear conscience1 as it was only a question of priority and did not involve any support of the system1 which he deemed far inferior to that of ,copernicus4 ,the following year saw friction between the two astronomers1 and we learn from ,kepler's abject letter of apology that he was entirely in the wrong4 ,it was about money matters1 which in one way or another embittered the rest of ,kepler's life1 and it arose during his absence from ,prague4 ,on his return in ,september1 #afja1 ,tycho presented him to the ,emperor1 who gave him the title of ,imperial ,mathematician1 on condition of assisting ,tycho in his calculations1 the very thing ,kepler was most anxious to be allowed to do3 for nowhere else in the world was there such a collection of good observations sufficient for his purpose of reforming the whole theory of astronomy4 ,the ,emperor's interest was still mainly with astrology1 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 ,castile1 and he made liberal promises to pay the expenses4 ,tycho's other principal assistant1 ,longomontanus1 did not stay long after giving up the ,mars observations to ,kepler1 but instead of working at the new lunar theory1 suddenly left to take up a professorship of astronomy in his native ,denmark4 ,very shortly afterwards ,tycho himself died of acute distemper2 ,kepler began to prepare the mass of manuscripts for publication1 but1 as everything was claimed by the ,brahe family1 he was not allowed to finish the work4 ,he succeeded to ,tycho's post of principal mathematician to the ,emperor1 at a reduced official salary1 which owing to the emptiness of the ,imperial treasury was almost always in arrear4 ,in order to meet his expenses he had recourse to the casting of nativities1 for which he gained considerable reputation and received very good pay4 ,he worked by the conventional rules of astrology1 and was quite prepared to take fees for so doing1 although he had very little faith in them1 preferring his own fanciful ideas4 ,in #afjd the constellation of ,cassiopeia was once more temporarily enriched by the appearance of a new star1 said by some to be brighter than ,tycho's nova1 and by others to be twice as bright as ,jupiter4 ,kepler at once wrote a short account of it1 from which may be gathered some idea of his attitude towards astrology4 ,contrasting the two novae1 he says3 ,yonder one chose for its appearance a time no way remarkable1 and came into the world quite unexpectedly1 like an enemy storming a town and breaking into the market-place before the citizens are aware of his approach2 but ours has come exactly in the year of which astrologers have written so much about the fiery trigon that happens in it2 just in the month in which 7according to ,cyprian71 ,mars comes up to a very perfect conjunction with the other two superior planets2 just in the day when ,mars has joined ,jupiter1 and just in the region where this conjunction has taken place4 ,therefore the apparition of this star is not like a secret hostile irruption1 as was that one of #aegb1 but the spectacle of a public triumph1 or the entry of a mighty potentate2 when the couriers ride in some time before to prepare his lodgings1 and the crowd of young urchins begin to think the time over long to wait1 then roll in1 one after another1 the ammunition and money1 and baggage waggons1 and presently the trampling of horse and the rush of people from every side to the streets and windows2 and when the crowd have gazed with their jaws all agape at the troops of knights2 then at last the trumpeters and archers and lackeys so distinguish the person of the monarch1 that there is no occasion to point him out1 but every one cries of his own accord--',here we have him'4 ,what it may portend is hard to determine1 and this much only is certain1 that it comes to tell mankind either nothing at all or high and mighty news1 quite beyond human sense and understanding4 ,it will have an important influence on political and social relations2 not indeed by its own nature1 but as it were accidentally through the disposition of mankind4 ,first1 it portends to the booksellers great disturbances and tolerable gains2 for almost every ,theologus1 ,philosophicus1 ,medicus1 and ,mathematicus1 or whoever else1 having no laborious occupation entrusted to him1 seeks his pleasure in studiis1 will make particular remarks upon it1 and will wish to bring these remarks to the light4 ,just so will others1 learned and unlearned1 wish to know its meaning1 and they will buy the authors who profess to tell them4 ,i mention these things merely by way of example1 because although thus much can be easily predicted without great skill1 yet may it happen just as easily1 and in the same manner1 that the vulgar1 or whoever else is of easy faith1 or1 it may be1 crazy1 may wish to exalt himself into a great prophet2 or it may even happen that some powerful lord1 who has good foundation and beginning of great dignities1 will be cheered on by this phenomenon to venture on some new scheme1 just as if ,god had set up this star in the darkness merely to enlighten them4 ,he made no secret of his views on conventional astrology1 as to which he claimed to speak with the authority of one fully conversant with its principles1 but he nevertheless expressed his sincere conviction that the conjunctions and aspects of the planets certainly did affect things on the earth1 maintaining that he was driven to this belief against his will by most unfailing experiences4 ,meanwhile the projected ,rudolphine ,tables were continually delayed by the want of money4 ,kepler's nominal salary should have been ample for his expenses1 increased though they were by his growing family1 but in the depleted state of the treasury there were many who objected to any payment for such unpractical purposes4 ,this particular attitude has not been confined to any special epoch or country1 but the obvious result in ,kepler's case was to compel him to apply himself to less expensive matters than the ,planetary ,tables1 and among these must be included not only the horoscopes or nativities1 which owing to his reputation were always in demand1 but also other writings which probably did not pay so well4 ,in #afjd he published ,a ,supplement to ,vitellion1 containing the earliest known reasonable theory of optics1 and especially of dioptrics or vision through lenses4 ,he compared the mechanism of the eye with that of ,porta's ,camera ,obscura1 but made no attempt to explain how the image formed on the retina is understood by the brain4 ,he went carefully into the question of refraction1 the importance of which ,tycho had been the first astronomer to recognise1 though he only applied it at low altitudes1 and had not arrived at a true theory or accurate values4 ,kepler wasted a good deal of time and ingenuity on trial theories4 ,he would invariably start with some hypothesis1 and work out the effect4 ,he would then test it by experiment1 and when it failed would at once recognise that his hypothesis was a priori bound to fail4 ,he rarely seems to have noticed the fatal objections in time to save himself trouble4 ,he would then at once start again on a new hypothesis1 equally gratuitous and equally unfounded4 ,it never seems to have occurred to him that there might be a better way of approaching a problem4 ,among the lines he followed in this particular investigation were1 first1 that refraction depends only on the angle of incidence1 which1 he says1 cannot be correct as it would thus be the same for all refracting substances2 next1 that it depended also on the density of the medium4 ,this was a good shot1 but he unfortunately assumed that all rays passing into a denser medium would apparently penetrate it to a depth depending only on the medium1 which means that there is a constant ratio between the tangents1 instead of the sines1 of the inclination of the incident and refracted rays to the normal4 ,experiment proved that this gave too high values for refraction near the vertical compared with those near the horizon1 so ,kepler went off at a tangent and tried a totally new set of ideas1 which all reduced to the absurdity of a refraction which vanished at the horizon4 ,these were followed by another set1 involving either a constant amount of refraction or one becoming infinite4 ,he then came to the conclusion that these geometrical methods must fail because the refracted image is not real1 and determined to try by analogy only1 comparing the equally unreal image formed by a mirror with that formed by refraction in water4 ,he noticed how the bottom of a vessel containing water appears to rise more and more away from the vertical1 and at once jumped to the analogy of a concave mirror1 which magnifies the image1 while a convex mirror was likened to a rarer medium4 ,this line of attack also failed him1 as did various attempts to find relations between his measurements of refraction and conic sections1 and he broke off suddenly with a diatribe against ,tycho's critics1 whom he likened to blind men disputing about colours4 ,not many years later ,snell discovered the true law of refraction1 but ,kepler's contribution to the subject1 though he failed to discover the actual law1 includes several of the adopted by-laws4 ,he noted that atmospheric refraction would alter with the height of the atmosphere and with temperature1 and also recognised the fact that rainbow colours depend on the angle of refraction1 whether seen in the rainbow itself1 or in dew1 glass1 water1 or any similar medium4 ,he thus came near to anticipating ,newton4 ,before leaving the subject of ,kepler's optics it will be well to recall that a few years later after hearing of ,galileo's telescope1 ,kepler suggested that for astronomical purposes two convex lenses should be used1 so that there should be a real image where measuring wires could be placed for reference4 ,he did not carry out the idea himself1 and it was left to the ,englishman ,gascoigne to produce the first instrument on this ,keplerian principle1 universally known as the ,astronomical ,telescope4 ,in #afjf came a second treatise on the new star1 discussing various theories to account for its appearance1 and refusing to accept the notion that it was a fortuitous concourse of atoms4 ,this was followed in #afjg by a treatise on comets1 suggested by the comet appearing that year1 known as ,halley's comet after its next return4 ,he regarded comets as planets moving in straight lines1 never having examined sufficient observations of any comet to convince himself that their paths are curved4 ,if he had not assumed that they were external to the system and so could not be expected to return1 he might have anticipated ,halley's discovery4 ,another suggestive remark of his was to the effect that the planets must be self-luminous1 as otherwise ,mercury and ,venus1 at any rate1 ought to show phases4 ,this was put to the test not long afterwards by means of ,galileo's telescope4 ,in #afjg ,kepler rushed into print with an alleged observation of ,mercury crossing the sun1 but after ,galileo's discovery of sun-spots1 ,kepler at once cheerfully retracted his observation of ,mercury1 and so far was he from being annoyed or bigoted in his views1 that he warmly adopted ,galileo's side1 in contrast to most of those whose opinions were liable to be overthrown by the new discoveries4 ,maestlin and others of ,kepler's friends took the opposite view4 ,,chapter ,v4 ,,kepler',s ,,laws4 ,when ,gilbert of ,colchester1 in his ,new ,philosophy1 founded on his researches in magnetism1 was dealing with tides1 he did not suggest that the moon attracted the water1 but that subterranean spirits and humours1 rising in sympathy with the moon1 cause the sea also to rise and flow to the shores and up rivers4 ,it appears that an idea1 presented in some such way as this1 was more readily received than a plain statement4 ,this so-called philosophical method was1 in fact1 very generally applied1 and ,kepler1 who shared ,galileo's admiration for ,gilbert's work1 adopted it in his own attempt to extend the idea of magnetic attraction to the planets4 ,the general idea of gravity opposed the hypothesis of the rotation of the earth on the ground that loose objects would fly off3 moreover1 the latest refinements of the old system of planetary motions necessitated their orbits being described about a mere empty point4 ,kepler very strongly combated these notions1 pointing out the absurdity of the conclusions to which they tended1 and proceeded in set terms to describe his own theory4 ,every corporeal substance1 so far forth as it is corporeal1 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 it4 ,gravity is a mutual affection between cognate bodies towards union or conjunction 7similar in kind to the magnetic virtue71 so that the earth attracts a stone much rather than the stone seeks the earth4 ,heavy bodies 7if we begin by assuming the earth to be in the centre of the world7 are not carried to the centre of the world in its quality of centre of the world1 but as to the centre of a cognate round body1 namely1 the earth2 so that wheresoever the earth may be placed1 or whithersoever it may be carried by its animal faculty1 heavy bodies will always be carried towards it4 ,if the earth were not round1 heavy bodies would not tend from every side in a straight line towards the centre of the earth1 but to different points from different sides4 ,if two stones were placed in any part of the world near each other1 and beyond the sphere of influence of a third cognate body1 these stones1 like two magnetic needles1 would come together in the intermediate point1 each approaching the other by a space proportional to the comparative mass of the other4 ,if the moon and earth were not retained in their orbits by their animal force or some other equivalent1 the earth would mount to the moon by a fifty-fourth part of their distance1 and the moon fall towards the earth through the other fifty-three parts1 and they would there meet1 assuming1 however1 that the substance of both is of the same density4 ,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 moon4 ,the sphere of the attractive virtue which is in the moon extends as far as the earth1 and entices up the waters2 but as the moon flies rapidly across the zenith1 and the waters cannot follow so quickly1 a flow of the ocean is occasioned in the torrid zone towards the westward4 ,if the attractive virtue of the moon extends as far as the earth1 it follows with greater reason that the attractive virtue of the earth extends as far as the moon and much farther2 and1 in short1 nothing which consists of earthly substance anyhow constituted although thrown up to any height1 can ever escape the powerful operation of this attractive virtue4 ,nothing which consists of corporeal matter is absolutely light1 but that is comparatively lighter which is rarer1 either by its own nature1 or by accidental heat4 ,and it is not to be thought that light bodies are escaping to the surface of the universe while they are carried upwards1 or that they are not attracted by the earth4 ,they are attracted1 but in a less degree1 and so are driven outwards by the heavy bodies2 which being done1 they stop1 and are kept by the earth in their own place4 ,but although the attractive virtue of the earth extends upwards1 as has been said1 so very far1 yet if any stone should be at a distance great enough to become sensible compared with the earth's diameter1 it is true that on the motion of the earth such a stone would not follow altogether2 its own force of resistance would be combined with the attractive force of the earth1 and thus it would extricate itself in some degree from the motion of the earth4 ,the above passage from the ,introduction to ,kepler's ,commentaries on the ,motion of ,mars1 always regarded as his most valuable work1 must have been known to ,newton1 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 ,gravitation4 ,kepler's glimpse at such a theory could have been no more than a glimpse1 for he went no further with it4 ,this seems a pity1 as it is far less fanciful than many of his ideas1 though not free from the virtues and animal faculties1 that correspond to ,gilbert's spirits and humours4 ,we must1 however1 proceed to the subject of ,mars1 which was1 as before noted1 the first important investigation entrusted to ,kepler on his arrival at ,prague4 ,the time taken from one opposition of ,mars to the next is decidedly unequal at different parts of his orbit1 so that many oppositions must be used to determine the mean motion4 ,the ancients had noticed that what was called the second inequality1 due as we now know to the orbital motion of the earth1 only vanished when earth1 sun1 and planet were in line1 i4e4 at the planet's opposition2 therefore they used oppositions to determine the mean motion1 but deemed it necessary to apply a correction to the true opposition to reduce to mean opposition1 thus sacrificing part of the advantage of using oppositions4 ,tycho and ,longomontanus had followed this method in their calculations from ,tycho's twenty years' observations4 ,their aim was to find a position of the equant1 such that these observations would show a constant angular motion about it2 and that the computed positions would agree in latitude and longitude with the actual observed positions4 ,when ,kepler arrived he was told that their longitudes agreed within a couple of minutes of arc1 but that something was wrong with the latitudes4 ,he found1 however1 that even in longitude their positions showed discordances ten times as great as they admitted1 and so1 to clear the ground of assumptions as far as possible1 he determined to use true oppositions4 ,to this ,tycho objected1 and ,kepler had great difficulty in convincing him that the new move would be any improvement1 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 arc1 while giving quite erroneous values of the latitudes and second inequalities4 ,to avoid the possibility of further objection he carried out this demonstration separately for each of the systems of ,ptolemy1 ,copernicus1 and ,tycho4 ,for the new method he noticed that great accuracy was required in the reduction of the observed places of ,mars to the ecliptic1 and for this purpose the value obtained for the parallax by ,tycho's assistants fell far short of the requisite accuracy4 ,kepler therefore was obliged to recompute the parallax from the original observations1 as also the position of the line of nodes and the inclination of the orbit4 ,the last he found to be constant1 thus corroborating his theory that the plane of the orbit passed through the sun4 ,he repeated his calculations no fewer than seventy times 7and that before the invention of logarithms71 and at length adopted values for the mean longitude and longitude of aphelion4 ,he found no discordance greater than two minutes of arc in ,tycho's observed longitudes in opposition1 but the latitudes1 and also longitudes in other parts of the orbit were much more discordant1 and he found to his chagrin that four years' work was practically wasted4 ,before making a fresh start he looked for some simplification of the labour2 and determined to adopt ,ptolemy's assumption known as the principle of the bisection of the excentricity4 ,hitherto1 since ,ptolemy had given no reason for this assumption1 ,kepler had preferred not to make it1 only taking for granted that the centre was at some point on the line called the excentricity 7see ,figs4 #a1 #b74 ,a marked improvement in residuals was the result of this step1 proving1 so far1 the correctness of ,ptolemy's principle1 but there still remained discordances amounting to eight minutes of arc4 ,copernicus1 who had no idea of the accuracy obtainable in observations1 would probably have regarded such an agreement as remarkably good2 but ,kepler refused to admit the possibility of an error of eight minutes in any of ,tycho's observations4 ,he thereupon vowed to construct from these eight minutes a new planetary theory that should account for them all4 ,his repeated failures had by this time convinced him that no uniformly described circle could possibly represent the motion of ,mars4 ,either the orbit could not be circular1 or else the angular velocity could not be constant about any point whatever4 ,he determined to attack the second inequality1 i4e4 the optical illusion caused by the earth's annual motion1 but first revived an old idea of his own that for the sake of uniformity the sun1 or as he preferred to regard it1 the earth1 should have an equant as well as the planets4 ,from the irregularities of the solar motion he soon found that this was the case1 and that the motion was uniform about a point on the line from the sun to the centre of the earth's orbit1 such that the centre bisected the distance from the sun to the ,equant2 this fully supported ,ptolemy's principle4 ,clearly then the earth's linear velocity could not be constant1 and ,kepler was encouraged to revive another of his speculations as to a force which was weaker at greater distances4 ,he found the velocity greater at the nearer apse1 so that the time over an equal arc at either apse was proportional to the distance4 ,he conjectured that this might prove to be true for arcs at all parts of the orbit1 and to test this he divided the orbit into #cfj equal parts1 and calculated the distances to the points of division4 ,archimedes had obtained an approximation to the area of a circle by dividing it radially into a very large number of triangles1 and ,kepler had this device in mind4 ,he found that the sums of successive distances from his #cfj points were approximately proportional to the times from point to point1 and was thus enabled to represent much more accurately the annual motion of the earth which produced the second inequality of ,mars1 to whose motion he now returned4 ,three points are sufficient to define a circle1 so he took three observed positions of ,mars and found a circle2 he then took three other positions1 but obtained a different circle1 and a third set gave yet another4 ,it thus began to appear that the orbit could not be a circle4 ,he next tried to divide into #cfj equal parts1 as he had in the case of the earth1 but the sums of distances failed to fit the times1 and he realised that the sums of distances were not a good measure of the area of successive triangles4 ,he noted1 however1 that the errors at the apses were now smaller than with a central circular orbit1 and of the opposite sign1 so he determined to try whether an oval orbit would fit better1 following a suggestion made by ,purbach in the case of ,mercury1 whose orbit is even more eccentric than that of ,mars1 though observations were too scanty to form the foundation of any theory4 ,kepler gave his fancy play in the choice of an oval1 greater at one end than the other1 endeavouring to satisfy some ideas about epicyclic motion1 but could not find a satisfactory curve4 ,he then had the fortunate idea of trying an ellipse with the same axis as his tentative oval4 ,mars now appeared too slow at the apses instead of too quick1 so obviously some intermediate ellipse must be sought between the trial ellipse and the circle on the same axis4 ,at this point the long arm of coincidence came into play4 ,half-way between the apses lay the mean distance1 and at this position the error was half the distance between the ellipse and the circle1 amounting to 4#jjdbi of a radius4 ,with these figures in his mind1 ,kepler looked up the greatest optical inequality of ,mars1 the angle between the straight lines from ,mars to the ,sun and to the centre of the circle4#c ,the secant of this angle was #a4jjdbi1 so that he noted that an ellipse reduced from the circle in the ratio of #a4jjdbi to #a would fit the motion of ,mars at the mean distance as well as the apses4 ,footnote #c3 ,this is clearly a maximum at ,,amc in ,fig4 #b1 when its tangent ,,ac_/,,cm the eccentricity4 ,it is often said that a coincidence like this only happens to somebody who deserves his luck1 but this simply means that recognition is essential to the coincidence4 ,in the same way the appearance of one of a large number of people mentioned is hailed as a case of the old adage ,talk of the devil1 etc41 ignoring all the people who failed to appear4 ,no one1 however1 will consider ,kepler unduly favoured4 ,his genius1 in his case certainly an infinite capacity for taking pains1 enabled him out of his medley of hypotheses1 mainly unsound1 by dint of enormous labour and patience1 to arrive thus at the first two of the laws which established his title of ,legislator of the ,heavens4 ,,figures ,,explanatory ,,of ,,kepler',s ,,theory ,,of ,,the ,,motion ,,of ,,mars4 ,illustration3 ,,fig4 #a4 _/ _/ ,q ,e ,c ,a ,p _/ _/ ,illustration3 ,,fig4 #b4 ,m _/ _/_/ ,n _/ ,q ,e ,c ,a ,p _/ _/_/ _/ ,transcriber's ,note3 ,approximate renditions of these figures are provided4 ,fig4 #a is a circle4 ,fig4 #b is a circle which contains an ellipse1 tangent to the circle at ,q and ,p4 ,line segments from ,m 7on the circle7 and ,n 7on the ellipse7 meet at point ,a4 ,,fig4 #a4--;,in ,ptolemy's excentric theory1 ,a may be taken to represent the earth1 ,c the centre of a planet's orbit1 and ,e the equant1 ,p 7perigee7 and ,q 7apogee7 being the apses of the orbit4 ,ptolemy's idea was that uniform motion in a circle must be provided1 and since the motion was not uniform about the earth1 ,a could not coincide with ,c2 and since the motion still failed to be uniform about ,a or ,c1 some point ,e must be found about which the motion should be uniform4 ,,fig4 #b4--,this is not drawn to scale1 but is intended to illustrate ,kepler's modification of ,ptolemy's excentric4 ,kepler found velocities at ,p and ,q proportional not to ,,ap and ,,aq but to ,,aq and ,,ap1 or to ,,ep and ,,eq if ,,ec ,,ca 7bisection of the excentricity74 ,the velocity at ,m was wrong1 and ,,am appeared too great4 ,kepler's first ellipse had ,m moved too near ,c4 ,the distance ,,ac is much exaggerated in the figure1 as also is ,,mn4 ,,an ,,cp1 the radius of the circle4 ,,mn should be 4#jjdbi of the radius1 and ,,mc_/,,nc should be #a4jjdbi4 ,the velocity at ,n appeared to be proportional to ,,en 7 ,,an74 ,kepler concluded that ,mars moved round ,,pnq1 so that the area described about ,a 7the sun7 was equal in equal times1 ,a being the focus of the ellipse ,,pnq4 ,the angular velocity is not quite constant about ,e1 the equant or empty focus1 but the difference could hardly have been detected in ,kepler's time4 ,kepler's improved determination of the earth's orbit was obtained by plotting the different positions of the earth corresponding to successive rotations of ,mars1 i4e4 intervals of #fhg days4 ,at each of these the date of the year would give the angle ,,mse 7,mars-,sun-,earth71 and ,tycho's observation the angle ,,mes4 ,so the triangle could be solved except for scale1 and the ratio of ,,se to ,,sm would give the distance of ,mars from the sun in terms of that of the earth4 ,measuring from a fixed position of ,mars 7e4g4 perihelion71 this gave the variation of ,,se1 showing the earth's inequality4 ,measuring from a fixed position of the earth1 it would give similarly a series of positions of ,mars1 which1 though lying not far from the circle whose diameter was the axis of ,mars' orbit1 joining perihelion and aphelion1 always fell inside the circle except at those two points4 ,it was a long time before it dawned upon ,kepler that the simplest figure falling within the circle except at the two extremities of the diameter1 was an ellipse1 and it is not clear why his first attempt with an ellipse should have been just as much too narrow as the circle was too wide4 ,the fact remains that he recognised suddenly that halving this error was tantamount to reducing the circle to the ellipse whose eccentricity was that of the old theory1 i4e4 that in which the sun would be in one focus and the equant in the other4 ,having now fitted the ends of both major and minor axes of the ellipse1 he leaped to the conclusion that the orbit would fit everywhere4 ,the practical effect of his clearing of the second inequality was to refer the orbit of ,mars directly to the sun1 and he found that the area between successive distances of ,mars from the sun 7instead of the sum of the distances7 was strictly proportional to the time taken1 in short1 equal areas were described in equal times 7#bnd ,law7 when referred to the sun in the focus of the ellipse 7#ast ,law74 ,he announced that 7#a7 ,the planet describes an ellipse1 the sun being in one focus2 and 7#b7 ,the straight line joining the planet to the sun sweeps out equal areas in any two equal intervals of time4 ,these are ,kepler's first and second ,laws though not discovered in that order1 and it was at once clear that ,ptolemy's bisection of the excentricity simply amounted to the fact that the centre of an ellipse bisects the distance between the foci1 the sun being in one focus and the angular velocity being uniform about the empty focus4 ,for so many centuries had the fetish of circular motion postponed discovery4 ,it was natural that ,kepler should assume that his laws would apply equally to all the planets1 but the proof of this1 as well as the reason underlying the laws1 was only given by ,newton1 who approached the subject from a totally different standpoint4 ,this commentary on ,mars was published in #afji1 the year of the invention of the telescope1 and ,kepler petitioned the ,emperor for further funds to enable him to complete the study of the other planets1 but once more there was delay2 in #afab ,rudolph died1 and his brother ,matthias who succeeded him1 cared very little for astronomy or even astrology1 though ,kepler was reappointed to his post of ,imperial ,mathematician4 ,he left ,prague to take up a permanent professorship at the ,university of ,linz4 ,his own account of the circumstances is gloomy enough4 ,he says1 ,in the first place ,i could get no money from the ,court1 and my wife1 who had for a long time been suffering from low spirits and despondency1 was taken violently ill towards the end of #afaj1 with the ,hungarian fever1 epilepsy and phrenitis4 ,she was scarcely convalescent when all my three children were at once attacked with smallpox4 ,leopold with his army occupied the town beyond the river just as ,i lost the dearest of my sons1 him whose nativity you will find in my book on the new star4 ,the town on this side of the river where ,i lived was harassed by the ,bohemian troops1 whose new levies were insubordinate and insolent2 to complete the whole1 the ,austrian army brought the plague with them into the city4 ,i went into ,austria and endeavoured to procure the situation which ,i now hold4 ,returning in ,june1 ,i found my wife in a decline from her grief at the death of her son1 and on the eve of an infectious fever1 and ,i lost her also within eleven days of my return4 ,then came fresh annoyance1 of course1 and her fortune was to be divided with my step-sisters4 ,the ,emperor ,rudolph would not agree to my departure2 vain hopes were given me of being paid from ,saxony2 my time and money were wasted together1 till on the death of the ,emperor in #afab1 ,i was named again by his successor1 and suffered to depart to ,linz4 ,being thus left a widower with a ten-year-old daughter ,susanna1 and a boy ,louis of half her age1 he looked for a second wife to take charge of them4 ,he has given an account of eleven ladies whose suitability he considered4 ,the first1 an intimate friend of his first wife1 ultimately declined2 one was too old1 another an invalid1 another too proud of her birth and quarterings1 another could do nothing useful1 and so on4 ,number eight kept him guessing for three months1 until he tired of her constant indecision1 and confided his disappointment to number nine1 who was not impressed4 ,number ten1 introduced by a friend1 ,kepler found exceedingly ugly and enormously fat1 and number eleven apparently too young4 ,kepler then reconsidered one of the earlier ones1 disregarding the advice of his friends who objected to her lowly station4 ,she was the orphan daughter of a cabinetmaker1 educated for twelve years by favour of the ,lady of ,stahrenburg1 and ,kepler writes of her3 ,her person and manners are suitable to mine2 no pride1 no extravagance2 she can bear to work2 she has a tolerable knowledge of how to manage a family2 middle-aged and of a disposition and capability to acquire what she still wants4 ,wine from the ,austrian vineyards was plentiful and cheap at the time of the marriage1 and ,kepler bought a few casks for his household4 ,when the seller came to ascertain the quantity1 ,kepler noticed that no proper allowance was made for the bulging parts1 and the upshot of his objections was that he wrote a book on a new method of gauging--one of the earliest specimens of modern analysis1 extending the properties of plane figures to segments of cones and cylinders as being incorporated circles4 ,he was summoned before the ,diet at ,ratisbon to give his opinion on the ,gregorian ,reform of the ,calendar1 and soon afterwards was excommunicated1 having fallen foul of the ,roman ,catholic party at ,linz just as he had previously at ,gratz1 the reason apparently being that he desired to think for himself4 ,meanwhile his salary was not paid any more regularly than before1 and he was forced to supplement it by publishing what he called a vile prophesying almanac which is scarcely more respectable than begging unless it be because it saves the ,emperor's credit1 who abandons me entirely1 and with all his frequent and recent orders in council1 would suffer me to perish with hunger4 ,in #afag he was invited to ,italy to succeed ,magini as ,professor of ,mathematics at ,bologna4 ,galileo urged him to accept the post1 but he excused himself on the ground that he was a ,german and brought up among ,germans with such liberty of speech as he thought might get him into trouble in ,italy4 ,in #afai ,matthias died and was succeeded by ,ferdinand ,,iii1 who again retained ,kepler in his post4 ,in the same year ,kepler reprinted his ,mysterium ,cosmographicum1 and also published his ,harmonics in five books dedicated to ,james ,i of ,england4 ,the first geometrical1 on the origin and demonstration of the laws of the figures which produce harmonious proportions2 the second1 architectonical1 on figurate geometry and the congruence of plane and solid regular figures2 the third1 properly ,harmonic1 on the derivation of musical proportions from figures1 and on the nature and distinction of things relating to song1 in opposition to the old theories2 the fourth1 metaphysical1 psychological1 and astrological1 on the mental essence of ,harmonics1 and of their kinds in the world1 especially on the harmony of rays emanating on the earth from the heavenly bodies1 and on their effect in nature and on the sublunary and human soul2 the fifth1 astronomical and metaphysical1 on the very exquisite ,harmonics of the celestial motions and the origin of the excentricities in harmonious proportions4 ,the extravagance of his fancies does not appear until the fourth book1 in which he reiterates the statement that he was forced to adopt his astrological opinions from direct and positive observation4 ,he despises ,the common herd of prophesiers who describe the operations of the stars as if they were a sort of deities1 the lords of heaven and earth1 and producing everything at their pleasure4 ,they never trouble themselves to consider what means the stars have of working any effects among us on the earth whilst they remain in the sky and send down nothing to us which is obvious to the senses1 except rays of light4 ,his own notion is ,like one who listens to a sweet melodious song1 and by the gladness of his countenance1 by his voice1 and by the beating of his hand or foot attuned to the music1 gives token that he perceives and approves the harmony3 just so does sublunary nature1 with the notable and evident emotion of the bowels of the earth1 bear like witness to the same feelings1 especially at those times when the rays of the planets form harmonious configurations on the earth1 and again ,the earth is not an animal like a dog1 ready at every nod2 but more like a bull or an elephant1 slow to become angry1 and so much the more furious when incensed4 ,he seems to have believed the earth to be actually a living animal1 as witness the following3 ,if anyone who has climbed the peaks of the highest mountains1 throw a stone down their very deep clefts1 a sound is heard from them2 or if he throw it into one of the mountain lakes1 which beyond doubt are bottomless1 a storm will immediately arise1 just as when you thrust a straw into the ear or nose of a ticklish animal1 it shakes its head1 or runs shudderingly away4 ,what so like breathing1 especially of those fish who draw water into their mouths and spout it out again through their gills1 as that wonderful tide6 ,for although it is so regulated according to the course of the moon1 that1 in the preface to my ',commentaries on ,mars1' ,i have mentioned it as probable that the waters are attracted by the moon1 as iron by the loadstone1 yet if anyone uphold that the earth regulates its breathing according to the motion of the sun and moon1 as animals have daily and nightly alternations of sleep and waking1 ,i shall not think his philosophy unworthy of being listened to2 especially if any flexible parts should be discovered in the depths of the earth1 to supply the functions of lungs or gills4 ,in the same book ,kepler enlarges again on his views in reference to the basis of astrology as concerned with nativities and the importance of planetary conjunctions4 ,he gives particulars of his own nativity4 ,jupiter nearest the nonagesimal had passed by four degrees the trine of ,saturn2 the ,sun and ,venus in conjunction were moving from the latter towards the former1 nearly in sextiles with both3 they were also removing from quadratures with ,mars1 to which ,mercury was closely approaching3 the moon drew near to the trine of the same planet1 close to the ,bull's ,eye even in latitude4 ,the #beth degree of ,gemini was rising1 and the #bbnd of ,aquarius culminating4 ,that there was this triple configuration on that day--namely the sextile of ,saturn and the ,sun1 the sextile of ,mars and ,jupiter1 and the quadrature of ,mercury and ,mars1 is proved by the change of weather2 for after a frost of some days1 that very day became warmer1 there was a thaw and a fall of rain4 ,this alleged proof is interesting as it relies on the same principle which was held to justify the correction of an uncertain birth-time1 by reference to illnesses1 etc41 met with later4 ,kepler however goes on to say1 ,if ,i am to speak of the results of my studies1 what1 ,i pray1 can ,i find in the sky1 even remotely alluding to it8 ,the learned confess that several not despicable branches of philosophy have been newly extricated or amended or brought to perfection by me3 but here my constellations were1 not ,mercury from the ,east in the angle of the seventh1 and in quadratures with ,mars1 but ,copernicus1 but ,tycho ,brahe1 without whose books of observations everything now set by me in the clearest light must have remained buried in darkness2 not ,saturn predominating ,mercury1 but my lords the ,emperors ,rudolph and ,matthias1 not ,capricorn the house of ,saturn but ,upper ,austria1 the house of the ,emperor1 and the ready and unexampled bounty of his nobles to my petition4 ,here is that corner1 not the western one of the horoscope1 but on the earth whither1 by permission of my ,imperial master1 ,i have betaken myself from a too uneasy ,court2 and whence1 during these years of my life1 which now tends towards its setting1 emanate these ,harmonics and the other matters on which ,i am engaged4 ,the fifth book contains a great deal of nonsense about the harmony of the spheres2 the notes contributed by the several planets are gravely set down1 that of ,mercury having the greatest resemblance to a melody1 though perhaps more reminiscent of a bugle-call4 ,yet the book is not all worthless for it includes ,kepler's ,third ,law1 which he had diligently sought for years4 ,in his own words1 ,the proportion existing between the periodic times of any two planets is exactly the sesquiplicate proportion of the mean distances of the orbits1 or as generally given1 the squares of the periodic times are proportional to the cubes of the mean distances4 ,kepler was evidently transported with delight and wrote1 ,what ,i prophesied two and twenty years ago1 as soon as ,i discovered the five solids among the heavenly orbits1--what ,i firmly believed long before ,i had seen ,ptolemy's ',harmonics'--what ,i had promised my friends in the title of this book1 which ,i named before ,i was sure of my discovery1--what sixteen years ago ,i urged as a thing to be sought1--that for which ,i joined ,tycho ,brahe1 for which ,i settled in ,prague1 for which ,i have devoted the best part of my life to astronomical computations1 at length ,i have brought to light1 and have recognised its truth beyond my most sanguine expectations4 ,great as is the absolute nature of ,harmonics1 with all its details as set forth in my third book1 it is all found among the celestial motions1 not indeed in the manner which ,i imagined 7that is not the least part of my delight71 but in another very different1 and yet most perfect and excellent4 ,it is now eighteen months since ,i got the first glimpse of light1 three months since the dawn1 very few days since the unveiled sun1 most admirable to gaze on1 burst out upon me4 ,nothing holds me2 ,i will indulge in my sacred fury2 ,i will triumph over mankind by the honest confession that ,i have stolen the golden vases of the ,egyptians to build up a tabernacle for my ,god far away from the confines of ,egypt4 ,if you forgive me1 ,i rejoice1 if you are angry1 ,i can bear it2 the die is cast1 the book is written2 to be read either now or by posterity1 ,i care not which2 it may well wait a century for a reader1 as ,god has waited six thousand years for an observer4 ,he gives the date #aeth ,may1 #afah1 for the completion of his discovery4 ,in his ,epitome of the ,copernican ,astronomy1 he gives his own idea as to the reason for this ,third ,law4 ,four causes concur for lengthening the periodic time4 ,first1 the length of the path2 secondly1 the weight or quantity of matter to be carried2 thirdly1 the degree of strength of the moving virtue2 fourthly1 the bulk or space into which is spread out the matter to be moved4 ,the orbital paths of the planets are in the simple ratio of the distances2 the weights or quantities of matter in different planets are in the subduplicate ratio of the same distances1 as has been already proved2 so that with every increase of distance a planet has more matter and therefore is moved more slowly1 and accumulates more time in its revolution1 requiring already1 as it did1 more time by reason of the length of the way4 ,the third and fourth causes compensate each other in a comparison of different planets2 the simple and subduplicate proportion compound the sesquiplicate proportion1 which therefore is the ratio of the periodic times4 ,the only part of this explanation that is true is that the paths are in the simple ratio of the distances1 the proof so confidently claimed being of the circular kind commonly known as begging the question4 ,it was reserved for ,newton to establish the ,laws of ,motion1 to find the law of force that would constrain a planet to obey ,kepler's first and second ,laws1 and to prove that it must therefore also obey the third4 ,,chapter ,,vi4 ,,closing ,,years4 ,soon after its publication ,kepler's ,epitome was placed along with the book of ,copernicus1 on the list of books prohibited by the ,congregation of the ,index at ,rome1 and he feared that this might prevent the publication or sale of his books in ,austria also1 but was told that though ,galileo's violence was getting him into trouble1 there would be no difficulty in obtaining permission for learned men to read any prohibited books1 and that he 7,kepler7 need fear nothing so long as he remained quiet4 ,in his various works on ,comets1 he adhered to the opinion that they travelled in straight lines with varying velocity4 ,he suggested that comets come from the remotest parts of ether1 as whales and monsters from the depth of the sea1 and that perhaps they are something of the nature of silkworms1 and are wasted and consumed in spinning their own tails4 ,napier's invention of logarithms at once attracted ,kepler's attention4 ,he must have regretted that the discovery was not made early enough to save him a vast amount of labour in computations1 but he managed to find time to compute some logarithm tables for himself1 though he does not seem to have understood quite what ,napier had done1 and though with his usual honesty he gave full credit to the ,scottish baron for his invention4 ,though ,eugenists may find a difficulty in reconciling ,napier's brilliancy with the extreme youth of his parents1 they may at any rate attribute ,kepler's occasional fits of bad temper to heredity4 ,his cantankerous mother1 ,catherine ,kepler1 had for some years been carrying on an action for slander against a woman who had accused her of administering a poisonous potion4 ,dame ,kepler employed a young advocate who for reasons of his own nursed the case so long that after five years had elapsed without any conclusion being reached another judge was appointed1 who had himself suffered from the caustic tongue of the prosecutrix1 and so was already prejudiced against her4 ,the defendant1 knowing this1 turned the tables on her opponent by bringing an accusation of witchcraft against her1 and ,catherine ,kepler was imprisoned and condemned to the torture in ,july1 #afbj4 ,kepler1 hearing of the sentence1 hurried back from ,linz1 and succeeded in stopping the completion of the sentence1 securing his mother's release the following year1 as it was made clear that the only support for the case against her was her own intemperate language4 ,kepler returned to ,linz1 and his mother at once brought another action for costs and damages against her late opponent1 but died before the case could be tried4 ,a few months before this ,sir ,henry ,wotton1 ,english ,ambassador to ,venice1 visited ,kepler1 and finding him as usual1 almost penniless1 urged him to go to ,england1 promising him a warm welcome there4 ,kepler1 however1 would not at that time leave ,germany1 giving several reasons1 one of which was that he dreaded the confinement of an island4 ,later on he expressed his willingness to go as soon as his ,rudolphine ,tables were published1 and lecture on them1 even in ,england1 if he could not do it in ,germany1 and if a good enough salary were forthcoming4 ,in #afbd he went to ,vienna1 and managed to extract from the ,treasury #fjjj florins on account of expenses connected with the ,tables1 but1 instead of a further grant1 was given letters to the ,states of ,swabia1 which owed money to the ,imperial treasury4 ,some of this he succeeded in collecting1 but the ,tables were still further delayed by the religious disturbances then becoming violent4 ,the ,jesuits contrived to have ,kepler's library sealed up1 and1 but for the ,imperial protection1 would have imprisoned him also2 moreover the peasants revolted and blockaded ,linz4 ,in #afbg1 however1 the long promised ,tables1 the first to discard the conventional circular motion1 were at last published at ,ulm in four parts4 ,two of these parts consisted of subsidiary ,tables1 of logarithms and other computing devices1 another contained ,tables of the elements of the sun1 moon1 and planets1 and the fourth gave the places of a thousand stars as determined by ,tycho1 with ,tycho's refraction ,tables1 which had the peculiarity of using different values for the refraction of the sun1 moon1 and stars4 ,from a map prefixed to some copies of the ,tables1 we may infer that ,kepler was one of the first1 if not actually the first1 to suggest the method of determining differences of longitude by occultations of stars at the moon's limb4 ,in an ,appendix1 he showed how his ,tables could be used by astrologers for their predictions1 saying ,astronomy is the daughter of ,astrology1 and this modern ,astrology again is the daughter of ,astronomy1 bearing something of the lineaments of her grandmother2 and1 as ,i have already said1 this foolish daughter1 ,astrology1 supports her wise but needy mother1 ,astronomy1 from the profits of a profession not generally considered creditable4 ,there is no doubt that ,kepler strongly resented having to depend so much for his income on such methods which he certainly did not consider creditable4 ,it was probably ,galileo whose praise of the new ,tables induced the ,grand ,duke of ,tuscany to send ,kepler a gold chain soon after their publication1 and we may perhaps regard it as a mark of favour from the ,emperor ,ferdinand that he permitted ,kepler to attach himself to the great ,wallenstein1 now ,duke of ,friedland1 and a firm believer in ,astrology4 ,the ,duke was a better paymaster than either of the three successive ,emperors4 ,he furnished ,kepler with an assistant and a printing press2 and obtained for him the ,professorship of ,astronomy at the ,university of ,rostock in ,mecklenburg4 ,apparently1 however1 the ,emperor could not induce ,wallenstein to take over the responsibility of the #hjjj crowns1 still owing from the ,imperial treasury on account of the ,rudolphine ,tables4 ,kepler made a last attempt to secure payment at ,ratisbon1 but his journey thither brought disappointment and fatigue and left him in such a condition that he rapidly succumbed to an attack of fever1 dying in ,november1 #afcj1 in his fifty-ninth year4 ,his body was buried at ,ratisbon1 but the tombstone was destroyed during the war then raging4 ,his daughter1 ,susanna1 the wife of ,jacob ,bartsch1 a physician who had helped ,kepler with his ,ephemeris1 lost her husband soon after her father's death1 and succeeded in obtaining part of ,kepler's arrears of salary by threatening to keep ,tycho's manuscripts1 but her stepmother was left almost penniless with five young children4 ,for their benefit ,louis ,kepler printed a ,dream of ,lunar ,astronomy1 which first his father and then his brother-in-law had been preparing for publication at the time of their respective deaths4 ,it is a curious mixture of saga and fairy tale with a little science in the way of astronomy studied from the moon1 and cast in the form of a dream to overcome the practical difficulties of the hypothesis of visiting the moon4 ,other writings in large numbers were left unpublished4 ,no attempt at a complete edition of ,kepler's works was made for a long time4 ,one was projected in #agad by his biographer1 ,hantsch1 but all that appeared was one volume of letters4 ,after various learned bodies had declined to move in the matter the manuscripts were purchased for the ,imperial ,russian library4 ,an edition was at length brought out at ,frankfort by ,c4 ,frisch1 in eight volumes1 appearing at intervals from #aheh-ahgj4 ,kepler's fame does not rest upon his voluminous works4 ,with his peculiar method of approaching problems there was bound to be an inordinate amount of chaff mixed with the grain1 and he used no winnowing machine4 ,his simplicity and transparent honesty induced him to include everything1 in fact he seemed to glory in the number of false trails he laboriously followed4 ,he was one who might be expected to find the proverbial needle in a haystack1 but unfortunately the needle was not always there4 ,delambre says1 ,ardent1 restless1 burning to distinguish himself by his discoveries he attempted everything1 and having once obtained a glimpse of one1 no labour was too hard for him in following or verifying it4 ,all his attempts had not the same success1 and in fact that was impossible4 ,those which have failed seem to us only fanciful2 those which have been more fortunate appear sublime4 ,when in search of that which really existed1 he has sometimes found it2 when he devoted himself to the pursuit of a chimera1 he could not but fail1 but even then he unfolded the same qualities1 and that obstinate perseverance that must triumph over all difficulties but those which are insurmountable4 ,berry1 in his ,short ,history of ,astronomy1 says as one reads chapter after chapter without a lucid1 still less a correct idea1 it is impossible to refrain from regrets that the intelligence of ,kepler should have been so wasted1 and it is difficult not to suspect at times that some of the valuable results which lie embedded in this great mass of tedious speculation were arrived at by a mere accident4 ,on the other hand it must not be forgotten that such accidents have a habit of happening only to great men1 and that if ,kepler loved to give reins to his imagination he was equally impressed with the necessity of scrupulously comparing speculative results with observed facts1 and of surrendering without demur the most beloved of his fancies if it was unable to stand this test4 ,if ,kepler had burnt three-quarters of what he printed1 we should in all probability have formed a higher opinion of his intellectual grasp and sobriety of judgment1 but we should have lost to a great extent the impression of extraordinary enthusiasm and industry1 and of almost unequalled intellectual honesty which we now get from a study of his works4 ,professor ,forbes is more enthusiastic4 ,in his ,history of ,astronomy1 he refers to ,kepler as the man whose place1 as is generally agreed1 would have been the most difficult to fill among all those who have contributed to the advance of astronomical knowledge1 and again propos of ,kepler's great book1 it must be obvious that he had at that time some inkling of the meaning of his laws--universal gravitation4 ,from that moment the idea of universal gravitation was in the air1 and hints and guesses were thrown out by many2 and in time the law of gravitation would doubtless have been discovered1 though probably not by the work of one man1 even if ,newton had not lived4 ,but1 if ,kepler had not lived1 who else could have discovered his ,laws8 ,,appendix ,i4 ,,list ,,of ,,dates4 ,johann ,kepler1 born #aega2 school at ,maulbronn1 #aehf2 ,university of ,tbingen1 #aehi2 ,m4,a4 of ,tbingen1 #aeia2 ,professor at ,gratz1 #aeid2 ,prodromus1 with ,mysterium ,cosmographicum1 published #aeif2 first marriage1 #aeig2 joins ,tycho ,brahe at ,prague1 #afjj2 death of ,tycho1 #afja2 ,kepler's optics1 #afjc2 ,nova1 #afjd2 on ,comets1 #afjg2 ,commentary on ,mars1 including ,first and ,second ,laws1 #afji2 ,professor at ,linz1 #afab2 second marriage1 #afac2 ,third ,law discovered1 #afah2 ,epitome of ,copernican ,astronomy1 #afah-afba2 ,rudolphine ,tables published1 #afbg2 died1 #afcj4 ,,appendix ,,ii4 ,,bibliography4 ,for a full account of the various systems of ,kepler and his predecessors the reader cannot do better than consult the ,history of the ,planetary ,systems1 from ,thales to ,kepler1 by ,dr4 ,j4,l4,e4 ,dreyer 7,cambridge ,univ4 ,press1 #aijf74 ,the same author's ,tycho ,brahe gives a wealth of detail about that ,phoenix of ,astronomers1 as ,kepler styles him4 ,a great proportion of the literature relating to ,kepler is ,german1 but he has his place in the histories of astronomy1 from ,delambre and the more modern ,r4 ,wolfs ,geschichte to those of ,a4 ,berry1 ,history of ,astronomy 7,university ,extension ,manuals1 ,murray1 #ahih71 and ,professor ,g4 ,forbes1 ,history of ,astronomy 7,history of ,science ,series1 ,watts1 #aiji74 ,,glossary4 ,apogee3 ,the point in the orbit of a celestial body when it is furthest from the earth4 ,apse3 ,an extremity of the major axis of the orbit of a body2 a body is at its greatest and least distances from the body about which it revolves1 when at one or other apse4 ,conjunction3 ,when a plane containing the earth's axis and passing through the centre of the sun also passes through that of the moon or a planet1 at the same side of the earth1 the moon or planet is in conjunction1 or if on opposite sides of the earth1 the moon or planet is in opposition4 ,mercury and ,venus cannot be in opposition1 but are in inferior or superior conjunction according as they are nearer or further than the sun4 ,deferent3 ,in the epicyclic theory1 uneven motion is represented by motion round a circle whose centre travels round another circle1 the latter is called the deferent4 ,ecliptic3 ,the plane of the earth's orbital motion about the sun1 which cuts the heavens in a great circle4 ,it is so called because obviously eclipses can only occur when the moon is also approximately in this plane1 besides being in conjunction or opposition with the sun4 ,epicycle3 ,a point moving on the circumference of a circle whose centre describes another circle1 traces an epicycle with reference to the centre of the second circle4 ,equant3 ,in ,ptolemy's excentric theory1 when a planet is describing a circle about a centre which is not the earth1 in order to satisfy the convention that the motion must be uniform1 a point was found about which the motion was apparently uniform1#d and this point was called the equant4 ,footnote #d3 ,i4e4 the angular motion about the equant was uniform4 ,equinox3 ,when the sun is in the plane of the earth's equator the lengths of day and night are equal4 ,this happens twice a year1 and the times when the sun passes the equator are called the vernal or spring equinox and the autumnal equinox respectively4 ,evection3 ,the second inequality of the moon1 which vanishes at new and full moon and is a maximum at first and last quarter4 ,excentric3 ,as an alternative to epicycles1 planets whose motion round the earth was not uniform could be represented as moving round a point some distance from the earth called the excentric4 ,geocentric3 ,referred to the centre of the earth2 e4g4 ,ptolemy's theory4 ,heliocentric3 ,referred to the centre of the sun2 e4g4 the theory commonly called ,copernican4 ,inequality3 ,the difference between the actual position of a planet and its theoretical position on the hypothesis of uniform circular motion4 ,node3 ,the points where the orbit of the moon or a planet intersect the plane of the ecliptic4 ,the ascending node is the one when the planet is moving northwards1 and the line of intersection of the orbital plane with the ecliptic is the line of nodes4 ,occultation3 ,usually means when a planet or star is hidden by the moon1 but it also includes occultation of a star by a planet or of a satellite by a planet or of one planet by another4 ,opposition v4 ,conjunction4 ,parallax3 ,the error introduced by observing from some point other than that required in theory1 e4g4 in geocentric places because the observations are made from the surface of the earth instead of the centre1 or in heliocentric places because observations are made from the earth and not from the sun4 ,perigee3 ,the point in the orbit of a celestial body when it is nearest to the earth4 ,precession3 ,owing to the slow motion of the earth's pole around the pole of the ecliptic1 the equator cuts the ecliptic a little earlier every year1 so that the equinox each year slightly precedes1 with reference to the stars1 that of the previous year4