A Short History of Astronomy (1898)/Chapter 7
135. John Kepler, or Keppler, was born in 1571, seven years after Galilei, at Weil in Würtemberg; his parents were in reduced circumstances, though his father had some claims to noble descent. Though Weil itself was predominantly Roman Catholic, the Keplers were Protestants, a fact which frequently stood in Kepler's way at various stages of his career. But the father could have been by no means zealous in his faith, for he enlisted in the army of the notorious Duke of Alva when it was engaged in trying to suppress the revolt of the Netherlands against Spanish persecution.
John Kepler's childhood was marked by more than the usual number of illnesses, and his bodily weaknesses, combined with a promise of great intellectual ability, seemed to point to the Church as a suitable career for him. After attending various elementary schools with great irregularity —due partly to ill-health, partly to the requirements of manual work at home—he was sent in 1584 at the public expense to the monastic school at Adelberg, and two years later to the more advanced school or college of the same kind at Maulbronn, which was connected with the University of Tübingen, then one of the great centres of Protestant theology.
In 1588 he obtained the B.A. degree, and in the following year entered the philosophical faculty at Tübingen.
There he came under the influence of Maestlin, the professor of mathematics, by whom he was in private taught the principles of the Coppernican system, though the professorial lectures were still on the traditional lines.
In 1591 Kepler graduated as M.A., being second out of fourteen candidates, and then devoted himself chiefly to the study of theology.
136. In 1594, however, the Protestant Estates of Styria applied to Tübingen for a lecturer on mathematics (including astronomy) for the high school of Gratz, and the appointment was offered to Kepler. Having no special knowledge of the subject and as yet no taste for it, he naturally hesitated about accepting the offer, but finally decided to do so, expressly stipulating, however, that he should not thereby forfeit his claims to ecclesiastical preferment in Würtemberg. The demand for higher mathematics at Gratz seems to have been slight; during his first year Kepler's mathematical lectures were attended by very few students, and in the following year by none, so that to prevent his salary from being wasted he was set to teach the elements of various other subjects. It was moreover one of his duties to prepare an annual almanack or calendar, which was expected to contain not merely the usual elementary astronomical information such as we are accustomed to in the calendars of to-day, but also astrological information of a more interesting character, such as predictions of the weather and of remarkable events, guidance as to unlucky and lucky times, and the like. Kepler's first calendar, for the year 1595, contained some happy weather-prophecies, and he acquired accordingly a considerable popular reputation as a prophet and astrologer, which remained throughout his life.
Meanwhile his official duties evidently left him a good deal of leisure, which he spent with characteristic energy in acquiring as thorough a knowledge as possible of astronomy, and in speculating on the subject.
According to his own statement, "there were three things in particular, viz. the number, the size, and the motion of the heavenly bodies, as to which he searched zealously for reasons why they were as they were and not otherwise"; and the results of a long course of wild speculation on the subject led him at last to a result with which he was immensely pleased—a numerical relation connecting the distances of the several planets from the sun with certain geometrical bodies known as the regular solids (of which the cube is the best known), a relation which is not very accurate numerically, and is of absolutely no significance or importance. This discovery, together with a detailed account of the steps which led to it, as well as of a number of other steps which led nowhere, was published in 1596 in a book a portion of the title of which may be translated as The Forerunner of Dissertations on the Universe, containing the Mystery of the Universe, commonly referred to as the Mysterium Cosmographicum. The contents were probably much more attractive and seemed more valuable to Kepler's contemporaries than to us, but even to those who were least inclined to attach weight to its conclusions, the book shewed evidence of considerable astronomical knowledge and very great ingenuity; and both Tycho Brahe and Galilei, to whom copies were sent, recognised in the author a rising astronomer likely to do good work.
137. In 1597 Kepler married. In the following year the religious troubles, which had for some years been steadily growing, were increased by the action of the Archduke Ferdinand of Austria (afterwards the Emperor Ferdinand II.), who on his return from a pilgrimage to Loretto started a vigorous persecution of Protestants in his dominions, one step in which was an order that all Protestant ministers and teachers in Styria should quit the country at once (1598). Kepler accordingly fled to Hungary, but returned after a few weeks by special permission of the Archduke, given apparently on the advice of the Jesuit party, who had hopes of converting the astronomer. Kepler's hearers had, however, mostly been scattered by the persecution, it became difficult to ensure regular payment of his stipend, and the rising tide of Catholicism made his position increasingly insecure. Tycho's overtures were accordingly welcome, and in 1600 he paid a visit to him, as already described (chapter v., § 108), at Benatek and Prague. He returned to Gratz in the autumn, still uncertain whether to accept Tycho's offer or not, but being then definitely dismissed from his position at Gratz on account of his Protestant opinions, he returned finally to Prague at the end of the year.
138. Soon after Tycho's death Kepler was appointed his successor as mathematician to the Emperor Rudolph (1602), but at only half his predecessor's salary, and even this was paid with great irregularity, so that complaints as to arrears and constant pecuniary difficulties played an important part in his future life, as they had done during the later years at Gratz. Tycho's instruments never passed into his possession, but as he had little taste or skill for observing, the loss was probably not great; fortunately, after some difficulties with the heirs, he secured control of the greater part of Tycho's incomparable series of observations, the working up of which into an improved theory of the solar system was the main occupation of the next 25 years of his life. Before, however, he had achieved any substantial result in this direction, he published several minor works—for example, two pamphlets on a new star which appeared in 1604, and a treatise on the applications of optics to astronomy (published in 1604 with a title beginning Ad Vitellionem Paralipomena quibus Astronoiniae Pars Optica Traditur . . .), the most interesting and important part of which was a considerable improvement in the theory of astronomical refraction (chapter ii., § 46, and chapter v., § 110). A later optical treatise (the Dioptrice of 1611) contained a suggestion for the construction of a telescope by the use of two convex lenses, which is the form now most commonly adopted, and is a notable improvement on Galilei's instrument (chapter vi., § 118), one of the lenses of which is concave; but Kepler does not seem himself to have had enough mechanical skill to actually construct a telescope on this plan, or to have had access to workmen capable of doing so for him; and it is probable that Galilei's enemy Scheiner (chapter vi., §§ 124, 125) was the first person to use (about 1613) an instrument of this kind.
139. It has already been mentioned (chapter v., § 108) that when Tycho was dividing the work of his observatory among his assistants he assigned to Kepler the study of the planet Mars, probably as presenting more difficulties than the subjects assigned to the others. It had been known since the time of Coppernicus that the planets, including the earth, revolved round the sun in paths that were at any rate not very different from circles, and that the deviations from uniform circular motion could be represented roughly by systems of eccentrics and epicycles. The deviations from uniform circular motion were, however, notably different in amount in different planets, being, for example, very small in the case of Venus, relatively large in the case of Mars, and larger still in that of Mercury. The Prussian Tables calculated by Reinhold on a Coppernican basis (chapter v., § 94) were soon found to represent the actual motions very imperfectly, errors of 4° and 5° having been noted by Tycho and Kepler, so that the principles on which the tables were calculated were evidently at fault.
The solution of the problem was clearly more likely to be found by the study of a planet in which the deviations from circular motion were as great as possible. In the case of Mercury satisfactory observations were scarce, whereas in the case of Mars there was an abundant series recorded by Tycho, and hence it was true insight on Tycho's part to assign to his ablest assistant this particular planet, and on Kepler's to continue the research with unwearied patience. The particular system of epicycles used by Coppernicus (chapter iv., § 87) having proved defective, Kepler set to work to devise other geometrical schemes, the results of which could be compared with observation. The places of Mars as seen on the sky being a combined result of the motions of Mars and of the earth in their respective orbits round the sun, the irregularities of the two orbits were apparently inextricably mixed up, and a great simplification was accordingly effected when Kepler succeeded, by an ingenious combination of observations taken at suitable times, in disentangling the irregularities due to the earth from those due to the motion of Mars itself, and thus rendering it possible to concentrate his attention on the latter. His fertile imagination suggested hypothesis after hypothesis, combination after combination of eccentric, epicycle, and equant; he calculated the results of each and compared them rigorously with observation; and at one stage he arrived at a geometrical scheme which was capable of representing the observations with errors not exceeding 8'. A man of less intellectual honesty, or less convinced of the necessity of subordinating theory to fact when the two conflict, might have rested content with this degree of accuracy, or might have supposed Tycho's refractory observations to be in error. Kepler, however, thought otherwise:—
140. He accordingly started afresh, and after trying a variety of other combinations of circles decided that the path of Mars must be an oval of some kind. At first he was inclined to believe in an egg-shaped oval, larger at one end than at the other, but soon had to abandon this idea. Finally he tried the simplest known oval curve, the ellipse, and found to his delight that it satisfied the conditions of the problem, if the sun were taken to be at a focus of the ellipse described by Mars.
It was further necessary to formulate the law of variation of the rate of motion of the planet in different parts of its orbit. Here again Kepler tried a number of hypotheses, in the course of which he fairly lost his way in the intricacies of the mathematical questions involved, but fortunately arrived, after a dubious process of compensation of errors, at a simple law which agreed with observation. He found that the planet moved fast when near the sun and slowly when distant from it, in such a way that the area described or swept out in any time by the line joining the sun to Mars was always proportional to the time. Thus in fig. 60 the motion of Mars is most rapid at the point a nearest to the focus s where the sun is, least rapid at a', and the shaded and unshaded portions of the figure represent equal areas each corresponding to the motion of the planet during a month. Kepler's triumph at arriving at this result is expressed by the figure of victory in the corner of the diagram (fig. 61) which was used in establishing the last stage of his proof.
141. Thus were established for the case of Mars the two important results generally known as Kepler's first two laws:—
1. The planet describes an ellipse, the sun being in one focus.
2. The straight line joining the planet to the sun sweeps out equal areas in any two equal intervals of time.
The full history of this investigation, with the results already stated and a number of developments and results of minor importance, together with innumerable digressions and quaint comments on the progress of the inquiry, was published in 1609 in a book of considerable length, the Commentaries on the Motions of Mars.
142. Although the two laws of planetary motion just given were only fully established for the case of Mars, Kepler stated that the earth's path also must be an oval of some kind, and was evidently already convinced—aided by his firm belief in the harmony of Nature—that all the planets moved in accordance with the same laws. This view is indicated in the dedication of the book to the Emperor Rudolph, which gives a fanciful account of the work as a
struggle against the rebellious War-God Mars, as the result of which he is finally brought captive to the feet of the Emperor and undertakes to live for the future as a loyal subject. As, however, he has many relations in the ethereal spaces—his father Jupiter, his grandfather Saturn, his dear sister Venus, his faithful brother Mercury—and he yearns for them and they for him on account of the similarity of their habits, he entreats the Emperor to send out an expedition as soon as possible to capture them also, and with that object to provide Kepler with the "sinews of war" in order that he may equip a suitable army.
Although the money thus delicately asked for was only supplied very irregularly, Kepler kept steadily in view the expedition for which it was to be used, or, in plainer words, he worked steadily at the problem of extending his elliptic theory to the other planets, and constructing the tables of the planetary motions, based on Tycho's observations, at which he had so long been engaged.
143. In 1611 his patron Rudolph was forced to abdicate the imperial crown in favour of his brother Matthias, who had little interest in astronomy, or even in astrology; and as Kepler's position was thus rendered more insecure than ever, he opened negotiations with the Estates of Upper Austria, as the result of which he was promised a small salary, on condition of undertaking the somewhat varied duties of teaching mathematics at the high school of Linz, the capital, of constructing a new map of the province, and of completing his planetary tables. For the present, however, he decided to stay with Rudolph.
In the same year Kepler lost his wife, who had long been in weak bodily and mental health.
In the following year (1612) Rudolph died, and Kepler then moved to Linz and took up his new duties there, though still holding the appointment of mathematician to the Emperor and occasionally even receiving some portion of the salary of the office. In 1613 he married again, after a careful consideration, recorded in an extraordinary but very characteristic letter to one of his friends, of the relative merits of eleven ladies whom he regarded as possible; and the provision of a proper supply of wine for his new household led to the publication of a pamphlet, of some mathematical interest, dealing with the proper way of measuring the contents of a cask with curved sides.
144. In the years 1618–1621, although in some ways the most disturbed years of his life, he published three books of importance—an Epitome of the Copernican Astronomy, the Harmony of the World, and a treatise on Comets.
The second and most important of these, published in 1619, though the leading idea in it was discovered early in 1618, was regarded by Kepler as a development of his early Mysterium Cosmographicum (§ 136). His speculative and mystic temperament led him constantly to search for relations between the various numerical quantities occurring in the solar system; by a happy inspiration he thought of trying to get a relation connecting the sizes of the orbits of the various planets with their times of revolution round the sun, and after a number of unsuccessful attempts discovered a simple and important relation, commonly known as Kepler's third law:—
The squares of the times of revolution of any two planets (including the earth) about the sun are proportional to the cubes of their mean distances from the sun.
If, for example, we express the times of revolution of the various planets in terms of any one, which may be conveniently taken to be that of the earth, namely a year, and in the same way express the distances in terms of the distance of the earth from the sun as a unit, then the times of revolution of the several planets taken in the order Mercury, Venus, Earth, Mars, Jupiter, Saturn are approximately ⋅24, ⋅615, 1, 1⋅88, 11⋅86, 29⋅457, and their distances from the sun are respectively ⋅387, ⋅723, 1, 1⋅524, 5⋅203, 9⋅539; if now we take the squares of the first series of numbers (the square of a number being the number multiplied by itself) and the cubes of the second series (the cube of a number being the number multiplied by itself twice, or the square multiplied again by the number), we get the two series of numbers given approximately by the table:—
Here it will be seen that the two series of numbers, in the upper and lower row respectively, agree completely for as many decimal places as are given, except in the cases of the two outer planets, where the lower numbers are slightly in excess of the upper. For this discrepancy Newton afterwards assigned a reason (chapter ix., § 186), but with the somewhat imperfect knowledge of the times of revolution and distances which Kepler possessed the discrepancy was barely capable of detection, and he was therefore justified—from his standpoint—in speaking of the law as "precise."
It should be noticed further that Kepler's law requires no knowledge of the actual distances of the several planets
from the sun, but only of their relative distances, i.e. the number of times farther off from the sun or nearer to the sun any planet is than any other. In other words, it is necessary to have or to be able to construct a map of the solar system correct in its proportions, but it is quite unnecessary for this purpose to know the scale of the map.
Although the Harmony of the World is a large book, there is scarcely anything of value in it except what has already been given. A good deal of space is occupied with repetitions of the earlier speculations contained in the Mysterium Cosmographicum, and most of the rest is filled with worthless analogies between the proportioas of the solar system and the relations between various musical scales.
He is bold enough to write down in black and white the "music of the spheres" (in the form shewn in fig. 62), while the nonsense which he was capable of writing may be further illustrated by the remark which occurs in the same part of the book: "The Earth sings the notes M I, F A, M I, so that you may guess from them that in this abode of ours MIsery (miseria) and FAmine (fames) prevail."
145. The Epitome of the Copernican Astronomy, which appeared in parts in 1618, 1620, and 1621, although there are no very striking discoveries in it, is one of the most attractive of Kepler's books, being singularly free from the extravagances which usually render his writings so tedious. It contains within moderately short compass, in the form of question and answer, an account of astronomy as known at the time, expounded from the Coppernican standpoint, and embodies both Kepler's own and Galilei's latest discoveries. Such a text-book supplied a decided want, and that this was recognised by enemies as well as by friends was shewn by its prompt appearance in the Roman Index of Prohibited Books (cf. chapter vi., §§ 126, 132). The Epitome contains the first clear statement that the two fundamental laws of planetary motion established for the case of Mars (§ 141) were true also for the other planets (no satisfactory proof being, however, given), and that they applied also to the motion of the moon round the earth, though in this case there were further irregularities which complicated matters. The theory of the moon is worked out in considerable detail, both evection (chapter ii., § 48) and variation (chapter iii., § 60; chapter v., § 111) being fully dealt with, though the "annual equation" which Tycho had just begun to recognise at the end of his life (chapter v., § 111) is not discussed. Another interesting development of his own discoveries is the recognition that his third law of planetary motion applied also to the movements of the four satellites round Jupiter, as recorded by Galilei and Simon Marius (chapter vi., § 118). Kepler also introduced in the Epitome a considerable improvement in the customary estimate of the distance of the earth from the sun, from which those of the other planets could at once be deduced.
If, as had been generally believed since the time of Hipparchus and Ptolemy, the distance of the sun were 1,200 times the radius of the earth, then the parallax (chapter ii., §§ 43, 49) of the sun would at times be as much as 3', and that of Mars, which in some positions is much nearer to the earth, proportionally larger. But Kepler had been unable to detect any parallax of Mars, and therefore inferred that the distances of Mars and of the sun must be greater than had been supposed. Having no exact data to go on, he produced out of his imagination and his ideas of the harmony of the solar system a distance about three times as great as the traditional one. He argued that, as the earth was the abode of measuring creatures, it was reasonable to expect that the measurements of the solar system would bear some simple relation to the dimensions of the earth. Accordingly he assumed that the volume of the sun was as many times greater than the volume of the earth as the distance of the sun was greater than the radius of the earth, and from this quaint assumption deduced the value of the distance already stated, which, though an improvement on the old value, was still only about one-seventh of the true distance.
The Epitome contains also a good account of eclipses both of the sun and moon, with the causes, means of predicting them, etc. The faint light (usually reddish) with which the face of the eclipsed moon often shines is correctly explained as being sunlight which has passed through the atmosphere of the earth, and has there been bent from a straight course so as to reach the moon, which the light of the sun in general is, owing to the interposition of the earth, unable to reach. Kepler mentions also a ring of light seen round the eclipsed sun in 1567, when the eclipse was probably total, not annular (chapter ii., § 43), and ascribes it to some sort of luminous atmosphere round the sun, referring to a description in Plutarch of the same appearance. This seems to have been an early observation, and a rational though of course very imperfect explanation, of that remarkable solar envelope known as the corona which has attracted so much attention in the last half-century (chapter xiii., § 301).
146. The treatise on Comets (1619) contained an account of a comet seen in 1607, afterwards famous as Halley's comet (chapter x., § 200), and of three comets seen in 1618. Following Tycho, Kepler held firmly the view that comets were celestial not terrestrial bodies, and accounted for their appearance and disappearance by supposing that they moved in straight lines, and therefore after having once passed near the earth receded indefinitely into space; he does not appear to have made any serious attempt to test this theory by comparison with observation, being evidently of opinion that the path of a body which would never reappear was not a suitable object for serious study. He agreed with the observation made by Fracastor and Apian (chapter iii., § 69) that comets' tails point away from the sun, and explained this by the supposition that the tail is formed by rays of the sun which penetrate the body of the comet and carry away with them some portion of its substance, a theory which, allowance being made for the change in our views as to the nature of light, is a curiously correct anticipation of modern theories of comets' tails (chapter xiii., § 304).
In a book intended to have a popular sale it was necessary to make the most of the "meaning" of the appearance of a comet, and of its influence on human affairs, and as Kepler was writing when the Thirty Years' War had just begun, while religious persecutions and wars had been going on in Europe almost without interruption during his lifetime, it was not difficult to find sensational events which had happened soon after or shortly before the appearance of the comets referred to. Kepler himself was evidently not inclined to attach much importance to such coincidences; he thought that possibly actual contact with a comet's tail might produce pestilence, but beyond that was not prepared to do more than endorse the pious if somewhat neutral opinion that one of the uses of a comet is to remind us that we are mortal. His belief that comets are very numerous is expressed in the curious form: "There are as many arguments to prove the annual motion of the earth round the sun as there are comets in the heavens."
147. Meanwhile Kepler's position at Linz had become more and more uncomfortable, owing to the rising tide of the religious and political disturbances which finally led to the outbreak of the Thirty Years' War in 1618; but notwithstanding this he had refused in 1617 an offer of a chair of mathematics at Bologna, partly through attachment to his native country and partly through a well-founded distrust of the Papal party in Italy. Three years afterwards he rejected also the overtures made by the English ambassador, with a view to securing him as an ornament to the court of James I., one of his chief grounds for refusal in this case being a doubt whether he would not suffer from being cooped up within the limits of an island. In 1619 the Emperor Matthias died, and was succeeded by Ferdinand II., who as Archduke had started the persecution of the Protestants at Gratz (§ 137) and who had few scientific interests. Kepler was, however, after some delay, confirmed in his appointment as Imperial Mathematician. In 1620 Linz was occupied by the Imperialist troops, and by 1626 the oppression of the Protestants by the Roman Catholics had gone so far that Kepler made up his mind to leave, and, after sending his family to Regensburg, went himself to Ulm.
148. At Ulm Kepler published his last great work. For more than a quarter of a century he had been steadily working out in detail, on the basis of Tycho's observations and of his own theories, the motions of the heavenly bodies, expressing the results in such convenient tabular form that the determination of the place of any body at any required time, as well as the investigation of other astronomical events such as eclipses, became merely a matter of calculation according to fixed rules; this great undertaking, in some sense the summing up of his own and of Tycho's work, was finally published in 1627 as the Rudolphine Tables (the name being given in honour of his former patron), and remained for something like a century the standard astronomical tables.
It had long been Kepler's intention, after finishing the tables, to write a complete treatise on astronomy, to be called the New Almagest; but this scheme was never fairly started, much less carried out.
149. After a number of unsuccessful attempts to secure the arrears of his salary, he was told to apply to Wallenstein, the famous Imperialist general, then established in Silesia in a semi-independent position, who was keenly interested in astrology and usually took about with him one or more representatives of the art. Kepler accordingly joined Wallenstein in 1628, and did astrology for him, in addition to writing some minor astronomical and astrological treatises. In 1630 he travelled to Regensburg, where the Diet was then sitting, to press in person his claims for various arrears of salary; but, worn out by anxiety and by the fatigues of the journey, he was seized by a fever a few days after his arrival, and died on November 15th (n.s.), 1630, in his 59th year.
The inventory of his property, made after his death, shews that he was in possession of a substantial amount, so that the effect of extreme poverty which his letters convey must have been to a considerable extent due to his over-anxious and excitable temperament.
150. In addition to the great discoveries already mentioned Kepler made a good many minor contributions to astronomy, such as new methods of finding the longitude, and various improvements in methods of calculation required for astronomical problems. He also made speculations of some interest as to possible causes underlying the known celestial motions. Whereas the Ptolemaic system required a number of motions round mere geometrical points, centres of epicycles or eccentrics, equants, etc., unoccupied by any real body, and many such motions were still required by Coppernicus, Kepler's scheme of the solar system placed a real body, the sun, at the most important point connected with the path of each planet, and dealt similarly with the moon's motion round the earth and with that of the four satellites round Jupiter. Motions of revolution came in fact to be associated not with some central point but with some central body, and it became therefore an inquiry of interest to ascertain if there were any connection between the motion and the central body. The property possessed by a magnet of attracting a piece of iron at some little distance from it suggested a possible analogy to Kepler, who had read with care and was evidently impressed by the treatise On the Magnet (De Magnete) published in 1600 by our countryman William Gilbert of Colchester (1540–1603). He suggested that the planets might thus be regarded as connected with the sun, and therefore as sharing to some extent the sun's own motion of revolution. In other words, a certain "carrying virtue" spread out from the sun, with or like the rays of light and heat, and tried to carry the planets round with the sun.
The annexed diagram is given by Kepler in illustration of this rather confused and vague theory.
He believed also in a more general "gravity," which he defined as "a mutual bodily affection between allied bodies tending towards their union or junction," and regarded the tides as due to an action of this sort between the moon and the water of the earth. But the speculative ideas thus thrown out, which it is possible to regard as anticipations of Newton's discovery of universal gravitation, were not in any way developed logically, and Kepler's mechanical ideas were too imperfect for him to have made real progress in this direction.
151. There are few astronomers about whose merits such different opinions have been held as about Kepler. There is, it is true, a general agreement as to the great importance of his three laws of planetary motion, and as to the substantial value of the Rudolphine Tables and of various minor discoveries. These results, however, fill but a small part of Kepler's voluminous writings, which are encumbered with masses of wild speculation, of mystic and occult fancies, of astrology, weather prophecies, and the like, which are not only worthless from the standpoint of modern astronomy, but which—unlike many erroneous or imperfect speculations—in no way pointed towards the direction in which the science was next to make progress, and must have appeared almost as unsound to sober-minded contemporaries like Galilei as to us. Hence as one reads chapter after chapter without a lucid still less a correct idea, it is impossible to refrain from regrets that the intelligence of Kepler should have been so wasted, and it is difficult not to suspect at times that some of the valuable results which lie imbedded in this great mass of tedious speculation were arrived at by a mere accident. On the other hand, it must not be forgotten that such accidents have a habit of happening only to great men, and that if Kepler loved to give reins to his imagination he was equally impressed with the necessity of scrupulously comparing speculative results with observed facts, and of surrendering without demur the most beloved of his fancies if it was unable to stand this test. If Kepler had burnt three-quarters of what he printed, we should in all probability have formed a higher opinion of his intellectual grasp and sobriety of judgment, but we should have lost to a great extent the impression of extraordinary enthusiasm and industry, and of almost unequalled intellectual honesty, which we now get from a study of his works.
- The astronomer appears to have used both spellings of his name almost indifferently. For example, the title-page of his most important book, the Commentaries on the Motions of Mars (§ 141), has the form Kepler, while the dedication of the same book is signed Keppler.
- The regular solids being taken in the order: cube, tetrahedron, dodecahedron, icosahedron, octohedron, and of such magnitude that a sphere can be circumscribed to each and at the same time inscribed in the preceding solid of the series, then the radii of the six spheres so obtained were shewn by Kepler to be approximately proportional to the distances from the sun of the six planets Saturn, Jupiter, Mars, Earth, Venus, and Mercury.
- Two stars 4' apart only just appear distinct to the naked eye of a person with average keenness of sight.
- Commentaries on the Motions of Mars, Part II., end of chapter xix.
- An ellipse is one of several curves, known as conic sections, which can be formed by taking a section of a cone, and may also be defined as a curve the sum of the distances of any point on which from two fixed points inside it, known as the foci, is always the same. Thus if, in the figure, s and h are the foci, and p, q are any two points on the curve, then the distances s p, h p added together are equal to the distances s q, q h added together, and each sum is equal to the length a a' of the ellipse. The ratio of the distance s h to the length a a' is known as the eccentricity, and is a convenient measure of the extent to which the ellipse differs from a circle.
- The ellipse is more elongated than the actual path of Mars, an accurate drawing of which would be undistinguishable to the eye from a circle. The eccentricity is 1 in the figure, that of Mars being 1.
- Astronomia Nova αἰτιολογητος sen Physica Coelestis, tradita Commentariis de Motibus Stellae Martis. Ex Observationibus G. V. Tychonis Brahe.
- It contains the germs of the method of infinitesimals.
- Harmonices Mundi Libri V.
- There may be some interest in Kepler's own statement of the law: "Res est certissima exactissimaque, quod proportionis quae est inter binorum quorumque planetarum tempora periodica, sit praecise sesquialtera proportionis mediarum distantiarum, id est orbium ipsorum."—Harmony of the World, Book V., chapter iii.
- Epitome, Book IV., Part 2.
- Introduction to the Commentaries on the Motions of Mars.