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question naturally suggested itself, what was the nature of these orbits? "Mæst lin, a zealous Copernican, supposed comets to revolve in a circle round the sun; and he explained the inequalities of their motion, by introducing an epicycle on which he made them move; but Tycho who did not admit the earth's motion round the sun, proposed to explain their irregularities by combining their own circular motion round the sun, with the revolution round the earth that he attributed to that body.
It is melancholy, after relating so many brilliant discoveries, to have to record that Tycho rejected the Copernican system of the world. Whether he was influenced by a wrong interpretation of some passages of Scripture, or the desire to attach his name to a new theory of the universe, or was really persuaded that the arrangement proposed by Copernicus was physically untenable, it is not easy to decide. It is certain that he generally speaks of this great astronomer, in terms, not of respect merely, but admiration: and he refutes himself the arguments urged against the diurnal motion of the earth by Ptolemy and others, remarking that the atmosphere would in this case revolve with the earth, nor would those absurd consequences result which had been supposed. Yet led by other reasons of no greater weight, he rejected this diurnal motion: and placed, with Aristotle and Ptolemy, the earth at rest in the centre of the universe. He supposed the sun to revolve round the earth; but his system differed from that of Ptolemy in this, that he made the five minor planets to revolve round the former, which carried them along with it in its annual course round the earth. It is not to be denied that, mathematically speaking, this system satisfies the phenomena observed; but it is so immeasurably inferior to that of Copernicus in simplicity, that it appears very extraordinary, it should have been imagined after the other was known. Holding, as the system of Tycho does, an intermediate place between those of Ptolemy and Copernicus, one would have expected that it should have been the first attempt to simplify the complication of the former. Indeed we may easily pass from the first to the second of these, by making, for the inferior planets, the deferent coincide with the annual orbit of the sun, and the epicycle, with the
Epist. Astron. p. 74.
planet's proper orbit; and in the ease of the superior planets, the converse.
It cannot but be interesting to know the reasons which the Danish philosopher urged in defence of his own theory against the adherents of Copernicus. We shall not notice the arguments drawn from the Scriptures, because it is now generally admitted, as indeed was asserted by the Copernicans of those days, that we are not to look in them for strictness of scientific expression on such subjects, since they naturally use only such language as would be intelligible to those to whom they were addressed*: It can scarcely be supposed that the Pope, to whom Copernicus dedicated his work, or the prelates who exhorted him to publish it, imagined that his doctrines contained anything contrary to the Scriptures. It was in later times that the cry of impiety was set up; and it was pretty evident that this clamour arose from the offended pride and prejudices of the Aristotelians, determined to punish as heresy what they could not refute as false philosophy. Tycho Brahé, indeed, was not an Aristotelian, and it is probable that he was sincere when he quoted certain phrases used by Moses, as hostile to the opinion of the earth's motion; but it is lamentable to see such a man leading the way in an opposition of this kind. He would probably have repented of it, could he have foreseen the consequences to which it led in the case of Galileo.
The great argument used by this eminent man against the earth's diurnal motion was this: if a stone be suffered to fall from the top of a high tower, it would not, as we see it does, fall at the foot of that tower; the earth's velocity of rotation being so great, that during the few seconds it took to fall, the tower itself would have passed through an are of several hundred feet, and the stone be left far behind ere it touched the ground. To this it was answered by his correspondent Rothmann, that every body on the earth's surface partakes of the earth's motion; that consequently the motion of a falling body is compounded of a rectilinear and circular motion; the former tending to the centre of the earth; the latter in the circumference of the circle described by the point from which it falls; and if the velocity with which this point revolves be so great, what then must be
• V. Rothmann in Epist. Astron. p. 130. Epist. Astron. p. 185.
that of the sphere of the fixed stars, if we, taking the other side, suppose the earth to be at rest.
The arguments of Tycho against the annual motion of the earth were much more weighty: the most important of them was drawn from the fact of the fixed stars having no annual parallax. If, as Copernicus teaches, the earth revolves round the sun in an orbit nearly circular, her places, at intervals of nearly six months, will be distant from each other by the whole diameter of the orbit. If then we suppose lines drawn from two places diametrically opposite to the nearest fixed star, these two lines will form an angle (called the annual parallax), which must be appreciable, unless the distance of the star is so great, that, compared with it, the diameter of the earth's orbit is insensible. But this diameter is in itself of such immense length, that, according to Tycho, the supposition just mentioned would be preposterous; and as it is established by observation that the parallax of the orbit is insensible, we must conclude that the earth does not move round the sun. This is the most specious of all the arguments urged against the Copernican system. We have now incontestable proofs of the earth's motion, and we know, astonishing as the fact may appear, that the distance of the fixed stars is infinitely great when compared, not merely with the diameter of the earth itself, but even with that of its orbit. But it is not surprising that in earlier ages men should have been reluctant to admit such a con clusion; however as there is no absurdity in it, they were not justified on this single ground in rejecting a system so simple and beautiful as that of Copernicus. Tycho Brahe has urged another argument, ingenious enough though founded on a mistake in facts. The imperfection of the instruments and methods of observations used in those times, led him to ascribe a sensible, though small, apparent diameter to some of the fixed stars. If then, he argued, the fixed stars be as distant as Copernicus supposes, in order to subtend a visible angie, the diameters, even of stars of the third magnitude, must be greater than that of the annual orbit. This reason, even if true, would not be conclusive; as we are in absolute ignorance of the real magnitudes of the fixed stars; nor can we take upon us to affirm that they may not extend even beyond the limits here mentioned: but the fact is, that, seen in
the best telescopes, they have no appreciable diameter, appearing simply as highly luminous points.
Reformation of the Calendar-Kepler.
WHEN Julius Cæsar, and his adviser
council of Nice, it became of importance to the Church, to fix definitively the place of the equinox in the calendar. A proof that religious and not civil considerations led to the reform may be found in the extent of the changes effected. It may be said that a certain degree of inconvenience would result to the public, from having a moveable instead of a fixed year; and though, in the Julian system, the anticipation of the seasons is so slow that the inconvenience must be nearly inappreciable; yet there could have been no objection to fixing permanently the different seasons, as it might have been effected without difficulty by adopting a different intercalation for the future. But on the other hand, the time at which the year shall be made to begin is entirely arbitrary, and in practice a matter of perfect indifference. In the age of Cæsar, the year began a few days after the winter solstice, and the vernal equinox fell on the 21st of March. In the age of Pope Gregory XIII., this equinox fell on the 10th: here it might have been fixed for the future without any inconvenience; but the Pope and his astronomers took the very unnecessary step of suppressing altogether eleven days in the year 1582, in order to bring the equinox to the 21st. This uncalled for measure had the inconvenience of introducing into Europe two styles, or modes of reckoning dates, as the new calendar was for a long time rejected by the Protestant states of Europe, and to this day has not been received in the empire of Russia. In the north of Germany it was not admitted till the year 1699, nor in England till 1751; 169 years after its publication by Gregory at Rome.
The equinox being once brought to the 21st of March, the object of those who effected the reform was to keep it as nearly as possible to that day by a proper system of intercalation, and to effect this the Julian calendar was modified in the following manner. It was arranged that, for the future, in the year concluding every century, which ought in the Julian system to be bissextile, the intercalary day should be suppressed; but that this should only be done for each three successive centuries, and not in the fourth. Thus the years 1700, 1800, 1900, are not bissextile, but the year 2000 is so again, in 2100, 2200, 2300, the intercalary day is suppressed, but re-established in 2400. La Place has remarked, that to give this intercalation
all the accuracy of which it is susceptible, it would be necessary, at the end of 4000 years, to render the secular year common instead of bissextile: that is to say, in this interval to intercalate only 969 instead of 970 times*.
It is a singular fact, that the Persians have been for several centuries in possession of a calendar constructed much more scientific principles, than Europe, with her superior knowledge, can boast of. It has been stated by La Place, Montucla, and Bailly, that the Persian intercalation consisted in inserting eight days in thirty-three years. This, if true, would at once be a much more accurate and simple method than the Gregorian; but the fact is, that the Persians combine two periods, each of considerable accuracy, the one erring a little in excess, the other in defect. The first period is one of twenty-nine years, in which they intercalate seven days: this is followed by four successive periods of thirty-three years, in each of which they intercalate eight times: forming a whole period of 161 years, which includes thirtynine intercalary days. To show the extreme accuracy of this method, it is only necessary to remark, that it supposes the length of the year to be 365 5h 48 49 1875, the real length being 365 54 48m 49.7, the difference is less than a second; while in the Gregorian calendar it amounts to more than twenty-one seconds. The first year of the Persian æra began with the vernal equinox (A.D. 1070); the astronomers of that country have very wisely avoided subjecting themselves to the unnecessary and embarrassing condition, that the equinox should always coincide with the first day of the year. However in their system it never can be far from it, while in the Gregorian, the real equinox which ought to fall on the 21st of March may sometimes fall on the 19th.
4' 29' 33 161 The Gregorian calendar intercalates 97 days in 400 years the fraction 2 does not appear among the convergents, and consequently the calculation is not as accurate as it might be: it supposes the length of the year equal to 365d. 2425, or too great by 0.0002586, that is, rather more than twenty-one seconds: this excess in 4000 years would amount to little more than a day.
This was first made known in a note from derne, vol. i. p. 81. Sédillot to Delambre, printed in the Astron. Mo
can then be little doubt that the Persian system is the most elegant and scientific of any that has hitherto been used: the principal objection to it is that the intercalations cannot follow a law so simple as those in the Gregorian Calendar; on the other hand it surpasses this latter in accuracy; as it does that adopted during the revolution in France, by being freed from the extreme complication, consequent on making the beginning of the year invariably coincide with the equi
When Tycho Brahé retired in disgust from his native country to Bohemia, a fortunate chance caused him to fall in with a young man of a genius the most opposite to his own, but also perhaps, on that very account, the best qualified to deduce novel and important truths from the observations he had been so long accumulating. Kepler, at a very early age, had distinguished himself by a work on the distances of the planets from the sun, full, it must be confessed, of fanciful and erroneous views, but still, in the opinion of Tycho, bearing the stamp of genius. The objects of this singular treatise, called the Mysterium Cosmographicum, was to point out a supposed relation between the magnitudes of the orbits of the five principal planets, and the five regular solids of geometry. His theory with regard to these was the following:-" Round the orbit of the earth circumscribe a dodecahedron, the circle comprising it will be that of Mars. Round Mars circumscribe a tetrahedron, the circle comprising it will be that of Jupiter. Round Jupiter circumscribe a cube-the circle comprising it will be that of Saturn: now within the earth inscribe an icosahedron, -the inscribed circle will be that of Venus: in Venus inscribe an octahedron -the circle inscribed in it will be that of Mercury." It is scarcely necessary to observe that these proportions are alto
The two fractions and are very near ap
proximate values of the part of a day, by which the year exceeds 365. The former of these is a little
too small, the latter rather too large; if then we
make these two periods alternate, the errors being
making on the whole
See the note to page 48.
died at Ratisbon in 1630.
Myst. Cosmograph. p. 10. Frankfort, 1621.
gether fanciful, and indeed incompatible with the very laws discovered subsequently by Kepler himself, but they are interesting, as these speculations, apparently so chimerical, led to one of the most brilliant discoveries ever made in astronomical science. It does credit to the sagacity of Tycho, himself eminently a practical man, and little disposed to admire such subtleties, that he seems to have perceived at once all the force of the genius thus misdirected; and did not rest till he succeeded in fixing the young author near his own person. Indeed the protection he extended to Kepler is the more remarkable, as the latter had been from his youth an ardent partisan of the Copernican theory, which Tycho pertinaciously, though vainly, endeavoured to supplant. His obstinacy on this point was carried so far, that he is said to have requested of Kepler on his death bed, whatever might be his real sentiments, to adopt in his published works the Tychonian system of the world. It was not to be expected that such an injunction should be complied with; and the person to whom the dying astronomer addressed this extraordinary request, was of all men least likely to sacrifice truth and reason at the shrine of authority.
It had been universally admitted, from Aristotle down to Copernicus and Tycho, that the orbits of the planets were circular, or rather formed by a combination of circles, and a variety of weak and vague reasons were adduced to account for the supposed fact. However, the hypothesis was natural, and its deviations from the truth were not much greater than the errors of the ancient observations: its inadequacy could not be detected, till a mass of more accurate observations had been collected, and then submitted to discussion by a philosopher of quick conception and unfettered judgment. These conditions were eminently satisfied in Kepler, and the observations of Tycho supplied him with the most valuable data. It was then found that the planets, supposed by Copernicus to revolve round the sun in circles, move in ellipses, of which that body occupies one of the foci; and their motion is such, that straight lines being drawn from the focus in question to any two points of the orbit, the area thus intercepted is proportional to the time employed by the planet in passing from one of these points to the other To these 'two most important
theorems a third was subsequently added, that the squares of the times of revolution of two planets are to each other as the cubes of the greater axes of their respective ellipses. These are the three laws which have immortalized the name of Kepler, and effected a revolution in astronomy. Beautiful and important as they are in themselves, it is impossible to appreciate their full value, without a knowledge of the sublime theory founded on them by Newton; nor can we appreciate, as we ought, the genius of their inventor, if we forget the immense strength of the prejudices that he dared to break through. Perhaps his contemporaries alone could properly estimate the courage required to discard the circular motions, which even that great reformer, Copernicus, considered solely admissible in the heavens; and to introduce squares and cubes into the propor. tions of revolution and distance: but they could not foresee the results to which these discoveries have led; and it may safely be affirmed that it is only since the publication of the Principia, that they have met with their deserved tribute of admiration. It is remarkable enough, that, whether from this cause, or from the fanciful speculations with which truth in Kepler's works is so often accompanied and obscured, his distinguished contemporary Galileo does not seem to have perceived the importance of the three famous laws; though his own knowledge qualified him better than any one to appreciate them justly, and they would have furnished him with some strong arguments in favour of the Copernican system.
The way in which Kepler was led to these great discoveries is one of the most interesting objects of study in the whole range of physical science. Supposing the planetary orbits circular, it is generally possible, by a proper combination of epicycles, to represent the various inequalities of their motions; but it is extremely difficult, in this hypothesis, to satisfy, at once, the variations of velocity and those of distance. Thus we have seen that Ptolemy, in representing the evection of the moon, was led to a theory which would give the variations of distance considerably greater than they really are; a fact which he seems to have overlooked, but which could not long escape the notice of astronomers. Mars, whose apparent diameter varies from 4" to 18", and whose extreme distances from the earth are and 2 parts
of the mean radius of the ecliptic, was one of the heavenly bodies, whose motion and distances were most difficult of conciliation. Tycho had already remarked that the annual orbit of Copernicus, or the epicycle of Ptolemy, was not always of the same magnitude with regard to the excentric; but that it produced a sensible alteration in the three superior planets, and that for Mars, the difference amounted to 1° 45'. This, however, is exaggerated, as the difference in question does not exceed 38'; but the statement of Tycho, combined with other observations, led Kepler to the first step in the progress of his discoveries. We have seen that Ptolemy had been induced to bisect the excentricity of the orbits of the planets, by placing the centre of equal motion as far from the centre of equal distances in one direction, as the earth was from it in that opposite. Kepler, who, following Copernicus, considered the earth itself as a planet, was led to extend this system to the terrestrial orbit, and by this he was enabled to represent better than had hitherto been done, what may be called the optical part of these inequalities: that is to say, the part depending on the motion being seen from a point not coincident with the centre of the circular orbit.
The next step was to represent the real inequality of the planet's motion more accurately than had hitherto been done by epicycles and excentrics. Kepler began by supposing the times occupied by the earth in equal parts of the excentric to be to each other in the proportion of the distances of those parts; and hence that the periodic time was to the sum of these distances as any given time to the sum of the distances to the corresponding arc*. This principle is only approximately true, and the calculation of the sum of the distances was involved in much difficulty. While meditating on the best method of surmounting these, Kepler was led to substitute, for the sum of the distances, the areas comprised between two positions of the radius vector; and he thus arrived at the first of his great laws, that the radius vector of a planet describes areas proportional to the times. The reasoning by which he was led to this substi
De Mot. Stellæ Martis. p. 192. Kepler was led to this hypothesis from observing that the law in question might be deduced as a consequence of the Ptolemaic theory when the planet was near its apogee or perigee. V. Op. Cit. p. 165.