His mechanical skill was very great, and in his workmanship he was satisfied with nothing but the best. He recognised the importance of rigidity in the instruments, and, whereas these had generally been made of wood, he designed them in metal. His instruments included armillae like those which had been used in Alexandria, and other armillae designed by himself--s.e.xtants, mural quadrants, large celestial globes and various instruments for special purposes. He lived before the days of telescopes and accurate clocks. He invented the method of sub-dividing the degrees on the arc of an instrument by transversals somewhat in the way that Pedro Nunez had proposed.
He originated the true system of observation and reduction of observations, recognising the fact that the best instrument in the world is not perfect; and with each of his instruments he set to work to find out the errors of graduation and the errors of mounting, the necessary correction being applied to each observation.
When he wanted to point his instrument exactly to a star he was confronted with precisely the same difficulty as is met in gunnery and rifle-shooting. The sights and the object aimed at cannot be in focus together, and a great deal depends on the form of sight. Tycho Brahe invented, and applied to the pointers of his instruments, an aperture-sight of variable area, like the iris diaphragm used now in photography. This enabled him to get the best result with stars of different brightness. The telescope not having been invented, he could not use a telescopic-sight as we now do in gunnery. This not only removes the difficulty of focussing, but makes the minimum visible angle smaller. Helmholtz has defined the minimum angle measurable with the naked eye as being one minute of arc. In view of this it is simply marvellous that, when the positions of Tycho"s standard stars are compared with the best modern catalogues, his probable error in right ascension is only 24", 1, and in declination only 25", 9.
Clocks of a sort had been made, but Tycho Brahe found them so unreliable that he seldom used them, and many of his position-measurements were made by measuring the angular distances from known stars.
Taking into consideration the absence of either a telescope or a clock, and reading his account of the labour he bestowed upon each observation, we must all agree that Kepler, who inherited these observations in MS., was justified, under the conditions then existing, in declaring that there was no hope of anyone ever improving upon them.
In the year 1572, on November 11th, Tycho discovered in Ca.s.siopeia a new star of great brilliance, and continued to observe it until the end of January, 1573. So incredible to him was such an event that he refused to believe his own eyes until he got others to confirm what he saw. He made accurate observations of its distance from the nine princ.i.p.al stars in Ca.s.seiopeia, and proved that it had no measurable parallax. Later he employed the same method with the comets of 1577, 1580, 1582, 1585, 1590, 1593, and 1596, and proved that they too had no measurable parallax and must be very distant.
The startling discovery that stars are not necessarily permanent, that new stars may appear, and possibly that old ones may disappear, had upon him exactly the same effect that a similar occurrence had upon Hipparchus 1,700 years before. He felt it his duty to catalogue all the princ.i.p.al stars, so that there should be no mistake in the future. During the construction of his catalogue of 1,000 stars he prepared and used accurate tables of refraction deduced from his own observations. Thus he eliminated (so far as naked eye observations required) the effect of atmospheric refraction which makes the alt.i.tude of a star seem greater than it really is.
Tycho Brahe was able to correct the lunar theory by his observations.
Copernicus had introduced two epicycles on the lunar orbit in the hope of obtaining a better accordance between theory and observation; and he was not too ambitious, as his desire was to get the tables accurate to ten minutes. Tycho Brahe found that the tables of Copernicus were in error as much as two degrees. He re-discovered the inequality called "variation" by observing the moon in all phases--a thing which had not been attended to. [It is remarkable that in the nineteenth century Sir George Airy established an altazimuth at Greenwich Observatory with this special object, to get observations of the moon in all phases.] He also discovered other lunar equalities, and wanted to add another epicycle to the moon"s...o...b..t, but he feared that these would soon become unmanageable if further observations showed more new inequalities.
But, as it turned out, the most fruitful work of Tycho Brahe was on the motions of the planets, and especially of the planet Mars, for it was by an examination of these results that Kepler was led to the discovery of his immortal laws.
After the death of King Frederick the observatories of Tycho Brahe were not supported. The gigantic power and industry displayed by this determined man were accompanied, as often happens, by an overbearing manner, intolerant of obstacles. This led to friction, and eventually the observatories were dismantled, and Tycho Brahe was received by the Emperor Rudolph II., who placed a house in Prague at his disposal.
Here he worked for a few years, with Kepler as one of his a.s.sistants, and he died in the year 1601.
It is an interesting fact that Tycho Brahe had a firm conviction that mundane events could be predicted by astrology, and that this belief was supported by his own predictions.
It has already been stated that Tycho Brahe maintained that observation must precede theory. He did not accept the Copernican theory that the earth moves, but for a working hypothesis he used a modification of an old Egyptian theory, mathematically identical with that of Copernicus, but not involving a stellar parallax. He says (_De Mundi_, etc.) that
the Ptolemean system was too complicated, and the new one which that great man Copernicus had proposed, following in the footsteps of Aristarchus of Samos, though there was nothing in it contrary to mathematical principles, was in opposition to those of physics, as the heavy and sluggish earth is unfit to move, and the system is even opposed to the authority of Scripture. The absence of annual parallax further involves an incredible distance between the outermost planet and the fixed stars.
We are bound to admit that in the circ.u.mstances of the case, so long as there was no question of dynamical forces connecting the members of the solar system, his reasoning, as we should expect from such a man, is practical and sound. It is not surprising, then, that astronomers generally did not readily accept the views of Copernicus, that Luther (Luther"s _Tischreden_, pp. 22, 60) derided him in his usual pithy manner, that Melancthon (_Initia doctrinae physicae_) said that Scripture, and also science, are against the earth"s motion; and that the men of science whose opinion was asked for by the cardinals (who wished to know whether Galileo was right or wrong) looked upon Copernicus as a weaver of fanciful theories.
Johann Kepler is the name of the man whose place, as is generally agreed, would have been the most difficult to fill among all those who have contributed to the advance of astronomical knowledge. He was born at Wiel, in the Duchy of Wurtemberg, in 1571. He held an appointment at Gratz, in Styria, and went to join Tycho Brahe in Prague, and to a.s.sist in reducing his observations. These came into his possession when Tycho Brahe died, the Emperor Rudolph entrusting to him the preparation of new tables (called the Rudolphine tables) founded on the new and accurate observations. He had the most profound respect for the knowledge, skill, determination, and perseverance of the man who had reaped such a harvest of most accurate data; and though Tycho hardly recognised the transcendent genius of the man who was working as his a.s.sistant, and although there were disagreements between them, Kepler held to his post, sustained by the conviction that, with these observations to test any theory, he would be in a position to settle for ever the problem of the solar system.
[Ill.u.s.tration: PORTRAIT OF JOHANNES KEPLER. By F. Wanderer, from Reitlinger"s "Johannes Kepler" (original in Stra.s.sburg).]
It has seemed to many that Plato"s demand for uniform circular motion (linear or angular) was responsible for a loss to astronomy of good work during fifteen hundred years, for a hundred ill-considered speculative cosmogonies, for dissatisfaction, amounting to disgust, with these _ priori_ guesses, and for the relegation of the science to less intellectual races than Greeks and other Europeans.
n.o.body seemed to dare to depart from this fetish of uniform angular motion and circular orbits until the insight, boldness, and independence of Johann Kepler opened up a new world of thought and of intellectual delight.
While at work on the Rudolphine tables he used the old epicycles and deferents and excentrics, but he could not make theory agree with observation. His instincts told him that these apologists for uniform motion were a fraud; and he proved it to himself by trying every possible variation of the elements and finding them fail. The number of hypotheses which he examined and rejected was almost incredible (for example, that the planets turn round centres at a little distance from the sun, that the epicycles have centres at a little distance from the deferent, and so on). He says that, after using all these devices to make theory agree with Tycho"s observations, he still found errors amounting to eight minutes of a degree. Then he said boldly that it was impossible that so good an observer as Tycho could have made a mistake of eight minutes, and added: "Out of these eight minutes we will construct a new theory that will explain the motions of all the planets." And he did it, with elliptic orbits having the sun in a focus of each.[2]
It is often difficult to define the boundaries between fancies, imagination, hypothesis, and sound theory. This extraordinary genius was a master in all these modes of attacking a problem. His a.n.a.logy between the s.p.a.ces occupied by the five regular solids and the distances of the planets from the sun, which filled him with so much delight, was a display of pure fancy. His demonstration of the three fundamental laws of planetary motion was the most strict and complete theory that had ever been attempted.
It has been often suggested that the revival by Copernicus of the notion of a moving earth was a help to Kepler. No one who reads Kepler"s great book could hold such an opinion for a moment. In fact, the excellence of Copernicus"s book helped to prolong the life of the epicyclical theories in opposition to Kepler"s teaching.
All of the best theories were compared by him with observation. These were the Ptolemaic, the Copernican, and the Tychonic. The two latter placed all of the planetary orbits concentric with one another, the sun being placed a little away from their common centre, and having no apparent relation to them, and being actually outside the planes in which they move. Kepler"s first great discovery was that the planes of all the orbits pa.s.s through the sun; his second was that the line of apses of each planet pa.s.ses through the sun; both were contradictory to the Copernican theory.
He proceeds cautiously with his propositions until he arrives at his great laws, and he concludes his book by comparing observations of Mars, of all dates, with his theory.
His first law states that the planets describe ellipses with the sun at a focus of each ellipse.
His second law (a far more difficult one to prove) states that a line drawn from a planet to the sun sweeps over equal areas in equal times. These two laws were published in his great work, _Astronomia Nova, sen. Physica Coelestis tradita commentariis de Motibus Stelloe; Martis_, Prague, 1609.
It took him nine years more[3] to discover his third law, that the squares of the periodic times are proportional to the cubes of the mean distances from the sun.
These three laws contain implicitly the law of universal gravitation. They are simply an alternative way of expressing that law in dealing with planets, not particles. Only, the power of the greatest human intellect is so utterly feeble that the meaning of the words in Kepler"s three laws could not be understood until expounded by the logic of Newton"s dynamics.
The joy with which Kepler contemplated the final demonstration of these laws, the evolution of which had occupied twenty years, can hardly be imagined by us. He has given some idea of it in a pa.s.sage in his work on _Harmonics_, which is not now quoted, only lest someone might say it was egotistical--a term which is simply grotesque when applied to such a man with such a life"s work accomplished.
The whole book, _Astronomia Nova_, is a pleasure to read; the ma.s.s of observations that are used, and the ingenuity of the propositions, contrast strongly with the loose and imperfectly supported explanations of all his predecessors; and the indulgent reader will excuse the devotion of a few lines to an example of the ingenuity and beauty of his methods.
It may seem a hopeless task to find out the true paths of Mars and the earth (at that time when their shape even was not known) from the observations giving only the relative direction from night to night. Now, Kepler had twenty years of observations of Mars to deal with. This enabled him to use a new method, to find the earth"s...o...b..t. Observe the date at any time when Mars is in opposition. The earth"s position E at that date gives the longitude of Mars M. His period is 687 days. Now choose dates before and after the princ.i.p.al date at intervals of 687 days and its multiples. Mars is in each case in the same position. Now for any date when Mars is at M and the earth at E the date of the year gives the angle ESM. And the observation of Tycho gives the direction of Mars compared with the sun, SEM. So all the angles of the triangle SEM in any of these positions of E are known, and also the ratios of SE, SE, SE, SE to SM and to each other.
For the orbit of Mars observations were chosen at intervals of a year, when the earth was always in the same place.
[Ill.u.s.tration]
But Kepler saw much farther than the geometrical facts. He realised that the orbits are followed owing to a force directed to the sun; and he guessed that this is the same force as the gravity that makes a stone fall. He saw the difficulty of gravitation acting through the void s.p.a.ce. He compared universal gravitation to magnetism, and speaks of the work of Gilbert of Colchester. (Gilbert"s book, _De Mundo Nostro Sublunari, Philosophia Nova_, Amstelodami, 1651, containing similar views, was published forty-eight years after Gilbert"s death, and forty-two years after Kepler"s book and reference. His book _De Magnete_ was published in 1600.)
A few of Kepler"s views on gravitation, extracted from the Introduction to his _Astronomia Nova_, may now be mentioned:--
1. Every body at rest remains at rest if outside the attractive power of other bodies.
2. Gravity is a property of ma.s.ses mutually attracting in such manner that the earth attracts a stone much more than a stone attracts the earth.
3. Bodies are attracted to the earth"s centre, not because it is the centre of the universe, but because it is the centre of the attracting particles of the earth.
4. If the earth be not round (but spheroidal?), then bodies at different lat.i.tudes will not be attracted to its centre, but to different points in the neighbourhood of that centre.
5. If the earth and moon were not retained in their orbits by vital force (_aut alia aligua aequipollenti_), the earth and moon would come together.
6. If the earth were to cease to attract its waters, the oceans would all rise and flow to the moon.
7. He attributes the tides to lunar attraction. Kepler had been appointed Imperial Astronomer with a handsome salary (on paper), a fraction of which was doled out to him very irregularly. He was led to miserable makeshifts to earn enough to keep his family from starvation; and proceeded to Ratisbon in 1630 to represent his claims to the Diet. He arrived worn out and debilitated; he failed in his appeal, and died from fever, contracted under, and fed upon, disappointment and exhaustion. Those were not the days when men could adopt as a profession the "research of endowment."
Before taking leave of Kepler, who was by no means a man of one idea, it ought to be here recorded that he was the first to suggest that a telescope made with both lenses convex (not a Galilean telescope) can have cross wires in the focus, for use as a pointer to fix accurately the positions of stars. An Englishman, Gascoigne, was the first to use this in practice.
From the all too brief epitome here given of Kepler"s greatest book, it must be obvious that he had at that time some inkling of the meaning of his laws--universal gravitation. From that moment the idea of universal gravitation was in the air, and hints and guesses were thrown out by many; and in time the law of gravitation would doubtless have been discovered, though probably not by the work of one man, even if Newton had not lived. But, if Kepler had not lived, who else could have discovered his laws?
FOOTNOTES:
[1] When the writer visited M. D"Arrest, the astronomer, at Copenhagen, in 1872, he was presented by D"Arrest with one of several bricks collected from the ruins of Uraniborg. This was one of his most cherished possessions until, on returning home after a prolonged absence on astronomical work, he found that his treasure had been tidied away from his study.
[2] An ellipse is one of the plane, sections of a cone. It is an oval curve, which may be drawn by fixing two pins in a sheet of paper at S and H, fastening a string, SPH, to the two pins, and stretching it with a pencil point at P, and moving the pencil point, while the string is kept taut, to trace the oval ellipse, APB. S and H are the _foci_. Kepler found the sun to be in one focus, say S. AB is the _major axis_. DE is the _minor axis_. C is the _centre_. The direction of AB is the _line of apses_. The ratio of CS to CA is the _excentricity_. The position of the planet at A is the _perihelion_ (nearest to the sun). The position of the planet at B is the _aphelion_ (farthest from the sun). The angle ASP is the _anomaly_ when the planet is at P. CA or a line drawn from S to D is the _mean distance_ of the planet from the sun.
[Ill.u.s.tration]
[3] The ruled logarithmic paper we now use was not then to be had by going into a stationer"s shop. Else he would have accomplished this in five minutes.