Pioneers of Science Part 5

The imagined discovery is purely fict.i.tious and accidental. First of all, eight planets are now known; and secondly, their real distances agree only very approximately with Kepler's hypothesis.

[Ill.u.s.tration: FIG. 28.--Frameworks with inscribed and circ.u.mscribed spheres, representing the five regular solids distributed as Kepler supposed them to be among the planetary orbits. (See "Summary" at beginning of this lecture, p. 57.)]

Nevertheless, the idea gave him great delight. He says:--"The intense pleasure I have received from this discovery can never be told in words.

I regretted no more the time wasted; I tired of no labour; I shunned no toil of reckoning, days and nights spent in calculation, until I could see whether my hypothesis would agree with the orbits of Copernicus, or whether my joy was to vanish into air."

He then went on to speculate as to the cause of the planets' motion.

The old idea was that they were carried round by angels or celestial intelligences. Kepler tried to establish some propelling force emanating from the sun, like the spokes of a windmill.

This first book of his brought him into notice, and served as an introduction to Tycho and to Galileo.

Tycho Brahe was at this time at Prague under the patronage of the Emperor Rudolph; and as he was known to have by far the best planetary observations of any man living, Kepler wrote to him to know if he might come and examine them so as to perfect his theory.

Tycho immediately replied, "Come, not as a stranger, but as a very welcome friend; come and share in my observations with such instruments as I have with me, and as a dearly beloved a.s.sociate." After this visit, Tycho wrote again, offering him the post of mathematical a.s.sistant, which after hesitation was accepted. Part of the hesitation Kepler expresses by saying that "for observations his sight was dull, and for mechanical operations his hand was awkward. He suffered much from weak eyes, and dare not expose himself to night air." In all this he was, of course, the antipodes of Tycho, but in mathematical skill he was greatly his superior.

On his way to Prague he was seized with one of his periodical illnesses, and all his means were exhausted by the time he could set forward again, so that he had to apply for help to Tycho.

It is clear, indeed, that for some time now he subsisted entirely on the bounty of Tycho, and he expresses himself most deeply grateful for all the kindness he received from that n.o.ble and distinguished man, the head of the scientific world at that date.

To ill.u.s.trate Tycho's kindness and generosity, I must read you a letter written to him by Kepler. It seems that Kepler, on one of his absences from Prague, driven half mad with poverty and trouble, fell foul of Tycho, whom he thought to be behaving badly in money matters to him and his family, and wrote him a violent letter full of reproaches and insults. Tycho's secretary replied quietly enough, pointing out the groundlessness and ingrat.i.tude of the accusation.

Kepler repents instantly, and replies:--

"MOST n.o.bLE TYCHO," (these are the words of his letter), "how shall I enumerate or rightly estimate your benefits conferred on me? For two months you have liberally and gratuitously maintained me, and my whole family; you have provided for all my wishes; you have done me every possible kindness; you have communicated to me everything you hold most dear; no one, by word or deed, has intentionally injured me in anything; in short, not to your children, your wife, or yourself have you shown more indulgence than to me. This being so, as I am anxious to put on record, I cannot reflect without consternation that I should have been so given up by G.o.d to my own intemperance as to shut my eyes on all these benefits; that, instead of modest and respectful grat.i.tude, I should indulge for three weeks in continual moroseness towards all your family, in headlong pa.s.sion and the utmost insolence towards yourself, who possess so many claims on my veneration, from your n.o.ble family, your extraordinary learning, and distinguished reputation. Whatever I have said or written against the person, the fame, the honour, and the learning of your excellency; or whatever, in any other way, I have injuriously spoken or written (if they admit no other more favourable interpretation), as, to my grief, I have spoken and written many things, and more than I can remember; all and everything I recant, and freely and honestly declare and profess to be groundless, false, and incapable of proof."

Tycho accepted the apology thus heartily rendered, and the temporary breach was permanently healed.

In 1601, Kepler was appointed "Imperial mathematician," to a.s.sist Tycho in his calculations.

The Emperor Rudolph did a good piece of work in thus maintaining these two eminent men, but it is quite clear that it was as astrologers that he valued them; and all he cared for in the planetary motions was limited to their supposed effect on his own and his kingdom's destiny.

He seems to have been politically a weak and superst.i.tious prince, who was letting his kingdom get into hopeless confusion, and entangling himself in all manner of political complications. While Bohemia suffered, however, the world has benefited at his hands; and the tables upon which Tycho was now engaged are well called the Rudolphine tables.

These tables of planetary motion Tycho had always regarded as the main work of his life; but he died before they were finished, and on his death-bed he intrusted the completion of them to Kepler, who loyally undertook their charge.

The Imperial funds were by this time, however, so taxed by wars and other difficulties that the tables could only be proceeded with very slowly, a staff of calculators being out of the question. In fact, Kepler could not get even his own salary paid: he got orders, and promises, and drafts on estates for it; but when the time came for them to be honoured they were worthless, and he had no power to enforce his claims.

So everything but brooding had to be abandoned as too expensive, and he proceeded to study optics. He gave a very accurate explanation of the action of the human eye, and made many hypotheses, some of them shrewd and close to the mark, concerning the law of refraction of light in dense media: but though several minor points of interest turned up, nothing of the first magnitude came out of this long research.

The true law of refraction was discovered some years after by a Dutch professor, Willebrod Snell.

We must now devote a little time to the main work of Kepler's life. All the time he had been at Prague he had been making a severe study of the motion of the planet Mars, a.n.a.lyzing minutely Tycho's books of observations, in order to find out, if possible, the true theory of his motion. Aristotle had taught that circular motion was the only perfect and natural motion, and that the heavenly bodies therefore necessarily moved in circles.

So firmly had this idea become rooted in men's minds, that no one ever seems to have contemplated the possibility of its being false or meaningless.

When Hipparchus and others found that, as a matter of fact, the planets did _not_ revolve in simple circles, they did not try other curves, as we should at once do now, but they tried combinations of circles, as we saw in Lecture I. The small circle carried by a bigger one was called an Epicycle. The carrying circle was called the Deferent. If for any reason the earth had to be placed out of the centre, the main planetary orbit was called an Excentric, and so on.

But although the planetary paths might be roughly represented by a combination of circles, their speeds could not, on the hypothesis of uniform motion in each circle round the earth as a fixed body. Hence was introduced the idea of an Equant, _i.e._ an arbitrary point, not the earth, about which the speed might be uniform. Copernicus, by making the sun the centre, had been able to simplify a good deal of this, and to abolish the equant.

But now that Kepler had the accurate observations of Tycho to refer to, he found immense difficulty in obtaining the true positions of the planets for long together on any such theory.

He specially attacked the motion of the planet Mars, because that was sufficiently rapid in its changes for a considerable collection of data to have acc.u.mulated with respect to it. He tried all manner of circular orbits for the earth and for Mars, placing them in all sorts of aspects with respect to the sun. The problem to be solved was to choose such an orbit and such a law of speed, for both the earth and Mars, that a line joining them, produced out to the stars, should always mark correctly the apparent position of Mars as seen from the earth. He had to arrange the size of the orbits that suited best, then the positions of their centres, both being supposed excentric with respect to the sun; but he could not get any such arrangement to work with uniform motion about the sun. So he reintroduced the equant, and thus had another variable at his disposal--in fact, two, for he had an equant for the earth and another for Mars, getting a pattern of the kind suggested in Fig. 29.

The equants might divide the line in any arbitrary ratio. All sorts of combinations had to be tried, the relative positions of the earth and Mars to be worked out for each, and compared with Tycho's recorded observations. It was easy to get them to agree for a short time, but sooner or later a discrepancy showed itself.

[Ill.u.s.tration: FIG. 29.--_S_ represents the sun; _EC_, the centre of the earth's...o...b..t, to be placed as best suited; _MC_, the same for Mars; _EE_, the earth's equant, or point about which the earth uniformly revolved (_i.e._ the point determining the law of speed about the sun), likewise to be placed anywhere, but supposed to be in the line joining _S_ to _EC_; _ME_, the same thing for Mars; with _?ME_ for an alternative hypothesis that perhaps Mars' equant was on line joining _EC_ with _MC_.]

I need not say that all these attempts and gropings, thus briefly summarized, entailed enormous labour, and required not only great pertinacity, but a most singularly const.i.tuted mind, that could thus continue groping in the dark without a possible ray of theory to illuminate its search. Grope he did, however, with unexampled diligence.

At length he hit upon a point that seemed nearly right. He thought he had found the truth; but no, before long the position of the planet, as calculated, and as recorded by Tycho, differed by eight minutes of arc, or about one-eighth of a degree. Could the observation be wrong by this small amount? No, he had known Tycho, and knew that he was never wrong eight minutes in an observation.

So he set out the whole weary way again, and said that with those eight minutes he would yet find out the law of the universe. He proceeded to see if by making the planet librate, or the plane of its...o...b..t tilt up and down, anything could be done. He was rewarded by finding that at any rate the plane of the orbit did not tilt up and down: it was fixed, and this was a simplification on Copernicus's theory. It is not an absolute fixture, but the changes are very small (see Laplace, page 266).

[Ill.u.s.tration: FIG. 30.--Excentric circle supposed to be divided into equal areas. The sun, _S_, being placed at a selected point, it was possible to represent the varying speed of a planet by saying that it moved from _A_ to _B_, from _B_ to _C_, and so on, in equal times.]

At last he thought of giving up the idea of _uniform_ circular motion, and of trying _varying_ circular motion, say inversely as its distance from the sun. To simplify calculation, he divided the orbit into triangles, and tried if making the triangles equal would do. A great piece of luck, they did beautifully: the rate of description of areas (not arcs) is uniform. Over this discovery he greatly rejoices. He feels as though he had been carrying on a war against the planet and had triumphed; but his gratulation was premature. Before long fresh little errors appeared, and grew in importance. Thus he announces it himself:--

"While thus triumphing over Mars, and preparing for him, as for one already vanquished, tabular prisons and equated excentric fetters, it is buzzed here and there that the victory is vain, and that the war is raging anew as violently as before. For the enemy left at home a despised captive has burst all the chains of the equations, and broken forth from the prisons of the tables."

Still, a part of the truth had been gained, and was not to be abandoned any more. The law of speed was fixed: that which is now known as his second law. But what about the shape of the orbit--Was it after all possible that Aristotle, and every philosopher since Aristotle, had been wrong? that circular motion was not the perfect and natural motion, but that planets might move in some other closed curve?

Suppose he tried an oval. Well, there are a great variety of ovals, and several were tried: with the result that they could be made to answer better than a circle, but still were not right.

Now, however, the geometrical and mathematical difficulties of calculation, which before had been tedious and oppressive, threatened to become overwhelming; and it is with a rising sense of despondency that Kepler sees his six years' unremitting labour leading deeper and deeper into complication.

One most disheartening circ.u.mstance appeared, viz. that when he made the circuit oval his law of equable description of areas broke down. That seemed to require the circular orbit, and yet no circular orbit was quite accurate.

While thinking and pondering for weeks and months over this new dilemma and complication of difficulties, till his brain reeled, an accidental ray of light broke upon him in a way not now intelligible, or barely intelligible. Half the extreme breadth intercepted between the circle and oval was 429/100,000 of the radius, and he remembered that the "optical inequality" of Mars was also about 429/100,000. This coincidence, in his own words, woke him out of sleep; and for some reason or other impelled him instantly to try making the planet oscillate in the diameter of its epicycle instead of revolve round it--a singular idea, but Copernicus had had a similar one to explain the motions of Mercury.

[Ill.u.s.tration: FIG. 31.--Mode of drawing an ellipse. The two pins _F_ are the foci.]

Away he started through his calculations again. A long course of work night and day was rewarded by finding that he was now able to hit off the motions better than before; but what a singularly complicated motion it was. Could it be expressed no more simply? Yes, the curve so described by the planet is a comparatively simple one: it is a special kind of oval--the ellipse. Strange that he had not thought of it before.

It was a famous curve, for the Greek geometers had studied it as one of the sections of a cone, but it was not so well known in Kepler's time.

The fact that the planets move in it has raised it to the first importance, and it is familiar enough to us now. But did it satisfy the law of speed? Could the rate of description of areas be uniform with it? Well, he tried the ellipse, and to his inexpressible delight he found that it did satisfy the condition of equable description of areas, if the sun was in one focus. So, moving the planet in a selected ellipse, with the sun in one focus, at a speed given by the equable area description, its position agreed with Tycho's observations within the limits of the error of experiment. Mars was finally conquered, and remains in his prison-house to this day. The orbit was found.

[Ill.u.s.tration: FIG. 32.]

In a paroxysm of delight Kepler celebrates his victory by a triumphant figure, sketched actually on his geometrical diagram--the diagram which proves that the law of equable description of areas can hold good with an ellipse. The above is a tracing of it.

Such is a crude and bald sketch of the steps by which Kepler rose to his great generalizations--the two laws which have immortalized his name.

All the complications of epicycle, equant, deferent, excentric, and the like, were swept at once away, and an orbit of striking and beautiful properties subst.i.tuted. Well might he be called, as he was, "the legislator," or law interpreter, "of the heavens."

[Ill.u.s.tration: FIG. 33.--If _S_ is the sun, a planet or comet moves from _P_ to _P_1_, from _P_2_ to _P_3_, and from _P_4_ to _P_5_ in the same time; if the shaded areas are equal.]

He concludes his book on the motions of Mars with a half comic appeal to the Emperor to provide him with the sinews of war for an attack on Mars's relations--father Jupiter, brother Mercury, and the rest--but the death of his unhappy patron in 1612 put an end to all these schemes, and reduced Kepler to the utmost misery. While at Prague his salary was in continual arrear, and it was with difficulty that he could provide sustenance for his family. He had been there eleven years, but they had been hard years of poverty, and he could leave without regret were it not that he should have to leave Tycho's instruments and observations behind him. While he was hesitating what best to do, and reduced to the verge of despair, his wife, who had long been suffering from low spirits and despondency, and his three children, were taken ill; one of the sons died of small-pox, and the wife eleven days after of low fever and epilepsy. No money could be got at Prague, so after a short time he accepted a professorship at Linz, and withdrew with his two quite young remaining children.

He provided for himself now partly by publishing a prophesying almanack, a sort of Zadkiel arrangement--a thing which he despised, but the support of which he could not afford to do without. He is continually attacking and throwing sarcasm at astrology, but it was the only thing for which people would pay him, and on it after a fashion he lived. We do not find that his circ.u.mstances were ever prosperous, and though 8,000 crowns were due to him from Bohemia he could not manage to get them paid.

About this time occurred a singular interruption to his work. His old mother, of whose fierce temper something has already been indicated, had been engaged in a law-suit for some years near their old home in Wurtemberg. A change of judge having in process of time occurred, the defendant saw his way to turn the tables on the old lady by accusing her of sorcery. She was sent to prison, and condemned to the torture, with the usual intelligent idea of extracting a "voluntary" confession.