Kepler had to hurry from Linz to interpose. He succeeded in saving her from the torture, but she remained in prison for a year or so. Her spirit, however, was unbroken, for no sooner was she released than she commenced a fresh action against her accuser. But fresh trouble was averted by the death of the poor old dame at the age of nearly eighty.

This narration renders the unflagging energy shown by her son in his mathematical wrestlings less surprising.

Interspersed with these domestic troubles, and with hara.s.sing and unsuccessful attempts to get his rights, he still brooded over his old problem of some possible connection between the distances of the planets from the sun and their times of revolution, _i.e._ the length of their years.

It might well have been that there was no connection, that it was purely imaginary, like his old idea of the law of the successive distances of the planets, and like so many others of the guesses and fancies which he entertained and spent his energies in probing. But fortunately this time there was a connection, and he lived to have the joy of discovering it.

The connection is this, that if one compares the distance of the different planets from the sun with the length of time they take to go round him, the cube of the respective distances is proportional to the square of the corresponding times. In other words, the ratio of r^3 to T^2 for every planet is the same. Or, again, the length of a planet's year depends on the 3/2th power of its distance from the sun.

Or, once more, the speed of each planet in its...o...b..t is as the inverse square-root of its distance from the sun. The product of the distance into the square of the speed is the same for each planet.

This (however stated) is called Kepler's third law. It welds the planets together, and shows them to be one system. His rapture on detecting the law was unbounded, and he breaks out into an exulting rhapsody:--

"What I prophesied two-and-twenty years ago, as soon as I discovered the five solids among the heavenly orbits--what I firmly believed long before I had seen Ptolemy's _Harmonies_--what I had promised my friends in the t.i.tle of this book, which I named before I was sure of my discovery--what sixteen years ago, I urged as a thing to be sought--that for which I joined Tycho Brahe, for which I settled in Prague, for which I have devoted the best part of my life to astronomical contemplations, at length I have brought to light, and recognized its truth beyond my most sanguine expectations. It is not eighteen months since I got the first glimpse of light, three months since the dawn, very few days since the unveiled sun, most admirable to gaze upon, burst upon me. Nothing holds me; I will indulge my sacred fury; I will triumph over mankind by the honest confession that I have stolen the golden vases of the Egyptians to build up a tabernacle for my G.o.d far away from the confines of Egypt. If you forgive me, I rejoice; if you are angry, I can bear it; the die is cast, the book is written, to be read either now or by posterity, I care not which; it may well wait a century for a reader, as G.o.d has waited six thousand years for an observer."

Soon after this great work his third book appeared: it was an epitome of the Copernican theory, a clear and fairly popular exposition of it, which had the honour of being at once suppressed and placed on the list of books prohibited by the Church, side by side with the work of Copernicus himself, _De Revolutionibus...o...b..um Coelestium_.

This honour, however, gave Kepler no satisfaction--it rather occasioned him dismay, especially as it deprived him of all pecuniary benefit, and made it almost impossible for him to get a publisher to undertake another book.

Still he worked on at the Rudolphine tables of Tycho, and ultimately, with some small help from Vienna, completed them; but he could not get the means to print them. He applied to the Court till he was sick of applying: they lay idle four years. At last he determined to pay for the type himself. What he paid it with, G.o.d knows, but he did pay it, and he did bring out the tables, and so was faithful to the behest of his friend.

This great publication marks an era in astronomy. They were the first really accurate tables which navigators ever possessed; they were the precursors of our present _Nautical Almanack_.

After this, the Grand Duke of Tuscany sent Kepler a golden chain, which is interesting inasmuch as it must really have come from Galileo, who was in high favour at the Italian Court at this time.

Once more Kepler made a determined attempt to get his arrears of salary paid, and rescue himself and family from their bitter poverty. He travelled to Prague on purpose, attended the imperial meeting, and pleaded his own cause, but it was all fruitless; and exhausted by the journey, weakened by over-study, and disheartened by the failure, he caught a fever, and died in his fifty-ninth year. His body was buried at Ratisbon, and a century ago a proposal was made to erect a marble monument to his memory, but nothing was done. It matters little one way or the other whether Germany, having almost refused him bread during his life, should, a century and a half after his death, offer him a stone.

[Ill.u.s.tration: FIG. 34.--Portrait of Kepler, older.]

The contiguity of the lives of Kepler and Tycho furnishes a moral too obvious to need pointing out. What Kepler might have achieved had he been relieved of those ghastly struggles for subsistence one cannot tell, but this much is clear, that had Tycho been subjected to the same misfortune, instead of being born rich and being a.s.sisted by generous and enlightened patrons, he could have accomplished very little. His instruments, his observatory--the tools by which he did his work--would have been impossible for him. Frederick and Sophia of Denmark, and Rudolph of Bohemia, are therefore to be remembered as co-workers with him.

Kepler, with his ill-health and inferior physical energy, was unable to command the like advantages. Much, nevertheless, he did; more one cannot but feel he might have done had he been properly helped. Besides, the world would have been free from the reproach of accepting the fruits of his bright genius while condemning the worker to a life of misery, relieved only by the beauty of his own thoughts and the ecstasy awakened in him by the harmony and precision of Nature.

Concerning the method of Kepler, the mode by which he made his discoveries, we must remember that he gives us an account of all the steps, unsuccessful as well as successful, by which he travelled. He maps out his route like a traveller. In fact he compares himself to Columbus or Magellan, voyaging into unknown lands, and recording his wandering route. This being remembered, it will be found that his methods do not differ so utterly from those used by other philosophers in like case. His imagination was perhaps more luxuriant and was allowed freer play than most men's, but it was nevertheless always controlled by rigid examination and comparison of hypotheses with fact.

Brewster says of him:--"Ardent, restless, burning to distinguish himself by discovery, he attempted everything; and once having obtained a glimpse of a clue, no labour was too hard in following or verifying it. A few of his attempts succeeded--a mult.i.tude failed. Those which failed seem to us now fanciful, those which succeeded appear to us sublime. But his methods were the same. When in search of what really existed he sometimes found it; when in pursuit of a chimaera he could not but fail; but in either case he displayed the same great qualities, and that obstinate perseverance which must conquer all difficulties except those really insurmountable."

To realize what he did for astronomy, it is necessary for us now to consider some science still in its infancy. Astronomy is so clear and so thoroughly explored now, that it is difficult to put oneself into a contemporary att.i.tude. But take some other science still barely developed: meteorology, for instance. The science of the weather, the succession of winds and rain, sunshine and frost, clouds and fog, is now very much in the condition of astronomy before Kepler.

We have pa.s.sed through the stage of ascribing atmospheric disturbances--thunderstorms, cyclones, earthquakes, and the like--to supernatural agency; we have had our Copernican era: not perhaps brought about by a single individual, but still achieved. Something of the laws of cyclone and anticyclone are known, and rude weather predictions across the Atlantic are roughly possible. Barometers and thermometers and anemometers, and all their tribe, represent the astronomical instruments in the island of Huen; and our numerous meteorological observatories, with their continual record of events, represent the work of Tycho Brahe.

Observation is heaped on observation; tables are compiled; volumes are filled with data; the hours of sunshine are recorded, the fall of rain, the moisture in the air, the kind of clouds, the temperature--millions of facts; but where is the Kepler to study and brood over them? Where is the man to spend his life in evolving the beginnings of law and order from the midst of all this chaos?

Perhaps as a man he may not come, but his era will come. Through this stage the science must pa.s.s, ere it is ready for the commanding intellect of a Newton.

But what a work it will be for the man, whoever he be that undertakes it--a fearful monotonous grind of calculation, hypothesis, hypothesis, calculation, a desperate and groping endeavour to reconcile theories with facts.

A life of such labour, crowned by three brilliant discoveries, the world owes (and too late recognizes its obligation) to the harshly treated German genius, Kepler.

SUMMARY OF FACTS FOR LECTURES IV AND V

In 1564, Michael Angelo died and Galileo was born; in 1642, Galileo died and Newton was born. Milton lived from 1608 to 1674.

For teaching the plurality of worlds, with other heterodox doctrines, and refusing to recant, Bruno, after six years' imprisonment in Rome, was burnt at the stake on the 16th of February, 1600 A.D. A "natural"

death in the dungeons of the Inquisition saved Antonio de Dominis, the explainer of the rainbow, from the same fate, but his body and books were publicly burned at Rome in 1624.

The persecution of Galileo began in 1615, became intense in 1632, and so lasted till his death and after.

Galileo Galilei, eldest son of Vincenzo de Bonajuti de Galilei, a n.o.ble Florentine, was born at Pisa, 18th of February, 1564. At the age of 17 was sent to the University of Pisa to study medicine. Observed the swing of a pendulum and applied it to count pulse-beats. Read Euclid and Archimedes, and could be kept at medicine no more. At 26 was appointed Lecturer in Mathematics at Pisa. Read Bruno and became smitten with the Copernican theory. Controverted the Aristotelians concerning falling bodies, at Pisa. Hence became unpopular and accepted a chair at Padua, 1592. Invented a thermometer. Wrote on astronomy, adopting the Ptolemaic system provisionally, and so opened up a correspondence with Kepler, with whom he formed a friendship. Lectured on the new star of 1604, and publicly renounced the old systems of astronomy. Invented a calculating compa.s.s or "Gunter's scale." In 1609 invented a telescope, after hearing of a Dutch optician's discovery. Invented the microscope soon after.

Rapidly completed a better telescope and began a survey of the heavens.

On the 8th of January, 1610, discovered Jupiter's satellites. Observed the mountains in the moon, and roughly measured their height. Explained the visibility of the new moon by _earth-shine_. Was invited to the Grand Ducal Court of Tuscany by Cosmo de Medici, and appointed philosopher to that personage. Discovered innumerable new stars, and the nebulae. Observed a triple appearance of Saturn. Discovered the phases of Venus predicted by Copernicus, and spots on the sun. Wrote on floating bodies. Tried to get his satellites utilized for determining longitude at sea.

Went to Rome to defend the Copernican system, then under official discussion, and as a result was formally forbidden ever to teach it. On the accession of Pope Urban VIII. in 1623, Galileo again visited Rome to pay his respects, and was well received. In 1632 appeared his "Dialogues" on the Ptolemaic and Copernican systems. Summoned to Rome, practically imprisoned, and "rigorously questioned." Was made to recant 22nd of June, 1633. Forbidden evermore to publish anything, or to teach, or receive friends. Retired to Arcetri in broken down health. Death of his favourite daughter, Sister Maria Celeste. Wrote and meditated on the laws of motion. Discovered the moon's libration. In 1637 he became blind. The rigour was then slightly relaxed and many visited him: among them John Milton. Died 8th of January, 1642, aged 78. As a prisoner of the Inquisition his right to make a will or to be buried in consecrated ground was disputed. Many of his ma.n.u.scripts were destroyed.

Galileo, besides being a singularly clear-headed thinker and experimental genius, was also something of a musician, a poet, and an artist. He was full of humour as well as of solid common-sense, and his literary style is brilliant. Of his scientific achievements those now reckoned most weighty, are the discovery of the Laws of Motion, and the laying of the foundations of Mechanics.

_Particulars of Jupiter's Satellites, Ill.u.s.trating their obedience to Kepler's third law._

-------------------------------------------------------------------------- | | | Distance| | | T^2 | | Time of | from | | | ---- Satellite.|Diameter revolution | Jupiter, | T^2 | d^3 | d^3 | miles.| in hours. |in Jovian | | | which is | miles | (T) | radii. | | |practically | | | (d) | | | constant.

----------|-------|------------|----------|---------|---------|----------- No. 1. | 2437 | 4247 | 6049 | 18037 | 22144 | 8149 No. 2. | 2188 | 8523 | 9623 | 72641 | 89111 | 8152 No. 3. | 3575 | 17772 | 15350 | 29488 | 39168 | 8153 No. 4. | 3059 | 40053 | 26998 |160426 |19679 | 8152 --------------------------------------------------------------------------

The diameter of Jupiter is 85,823 miles.

_Falling Bodies._

Since all bodies fall at the same rate, except for the disturbing effect of the resistance of the air, a statement of their rates of fall is of interest. In one second a freely falling body near the earth is found to drop 16 feet. In two seconds it drops 64 feet altogether, viz. 16 feet in the first, and 48 feet in the next second; because at the beginning of every second after the first it has the acc.u.mulated velocity of preceding seconds. The height fallen by a dropped body is not proportional to the time simply, but to what is rather absurdly called the square of the time, _i.e._ the time multiplied by itself.

For instance, in 3 seconds it drops 9 16 = 144 feet; in 4 seconds 16 16, or 256 feet, and so on. The distances travelled in 1, 2, 3, 4, &c., seconds by a body dropped from rest and not appreciably resisted by the air, are 1, 4, 9, 16, 25, &c., respectively, each multiplied by the constant 16 feet.

Another way of stating the law is to say that the heights travelled in successive seconds proceed in the proportion 1, 3, 5, 7, 9, &c.; again multiplied by 16 feet in each case.

[Ill.u.s.tration: FIG. 35.--Curve described by a projectile, showing how it drops from the line of fire, _O D_, in successive seconds, the same distances _AP_, _BQ_, _CR_, &c., as are stated above for a dropped body.]

All this was experimentally established by Galileo.

A body takes half a second to drop 4 feet; and a quarter of a second to drop 1 foot. The easiest way of estimating a quarter of a second with some accuracy is to drop a bullet one foot.

A bullet thrown or shot in any direction falls just as much as if merely dropped; but instead of falling from the starting-point it drops vertically from the line of fire. (See fig. 35).

The rate of fall depends on the intensity of gravity; if it could be doubled, a body would fall twice as far in the same time; but to make it fall a given distance in half the time the intensity of gravity would have to be quadrupled. At a place where the intensity of gravity is 1/3600 of what it is here, a body would fall as far in a minute as it now falls in a second. Such a place occurs at about the distance of the moon (_cf._ page 177).

The fact that the height fallen through is proportional to the square of the time proves that the attraction of the earth or the intensity of gravity is sensibly constant throughout ordinary small ranges. Over great distances of fall, gravity cannot be considered constant; so for things falling through great s.p.a.ces the Galilean law of the square of the time does not hold.