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.
[Illustration: FIG. 29.--_S_ represents the sun; _EC_, the centre of the
earth's orbit, 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 constituted 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 orbit 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).
[Illustration: 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 circumstance 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.
[Illustration: 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.
[Illustration: 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 substituted. Well might he be called, as he was, "the
legislator," or law interpreter, "of the heavens."
[Illustration: 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 circumstances 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
Wuertemberg. 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.
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 harassing 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 orbit 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 title 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 God 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 God 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 Orbium 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, God 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.
[Illustration: 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 assisted 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 multitude 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 attitude. 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 passed 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 pass, 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 noble
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
compass 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 manuscripts 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,
Illustrating 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 | 42.47 | 6.049 | 1803.7 | 221.44 | 8.149
No. 2. | 2188 | 85.23 | 9.623 | 7264.1 | 891.11 | 8.152
No. 3. | 3575 | 177.72 | 15.350 | 29488. | 3916.8 | 8.153
No. 4. | 3059 | 400.53 | 26.998 |160426. |19679. | 8.152
--------------------------------------------------------------------------
The diameter of Jupiter is 85,823 miles.
_Falling Bodies._