insolence. Tycho was fiery, no doubt, but I think we should wrong him
if we considered him insolent. Most of the nobles of his day were
haughty persons, accustomed to deal with serfs, and very likely to sneer
at and trample on any meek man of science whom they could easily
despise. So Tycho was not meek; he stood up for the honour of his
science, and paid them back in their own coin, with perhaps a little
interest. That this behaviour was not worldly-wise is true enough, but I
know of no commandment enjoining us to be worldly-wise.
If we knew more about his so-called imbecile _protege_ we should
probably find some reason for the interest which Tycho took in him.
Whether he was what is now called a "clairvoyant" or not, Tycho
evidently regarded his utterances as oracular, and of course when one is
receiving what may be a revelation from heaven it is natural to suppress
ordinary conversation.
Among the noble visitors whom he received and entertained, it is
interesting to notice James I. of England, who spent eight days at
Uraniburg on the occasion of his marriage with Anne of Denmark in 1590,
and seems to have been deeply impressed by his visit.
Among other gifts, James presented Tycho with a dog (depicted in Fig.
24), and this same animal was subsequently the cause of trouble. For it
seems that one day the Chancellor of Denmark, Walchendorf, brutally
kicked the poor beast; and Tycho, who was very fond of animals, gave him
a piece of his mind in no measured language. Walchendorf went home
determined to ruin him. King Frederick, however, remained his true
friend, doubtless partly influenced thereto by his Queen Sophia, an
enlightened woman who paid many visits to Uraniburg, and knew Tycho
well. But unfortunately Frederick died; and his son, a mere boy, came to
the throne.
Now was the time for the people whom Tycho had offended, for those who
were jealous of his great fame and importance, as well as for those who
cast longing eyes on his estate and endowments. The boy-king, too,
unfortunately paid a visit to Tycho, and, venturing upon a decided
opinion on some recondite subject, received a quiet setting down which
he ill relished.
Letters written by Tycho about this time are full of foreboding. He
greatly dreads having to leave Uraniburg, with which his whole life has
for twenty years been bound up. He tries to comfort himself with the
thought that, wherever he is sent, he will have the same heavens and the
same stars over his head.
Gradually his Norwegian estate and his pension were taken away, and in
five years poverty compelled him to abandon his magnificent temple, and
to take a small house in Copenhagen.
Not content with this, Walchendorf got a Royal Commission appointed to
inquire into the value of his astronomical labours. This sapient body
reported that his work was not only useless, but noxious; and soon after
he was attacked by the populace in the public street.
Nothing was left for him now but to leave the country, and he went into
Germany, leaving his wife and instruments to follow him whenever he
could find a home for them.
His wanderings in this dark time--some two years--are not quite clear;
but at last the enlightened Emperor of Bohemia, Rudolph II., invited him
to settle in Prague. Thither he repaired, a castle was given him as an
observatory, a house in the city, and 3000 crowns a year for life. So
his instruments were set up once more, students flocked to hear him and
to receive work at his hands--among them a poor youth, John Kepler, to
whom he was very kind, and who became, as you know, a still greater man
than his master.
But the spirit of Tycho was broken, and though some good work was done
at Prague--more observations made, and the Rudolphine tables begun--yet
the hand of death was upon him. A painful disease seized him, attended
with sleeplessness and temporary delirium, during the paroxysms of
which he frequently exclaimed, _Ne frustra vixisse videar_. ("Oh that it
may not appear that I have lived in vain!")
Quietly, however, at last, and surrounded by his friends and relatives,
this fierce, passionate soul passed away, on the 24th of October, 1601.
His beloved instruments, which were almost a part of himself, were
stored by Rudolph in a museum with scrupulous care, until the taking of
Prague by the Elector Palatine's troops. In this disturbed time they got
smashed, dispersed, and converted to other purposes. One thing only was
saved--the great brass globe, which some thirty years after was
recognized by a later king of Denmark as having belonged to Tycho, and
deposited in the Library of the Academy of Sciences at Copenhagen, where
I believe it is to this day.
The island of Huen was overrun by the Danish nobility, and nothing now
remains of Uraniburg but a mound of earth and two pits.
As to the real work of Tycho, that has become immortal enough,--chiefly
through the labours of his friend and scholar whose life we shall
consider in the next lecture.
SUMMARY OF FACTS FOR LECTURE III
_Life and work of Kepler._ Kepler was born in December, 1571, at Weil in
Wuertemberg. Father an officer in the duke's army, mother something of a
virago, both very poor. Kepler was utilized as a tavern pot-boy, but
ultimately sent to a charity school, and thence to the University of
Tuebingen. Health extremely delicate; he was liable to violent attacks
all his life. Studied mathematics, and accepted an astronomical
lectureship at Graz as the first post which offered. Endeavoured to
discover some connection between the number of the planets, their times
of revolution, and their distances from the sun. Ultimately hit upon his
fanciful regular-solid hypothesis, and published his first book in 1597.
In 1599 was invited by Tycho to Prague, and there appointed Imperial
mathematician, at a handsome but seldom paid salary. Observed the new
star of 1604. Endeavoured to find the law of refraction of light from
Vitellio's measurements, but failed. Analyzed Tycho's observations to
find the true law of motion of Mars. After incredible labour, through
innumerable wrong guesses, and six years of almost incessant
calculation, he at length emerged in his two "laws"--discoveries which
swept away all epicycles, deferents, equants, and other remnants of the
Greek system, and ushered in the dawn of modern astronomy.
LAW I. _Planets move in ellipses, with the Sun in one focus._
LAW II. _The radius vector (or line joining sun and planet) sweeps out
equal areas in equal times._
Published his second book containing these laws in 1609. Death of
Rudolph in 1612, and subsequent increased misery and misfortune of
Kepler. Ultimately discovered the connection between the times and
distances of the planets for which he had been groping all his mature
life, and announced it in 1618:--
LAW III. _The square of the time of revolution (or year) of each planet
is proportional to the cube of its mean distance from the sun._
The book in which this law was published ("On Celestial Harmonies") was
dedicated to James of England. In 1620 had to intervene to protect his
mother from being tortured for witchcraft. Accepted a professorship at
Linz. Published the Rudolphine tables in 1627, embodying Tycho's
observations and his own theory. Made a last effort to overcome his
poverty by getting the arrears of his salary paid at Prague, but was
unsuccessful, and, contracting brain fever on the journey, died in
November, 1630, aged 59.
A man of keen imagination, indomitable perseverance, and uncompromising
love of truth, Kepler overcame ill-health, poverty, and misfortune, and
placed himself in the very highest rank of scientific men. His laws, so
extraordinarily discovered, introduced order and simplicity into what
else would have been a chaos of detailed observations; and they served
as a secure basis for the splendid erection made on them by Newton.
_Seven planets of the Ptolemaic system--_
Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn.
_Six planets of the Copernican system--_
Mercury, Venus, Earth, Mars, Jupiter, Saturn.
_The five regular solids, in appropriate order--_
Octahedron, Icosahedron, Dodecahedron, Tetrahedron, Cube.
_Table illustrating Kepler's third law._
+---------+---------------+-----------+---------------+----------------+
| | Mean distance | Length | Cube of the | Square of the |
| Planet. | from Sun. | of Year. | Distance. | Time. |
| | D | T | D^3 | T^2 |
+---------+---------------+-----------+---------------+----------------+
| Mercury | .3871 | .24084 | .05801 | .05801 |
| Venus | .7233 | .61519 | .37845 | .37846 |
| Earth | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
| Mars | 1.5237 | 1.8808 | 3.5375 | 3.5375 |
| Jupiter | 5.2028 | 11.862 | 140.83 | 140.70 |
| Saturn | 9.5388 | 29.457 | 867.92 | 867.70 |
+---------+---------------+-----------+---------------+----------------+
The length of the earth's year is 365.256 days; its mean distance from
the sun, taken above as unity, is 92,000,000 miles.
LECTURE III
KEPLER AND THE LAWS OF PLANETARY MOTION
It is difficult to imagine a stronger contrast between two men engaged
in the same branch of science than exists between Tycho Brahe, the
subject of last lecture, and Kepler, our subject on the present
occasion.
The one, rich, noble, vigorous, passionate, strong in mechanical
ingenuity and experimental skill, but not above the average in
theoretical and mathematical power.
The other, poor, sickly, devoid of experimental gifts, and unfitted by
nature for accurate observation, but strong almost beyond competition in
speculative subtlety and innate mathematical perception.
The one is the complement of the other; and from the fact of their
following each other so closely arose the most surprising benefits to
science.
The outward life of Kepler is to a large extent a mere record of poverty
and misfortune. I shall only sketch in its broad features, so that we
may have more time to attend to his work.
He was born (so his biographer assures us) in longitude 29 deg. 7', latitude
48 deg. 54', on the 21st of December, 1571. His parents seem to have been of
fair condition, but by reason, it is said, of his becoming surety for a
friend, the father lost all his slender income, and was reduced to
keeping a tavern. Young John Kepler was thereupon taken from school,
and employed as pot-boy between the ages of nine and twelve. He was a
sickly lad, subject to violent illnesses from the cradle, so that his
life was frequently despaired of. Ultimately he was sent to a monastic
school and thence to the University of Tuebingen, where he graduated
second on the list. Meanwhile home affairs had gone to rack and ruin.
His father abandoned the home, and later died abroad. The mother
quarrelled with all her relations, including her son John; who was
therefore glad to get away as soon as possible.
All his connection with astronomy up to this time had been the hearing
the Copernican theory expounded in University lectures, and defending it
in a college debating society.
An astronomical lectureship at Graz happening to offer itself, he was
urged to take it, and agreed to do so, though stipulating that it should
not debar him from some more brilliant profession when there was a
chance.
For astronomy in those days seems to have ranked as a minor science,
like mineralogy or meteorology now. It had little of the special dignity
with which the labours of Kepler himself were destined so greatly to aid
in endowing it.
Well, he speedily became a thorough Copernican, and as he had a most
singularly restless and inquisitive mind, full of appreciation of
everything relating to number and magnitude--was a born speculator and
thinker just as Mozart was a born musician, or Bidder a born
calculator--he was agitated by questions such as these: Why are there
exactly six planets? Is there any connection between their orbital
distances, or between their orbits and the times of describing them?
These things tormented him, and he thought about them day and night. It
is characteristic of the spirit of the times--this questioning why there
should be six planets. Nowadays, we should simply record the fact and
look out for a seventh. Then, some occult property of the number six was
groped for, such as that it was equal to 1 + 2 + 3 and likewise equal to
1 x 2 x 3, and so on. Many fine reasons had been given for the seven
planets of the Ptolemaic system (see, for instance, p. 106), but for
the six planets of the Copernican system the reasons were not so cogent.
Again, with respect to their successive distances from the sun, some law
would seem to regulate their distance, but it was not known.
(Parenthetically I may remark that it is not known even now: a crude
empirical statement known as Bode's law--see page 294--is all that has
been discovered.)
Once more, the further the planet the slower it moved; there seemed to
be some law connecting speed and distance. This also Kepler made
continual attempts to discover.
[Illustration: FIG. 26.--Orbits of some of the planets drawn to scale:
showing the gap between Mars and Jupiter.]
One of his ideas concerning the law of the successive distances was
based on the inscription of a triangle in a circle. If you inscribe in a
circle a large number of equilateral triangles, they envelop another
circle bearing a definite ratio to the first: these might do for the
orbits of two planets (see Fig. 27). Then try inscribing and
circumscribing squares, hexagons, and other figures, and see if the
circles thus defined would correspond to the several planetary orbits.
But they would not give any satisfactory result. Brooding over this
disappointment, the idea of trying solid figures suddenly strikes him.
"What have plane figures to do with the celestial orbits?" he cries out;
"inscribe the regular solids." And then--brilliant idea--he remembers
that there are but five. Euclid had shown that there could be only five
regular solids.[4] The number evidently corresponds to the gaps between
the six planets. The reason of there being only six seems to be
attained. This coincidence assures him he is on the right track, and
with great enthusiasm and hope he "represents the earth's orbit by a
sphere as the norm and measure of all"; round it he circumscribes a
dodecahedron, and puts another sphere round that, which is approximately
the orbit of Mars; round that, again, a tetrahedron, the corners of
which mark the sphere of the orbit of Jupiter; round that sphere, again,
he places a cube, which roughly gives the orbit of Saturn.
[Illustration: FIG. 27.--Many-sided polygon or approximate circle
enveloped by straight lines, as for instance by a number of equilateral
triangles.]
On the other hand, he inscribes in the sphere of the earth's orbit an
icosahedron; and inside the sphere determined by that, an octahedron;
which figures he takes to inclose the spheres of Venus and of Mercury
respectively.
The imagined discovery is purely fictitious and accidental. First of
all, eight planets are now known; and secondly, their real distances
agree only very approximately with Kepler's hypothesis.
[Illustration: FIG. 28.--Frameworks with inscribed and circumscribed
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 associate." After this visit,
Tycho wrote again, offering him the post of mathematical assistant,
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 noble and distinguished man, the head
of the scientific world at that date.
To illustrate 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 ingratitude of the accusation.
Kepler repents instantly, and replies:--
"MOST NOBLE 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 God to my own
intemperance as to shut my eyes on all these benefits; that,
instead of modest and respectful gratitude, I should indulge for
three weeks in continual moroseness towards all your family, in
headlong passion and the utmost insolence towards yourself, who
possess so many claims on my veneration, from your noble 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 assist 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 superstitious 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, analyzing 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 accumulated 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.