WHEN presenting this celebrated book to the emperor, Kepler gave notice

that he contemplated a farther attack upon Mars's relations, father

Jupiter, brother Mercury, and the rest; and promised that he would be

successful, provided the emperor would not forget the sinews of war, and

order him to be furnished anew with means for recruiting his army. The

death of his unhappy patron, the Emperor Rodolph, which happened in

1612, barely in time to save him from the last disgrace of deposition

from the Imperial throne, seemed to put additional difficulties in the

way of Kepler's receiving the arrears so unjustly denied to him; but on

the accession of Rodolph's brother, Matthias, he was again named to his

post of Imperial Mathematician, and had also a permanent professorship

assigned to him in the University of Linz. He quitted Prague without

much regret, where he had struggled against poverty during eleven years.

Whatever disinclination he might feel to depart, arose from his

unwillingness to loosen still more the hold he yet retained upon the

wreck of Tycho Brahe's instruments and observations. Tengnagel,

son-in-law of Tycho, had abandoned astronomy for a political career, and

the other members of his family, who were principally females, suffered

the costly instruments to lie neglected and forgotten, although they had

obstructed with the utmost jealousy Kepler's attempts to continue their

utility. The only two instruments Kepler possessed of his own property,

were "An iron sextant of 2½ feet diameter, and a brass azimuthal

quadrant, of 3½ feet diameter, both divided into minutes of a degree."

These were the gift of his friend and patron, Hoffman, the President of

Styria, and with these he made all the observations which he added to

those of Tycho Brahe. His constitution was not favourable to these

studies, his health being always delicate, and suffering much from

exposure to the night air; his eyes also were very weak, as he mentions

himself in several places. In the summary of his character which he drew

up when proposing to become Tycho Brahe's assistant, he describes

himself as follows:—"For observations my sight is dull; for mechanical

operations my hand is awkward; in politics and domestic matters my

nature is troublesome and choleric; my constitution will not allow me,

even when in good health, to remain a long time sedentary (particularly

for an extraordinary time after dinner); I must rise often and walk

about, and in different seasons am forced to make corresponding changes

in my diet."

The year preceding his departure to Linz was denounced by him as

pregnant with misfortune and misery. "In the first place I could get no

money from the court, and my wife, who had for a long time been

suffering under low spirits and despondency, was taken violently ill

towards the end of 1610, with the Hungarian fever, epilepsy, and

phrenitis. She was scarcely convalescent when all my three children were

at once attacked with small-pox. Leopold with his army occupied the town

beyond the river, just as I lost the dearest of my sons, him whose

nativity you will find in my book on the new star. The town on this side

of the river where I lived was harassed by the Bohemian troops, whose

new levies were insubordinate and insolent: to complete the whole, the

Austrian army brought the plague with them into the city. I went into

Austria, and endeavoured to procure the situation which I now hold.

Returning in June, I found my wife in a decline from her grief at the

death of her son, and on the eve of an infectious fever; and I lost her

also, within eleven days after my return. Then came fresh annoyance, of

course, and her fortune was to be divided with my step-sisters. The

Emperor Rodolph would not agree to my departure; vain hopes were given

me of being paid from Saxony; my time and money were wasted together,

till on the death of the emperor, in 1612, I was named again by his

successor, and suffered to depart to Linz. These, methinks, were reasons

enough why I should have overlooked not only your letters, but even

astronomy itself."

Kepler's first marriage had not been a happy one; but the necessity in

which he felt himself of providing some one to take charge of his two

surviving children, of whom the eldest, Susanna, was born in 1602, and

Louis in 1607, determined him on entering a second time into the married

state. The account he has left us of the various negotiations which

preceded his final choice, does not, in any point, belie the oddity of

his character. His friends seem to have received a general commission to

look out for a suitable match, and in a long and most amusing letter to

the Baron Strahlendorf, we are made acquainted with the pretensions and

qualifications of no less than eleven ladies among whom his inclinations

wavered.

The first on the list was a widow, an intimate friend of his first

wife's, and who, on many accounts, appeared a most eligible match. "At

first she seemed favourably inclined to the proposal; it is certain that

she took time to consider it, but at last she very quietly excused

herself." It must have been from a recollection of this lady's good

qualities that Kepler was induced to make his offer; for we learn rather

unexpectedly, after being informed of her decision, that when he soon

afterwards paid his respects to her, it was for the first time that he

had seen her during the last six years; and he found, to his great

relief, that "there was no single pleasing point about her." The truth

seems to be that he was nettled by her answer, and he is at greater

pains than appear necessary, considering this last discovery, to

determine why she would not accept his offered hand. Among other reasons

he suggested her children, among whom were two marriageable daughters;

and it is diverting afterwards to find them also in the catalogue which

Kepler appeared to be making of all his female acquaintance. He seems to

have been much perplexed in attempting to reconcile his astrological

theory with the fact of his having taken so much trouble about a

negotiation not destined to succeed. "Have the stars exercised any

influence here? For just about this time the direction of the Mid-Heaven

is in hot opposition to Mars, and the passage of Saturn, through the

ascending point of the zodiac, in the scheme of my nativity, will happen

again next November and December. But if these are the causes, how do

they act? Is that explanation the true one which I have elsewhere given?

For I can never think of handing over to the stars the office of deities

to produce effects. Let us therefore suppose it accounted for by the

stars, that at this season I am violent in my temper and affections, in

rashness of belief, in a shew of pitiful tender-heartedness; in catching

at reputation by new and paradoxical notions, and the singularity of my

actions; in busily inquiring into, and weighing and discussing, various

reasons; in the uneasiness of my mind with respect to my choice. I thank

God that that did not happen which might have happened; that this

marriage did not take place: now for the others." Of these others, one

was too old, another in bad health, another too proud of her birth and

quarterings; a fourth had learned nothing but shewy accomplishments,

"not at all suitable to the sort of life she would have to lead with

me." Another grew impatient, and married a more decided admirer, whilst

he was hesitating. "The mischief (says he) in all these attachments was,

that whilst I was delaying, comparing, and balancing conflicting

reasons, every day saw me inflamed with a new passion." By the time he

reached the eighth, he found his match in this respect. "Fortune at

length has avenged herself on my doubtful inclinations. At first she was

quite complying, and her friends also: presently, whether she did or did

not consent, not only I, but she herself did not know. After the lapse

of a few days, came a renewed promise, which however had to be confirmed

a third time; and four days after that, she again repented her

confirmation, and begged to be excused from it. Upon this I gave her up,

and this time all my counsellors were of one opinion." This was the

longest courtship in the list, having lasted three whole months; and

quite disheartened by its bad success, Kepler's next attempt was of a

more timid complexion. His advances to No. 9, were made by confiding to

her the whole story of his recent disappointment, prudently determining

to be guided in his behaviour, by observing whether the treatment he had

experienced met with a proper degree of sympathy. Apparently the

experiment did not succeed; and almost reduced to despair, Kepler betook

himself to the advice of a friend, who had for some time past complained

that she was not consulted in this difficult negotiation. When she

produced No. 10, and the first visit was paid, the report upon her was

as follows:—"She has, undoubtedly, a good fortune, is of good family,

and of economical habits: but her physiognomy is most horribly ugly; she

would be stared at in the streets, not to mention the striking

disproportion in our figures. I am lank, lean, and spare; she is short

and thick: in a family notorious for fatness she is considered

superfluously fat." The only objection to No. 11 seems to have been her

excessive youth; and when this treaty was broken of on that account,

Kepler turned his back upon all his advisers, and chose for himself one

who had figured as No. 5 in the list, to whom he professes to have felt

attached throughout, but from whom the representations of his friends

had hitherto detained him, probably on account of her humble station.

The following is Kepler's summary of her character. "Her name is

Susanna, the daughter of John Reuthinger and Barbara, citizens of the

town of Eferdingen; the father was by trade a cabinet-maker, but both

her parents are dead. She has received an education well worth the

largest dowry, by favour of the Lady of Stahrenberg, the strictness of

whose household is famous throughout the province. Her person and

manners are suitable to mine; no pride, no extravagance; she can bear to

work; she has a tolerable knowledge how to manage a family; middle-aged,

and of a disposition and capability to acquire what she still wants. Her

I shall marry by favour of the noble baron of Stahrenberg at twelve

o'clock on the 30th of next October, with all Eferdingen assembled to

meet us, and we shall eat the marriage-dinner at Maurice's at the Golden

Lion."

Hantsch has made an absurd mistake with regard to this marriage, in

stating that the bride was only twelve years old. Kästner and other

biographers have been content to repeat the same assertion without any

comment, notwithstanding its evident improbability. The origin of the

blunder is to be found in Kepler's correspondence with Bernegger, to

whom, speaking of his wife, he says "She has been educated for twelve

years by the Lady of Stahrenberg." This is by no means a single instance

of carelessness in Hantsch; Kästner has pointed out others of greater

consequence. It was owing to this marriage, that Kepler took occasion to

write his new method of gauging, for as he tells us in his own peculiar

style "last November I brought home a new wife, and as the whole course

of Danube was then covered with the produce of the Austrian vineyards,

to be sold at a reasonable rate, I purchased a few casks, thinking it my

duty as a good husband and a father of a family, to see that my

household was well provided with drink." When the seller came to

ascertain the quantity, Kepler objected to his method of gauging, for

he allowed no difference, whatever might be the proportion of the

bulging parts. The reflections to which this incident gave rise,

terminated in the publication of the above-mentioned treatise, which

claims a place among the earliest specimens of what is now called the

modern analysis. In it he extended several properties of plane figures

to segments of cones and cylinders, from the consideration that "these

solids are incorporated circles," and, therefore, that those properties

are true of the whole which belong to each component part. That the book

might end as oddly as it began, Kepler concluded it with a parody of

Catullus:

"Et cum pocula mille mensi erîmus

Conturbabimus illa, ne sciamus."

His new residence at Linz was not long undisturbed. He quarrelled there,

as he had done in the early part of his life at Gratz, with the Roman

Catholic party, and was excommunicated. "Judge," says he to Peter

Hoffman, "how far I can assist you, in a place where the priest and

school-inspector have combined to brand me with the public stigma of

heresy, because in every question I take that side which seems to me to

be consonant with the word of God." The particular dogma which

occasioned his excommunication, was connected with the doctrine of

transubstantiation. He published his creed in a copy of Latin verses,

preserved by his biographer Hantsch.

Before this occurrence, Kepler had been called to the diet at Ratisbon

to give his opinion on the propriety of adopting the Gregorian

reformation of the calendar, and he published a short essay, pointing

out the respective convenience of doing so, or of altering the old

Julian Calendar in some other manner. Notwithstanding the readiness of

the diet to avail themselves of his talents for the settlement of a

difficult question, the arrears of his salary were not paid much more

regularly than they had been in Rodolph's time, and he was driven to

provide himself with money by the publication of his almanac, of which

necessity he heavily and justly complained. "In order to pay the expense

of the Ephemeris for these two years, I have also written a vile

prophesying almanac, which is scarcely more respectable than begging;

unless it be because it saves the emperor's credit, who abandons me

entirely; and with all his frequent and recent orders in council, would

suffer me to perish with hunger." Kepler published this Ephemeris

annually till 1620; ten years later he added those belonging to the

years from 1620 to 1628.

In 1617 Kepler was invited into Italy, to succeed Magini as Professor of

Mathematics at Bologna. The offer tempted him; but, after mature

consideration, he rejected it, on grounds which he thus explained to

Roffini:—"By birth and spirit I am a German, imbued with German

principles, and bound by such family ties, that even if the emperor

should consent, I could not, without the greatest difficulty, remove my

dwelling-place from Germany into Italy. And although the glory of

holding so distinguished a situation among the venerable professors of

Bologna stimulates me, and there appears great likelihood of notably

increasing my fortune, as well from the great concourse to the public

lectures, as from private tuition; yet, on the other hand, that period

of my life is past which was once excited by novelty, or which might

promise itself a long enjoyment of these advantages. Besides, from a boy

up to my present years, living a German among Germans, I am accustomed

to a degree of freedom in my speech and manners, which, if persevered in

on my removal to Bologna, seems likely to draw upon me, if not danger,

at least notoriety, and might expose me to suspicion and party malice.

Notwithstanding this answer, I have yet hopes that your most honourable

invitation will be of service to me, and may make the imperial treasurer

more ready than he has hitherto been to fulfil his master's intentions

towards me. In that case I shall the sooner be able to publish the

Rudolphine Tables and the Ephemerides, of which you had the scheme so

many years back; and in this manner you and your advisers may have no

reason to regret this invitation, though for the present it seems

fruitless."

In 1619, the Emperor Matthias died, and was succeeded by Ferdinand III.,

who retained Kepler in the post he had filled under his two predecessors

on the imperial throne. Kästner, in his "History of Mathematics," has

corrected a gross error of Hantsch, in asserting that Kepler

prognosticated Matthias's death. The letter to which Hantsch refers, in

support of his statement, does indeed mention the emperor's death, but

merely as a notorious event, for the purpose of recalling a date to the

memory of his correspondent.

CHAPTER VII.

_Kepler publishes his Harmonics—Account of his Astrological

Opinions and Discovery of the Law of the Periods of the Planetary

Revolutions—Sketch of Newton's proof of Kepler's Laws._

THE "Cosmographical Mystery" was written, as has been already mentioned,

when Kepler was only twenty-six, and the wildness of its theories might

be considered as due merely to the vivacity of a young man; but as if

purposely to shew that his maturer age did not renounce the creations of

his youthful fancy, he reprinted the "Mystery" in 1619, nearly at the

same time when he published his celebrated work on Harmonics; and the

extravagance of the latter publication does not at all lose in

comparison with its predecessor. It is dedicated to James I. of England,

and divided into five books: "The first, Geometrical, on the origin and

demonstration of the laws of the figures which produce harmonious

proportions;—the second, Architectonical, on figurate geometry, and the

congruence of plane and solid regular figures;—the third, properly

Harmonic, on the derivation of musical proportions from figures, and on

the nature and distinction of things relating to song, in opposition to

the old theories;—the fourth, Metaphysical, Psychological, and

Astrological, on the mental essence of harmonies, and of their kinds in

the world, especially on the harmony of rays emanating on the earth from

the heavenly bodies, and on their effect in nature, and on the sublunary

and human soul;—the fifth, Astronomical and Metaphysical, on the very

exquisite harmonies of the celestial motions, and the origin of the

excentricities in harmonious proportions."

The two first books are almost strictly, as Kepler styles them,

geometrical, relating in great measure to the inscription of regular

polygons in a circle. The following passage is curious, presenting an

analogous idea to that contained in one of the extracts already given

from the Commentaries on Mars. "The heptagon, and all other polygons and

stars beyond it, which have a prime number of sides, and all other

figures derived from them, cannot be inscribed geometrically in a

circle; although their sides have a necessary magnitude, it is equally a

matter of necessity that we remain ignorant of it. This is a question of

great importance, for on this account is it that the heptagon, and other

figures of this kind, have not been employed by God in the adornment of

the world, as the other intelligible figures are employed which have

been already explained." Kepler then introduces the algebraical

equation, on the solution of which this problem depends, and makes a

remark which is curious at this period of the history of algebra—that

the root of an equation which cannot be accurately found, may yet be

found within any degree of approximation by an expert calculator. In

conclusion he again remarks that "the side of the heptagon has no place

among scientific existences, since its formal description is impossible,

and therefore it cannot be known by the human mind, since the

possibility of description precedes the possibility of knowledge; nor is

it known even by the simple eternal act of an omniscient mind, because

its nature belongs to things which cannot be known. And yet this

scientific nonentity has some scientific properties, for if a heptagon

were described in a circle, the proportion of its sides would have

analogous proportions."

The third book is a treatise on music, in the confined and ordinary

sense in which we now use that word, and apparently a sober and rational

one, at least as nearly so as Kepler could be trusted to write on a

subject so dangerous to his discretion. All the extravagance of the work

seems reserved for the fourth book, the title of which already conveys

some notion of the nature of its contents. In this book he has collected

the substance of the astrological opinions scattered through his other

works. We shall content ourselves with merely citing his own words,

without any attempt to explain the difference between the astrology

which he believed, and that which he contemptuously rejected. The

distinctive line seems very finely drawn, and as both one and the other

are now discarded by all who enjoy the full use of their reasoning

powers, it is not of much consequence that it should be accurately

traced.

It is to be observed, that he does not in this treatise modify or recant

anything of his earlier opinions, but refers to the favourable judgment

of his contemporary philosophers as a reason for embodying them in a

regular form. "Since many very celebrated professors of philosophy and

medicine are of opinion that I have created a new and most true

philosophy, this tender plant, like all novelties, ought to be carefully

nursed and cherished, so that it may strike root in the minds of

philosophers, and not be choked by the excessive humours of vain

sophistications, or washed away by the torrents of vulgar prejudices, or

frozen by the chill of public neglect; and if I succeed in guarding it

from these dangers, I have no fear that it will be crushed by the storms

of calumny, or parched by the sun of sterling criticism."

One thing is very remarkable in Kepler's creed, that he whose candour is

so indisputable in every other part of his conduct, professed to have

been forced to adopt his astrological opinions from direct and positive

observation.—"It is now more than twenty years since I began to

maintain opinions like these on the predominant nature of the elements,

which, adopting the common name, I call sublunary. I have been driven to

this not by studying or admiring Plato, but singly and solely by

observing seasons, and noting the aspects by which they are produced. I

have seen the state of the atmosphere almost uniformly disturbed as

often as the planets are in conjunction, or in the other configurations

so celebrated among astrologers. I have noticed its tranquil state,

either when there are none or few such aspects, or when they are

transitory and of short duration. I have not formed an opinion on this

matter without good grounds, like the common herd of prophesiers, who

describe the operations of the stars as if they were a sort of deities,

the lords of heaven and earth, and producing everything at their

pleasure. They never trouble themselves to consider what means the stars

have of working any effects among us on the earth, whilst they remain in

the sky, and send down nothing to us which is obvious to the senses

except rays of light. This is the principal source of the filthy

astrological superstitions of that vulgar and childish race of dreamers,

the prognosticators."

The real manner in which the configurations of the stars operate,

according to Kepler, is as follows:—"Like one who listens to a sweet

melodious song, and by the gladness of his countenance, by his voice,

and by the beating of his hand or foot attuned to the music, gives token

that he perceives and approves the harmony: just so does sublunary

nature, with the notable and evident emotion of the bowels of the earth,

bear like witness to the same feelings, especially at those times when

the rays of the planets form harmonious configurations on the

earth."—"I have been confirmed in this theory by that which might have

deterred others; I mean, by observing that the emotions do not agree

nicely with the instants of the configurations; but the earth sometimes

appears lazy and obstinate, and at another time (after important and

long-continued configurations) she becomes exasperated, and gives way to

her passion, even without the continuation of aspects. For in fact the

earth is not an animal like a dog, ready at every nod; but more like a

bull, or an elephant, slow to become angry, and so much the more furious

when incensed."

This singular doctrine must not be mistaken for one of Kepler's

favourite allegories; he actually and literally professed to believe

that the earth was an enormous living animal; and he has enumerated,

with a particularity of details into which we forbear to follow him, the

analogies he recognized between its habits and those of men and other

animals. A few samples of these may speak for the rest. "If any one who

has climbed the peaks of the highest mountains throw a stone down their

very deep clefts, a sound is heard from them; or if he throw it into one

of the mountain lakes, which beyond doubt are bottomless, a storm will

immediately arise, just as when you thrust a straw into the ear or nose

of a ticklish animal, it shakes its head, or runs shuddering away. What

so like breathing, especially of those fish who draw water into their

mouths and spout it out again through their gills, as that wonderful

tide! For although it is so regulated according to the course of the

moon, that, in the preface to my 'Commentaries on Mars,' I have

mentioned it as probable that the waters are attracted by the moon as

iron is by the loadstone; yet, if any one uphold that the earth

regulates its breathing according to the motion of the sun and moon, as

animals have daily and nightly alternations of sleep and waking, I shall

not think his philosophy unworthy of being listened to; especially if

any flexible parts should be discovered in the depths of the earth to

supply the functions of lungs or gills."

From the next extract, we must leave the reader to learn as well as he

may, how much Kepler did, and how much he did not believe on the

subject of genethliac astrology.—"Hence it is that human spirits, at

the time of celestial aspects, are particularly urged to complete the

matters which they have in hand. What the goad is to the ox, what the

spur or the rowel is to the horse, to the soldier the bell and trumpet,

an animated speech to an audience, to a crowd of rustics a performance

on the fife and bagpipes, that to all, and especially in the aggregate,

is a heavenly configuration of suitable planets; so that every single

one is excited in his thoughts and actions, and all become more ready to

unite and associate their efforts. For instance, in war you may see that

tumults, battles, fights, invasions, assaults, attacks, and panic fears,

generally happen at the time of the aspects of Mars and Mercury, Mars

and Jupiter, Mars and the Sun, Mars and Saturn, &c. In epidemic

diseases, a greater number of persons are attacked at the times of the

powerful aspects, they suffer more severely, or even die, owing to the

failure of nature in her strife with the disease, which strife (and not

the death) is occasioned by the aspect. It is not the sky which does all

these things immediately, but the faculty of the vital soul, associating

its operation with the celestial harmonies, is the principal agent in

this so-called influence of the heavens. Indeed this word influence has

so fascinated some philosophers that they prefer raving with the

senseless vulgar, to learning the truth with me. This essential property

is the principal foundation of that admirable genethliac art. For when

anything begins to have its being when that is working harmonies, the

sensible harmony of the rays of the planets has peculiar influence on

it. This then is the cause why those who are born under a season of many

aspects among the planets, generally turn out busy and industrious,

whether they accustom themselves from childhood to amass wealth, or are

born or chosen to direct public affairs, or finally, have given their

attention to study. If any one think that I might be taken as an

instance of this last class, I do not grudge him the knowledge of my

nativity. I am not checked by the reproach of boastfulness,

notwithstanding those who, by speech or conduct, condemn as folly all

kinds of writing on this subject; the idiots, the half-learned, the

inventors of titles and trappings, to throw dust in the eyes of the

people, and those whom Picus calls the plebeian theologians: among the

true lovers of wisdom, I easily clear myself of this imputation, by the

advantage of my reader; for there is no one whose nativity or whose

internal disposition and temper I can learn so well as I know my own.

Well then, Jupiter nearest the nonagesimal had passed by four degrees

the trine of Saturn; the Sun and Venus, in conjunction, were moving from

the latter towards the former, nearly in sextiles with both: they were

also removing from quadratures with Mars, to which Mercury was closely

approaching: the moon drew near the trine of the same planet, close to

the Bull's Eye, even in latitude. The 25th degree of Gemini was rising,

and the 22d of Aquarius culminating. That there was this triple

configuration on that day—namely, the sextile of Saturn and the Sun,

the sextile of Mars and Jupiter, the quadrature of Mercury and Mars, is

proved by the change of weather; for, after a frost of some days, that

very day became warmer, there was a thaw and a fall of rain.[192]

"I do not wish this single instance to be taken as a defence and proof

of all the aphorisms of astrologers, nor do I attribute to the heavens

the government of human affairs: what a vast interval still separates

these philosophical observations from that folly or madness as it should

rather be called. For, following up this example, I knew a lady[193],

born under nearly the same aspects, whose disposition, indeed, was

exceedingly restless, but who not only makes no progress in literature

(that is not strange in a woman), but troubles her whole family, and is

the cause to herself of deplorable misery. What, in my case, assisted

the aspects was—firstly, the fancy of my mother when pregnant with me,

a great admirer of her mother-in-law, my grandmother, who had some

knowledge of medicine, my grandfather's profession; a second cause is,

that I was born a male, and not a female, for astrologers have sought

in vain to distinguish sexes in the sky; thirdly, I derive from my

mother a habit of body, more fit for study than other kinds of life;

fourthly, my parents' fortune was not large, and there was no landed

property to which I might succeed and become attached; fifthly, there

were the schools, and the liberality of the magistracy towards such boys

as were apt for learning. But now if I am to speak of the result of my

studies, what I pray can I find in the sky, even remotely alluding to

it. The learned confess that several not despicable branches of

philosophy have been newly extricated or amended or brought to

perfection by me: but here my constellations were, not Mercury from the

east, in the angle of the seventh, and in quadratures with Mars, but

Copernicus, but Tycho Brahe, without whose books of observations

everything now set by me in the clearest light must have remained buried

in darkness; not Saturn predominating Mercury, but my Lords the Emperors

Rodolph and Matthias; not Capricorn, the house of Saturn, but Upper

Austria, the home of the Emperor, and the ready and unexampled bounty of

his nobles to my petition. Here is that corner, not the western one of

the horoscope, but on the Earth, whither, by permission of my imperial

master, I have betaken myself from a too uneasy court; and whence,

during these years of my life, which now tends towards its setting,

emanate these Harmonies, and the other matters on which I am engaged.

"However, it may be owing to Jupiter's ascendancy that I take greater

delight in the application of geometry to physics, than in that abstract

pursuit which partakes of the dryness of Saturn; and it is perhaps the

gibbous moon, in the bright constellation of the Bull's forehead, which

fills my mind with fantastic images."

The most remarkable thing contained in the 5th Book, is the announcement

of the celebrated law connecting the mean distances of the planets with

the periods of their revolution about the Sun. This law is expressed in

mathematical language, by saying that the squares of the times vary as

the cubes of the distances.[194] Kepler's rapture on detecting it was

unbounded, as may be seen from the exulting rhapsody with which he

announced it. "What I prophecied 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 'Harmonics'—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 have

recognized its truth beyond my most sanguine expectations. Great as is

the absolute nature of Harmonics with all its details, as set forth in

my third book, it is all found among the celestial motions, not indeed

in the manner which I imagined, (that is not the least part of my

delight,) but in another very different, and yet most perfect and

excellent. It is now 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 on, burst out upon me. Nothing holds me; I

will indulge in my sacred fury; I will triumph over mankind by the

honest confession, that I have stolen the golden vases of the

Egyptians[195], 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."

He has told, with his usual particularity, the manner and precise moment

of the discovery. "Another part of my 'Cosmographical Mystery,'

suspended twenty-two years ago, because it was then undetermined, is

completed and introduced here, after I had discovered the true intervals

of the orbits, by means of Brahe's observations, and had spent the

continuous toil of a long time in investigating the true proportion of

the periodic times to the orbits,

Sera quidem respexit inertem,

Respexit tamen, et longo post tempore venit.

If you would know the precise moment, the first idea came across me on

the 8th March of this year, 1618; but chancing to make a mistake in the

calculation, I rejected it as false. I returned again to it with new

force on the 15th May, and it has dissipated the darkness of my mind by

such an agreement between this idea and my seventeen years' labour on

Brahe's observations, that at first I thought I must be dreaming, and

had taken my result for granted in my first assumptions. But the fact is

perfect, the fact is certain, that the proportion existing between the

periodic times of any two planets is exactly the sesquiplicate

proportion of the mean distances of the orbits."

There is high authority for not attempting over anxiously to understand

the rest of the work. Delambre sums it up as follows:—"In the music of

the celestial bodies it appears that Saturn and Jupiter take the bass,

Mars the tenor, the Earth and Venus the counter-tenor, and Mercury the

treble." If the patience of this indefatigable historian gave way, as he

confesses, in the perusal, any further notice of it here may be well

excused. Kepler became engaged, in consequence of this publication, in

an angry controversy with the eccentric Robert Fludd, who was at least

Kepler's match in wild extravagance and mysticism, if far inferior to

him in genius. It is diverting to hear each reproaching the other with

obscurity.

In the "Epitome of the Copernican Astronomy," which Kepler published

about the same time, we find the manner in which he endeavoured to

deduce the beautiful law of periodic times, from his principles of

motion and radiation of whirling forces. This work is in fact a summary

of all his astronomical opinions, drawn up in a popular style in the

form of question and answer. We find there a singular argument against

believing, as some did, that each planet is carried round by an angel,

for in that case, says Kepler, "the orbits would be perfectly circular;

but the elliptic form, which we find in them, rather smacks of the

nature of the lever and material necessity."

The investigation of the relation between the periodic times and

distances of the planets is introduced by a query whether or not they

are to be considered heavy. The answer is given in the following

terms:—"Although none of the celestial globes are heavy, in the sense

in which we say on earth that a stone is heavy, nor light as fire is

light with us, yet have they, by reason of their materiality, a natural

inability to move from place to place: they have a natural inertness or

quietude, in consequence of which they remain still in every situation

where they are placed alone.

"_P._ Is it then the sun, which by its turning carries round the

planets? How can the sun do this, having no hands to seize the planet at

so great a distance, and force it round along with itself?—Its bodily

virtue, sent forth in straight lines into the whole space of the world,

serves instead of hands; and this virtue, being a corporeal species,

turns with the body of the sun like a very rapid vortex, and travels

over the whole of that space which it fills as quickly as the sun

revolves in its very confined space round the centre.

"_P._ Explain what this virtue is, and belonging to what class of

things?—As there are two bodies, the mover and the moved, so are there

two powers by which the motion is obtained. The one is passive, and

rather belonging to matter, namely, the resemblance of the body of the

planet to the body of the sun in its corporeal form, and so that part of

the planetary body is friendly, the opposite part hostile to the sun.

The other power is active, and bearing more relation to form, namely,

the body of the sun has a power of attracting the planet by its friendly

part, of repelling it by the hostile part, and finally, of retaining it

if it be placed so that neither the one nor the other be turned directly

towards the sun.

"_P._ How can it be that the whole body of the planet should be like or

cognate to the body of the sun, and yet part of the planet friendly,

part hostile to the sun?—Just as when one magnet attracts another, the

bodies are cognate; but attraction takes place only on one side,

repulsion on the other.

"_P._ Whence, then, arises that difference of opposite parts in the same

body?—In magnets the diversity arises from the situation of the parts

with respect to the whole. In the heavens the matter is a little

differently arranged, for the sun does not, like the magnet, possess

only on one side, but in all the parts of its substance, this active and

energetic faculty of attracting, repelling, or retaining the planet. So

that it is probable that the centre of the solar body corresponds to one

extremity or pole of the magnet, and its whole surface to the other

pole.

"_P._ If this were so, all the planets would be restored[196] in the

same time with the sun?—True, if this were all: but it has been said

already that, besides this carrying power of the sun, there is also in

the planets a natural inertness to motion, which causes that, by reason

of their material substance, they are inclined to remain each in its

place. The carrying power of the sun, and the impotence or material

inertness of the planet, are thus in opposition. Each shares the

victory; the sun moves the planet from its place, although in some

degree it escapes from the chains with which it was held by the sun, and

so is taken hold of successively by every part of this circular virtue,

or, as it may be called, solar circumference, namely, by the parts which

follow those from which it has just extricated itself.

"_P._ But how does one planet extricate itself more than another from

this violence—First, because the virtue emanating from the sun has the

same degree of weakness at different distances, as the distances or the

width of the circles described on these distances.[197] This is the

principal reason. Secondly, the cause is partly in the greater or less

inertness or resistance of the planetary globes, which reduces the

proportions to one-half; but of this more hereafter.

"_P._ How can it be that the virtue emanating from the sun becomes

weaker at a greater distance? What is there to hurt or weaken

it?—Because that virtue is corporeal, and partaking of quantity, which

can be spread out and rarefied. Then, since there is as much virtue

diffused in the vast orb of Saturn as is collected in the very narrow

one of Mercury, it is very rare and therefore weak in Saturn's orbit,

very dense and therefore powerful at Mercury.

"_P._ You said, in the beginning of this inquiry into motion, that the

periodic times of the planets are exactly in the sesquiplicate

proportion of their orbits or circles: pray what is the cause of

this?—Four causes concur for lengthening the periodic time. First, the

length of the path; secondly, the weight or quantity of matter to be

carried; thirdly, the degree of strength of the moving virtue; fourthly,

the bulk or space into which is spread out the matter to be moved. The

circular paths of the planets are in the simple ratio of the distances;

the weights or quantities of matter in different planets are in the

subduplicate ratio of the same distances, as has been already proved; so

that with every increase of distance, a planet has more matter, and

therefore is moved more slowly, and accumulates more time in its

revolution, requiring already as it did more time by reason of the

length of the way. The third and fourth causes compensate each other in

a comparison of different planets: the simple and subduplicate

proportion compound the sesquiplicate proportion, which therefore is the

ratio of the periodic times."

Three of the four suppositions here made by Kepler to explain the

beautiful law he had detected, are now indisputably known to be false.

Neither the weights nor the sizes of the different planets observe the

proportions assigned by him, nor is the force by which they are retained

in their orbits in any respect similar in its effects to those

attributed by him to it. The wonder which might naturally be felt that

he should nevertheless reach the desired conclusion, will be

considerably abated on examining the mode in which he arrived at and

satisfied himself of the truth of these three suppositions. It has been

already mentioned that his notions on the existence of a whirling force

emanating from the sun, and decreasing in energy at increased distances,

are altogether inconsistent with all the experiments and observations we

are able to collect. His reason for asserting that the sizes of the

different planets are proportional to their distances from the sun, was

simply because he chose to take for granted that either their

solidities, surfaces, or diameters, must necessarily be in that

proportion, and of the three, the solidities appeared to him least

liable to objection. The last element of his precarious reasoning rested

upon equally groundless assumptions. Taking as a principle, that where

there is a number of different things they must be different in every

respect, he declared that it was quite unreasonable to suppose all the

planets of the same density. He thought it indisputable that they must

be rarer as they were farther from the sun, "and yet not in the

proportion of their distances, for thus we should sin against the law of

variety in another way, and make the quantity of matter (according to

what he had just said of their bulk) the same in all. But if we assume

the ratio of the quantities of matter to be half that of the distances,

we shall observe the best mean of all; for thus Saturn will be half as

heavy again as Jupiter, and Jupiter half again as dense as Saturn. And

the strongest argument of all is, that unless we assume this proportion

of the densities, the law of the periodic times will not answer." This

is the _proof_ alluded to, and it is clear that by such reasoning any

required result might be deduced from any given principles.

It may not be uninstructive to subjoin a sketch of the manner in which

Newton established the same celebrated results, starting from principles

of motion diametrically opposed to Kepler's, and it need scarcely be

added, reasoning upon them in a manner not less different. For this

purpose, a very few prefatory remarks will be found sufficient.

The different motions seen in nature are best analysed and classified by

supposing that every body in motion, if left to itself, will continue to

move forward at the same rate in a straight line, and by considering all

the observed deviations from this manner of moving, as exceptions and

disturbances occasioned by some external cause. To this supposed cause

is generally given the name of Force, and it is said to be the first law

of motion, that, unless acted on by some force, every body at rest

remains at rest, and every body in motion proceeds uniformly in a

straight line. Many employ this language, without perceiving that it

involves a definition of force, on the admission of which, it is reduced

to a truism. We see common instances of force in a blow, or a pull from

the end of a string fastened to the body: we shall also have occasion

presently to mention some forces where no visible connexion exists

between the moving body and that towards which the motion takes place,

and from which the force is said to proceed.

[Illustration:

_c_ C

+-------------+

\ / \

\ / \

\ / \

\ / \

\ / \

\ / \

+-------------+

B C' ]

A second law of motion, founded upon experiment, is this: if a body have

motion communicated to it in two directions, by one of which motions

alone it would have passed through a given space in a given time, as for

instance, through BC´ in one second, and by the other alone through any

other space B_c_ in the same time, it will, when both are given to it at

the same instant, pass in the same time (in the present instance in one

second) through BC the diagonal of the parallelogram of which BC´ and

B_c_ are sides.

[Illustration:

/ S \

/ /|\ \

/ / | \ \

/ / | \ \

/ / | \ \

/ / | \ \

/ / | \ \

--------+-+------+------+-+

A B C D E ]

Let a body, acted upon by no force, be moving along the line AE; that

means, according to what has been said, let it pass over the equal

straight lines AB, BC, CD, DE, &c., in equal times. If we take any point

S not in the line AE, and join AS, BS, &c., the triangles ASB, BSC, &c.

are also equal, having a common altitude and standing on equal bases, so

that if a string were conceived reaching from S to the moving body

(being lengthened or shortened in each position to suit its distance

from S), this string, as the body moved along AE, would sweep over equal

triangular areas in equal times.

[Illustration]

Let us now examine how far these conclusions will be altered if the body

from time to time is forced towards S. We will suppose it moving

uniformly from A to B as before, no matter for the present how it got to

A, or into the direction AB. If left to itself it would, in an equal

time (say 1´´) go through BC´ in the same straight line with and equal

to AB. But just as it reaches B, and is beginning to move along BC´, let

it be suddenly pulled towards S with a motion which, had it been at

rest, would have carried it in the same time, 1´´ through any other

space B_c_. According to the second law of motion, its direction during

this 1´´, in consequence of the two motions combined, will be along BC,

the diagonal of the parallelogram of which BC´, B_c_, are sides. In

this case, as this figure is drawn, BC, though passed in the same time,

is longer than AB; that is to say, the body is moving quicker than at

first. How is it with the triangular areas, supposed as before to be

swept by a string constantly stretched between S and the body? It will

soon be seen that these still remain equal, notwithstanding the change

of direction, and increased swiftness. For since CC´ is parallel to

B_c_, the triangles SCB, SC´B are equal, being on the same base SB, and

between the same parallels SB, CC´, and SC´B is equal to SBA as before,

therefore SCB, SBA are equal. The body is now moving uniformly (though

quicker than along AB) along BC. As before, it would in a time equal to

the time of passing along BC, go through an equal space CD´ in the same

straight line. But if at C it has a second pull towards S, strong enough

to carry it to _d_ in the same time, its direction will change a second

time to CD, the diagonal of the parallelogram, whose sides are CD´,

C_d_; and the circumstances being exactly similar to those at the first

pull, it is shewn in the same manner that the triangular area SDC = SCB

= SBA.

Thus it appears, that in consequence of these intermitting pulls towards

S, the body may be moving round, sometimes faster, sometimes slower, but

that the triangles formed by any of the straight portions of its path

(which are all described in equal times), and the lines joining S to the

ends of that portion, are all equal. The path it will take depends of

course, in other respects, upon the frequency and strength of the

different pulls, and it might happen, if they were duly proportionate,

that when at H, and moving off in the direction HA´, the pull H_a_

might be such as just to carry the body back to A, the point from which

it started, and with such a motion, that after one pull more, A_b_, at

A, it might move along AB as it did at first. If this were so, the body

would continue to move round in the same polygonal path, alternately

approaching and receding from S, as long as the same pulls were repeated

in the same order, and at the same intervals.

It seems almost unnecessary to remark, that the same equality which

subsists between any two of these triangular areas subsists also between

an equal number of them, from whatever part of the path taken; so that,

for instance, the four paths AB, BC, CD, DE, corresponding to the four

areas ASB, BSC, CSD, DSE, that is, to the area ABCDES, are passed in the

same time as the four EF, FG, GH, HA, corresponding to the equal area

EFGHAS. Hence it may be seen, if the whole time of revolution from A

round to A again be called a year, that in half a year the body will

have got to E, which in the present figure is more than half way round,

and so of any other periods.

The more frequently the pulls are supposed to recur, the more frequently

will the body change its direction; and if the pull were supposed

constantly exerted in the direction towards S, the body would move in a

curve round S, for no three successive positions of it could be in a

straight line. Those who are not familiar with the methods of measuring

curvilinear spaces must here be contented to observe, that the law

holds, however close the pulls are brought together, and however closely

the polygon is consequently made to resemble a curve: they may, if they

please, consider the minute portions into which the curve is so divided,

as differing insensibly from little rectilinear triangles, any equal

number of which, according to what has been said above, wherever taken

in the curve, would be swept in equal times. The theorem admits, in this

case also, a rigorous proof; but it is not easy to make it entirely

satisfactory, without entering into explanations which would detain us

too long from our principal subject.

The proportion in which the pull is strong or weak at different

distances from the central spot, is called "_the law of the central or

centripetal force_," and it may be observed, that after assuming the

laws of motion, our investigations cease to have anything hypothetical

or experimental in them; and that if we wish, according to these

principles of motion, to determine the law of force necessary to make a

body move in a curve of any required form, or conversely to discover the

form of the curve described, in consequence of any assumed law of force,

the inquiry is purely geometrical, depending upon the nature and

properties of geometrical quantities only. This distinction between what

is hypothetical, and what necessary truth, ought never to be lost sight

of.

As the object of the present treatise is not to teach geometry, we shall

describe, in very general terms, the manner in which Newton, who was

the first who systematically extended the laws of motion to the heavenly

bodies, identified their results with the two remaining laws of Kepler.

His "Principles of Natural Philosophy" contain general propositions with

regard to any law of centripetal force, but that which he supposed to be

the true one in our system, is expressed in mathematical language, by

saying that the centripetal force varies inversely as the square of the

distance, which means, that if the force at any distance be taken for

the unit of force, at half that distance, it is two times twice, or four

times as strong; at one-third the distance, three times thrice, or nine

times as strong, and so for other distances. He shewed the probability

of this law in the first instance by comparing the motion of the moon

with that of heavy bodies at the surface of the earth. Taking LP to

represent part of the moon's orbit described in one minute, the line PM

between the orbit and the tangent at L would shew the space through

which the central force at the earth (assuming the above principles of

motion to be correct) would draw the moon. From the known distance and

motion of the moon, this line PM is found to be about sixteen feet. The

distance of the moon is about sixty times the radius of the earth, and

therefore if the law of the central force in this instance were such as

has been supposed, the force at the earth's surface would be 60 times

60, or 3600 times stronger, and at the earth's surface, the central

force would make a body fall through 3600 times 16 feet in one minute.

Galileo had already taught that the spaces through which a body would be

made to fall, by the constant action of the same unvarying force, would

be proportional to the squares of the times during which the force was

exerted, and therefore according to these laws, a body at the earth's

surface ought (since there are sixty seconds in a minute) to fall

through 16 feet in one second, which was precisely the space previously

established by numerous experiments.

[Illustration]

With this confirmation of the supposition, Newton proceeded to the

purely geometrical calculation of the law of centripetal[198] force

necessary to make a moving body describe an ellipse round its focus,

which Kepler's observations had established to be the form of the orbits

of the planets round the sun. The result of the inquiry shewed that this

curve required the same law of the force, varying inversely as the

square of the distance, which therefore of course received additional

confirmation. His method of doing this may, perhaps, be understood by

referring to the last figure but one, in which C_d_, for instance,

representing the space fallen from any point C towards S, in a given

time, and the area CSD being proportional to the corresponding time, the

space through which the body would have fallen at C in any other time

(which would be greater, by Galileo's law, in proportion to the squares

of the times), might be represented by a quantity varying directly as

C_d_, and inversely in the duplicate proportion of the triangular area

CSD, that is to say, proportional to C_d_/(SC × D_k_)², if D_k_ be drawn

from D perpendicular on SC. If this polygon represent an ellipse, so

that CD represents a small arc of the curve, of which S is the focus, it

is found by the nature of that curve, that C_d_/(D_k_)² is the same at

all points of the curve, so that the law of variation of the force in

the same ellipse is represented solely by 1/(SC)². If C_d_, &c. are

drawn so that C_d_/(D_k_)² is not the same at every point, the curve

ceases to be an ellipse whose focus is at S, as Newton has shewn in the

same work. The line to which (Dk)²/Cd is found to be equal, is one drawn

through the focus at right angles to the longest axis of the ellipse

till it meets the curve;—this line is called the _latus rectum_, and is

a third proportional to the two principal axes.

Kepler's third law follows as an immediate consequence of this

determination; for, according to what has been already shown, the time

of revolution round the whole ellipse, or, as it is commonly called,

the periodic time, bears the same ratio to the unit of time as the whole

area of the ellipse does to the area described in that unit. The area of

the whole ellipse is proportional in different ellipses to the rectangle

contained by the two principal axes, and the area described in an unit

of time is proportional to SC × D_k_, that is to say, is in the

subduplicate ratio of SC² × D_k_², or D_k_²/C_d_, when the force varies

inversely as the square of the distance SC; and in the ellipse, as we

have said already, this is equal to a third proportional to the

principal axes; consequently the periodic times in different ellipses,

which are proportional to the whole areas of the ellipses directly, and

the areas described in the unit of time inversely, are in the compound

ratio of the rectangle of the axes directly, and subduplicately as a

third proportional to the axes inversely; that is to say, the squares of

these times are proportional to the cubes of the longest axes, which is

Kepler's law.

FOOTNOTES:

[192] This mode of verifying configurations, though something of the

boldest, was by no means unusual. On a former occasion Kepler, wishing

to cast the nativity of his friend Zehentmaier, and being unable to

procure more accurate information than that he was born about three

o'clock in the afternoon of the 21st of October, 1751, supplied the

deficiency by a record of fevers and accidents at known periods of his

life, from which he deduced a more exact horoscope.

[193] Kepler probably meant his own mother, whose horoscope he in many

places declared to be nearly the same as his own.

[194] See Preliminary Treatise, p. 13.

[195] In allusion to the Harmonics of Ptolemy.

[196] This is a word borrowed from the Ptolemaic astronomy, according to

which the sun and planets are hurried from their places by the daily

motion of the _primum mobile_, and by their own peculiar motion seek to

regain or be restored to their former places.

[197] In other parts of his works, Kepler assumes the diminution to be

proportional to the circles themselves, not to the diameters.

[198] In many curves, as in the circle and ellipse, there is a point to

which the name of centre is given, on account of peculiar properties

belonging to it: but the term "centripetal force" always refers to the

place towards which the force is directed, whether or not situated in

the centre of the curve.

CHAPTER VIII.

_The Epitome prohibited at Rome—Logarithmic Tables—Trial of

Catharine Kepler—Kepler invited to England—Rudolphine

Tables—Death—Conclusion._

KEPLER'S "Epitome," almost immediately on its appearance, enjoyed the

honour of being placed by the side of the work of Copernicus, on the

list of books prohibited by the congregation of the Index at Rome. He

was considerably alarmed on receiving this intelligence, anticipating

that it might occasion difficulties in publishing his future writings.

His words to Remus, who had communicated the news to him, are as

follows:—"I learn from your letter, for the first time, that my book is

prohibited at Rome and Florence. I particularly beg of you, to send me

the exact words of the censure, and that you will inform me whether that

censure would be a snare for the author, if he were caught in Italy, or

whether, if taken, he would be enjoined a recantation. It is also of

consequence for me to know whether there is any chance of the same

censure being extended into Austria. For if this be so, not only shall I

never again find a printer there, but also the copies which the

bookseller has left in Austria at my desire will be endangered, and the

ultimate loss will fall upon me. It will amount to giving me to

understand, that I must cease to profess Astronomy, after I have grown

old in the belief of these opinions, having been hitherto gainsayed by

no one,—and, in short, I must give up Austria itself, if room is no

longer to be left in it for philosophical liberty." He was, however,

tranquillized, in a great degree, by the reply of his friend, who told

him that "the book is only prohibited as contrary to the decree

pronounced by the holy office two years ago. This has been partly

occasioned by a Neapolitan monk (Foscarini), who was spreading these

notions by publishing them in Italian, whence were arising dangerous

consequences and opinions: and besides, Galileo was at the same time

pleading his cause at Rome with too much violence. Copernicus has been

corrected in the same manner for some lines, at least in the beginning

of his first book. But by obtaining a permission, they may be read (and,

as I suppose, this "Epitome" also) by the learned and skilful in this

science, both at Rome and throughout all Italy. There is therefore no

ground for your alarm, either in Italy or Austria; only keep yourself

within bounds, and put a guard upon your own passions."

We shall not dwell upon Kepler's different works on comets, beyond

mentioning that they were divided, on the plan of many of his other

publications, into three parts, Astronomical, Physical, and

Astrological. He maintained that comets move in straight lines, with a

varying degree of velocity. Later theories have shewn that they obey the

same laws of motion as the planets, differing from them only in the

extreme excentricity of their orbits. In the second book, which contains

the Physiology of Comets, there is a passing remark that comets come out

from the remotest parts of ether, as whales and monsters from the depth

of the sea; and the suggestion is thrown out that perhaps comets are

something of the nature of silkworms, and are wasted and consumed in

spinning their own tails.

Among his other laborious employments, Kepler yet found time to

calculate tables of logarithms, he having been one of the first in

Germany to appreciate the full importance of the facilities they afford

to the numerical calculator. In 1618 he wrote to his friend Schickhard:

"There is a Scottish Baron (whose name has escaped my memory), who has

made a famous contrivance, by which all need of multiplication and

division is supplied by mere addition and subtraction; and he does it

without sines. But even he wants a table of tangents[199], and the

variety, frequency, and difficulty of the additions and subtractions, in

some cases, is greater than the labour of multiplying and dividing."

Kepler dedicated his "Ephemeris" for 1620 to the author of this

celebrated invention, Baron Napier, of Merchistoun; and in 1624,

published what he called "Chilias Logarithmorum," containing the

Napierian logarithms of the quotients of 100,000 divided by the first

ten numbers, then proceeding by the quotients of every ten to 100, and

by hundreds to 100,000. In the supplement published the following year,

is a curious notice of the manner in which this subtle contrivance was

at first received: "In the year 1621, when I had gone into Upper

Austria, and had conferred everywhere with those skilled in mathematics,

on the subject of Napier's logarithms, I found that those whose prudence

had increased, and whose readiness had diminished, through age, were

hesitating whether to adopt this new sort of numbers, instead of a table

of sines; because they said it was disgraceful to a professor of

mathematics to exult like a child at some compendious method of working,

and meanwhile to admit a form of calculation, resting on no legitimate

proof, and which at some time might entangle us in error, when we least

feared it. They complained that Napier's demonstration rested on a

fiction of geometrical motion, too loose and slippery for a sound method

of reasonable demonstration to be founded on it.[200] "This led me

forthwith to conceive the germ of a legitimate demonstration, which

during that same winter I attempted, without reference to lines or

motion, or flow, or any other which I may call sensible quality.

"Now to answer the question; what is the use of logarithms? Exactly what

ten years ago was announced by their author, Napier, and which may be

told in these words.—Wheresoever in common arithmetic, and in the Rule

of Three, come two numbers to be multiplied together, there the sum of

the logarithms is to be taken; where one number is to be divided by

another, the difference; and the number corresponding to this sum or

difference, as the case may be, will be the required product or

quotient. This, I say, is the use of logarithms. But in the same work in

which I gave the demonstration of the principles, I could not satisfy

the unfledged arithmetical chickens, greedy of facilities, and gaping

with their beaks wide open, at the mention of this use, as if to bolt

down every particular gobbet, till they are crammed with my

precepticles."

The year 1622 was marked by the catastrophe of a singular adventure

which befell Kepler's mother, Catharine, then nearly seventy years old,

and by which he had been greatly harassed and annoyed during several

years. From her youth she had been noted for a rude and passionate

temper, which on the present occasion involved her in serious

difficulties. One of her female acquaintance, whose manner of life had

been by no means unblemished, was attacked after a miscarriage by

violent headaches, and Catharine, who had often taken occasion to sneer

at her notorious reputation, was accused with having produced these

consequences, by the administration of poisonous potions. She repelled

the charge with violence, and instituted an action of scandal against

this person, but was unlucky (according to Kepler's statement) in the

choice of a young doctor, whom she employed as her advocate. Considering

the suit to be very instructive, he delayed its termination during five

years, until the judge before whom it was tried was displaced. He was

succeeded by another, already indisposed against Catharine Kepler, who

on some occasion had taunted him with his sudden accession to wealth

from a very inferior situation. Her opponent, aware of this advantage,

turned the tables on her, and in her turn became the accuser. The end

of the matter was, that in July, 1620, Catharine was imprisoned, and

condemned to the torture. Kepler was then at Linz, but as soon as he

learned his mother's danger, hurried to the scene of trial. He found the

charges against her supported only by evidence which never could have

been listened to, if her own intemperate conduct had not given advantage

to her adversaries. He arrived in time to save her from the question,

but she was not finally acquitted and released from prison till November

in the following year. Kepler then returned to Linz, leaving behind him

his mother, whose spirit seemed in no degree broken by the unexpected

turn in the course of her litigation. She immediately commenced a new

action for costs and damages against the same antagonist, but this was

stopped by her death, in April 1622, in her seventy-fifth year.

In 1620 Kepler was visited by Sir Henry Wotton, the English ambassador

at Venice, who finding him, as indeed he might have been found at every

period of his life, oppressed by pecuniary difficulties, urged him to go

over to England, where he assured him of a welcome and honourable

reception; but Kepler could not resolve upon the proposed journey,

although in his letters he often returned to the consideration of it. In

one of them, dated a year later, he says, "The fires of civil war are

raging in Germany—they who are opposed to the honour of the empire are

getting the upper hand—everything in my neighbourhood seems abandoned

to flame and destruction. Shall I then cross the sea, whither Wotton

invites me? I, a German? a lover of firm land? who dread the confinement

of an island? who presage its dangers, and must drag along with me my

little wife and flock of children? Besides my son Louis, now thirteen

years old, I have a marriageable daughter, a two-year old son by my

second marriage, an infant daughter, and its mother but just recovering

from her confinement." Six years later, he says again,—"As soon as the

Rudolphine Tables are published, my desire will be to find a place where

I can lecture on them to a considerable assembly; if possible, in

Germany; if not, why then in Italy, France, the Netherlands, or England,

provided the salary is adequate for a traveller."

In the same year in which he received this invitation an affront was put

upon Kepler by his early patrons, the States of Styria, who ordered all

the copies of his "Calendar," for 1624, to be publicly burnt. Kepler

declares that the reason of this was, that he had given precedence in

the title-page to the States of Upper Ens, in whose service he then was,

above Styria. As this happened during his absence in Wirtemberg, it was

immediately coupled by rumour with his hasty departure from Linz: it was

said that he had incurred the Emperor's displeasure, and that a large

sum was set upon his head. At this period Matthias had been succeeded by

Ferdinand III., who still continued to Kepler his barren title of

imperial mathematician.

In 1624 Kepler went to Vienna, in the hopes of getting money to complete

the Rudolphine Tables, but was obliged to be satisfied with the sum of

6000 florins and with recommendatory letters to the States of Suabia,

from whom he also collected some money due to the emperor. On his return

he revisited the University of Tubingen, where he found his old

preceptor, Mästlin, still alive, but almost worn out with old age.

Mästlin had well deserved the regard Kepler always appears to have

entertained for him; he had treated him with great liberality whilst at

the University, where he refused to receive any remuneration for his

instruction. Kepler took every opportunity of shewing his gratitude;

even whilst he was struggling with poverty he contrived to send his old

master a handsome silver cup, in acknowledging the receipt of which

Mästlin says,—"Your mother had taken it into her head that you owed me

two hundred florins, and had brought fifteen florins and a chandelier

towards reducing the debt, which I advised her to send to you. I asked

her to stay to dinner, which she refused: however, we handselled your

cup, as you know she is of a thirsty temperament."

The publication of the Rudolphine Tables, which Kepler always had so

much at heart, was again delayed, notwithstanding the recent grant, by

the disturbances arising out of the two parties into which the

Reformation had divided the whole of Germany. Kepler's library was

sealed up by desire of the Jesuits, and nothing but his connexion with

the Imperial Court secured to him his own personal indemnity. Then

followed a popular insurrection, and the peasantry blockaded Linz, so

that it was not until 1627 that these celebrated tables finally made

their appearance, the earliest calculated on the supposition that the

planets move in elliptic orbits. Ptolemy's tables had been succeeded by

the "Alphonsine," so called from Alphonso, King of Castile, who, in the

thirteenth century, was an enlightened patron of astronomy. After the

discoveries of Copernicus, these again made way for the Prussian, or

Prutenic tables, calculated by his pupils Reinhold and Rheticus. These

remained in use till the observations of Tycho Brahe showed their

insufficiency, and Kepler's new theories enabled him to improve upon

them. The necessary types for these tables were cast at Kepler's own

expense. They are divided into four parts, the first and third

containing a variety of logarithmic and other tables, for the purpose of

facilitating astronomical calculations. In the second are tables of the

elements of the sun, moon, and planets. The fourth gives the places of

1000 stars as determined by Tycho, and also at the end his table of

refractions, which appears to have been different for the sun, moon, and

stars. Tycho Brahe assumed the horizontal refraction of the sun to be 7´

30´´, of the moon 8´, and of the other stars 3´. He considered all

refraction of the atmosphere to be insensible above 45° of altitude, and

even at half that altitude in the case of the fixed stars. A more

detailed account of these tables is here obviously unsuitable: it will

be sufficient to say merely, that if Kepler had done nothing in the

course of his whole life but construct these, he would have well earned

the title of a most useful and indefatigable calculator.

Some copies of these tables have prefixed to them a very remarkable map,

divided by hour lines, the object of which is thus explained:—

"The use of this nautical map is, that if at a given hour the place of

the moon is known by its edge being observed to touch any known star, or

the edges of the sun, or the shadow of the earth; and if that place

shall (if necessary) be reduced from apparent to real by clearing it of

parallax; and if the hour at Uraniburg be computed by the Rudolphine

tables, when the moon occupied that true place, the difference will show

the observer's meridian, whether the picture of the shores be accurate

or not, for by this means it may come to be corrected."

This is probably one of the earliest announcements of the method of

determining longitudes by occultations; the imperfect theory of the moon

long remained a principal obstacle to its introduction in practice.

Another interesting passage connected with the same object may be

introduced here. In a letter to his friend Cruger, dated in 1616, Kepler

says: "You propose a method of observing the distances of places by

sundials and automata. It is good, but needs a very accurate practice,

and confidence in those who have the care of the clocks. Let there be

only one clock, and let it be transported; and in both places let

meridian lines be drawn with which the clock may be compared when

brought. The only doubt remaining is, whether a greater error is likely

from the unequal tension in the automaton, and from its motion, which

varies with the state of the air, or from actually measuring the

distances. For if we trust the latter, we can easily determine the

longitudes by observing the differences of the height of the pole."

In an Appendix to the Rudolphine Tables, or, as Kepler calls it, "an

alms doled out to the nativity casters," he has shown how they may use

his tables for their astrological predictions. Everything in his hands

became an allegory; and on this occasion he says,—"Astronomy is the

daughter of Astrology, and this modern Astrology, again, is the daughter

of Astronomy, bearing something of the lineaments of her grandmother;

and, as I have already said, this foolish daughter, Astrology, supports

her wise but needy mother, Astronomy, from the profits of a profession

not generally considered creditable."

Soon after the publication of these tables, the Grand Duke of Tuscany

sent him a golden chain; and if we remember the high credit in which

Galileo stood at this time in Florence, it does not seem too much to

attribute this honourable mark of approbation to his representation of

the value of Kepler's services to astronomy. This was soon followed by a

new and final change in his fortunes. He received permission from the

emperor to attach himself to the celebrated Duke of Friedland, Albert

Wallenstein, one of the most remarkable men in the history of that

time. Wallenstein was a firm believer in astrology, and the reception

Kepler experienced by him was probably due, in great measure, to his

reputation in that art. However that may be, Kepler found in him a more

munificent patron than any one of his three emperors; but he was not

destined long to enjoy the appearance of better fortune. Almost the last

work which he published was a commentary on the letter addressed, by the

missionary Terrentio, from China, to the Jesuits at Ingolstadt. The

object of this communication was to obtain from Europe means for

carrying into effect a projected scheme for improving the Chinese

calendar. In this essay Kepler maintains the opinion, which has been

discussed with so much warmth in more modern times, that the pretended

ancient observations of the Chinese were obtained by computing them

backwards from a much more recent date. Wallenstein furnished him with

an assistant for his calculations, and with a printing press; and

through his influence nominated him to the professorship in the

University of Rostoch, in the Duchy of Mecklenburg. His claims on the

imperial treasury, which amounted at this time to 8000 crowns, and which

Ferdinand would gladly have transferred to the charge of Wallenstein,

still remained unsatisfied. Kepler made a last attempt to obtain them at

Ratisbon, where the imperial meeting was held, but without success. The

fatigue and vexation occasioned by his fruitless journey brought on a

fever, which unexpectedly put an end to his life, in the early part of

November, 1630, in his fifty-ninth year. His old master, Mästlin,

survived him for about a year, dying at the age of eighty-one.

Kepler left behind him two children by his first wife, Susanna and

Louis; and three sons and two daughters, Sebald, Cordelia, Friedman,

Hildebert, and Anna Maria, by his widow. Susanna married, a few months

before her father's death, a physician named Jacob Bartsch, the same who

latterly assisted Kepler in preparing his "Ephemeris." He died very

shortly after Kepler himself. Louis studied medicine, and died in 1663,

whilst practising as a physician at Konigsberg. The other children died

young.

Upon Kepler's death the Duke of Friedland caused an inventory to be

taken of his effects, when it appeared that near 24,000 florins were due

to him, chiefly on account of his salary from the emperor. His daughter

Susanna, Bartsch's widow, managed to obtain a part of these arrears by

refusing to give up Tycho Brahe's observations till her claims were

satisfied. The widow and younger children were left in very straightened

circumstances, which induced Louis, Kepler's eldest son, to print, for

their relief, one of his father's works, which had been left by him

unpublished. It was not without much reluctance, in consequence of a

superstitious feeling which he did not attempt to conceal or deny.

Kepler himself, and his son-in-law, Bartsch, had been employed in

preparing it for publication at the time of their respective deaths; and

Louis confessed that he did not approach the task without apprehension

that he was incurring some risk of a similar fate. This little rhapsody

is entitled a "Dream on Lunar Astronomy;" and was intended to illustrate

the appearances which would present themselves to an astronomer living

upon the moon.

The narrative in the dream is put into the mouth of a personage, named

Duracoto, the son of an Icelandic enchantress, of the name of

Fiolxhildis. Kepler tells us that he chose the last name from an old map

of Europe in his house, in which Iceland was called Fiolx: Duracoto

seemed to him analogous to the names he found in the history of

Scotland, the neighbouring country. Fiolxhildis was in the habit of

selling winds to mariners, and used to collect herbs to use in her

incantations on the sides of Mount Hecla, on the Eve of St. John.

Duracoto cut open one of his mother's bags, in punishment of which she

sold him to some traders, who brought him to Denmark, where he became

acquainted with Tycho Brahe. On his return to Iceland, Fiolxhildis

received him kindly, and was delighted with the progress he had made in

astronomy. She then informed him of the existence of certain spirits, or

demons, from whom, although no traveller herself, she acquired a

knowledge of other countries, and especially of a very remarkable

country, called Livania. Duracoto requesting further information, the

necessary ceremonies were performed for invoking the demon; Duracoto and

his mother enveloped their heads in their clothing, and presently "the

screaking of a harsh dissonant voice began to speak in the Icelandic

tongue." The island of Livania is situated in the depths of ether, at

the distance of about 250000 miles; the road thence or thither is very

seldom open, and even when it is passable, mankind find the journey a

most difficult and dangerous one. The demon describes the method

employed by his fellow spirits to convey such travellers as are thought

fit for the undertaking: "We bring no sedentary people into our company,

no corpulent or delicate persons; but we pick out those who waste their

life in the continual use of post-horses, or who sail frequently to the

Indies; who are accustomed to live upon biscuit, garlic, dried fish, and

such abominable feeding. Those withered old hags are exactly fit for us,

of whom the story is familiar that they travel immense distances by

night on goats, and forks, and old petticoats. The Germans do not suit

us at all; but we do not reject the dry Spaniards." This extract will

probably be sufficient to show the style of the work. The inhabitants of

Livania are represented to be divided into two classes, the Privolvans

and Subvolvans, by whom are meant those supposed to live in the

hemisphere facing the earth, which is called the Volva, and those on the

opposite half of the moon: but there is nothing very striking in the

account given of the various phenomena as respects these two classes. In

some notes which were added some time after the book was first written,

are some odd insights into Kepler's method of composing. Fiolxhildis had

been made to invoke the dæmon with twenty-one characters; Kepler

declares, in a note, that he cannot remember why he fixed on this

number, "except because that is the number of letters in _Astronomia

Copernicana_, or because there are twenty-one combinations of the

planets, two together, or because there are twenty-one different throws

upon two dice." The dream is abruptly terminated by a storm, in which,

says Kepler, "I suddenly waked; the Demon, Duracoto, and Fiolxhildis

were gone, and instead of their covered heads, I found myself rolled up

among the blankets."

Besides this trifle, Kepler left behind him a vast mass of unpublished

writings, which came at last into the hands of his biographer, Hantsch.

In 1714, Hantsch issued a prospectus for publishing them by

subscription, in twenty-two folio volumes. The plan met no

encouragement, and nothing was published but a single folio volume of

letters to and from Kepler, which seem to have furnished the principal

materials for the memoir prefixed to them. After various unavailing

attempts to interest different learned bodies in their appearance, the

manuscripts were purchased for the library at St. Petersburg, where

Euler, Lexell, and Kraft, undertook to examine them, and select the most

interesting parts for publication. The result of this examination does

not appear.

Kepler's body was buried in St. Peter's churchyard at Ratisbon, and a

simple inscription was placed on his tombstone. This appears to have

been destroyed not long after, in the course of the wars which still

desolated the country. In 1786, a proposal was made to erect a marble

monument to his memory, but nothing was done. Kästner, on whose

authority it is mentioned, says upon this, rather bitterly, that it

matters little whether or not Germany, having almost refused him bread

during his life, should, a century and a half after his death, offer him

a stone.

Delambre mentions, in his History of Astronomy, that this design was

resumed in 1803 by the Prince Bishop of Constance, and that a monument

has been erected in the Botanical Garden at Ratisbon, near the place of

his interment. It is built in the form of a temple, surmounted by a

sphere; in the centre is placed a bust of Kepler, in Carrara marble.

Delambre does not mention the original of the bust; but says it is not

unlike the figure engraved in the frontispiece of the Rudolphine Tables.

That frontispiece consists of a portico of ten pillars, supporting a

cupola covered with astronomical emblems. Copernicus, Tycho Brahe,

Ptolemy, Hipparchus, and other astronomers, are seen among them. In one

of the compartments of the common pedestal is a plan of the observatory

at Uraniburg; in another, a printing press; in a third is the figure of

a man, meant for Kepler, seated at a table. He is identified by the

titles of his works, which are round him; but the whole is so small as

to convey very little idea of his figure or countenance. The only

portrait known of Kepler was given by him to his assistant Gringallet,

who presented it to Bernegger; and it was placed by the latter in the

library at Strasburg. Hantsch had a copy taken for the purpose of

engraving it, but died before it was completed. A portrait of Kepler is

engraved in the seventh part of Boissard's Bibliotheca Chalcographica.

It is not known whence this was taken, but it may, perhaps, be a copy of

that which was engraved by desire of Bernegger in 1620. The likeness is

said not to have been well preserved. "His heart and genius," says

Kästner, "are faithfully depicted in his writings; and that may console

us, if we cannot entirely trust his portrait." In the preceding pages,

it has been endeavoured to select such passages from his writings as

might throw the greatest light on his character, with a subordinate

reference only to the importance of the subjects treated. In conclusion,

it may be well to support the opinion which has been ventured on the

real nature of his triumphs, and on the danger of attempting to follow

his method in the pursuit of truth, by the judgment pronounced by

Delambre, as well on his failures as on his success. "Considering these

matters in another point of view, it is not impossible to convince

ourselves that Kepler may have been always the same. Ardent, restless,

burning to distinguish himself by his discoveries, he attempted

everything; and having once obtained a glimpse of one, no labour was too

hard for him in following or verifying it. All his attempts had not the

same success, and, in fact, that was impossible. Those which have failed

seem to us only fanciful; those which have been more fortunate appear

sublime. When in search of that which really existed, he has sometimes

found it; when he devoted himself to the pursuit of a chimera, he could

not but fail; but even there he unfolded the same qualities, and that

obstinate perseverance that must triumph over all difficulties but those

which are insurmountable."[201]

_List of Kepler's published Works._

Ein Calender _Gratz_, 1594

Prodromus Dissertat. Cosmograph. _Tubingæ_, 1596, 4to.

De fundamentis Astrologiæ _Pragæ_, 1602, 4to.

Paralipomena ad Vitellionem _Francofurti_, 1604, 4to.

Epistola de Solis deliquio 1605

De stellâ novâ _Pragæ_, 1606, 4to.

Vom Kometen _Halle_, 1608, 4to.

Antwort an Röslin _Pragæ_, 1609, 4to.

Astronomia Nova _Pragæ_, 1609, fol.

Tertius interveniens _Frankfurt_, 1610, 4to.

Dissertatio cum Nuncio Sidereo _Francofurti_, 1610, 4to.

Strena, seu De dive sexangulâ _Frankfurt_, 1611, 4to.

Dioptrica _Francofurti_, 1611, 4to.

Vom Geburts Jahre des Heylandes _Strasburg_, 1613, 4to.

Respons. ad epist S. Calvisiii _Francofurti_, 1614, 4to.

Eclogæ Chronicæ _Frankfurt_, 1615, 4to.

Nova Stereometria _Lincii_, 1615, 4to.

Ephemerides 1617-1620 _Lincii_, 1616, 4to.

Epitomes Astron. Copern. Libri i. ii. iii. _Lentiis_, 1618, 8vo.

De Cometis _Aug. Vindelic._ 1619, 4to.

Harmonice Mundi _Lincii_, 1619, fol.

Kanones Pueriles _Ulmæ_, 1620

Epitomes Astron. Copern. Liber iv. _Lentiis_, 1622, 8vo.

Epitomes Astron. Copern. Libri v. vi. vii. _Francofurti_, 1622, 8vo.

Discurs von der grossen Conjunction _Linz._ 1623, 4to.

Chilias Logarithmorum _Marpurgi_, 1624, fol.

Supplementum _Lentiis_, 1625, 4to.

Hyperaspistes _Francofurti_, 1625, 8vo.

Tabulæ Rudolphinæ _Ulmæ_, 1627, fol.

Resp. ad epist. J. Bartschii _Sagani_, 1629, 4to.

De anni 1631 phænomenis _Lipsæ_, 1629, 4to.

Terrentii epistolium cum commentatiunculâ _Sagani_, 1630, 4to.

Ephemerides. _Sagani_, 1630, 4to.

Somnium _Francofurti_, 1634, 4to.

Tabulæ mannales _Argentorati_, 1700, 12mo.

FOOTNOTES:

[199] The meaning of this passage is not very clear: Kepler evidently

had seen and used logarithms at the time of writing this letter; yet

there is nothing in the method to justify this expression,—"_At tamen

opus est ipsi Tangentium canone._"

[200] This was the objection originally made to Newton's "Fluxions," and

in fact, Napier's idea of logarithms is identical with that method of

conceiving quantities. This may be seen at once from a few of his

definitions,

1 Def. A line is said to increase uniformly, when the point by which

it is described passes through equal intervals, in equal times.

2 Def. A line is said to diminish to a shorter one proportionally,

when the point passing along it cuts off in equal times segments

proportional to the remainder.

6 Def. The logarithm of any sine is the number most nearly denoting

the line, which has increased uniformly, whilst the radius has

diminished to that sine proportionally, the initial velocity being

the same in both motions. (Mirifici logarithmorum canonis

descriptio, Edinburgi 1614.)

This last definition contains what we should now call the differential

equation between a number and the logarithm of its reciprocal.

[201] Histoire del'Astronomie Moderne, Paris, 1821.

Transcriber's Notes.

Corrections.

The first line indicates the original, the second the correction.

Life of Galileo Galilei

p. 20:

success very inadeqnate to the zeal

success very inadequate to the zeal

p. 20:

"New method of Guaging,

"New method of Gauging,

p. 23:

the knowlege, if it existed

the knowledge, if it existed

p. 30, note:

to represent terrestial objects correctly.

to represent terrestrial objects correctly.

p. 64:

the palace of the Archishop Piccolomini

the palace of the Archbishop Piccolomini

p. 68:

that ladies ringlets

that ladies' ringlets

p. 69:

For hitherto I have never happened to see the terrestial earth

For hitherto I have never happened to see the terrestrial earth

p. 106:

80 1 50, _for_ any _read_ an indefinitely small.

80 2 44, _for_ any _read_ an indefinitely small.

Life of Kepler

p. 6:

Now, inscribe in the Earth an icosaedron, the circle inscribed in it

will be Venus.

Now, inscribe in the Earth an icosahedron, the circle inscribed in

it will be Venus.

Inscribe an octaedron in Venus, the circle inscribed in it will be

Mercury.

Inscribe an octahedron in Venus, the circle inscribed in it will be

Mercury.

p. 32:

Butthere are no such means

But there are no such means

p. 48:

the compound ratio of the rectangle of the axes directly, and

subduplicatly

the compound ratio of the rectangle of the axes directly, and

subduplicately

p. 52:

and was in-intended to illustrate the appearances

and was intended to illustrate the appearances