Imágenes de páginas

culation ;, but if any one will reflect portion between these two circles struck that two things were wanting to me, the eye as almost identical with that first, that I did not know the size of the between Saturn and Jupiter, and the Sinus Totus, that is, the radius of the triangle is the first figure, just as Saproposed quadrant; secondly, that the turn and Jupiter are the first planets. energies of the motions were not thus On the spot I tried the second distance expressed otherwise than in relation one between Jupiter and Mars with a square, to another ; whoever, I say, well consi- the third with a pentagon, the fourth ders this, will doubt, not without reason, with a hexagon. And as the eye again as to the progress I was likely to make cried out against the second distance in this difficult course. And yet, with between Jupiter and Mars, I combined unremitting labour, and an infinite re- the square with a triangle and a penciprocation of sines and arcs, I did tagon. There would be no end of menget so far as to be convinced that this tioning every trial. The failure of this theory could not hold.

fruitless attempt was the beginning of “ Álmost the whole summer was lost the last fortunate one ; for I reflected, in these annoying labours ; at last, by a that in this way I should never reach trifling accident, I lighted more nearly the sun, if I wished to observe the same on the truth. I looked on it as an in- rule throughout; nor should I have terposition of Providence, that I should any reason why there were six, rather obtain by chance, what I had failed to than twenty or a hundred moveable discover with my utmost exertions; and orbits. And yet figures pleased me, as I believed this the more, because I being quantities, and as having existed prayed constantly that I might succeed, before the heavens; for quantity was if Copernicus had really spoken the created with matter, and the heavens truth. It happened on the 9th or 19th* afterwards. But if (this was the current day of July, in the year 1595, that, of my thoughts), in relation to the quanhaving occasion to show, in my lecture- tity and proportion of the six orbits, as room, the passages of the great con- Copernicus has determined them among junctions through eight signs, and how the infinite other figures, five only could they pass gradually from one trine as- be found having peculiar properties above pect to another, I inscribed in a circle the rest, my business would be done.

And then again it struck me, what have plane figures to do among solid orbits ? Solid bodies ought rather to be introduced. This, reader, is the invention and the whole substance of this little work; for if any one, though but moderately skilled in geometry, should hear these words hinted, the five regular solids will directly occur to him with the proportions of their circumscribed and inscribed spheres: he has immediately before his eyes that scholium of Euclid to the 18th proposition of his 13th Book, in which it is proved to be impossible that there should be, or be imagined, more than five regular bodies.

“What is worthy of admiration (since

I had then no proof of any prerogatives a great number of triangles, or quasi- of the bodies with regard to their order) triangles, so that the end of one was is, that employing a conjecture which made the beginning of another. In this was far from being subtle, derived from manner a smaller circle was shadowed the distances of the planets, I should at out by the points in which the lines once attain my end so happily in arrangcrossed each other.

ing them, that I was not able to change “ The radius of a circle inscribed in anything afterwards with the utmost exa triangle is half the radius of that ercise of my reasoning powers. In medescribed about it; therefore the pro- mory of the event, I write down here for

you the sentence, just as it fell from me, This inconvenient mode of dating was neces and in the words in which it was that sary before the new or Gregorian style was uni. versally adopted.

moment conceived :- The Earth is the

A Scheme of the great Conjunctions of SATURN & JUPITER, their leaps through eight Signs, and their passages through all the four Triplicities

of the Zodiac.

circle, the measurer of all; round it de- this will be Saturn. Now, inscribe in seribe a dodecahedron, the circle in- the Earth an icosaedron, the circle including this will be Mars. Round Mars scribed in it will be Venus. Inscribe an describe a tetrahedron, the circle includ- octaedron in Venus, the circle inscribed ing this will be Jupiter. Describe a in it will be Mercury. This is the reason cube round Jupiter, the circle including of the number of the planets.

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“This was the cause, and such the suc- you acknowledge this feeling, and are cess, of my labour: now read my propo- candid, you will refrain from blame, such sitions in this book. The intense plea- as not without cause I anticipate ; but sure I have received from this discovery if, leaving that to itself, you fear lest never can be told in words. I regretted these things be not ascertained, and no more the time wasted; I tired of no that I have shouted triumph before viclabour; I shunned no toil of reckoning; tory, at least approach these pages, and days and nights I spent in calculations, learn the matter in consideration : you until I could see whether this opinion will not find, as just now, new and unwould agree with the orbits of Coper. known planets interposed; that boldness nicus, or whether my joy was to vanish of mine is not approved, but those old into air. I willingly subjoin that senti- ones very little loosened, and so furnished ment of Archytas, as given by Cicero: by the interposition (however absurd you .If I could mount up into heaven, and may think it) of rectilinear figures, that thoroughly perceive the nature of the in future you may give a reason to the world, and beauty of the stars, that ad- rustics when they ask for the hooks miration would be without a charm for which keep the skies from falling. me, unless I had some one like you, Farewell." reader, candid, attentive, and eager for In the third chapter Kepler mentions, knowledge, to whom to describe it. If that a thickness must be allowed to

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21, 22

one of

each orb sufficient to include the greatest parison with the real distances are as and least distance of the planet from the follows:sun. The form and result of his com.

Book v. If the Saturn

be taken at
= 577

635 Ch. 9. inner Jupiter

Mars = 333

According to 333 14 surface Mars

1000, then

= 795

Coperuicus 757 19 of the

the outer Earth

Venus = 795

they are 794 orbit of Venus Mercury = 577

723 27 It will be observed, that Kepler's re- planet placed above it, and underneath sults were far from being entirely satis- the days due to the other inferior plafactory; but he seems to have flattered nets, if they observed the proportion of himself, that the differences might be distance. Hence it appears that this attributed to erroneous measurements. proportion in every case gives a time Indeed, the science of observation was greater than the truth; as for instance, then so much in its infancy, that such if the earth's rate of revolution were to an assertion might be made without in. Jupiter's in the proportion of their discurring much risk of decisive refutation. tances, the second column shows that

Kepler next endeavo red to deter the time of her period would be 843 inmine why the regular solids followed in stead of 365 days; so of the rest. His this rather than any other order; and next attempt was to compare them by his imagination soon created a variety of two by two, in which he found that he essential distinctions between the cube, arrived at a proportion something like pyi amid, and dodecahedron, belonging the proportion of the distances, although to the superior planets, and the other two. as yet far from obtaining it exactly. This

The next question examined in the process amounts to taking the quotients book, is the reason why the zodiac is obtained by dividing the period of each divided into 360 degrees;; and on this planet by the period of the one next subject, he soon becomes enveloped in beyond. a variety of subtle considerations, (not the h 10759.27 be successively

4 403 very intelligible in the original, and still

2 4332.37

159 more difficult to explain shortly to others

1000 equal parts,

the periods of 0532 unacquainted with it,) in relation to the

the planet next divisions of the musical scale; the origin


2 오 of which he identifies with his five fa

of those parts in

$392 vourite solids. The twentieth chapter

But if the distance of each planet in 4 572 is appropriated to a more interesting

~ 290

1000 equal purts, the distance of $ 658 inquiry, containing the first traces of the next below will contain, ac $ 719 his finally successful researches into the

cording to Copernicus, in

$ 500 proportion between the distances of the From this table he argued that to make planets, and the times of their motions the proportions agree, we must assume round the sun. He begins with the one of two things, “ either that the generally admitted fact, that the more moving intelligences of the planets are distant planets move more slowly; but weakest in those which are farthest from in order to show that the proportion, the Sun, or that there is one moving whatever it may be, is not the simple intelligence in the Sun, the common one of the distances, he exhibits the centre forcing them all round, but those following little Table:

most violently which are nearest, and h

that it languishes in some sort, and 24

grows weaker at the most distant, be

cause of the remoteness and the atte. 3 10759.12 D. Scr.

nuation of the virtue." 46159

We stop here to insert a note added

by Kepler to the later editions, and q 위

shall take advantage of the same in#1174

terruption to warn the reader not to confound this notion of Kepler with the

theory of a gravitating force towards the 167 115

Sun, in the sense in which we now use At the head of each vertical column those words. According to our theory, is placed the real time (in days and sex- the effect of the presence of the Sun agesimal parts) of the revolution of the upon the planet is to pull it towards the

taken to consist of

the Periods of



below will contain

succession be taken to consist of

D. Scr.

4332.37 D.Scr.



686.39 D.Scr.



365.15 D.Scr.



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in 7 574

But accord 572

The distance
the distance




centre in a straight line, and the effect of and conversely the increase of the pe-
the motion thus produced combined with riods will be double the difference of the
the motion of the planet, which if un- distances. Therefore, half the incre-
disturbed would be in a straight line ment added to the shorter period ought
inclined to the direction of the radius, is, to give the true proportion of the dis-
that it describes a curve round the Sun. tances, so that the sum should represent
Kepler considered his planets as per- the distance of the superior planet, on
fectly quiet and unwilling to move when the same scale on which the shorter
left alone; and that this virtue supposed period represents the distance of the in-
by him to proceed in every direction out ferior one. For instance, the period of
of the Sun, swept them round, just as the Mercury is nearly 88 days; that of Ve-
sails of a windmill would carry round nus is 224], the difference is 136}: half
anything which became entangled in of this is 68}, which, added to 88, gives
them. In other parts of his works 156. The mean distance of Venus
Kepler mentions having speculated on ought, therefore, to be, in proportion to
á real attractive force in the centre; but that of Mercury, as 1564 to 88. If this be
as he knew that the planets are not done with all the planets, we get the fol-
always at the same distance from the lowing results, taking successively, as be-
Sun, and conceived erroneously, that to fore, the distance of each planet at 1000.
remove them from their least to their
greatest distance a repulsive force must

of which 274

ing to Co-290 be supposed alternating with an attrac

pernicus the next superior 694

they are tive one, he laid aside this notion as

contains 762

respectively 719 improbable. In a note he acknowledges 1000, is at

$ 563 that when he wrote the passage just As you see, we have now got nearer quoted, imbued as he then was with the truth." Scaliger's notions on moving intelli Finding that this theory of the rate gences, he literally believed “ that each of diminution would not bring him quite planet was moved by a living spirit, but close to the result he desired to find, afterwards came to look on the moving Kepler immediately imagined another, cause as a corporeal though immaterial This latter occasioned him a great deal substance, something in the nature of of perplexity, and affords another of light which is observed to diminish simi- the frequently recurring instances of larly at increased distances.". He then the waste of time and ingenuity occaproceeds as follows in the original text. sioned by his impetuous and precipitate

“ Let us then assume, as is very pro- temperament. Assuming the distance bable, that motion is dispensed by the of any planet, as for instance of Mars, sun in the same manner as light. The to be the unit of space, and the virtue at proportion in which light emanating that distance to be the unit of force, he from a centre is diminished, is taught supposed that as many particles as the by optical writers: for there is the same virtue at the Earth gained upon that of quantity of light, or of the solar rays, in Mars, so many particles of distance did the small circles as in the large; and the Earth lose. He endeavoured to detherefore, as it is more condensed in the termine the respective positions of the former, more attenuated in the latter, a planets upon this theory, by the rules of measure of the attenuation may be de. false position, but was much astonished rived from the proportion of the circles at finding the same exactly as on his themselves, both in the case of light and former hypothesis. The fact was, as he of the moving virtue. Therefore, by how himself discovered, although not until much the orbit of Venus is greater than after several years, that he had become that of Mercury, in the same proportion confused in his calculation; and when will the motion of the latter be stronger, half through the process, had retraced or more hurried, or more swift, or more his steps so as of course to arrive again powerful, or by whatever other word at the numbers from which he started, you like to express the fact, than that of and which he had taken from his former the former. But a larger orbit would results. This was the real secret of the require a proportionably longer time of identity of the two methods; and if, revolution, even though the moving force when he had taken the distance of Mars were the same. Hence it follows that at 1000, instead of assuming the distance the one cause of a greater distance of of the earth at 694, as he did, he had the planet from the Sun, produces a taken any other number, and operated double effect in increasing the period, upon it in the same manner, he would


have had the same reason for relying on dresses, in 1596, to Barbara Muller von the accuracy of his supposition. As it Muhleckh. This lady was already a was, the result utterly confounded him; widow for the second time, although two and he was obliged to leave it with the years younger than Kepler himself. On remark, that “ the two theories are thus occasion of this alliance he was required proved to be the same in fact, and only to prove the nobility of his family, and different in form; although how that the delay consequent upon the inquiry can possibly be, I have never to this postponed the marriage till the followday been able to understand.”—Hising year. He soon became involved perplexity was very reasonable; they in difficulties in consequence of this are by no means the same; it was only inconsiderate engagement: his wife's his method of juggling with the figures fortune was less than he had been led which seemed to connect them.

to expect, and he became embroiled on Notwithstanding all its faults, the that account with her relations. Still genius and unwearied perseverance dis- more serious inconvenience resulted to played by Kepler in this book, immedi- him from the troubled state in which the ately ranked him among astronomers of province of Styria was at that time, the first class; and he received the most arising out of the disputes in Boheflattering encomiums from many of the mia and the two great religious parties most celebrated; among others, from into which the empire was now divided, Galileo and Tycho Brahe, whose opinion the one headed by Rodolph, the feeble he invited upon his performance. Galileo minded emperor,--the other by Matthias, contented himself with praising in ge- his ambitious and enterprising brother. neral terms the ingenuity and good faith In the year following his marriage, he which appeared so conspicuously in it. thought it prudent, on account of some Tycho Brahe entered into a more de- opinions he had unadvisedly promultailed criticism of the work, and, as gated, (of what nature does not very Kepler shrewdly remarked, showed how distinctly appear,) to withdraw himself highly he thought of it by advising him from Gratz into Hungary. Thence he to try to adapt something of the same transmitted several short treatises to his kind to the Tychonic system. Kepler friend Zehentmaier, at Tubingen—"On also sent a copy of his book to the the Magnet,” “On the Cause of the imperial astronomer, Raimar, with a Obliquity of the Ecliptic,” and “ On the complimentary letter, in which he exalted Divine Wisdom, as shown in the Creahim above all other astronomers of the tion." Little is known of these works age. Raimar had surreptitiously ac- beyond the notice taken of them in Zequired a notion of Tycho Brahe's theory, hentmaier's answers. Kepler ha's himself and published it as his own; and Tycho, told us, that his magnetic philosophy in his letter, complained of Kepler's ex was built upon the investigations of travagant flattery. This drew a long Gilbert, of whom he always justly spoke apologetical reply from Kepler, in which with the greatest respect. he attributed the admiration he had ex About the same time a more violent pressed of Raimar to his own want of persecution had driven Tycho Brahe from information at that time, having since his observatory of Uraniburg, in the little met with many things in Euclid and island of Hueen, at the entrance of the Regiomontanus, which he then believed Baltic. This had been bestowed on him original in Raimar. With this explana- by the munificence of Frederick I. of tion, Tycho professed himself perfectly Denmark, who liberally furnished him satisfied.

with every means of prosecuting his

astronomical observations. After FreCHAPTER II.

derick's death, Tycho found himself unKepler's Marriage-He joins Tycho

able to withstand the party which had Brahe at Prague--Is appointed Im- constantly opposed him, and was forced,

at perial Mathematician-Treatise on

a great loss and much inconvenience, the New Star.

to quit his favourite island. On the in

vitation of the emperor, Rudolph II., The publication of this extraordinary he then betook himself, after a short book, early as it occurs in the history stay at Hamburg, to the castle of Beof Kepler's life, was yet preceded by his nach, near Prague, which was assigned marriage. He had contemplated this to him with an annual pension of three step so early as 1592; but that suit thousand florins, a truly munificent prohaving been broken off, he paid his ad- vision in those times and that country.

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