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was born a male, and not a female, for seen from the exulting rhapsody with astrologers have sought in vain to dis- which he announced it.

“ What I protinguish sexes in the sky; thirdly, I des phecied two-and-twenty years ago, as rive from my mother a habit of body, soon as I discovered the five solids more fit for study than other kinds of among the heavenly orbits what I life · fourthly, my parents' fortune was firmly believed long before I had seen not large, and there was no landed pro. Ptolemy's • Harmonics '-- what I had perty to which I might succeed and be- promised my friends in the title of this come attached ; fifthly, there were the book, which I named before I was sure of schools, and the liberality of the magis- my discovery-what, sixteen years ago, I tracy towards such boys as were apt urged as a thing to be sought-that for for learning. But now if I am to which I joined Tycho Brahe, for which speak of the result of my studies, what I settled in Prague, for which have I pray can I find in the sky, even re- devoted the best part of my life to astro motely alluding to it. The learned con- nomical contemplations, at length I fess that several not despicable branches have brought to light, and have recogof philosophy have been nely extri- nized its truth beyond my most sancated or amended or brought to per- guine expectations. Great as is the fection by me: but here my constella- absolute nature of Harmonics with all tions were, not Mercury from the east, its details, as set forth in my third book, in the angle of the seventh, and in it is all found among the celestial moquadratures with Mars, but Copernicus, tions, not indeed in the manner which but Tycho Brahe, without whose books I imagined, (that is not the least part of of observations everything now set by my delight,) but in another very differme in the clearest light must have re- ent, and yet most perfect and excellent. mained buried in darkness ; not Saturn It is now eighteen months since I got predominating Mercury, but my Lords the first glimpse of light, three months the Emperors Rodolph and Matthias; since the dawn, very few days since the not Capricorn, the house of Saturn, but unveiled sun, most admirable to gaze Upper Austria, the home of the Em. on, burst out upon me. Nothing holds peror, and the ready and unexampled me; I will indulge in my sacred fury; bounty of his nobles to my petition. I will triumph over mankind by the Here is that corner, not the western one honest confession, that I have stolen of the horoscope, but on the Earth, the golden vases of the Egyptians*, to whither, by permission of my imperial build up a tabernacle for my God far master, I have betaken myself from a away from the confines of Egypt. If too uneasy court; and whence, during you forgive me, I rejoice; if you are these years of my life, which now tends angry, I can bear it: the die is cast, towards its setting, emanate these Har- the book is written; to be read either monies, and the other matters on which now or by posterity, I care not which: I am engaged."

it may well wait a century for a reader, “ However, it may be owing to Ju- as God has waited six thousand years piter's ascendancy that I take greater for an observer." delight in the application of geometry

He has told, with his usual particuto physics, than in that abstract pursuit larity, the manner and precise moment which partakes of the dryness of Saturn; of the discovery. “ Another part of my and it is perhaps the gibbous moon, in Cosmographical Mystery,' suspended the bright constellation of the Bull's twenty-two years ago, because it was forehead, which fills my mind with fan- then undetermined, is completed and intastic images."

troduced here, after I had discovered The most remarkable thing contained the true intervals of the orbits, by means in the 5th Book, is the announcement of Brahe's observations, and had spent of the celebrated law connecting the the continuous toil of a long time in inmean distances of the planets with the vestigating the true proportion of the periods of their revolution about the periodic times to the orbits, Sun. This law is expressed in mathe- Sera quidem respexit inertem, matical language, by saying that the Respexit tamen, et longo post tempore venit. squares of the times vary as the cubes If you would know the precise moment, of the distances*. Kepler's rapture on the first idea came across me on the 8th detecting it was unbounded, as may be March of this year, 1618; but chancing * See Preliminary Treatise, p. 13.

In allusion to the Harmonics of Ptolemy.

to make a mistake in the calculation, I riality, a natural inability to move from rejected it as false. I returned again to place to place : they have a natural init with new force on the 15th May, and ertness or quietude, in consequence of it has dissipated the darkness of my which they remain still in every situation mind by such an agreement between where they are placed alone." this idea and my seventeen years' labour P. Is it then the sun, which by its on Brahe's observations, that at first I turning carries round the planets ? How thought I must be dreaming, and had can the sun do this, having no hands to taken my result for granted in my first seize the planet at so great a distance, assumptions. But the fact is perfect, and force it round along with itself ?the fact is certain, that the proportion Its bodily virtue, sent forth in straight existing between the periodic times of lines into the whole space of the world, any two planets is exactly the sesquipli- serves instead of hands; and this virtue, cate proportion of the mean distances of being a corporeal species, turns with the the orbits.".

body of the sun like a very rapid vortex, There is high authority for not attempt- and travels over the whole of that space ing over anxiously to understand the which it fills as quickly as the sun rerest of the work. Delambre sums it up volves in its very confined space round as follows:-"In the music of the ce- the centre. lestial bodies it appears that Saturn and “ P. Explain what this virtue is, and Jupiter take the bass, Mars the tenor, belonging to what class of things ? the Earth and Venus the counter-tenor, As there are two bodies, the mover and and Mercury the treble." If the patience the moved, so are there two powers by of this indefatigable historian gave way, which the motion is obtained." The one as he confesses, in the perusal, any is passive, and rather belonging to further notice of it here may be well matter, namely, the resemblance of the excused. Kepler became engaged, in body of the planet to the body of the consequence of this publication, in an sun in its corporeal form, and so that angry controversy with the eccentric part of the planetary body is friendly, the Robert Fludd, who was at least Kepler's opposite part hostile to the sun. The match in wild extravagance and mysti- other power is active, and bearing more cism, if far inferior to him in genius. It relation to form, namely, the body of is diverting to hear each reproaching the the sun has a power of attracting the other with obscurity.

planet by its friendly part, of repelling In the “Epitome of the Copernican it by the hostile part, and finally, of reAstronomy," which Kepler published taining it if it be placed so that neither about the same time, we find the manner the one nor the other be turned directly in which he endeavoured to deduce the towards the sun. beautiful law of periodic times, from P. How can it be that the whole body his principles of motion and radiation of the planet should be like or cognate to of whirling forces. This work is in the body of the sun, and yet part of the fact a summary of all his astronomi- planet friendly, part hostile to the sun ? cal opinions, drawn up in a popular -Just as when one magnet attracts style in the form of question and an- another, the bodies are cognate; but at. swer. We find there a singular argu- traction takes place only on one side, rement against believing, as some did, pulsion on the other. that each planet is carried round by an P. Whence, then, arises that differangel, for in that case, says Kepler, ence of opposite parts in the same body? "the orbits would be perfectly circular; -In magnets the diversity arises from but the elliptic form, which we find in the situation of the parts with respect to them, rather smacks of the nature of the whole. In the heavens the matter is the lever and material necessity.” a little differently arranged, for the sun

The investigation of the relation be- does not, like the magnet, possess only tween the periodic times and distances on one side, but in all the parts of its of the planets is introduced by a query substance, this active and energetic fawhether or not they are to be considered culty of attracting, repelling, or retainheavy. The answer is given in the fol- ing the planet. So that it is probable lowing terms :-“ Although none of the that the centre of the solar body correcelestial globes are heavy, in the sense sponds to one extremity or pole of the in which we say on earth that a stone is magnet, and its whole surface to the heavy, nor light as fire is light with us, other pole. yet have they, by reason of their mate- P. If this were so, all the planets

would be restored* in the same time with The circular paths of the planets are in the sun ?-True, if this were all : but it the simple ratio of the distances; the has been said already that, besides this weights or quantities of matter in diffecarrying power of the sun, there is also in rent planets are in the subduplicate ratio the planets a natural inertness to motion, of the same distances, as has been which causes that, by reason of their already proved; so that with every inmaterial substance, they are inclined to crease of distance, a planet has more remain each in its place. The carrying matter, and therefore is moved more power of the sun, and the impotence or slowly, and accumulates more time in its material inertness of the planet

, are thus revolution, requiring already as it did in opposition. Each shares the victory; more time by reason of the length of the the sun moves the planet from its place, way. The third and fourth causes comalthough in some degree it escapes from pensate each other in a comparison of the chains with which it was held by the different planets: the simple and subsun, and so is taken hold of successively duplicate proportion compound the sesby every part of this circular virtue, or, quiplicate proportion, which therefore is as it may be called, solar circumference, the ratio of the periodic times.". namely, by the parts which follow those Three of the four suppositions here from which it has just extricated itself. made by Kepler to explain the beautiful

“P. But how does one planet extricate law he had detected, are now indisputaitself more than another from this vio- bly known to be false, Neither the lence-First, because the virtue emana- weights nor the sizes of the different ting from the sun has the same degree of planets observe the proportions assigned weakness at different distances, as the by him, nor is the force by which they distances or the width of the circles de- are retained in their orbits in any respect scribed on these distancest. This is the similar in its effects to those attributed principal reason. Secondly, the cause by him to it. The wonder which might is partly in the greater or less inertness naturally be felt that he should never. or resistance of the planetary globes, theless reach the desired conclusion, will which reduces the proportions to one- be considerably abated on examining the half; but of this more hereafter.

mode in which he arrived at and satisfied P. How can it be that the virtue ema- himself of the truth of these three supnating from the sun becomes weaker at positions. It has been already mentioned a greater distance? What is there to that his notions on the existence of a hurt or weaken it? Because that whirling force emanating from the sun, virtue is corporeal, and, partaking of and decreasing in energy at increased quantity, which can be spread out and distances, are altogether inconsistent rarefied. Then, since there is as much with all the experiments and observavirtue diffused in the vast orb of Sa- tions we are able to collect. His reason turn as is collected in the very narrow for asserting that the sizes of the difone of Mercury, it is very rare and there- ferent planets are proportional to their fore weak in Saturn's orbit, very dense distances from the sun, was simply beand therefore powerful at Mercury. cause he chose to take for granted that

P. You said, in the beginning of this either their solidities, surfaces, or diainquiry into motion, that the periodic meters, must necessarily be in that times of the planets are exactly in the proportion, and of the three, the solidities sesquiplicate proportion of their orbits or appeared to him least liable to objection. circles : pray what is the cause of this ? The last element of his precarious rea-Four causes concur for lengthening soning rested upon equally groundless the periodic time. First, the length of assumptions. Taking as a principle, that the path; secondly, the weight or quan- where there is a number of different tity of matter to be carried ; thirdly, the things they must be different in every degree of strength of the moving virtue; respect, he declared that it was quite fourthly, the bulk or space into which unreasonable to suppose all the planets is spread out the matter to be moved.

of the same density. He thought it in• This is a word borrowed from the Ptolemaic disputable that they must be rarer as they astronomy, according to which the sun

were farther from the sun, "and yet not planets are hurried from their places by the daily in the proportion of their distances, for motion of the primum mobile, and by their own

thus we should sin against the law of peculiar motion seek to regain or be restored to their former places.

variety in another way, and make the + In other parts of his works Kepler assumes quantity of matter (according to what he the diminution to be proportional to the circles themselves, not to the diameters,

had just said of their bulk) the same in


all. But if we assume the ratio of the diagonal of the parallelogram of which quantities of matter to be half that of the B C'and B care sides. distances, we shall observe the best mean Let a body, acted upon by no force, of all; for thus Saturn will be half as be moving along the line AE; that heavy again as Jupiter, and Jupiter half

S 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

А в с D Ε from any given principles.

It may not be uninstructive to subjoin means, according to what has been said, a sketch of the manner in which Newton let it pass over the equal straight lines established the same celebrated results, AB, BC, CD, DE, &c., in equal times. starting from principles of motion dia- If we take any point S not in the line metrically opposed to Kepler's, and it A E, and join A S, B S, &c., the triangles need scarcely be added, reasoning upon AS B, BSC, &c. are also equal, having them in a manner not less different. a common altitude and standing on For this purpose, a very few prefatory equal bases, so that if a string were conremarks will be found sufficient.

ceived reaching from s to the moving The different motions seen in nature body (being lengthened or shortened in are best analysed and classified by sup- each position to suit its distance from posing that every body in motion, if left S), this string, as the body moved along to itself, will continue to move forward A E, would sweep over equal trianat the same rate in a straight line, and gular areas in equal times. by considering all the observed devia- Let us now examine how far these tions from this manner of moving, as

F 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

H 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

B a string fastened to the body: we shall

A also have occasion presently to mention conclusions will be altered if the body some forces where no visible connexion from time to time is forced towards s. exists between the moving body and we will suppose it moving uniformly that towards which the motion takes from A to B as before, no matter for the place, and from which the force is said present how it got to A, or into the to proceed.

direction A B. If left to itself it would, A second law of motion, founded upon in an equal time (say 1").go through experiment, is this: if a body have mo- B C' in the same straight line with and tion communicated to it in two directions, equal to AB. But just as it reaches by one of which motions alone it would B, and is beginning to move along BC", have passed through a given space in a let it be suddenly pulled towards S with given time, as for instance, through B C a motion which, had it been at rest, in one second, and by the other alone would have carried it in the same time, through any other space Bc in the same l", through any other space B c. Ac

time, it will, when both are cording to the second law of motion, its given to it at the same in direction during this l", in consequence

stant, pass in the same of the two motions combined, will be B

time (in the present in- along B C, the diagonal of the parallelostance in one second) through BC the gram of which B C', B e, are sides. In

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this case, as this figure is drawn, BC, responding to _the_four areas ASB,
though passed in the same time, is longer BSC, CSD, DS E, that is, to the area
than AB; that is to say, the body is ABCDES, are passed in the same
moving quicker than at first. How is it time as the four E F, FG, GH, H A, cor-
with the triangular areas, supposed as responding to the equal area E FGHAS.
before to be swept by a string constantly Hence it may be seen, if the whole
stretched between S and the body? It time of revolution from A round to A
will soon be seen that these still remain again be called a year, that in half a
equal, notwithstanding the change of year the body will have got to E, which
direction, and increased swiftness. For in the present figure is more than half
since CC' is parallel to B c, the tri- way round, and so of any other pe-
angles SCB, SCB are equal, being riods.
on the same base S B, and between The more frequently the pulls are
the same parallels SB, C C', and S C'B supposed to recur, the more frequently
is equal to SBA as before, therefore will the body change its direction, and if
SCB, S B A are equal. The body is the pull were supposed constantly ex-
now moving uniformly (though quicker erted in the direction towards S, the body
than along AB) along B C. As before, would move in a curve round S, for no
it would in a time equal to the time of three successive positions of it could be
passing along B C, go through an equal in a straight line. Those who are not
space CD' in the same straight line, familiar with the methods of measuring
But if at Cit has a second pull towards curvilinear spaces must here be con-
S, strong enough to carry it to d in the tented to observe, that the law holds,
same time, its direction will change a however close the pulls are brought to-
second time to CD, the diagonal of the gether, and however closely the polygon
parallelogram, whose sides are CD', Cd; is consequently made to resemble a
and the circumstances being exactly curve: they may, if they please, consider
similar to those at the first pull, it is the minute portions into which the curve
shewn in the same manner that the is so divided, as differing insensibly
triangular area SDC=SC B=S BA. from little rectilinear triangles, any equal

Thus it appears, that in consequence number of which, according to what has
of these intermitting pulls towards S, been said above, wherever taken in the
the body may be moving round, some- curve, would be swept in equal times.
times faster, sometimes slower, but that The theorem admits, in this case also,
the triangles formed by any of the a rigorous proof; but it is not easy to
straight portions of its path (which are make it entirely satisfactory, without
all described in equal times), and the entering into explanations which would
lines joining S to the ends of that por- detain us too long from our principal
tion, are all equal. The path it will take subject.
depends of course, in other respects, The proportion in which the pull
upon the frequency and strength of the is strong or weak at different dis-
different pulls, and it might happen, if tances from the central spot, is called
they were duly proportionate, that when the law of the central or centripetal
at H, and moving off in the direction force," and it may be observed, that
HA', the pull H a might be such as just after assuming the laws of motion, our
to carry the body back to A, the point investigations cease to have anything
from which it started, and with such a hypothetical or experimental in them;
motion, that after one pull more, Ab, at and that if we wish, according to these
A, it might move along A B as it did at principles of motion, to determine the
first. If this were so, the body would law of force necessary to make a body
continue to move round in the same move in a curve of any required form,
polygonal path, alternately approaching or conversely to discover the form of
and receding from S, as long as the the curve described, in consequence of
same pulls were repeated in the same any assumed law of force, the inquiry
order, and at the same intervals. is purely geometrical, depending upon

It seems almost unnecessary to re- the nature and properties of geometrical mark, that the same equality which sub- quantities only. This distinction besists between any two of these triangular tween what is hypothetical, and what areas subsists also between an equal necessary truth, ought never to be lost number of them, from whatever part of sight of. the path taken; so that, for instance, the As the object of the present treatise four paths AB, BC, CD, DE, coris not to teach geometry, we shall de

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