Epitome Astronomiae Copernicanae: Liber IV, part 9
Set forth the evident hypothesis, how any planet completes its circuits and meanwhile is drawn and thrust. Let us begin from the moment when the magnetic fibres present their side to the Sun, so that both ends of the fibres are equally distant from it, and this is at the greatest distance of all, A: then the Sun neither expels the planet nor allures it, but, as it were doubtful between the two, yet holds it, and by the rotation of its body and of the emitted form advances what it holds from A toward B, overcoming its resistance, and overcome in turn by it, so that it loses it as if from its hands, that is, from the rays A which precede of the power-bearing form, and receives it in the following ones H, and that in a certain proportion of the power of the form at that interval. The planet being thus advanced, while meanwhile the magnetic fibres by force of direction look to the same region of the world, it comes about that the region friendly to the Sun is little by little turned toward the Sun, and the discordant refuses the Sun: then therefore the globe begins to be drawn by the Sun, little, if the distances of the extremities from the Sun differ little from each other; by which drawing the planet from the breadth of the circle begun in A is gradually received inward about B toward the Sun, as into a narrower compass and into a stronger holding power, since denser, from which it therefore extricates itself less and is carried off the more swiftly. This attraction, very slow at first near A, is most rapid when the Sun has in view the whole friendly hemisphere of the planetary body, while the discordant is wholly hidden behind the body of the planet, that is, when the magnetic fibres are directed straight at the Sun itself, which happens about C, a quadrant of the whole circular compass; thence toward D this attraction again becomes slacker toward the Sun, but the swiftness of progress in the circle continues to grow, since the interval between the planet and the Sun is still decreasing (through the attraction). This slackening of attraction, after C, is at first almost nothing, soon felt more and more, as the part hostile to the planet puts itself forward and presents itself to the Sun to be seen, toward D, until, half the circuit being completed, at E again each hemisphere of the globe flying across looks equally at the Sun; for then all attraction ceases and the planet is nearest the Sun and therefore swiftest, as one that contends with the densest and therefore strongest holding power and extricates itself least from it as it goes round.
Immediately, however, the globe having passed this place E of its orbit toward F, because now the discordant hemisphere becomes nearer the Sun than the other, the friendly, and inclines more and more to the Sun, the planet also begins to be thrust out from the Sun, as from the narrower and denser orb of the solar form into a wider, rarer and weaker, whence the decrements of its motion also follow, and that in the contrary order, first more slowly, after E toward F, then, where the whole discordant hemisphere or region of the fibres is directed straight at the Sun, while the friendly region is turned away from the Sun, the planet is expelled most rapidly, but its motion has now again languished to a mean. This again happens about G, the other quadrant of the circuit. When the planet is advanced beyond toward H, this expulsion again slackens, until it wholly vanishes in A, the planet being restored to its former place and expelled farthest from the Sun.
But it is incredible that the planet, this liberty being allowed, should, the return being completed, be restored most exactly to the same interval. Here at last is the genuine place for that excuse of Ptolemy described above (p. 337), warning us that nothing occurs in heaven that hinders the motions natural to each body or makes them as it were wander from their paths. And so if laws of motions are such as nature has established, that the planet returns into itself most exactly, this also will happen most certainly, although without the fetters of orbs, in the free ether. But the laws which we have described are so framed. For the halves of the circuit are equal to each other, the one in which the planet is attracted, the other in which it is expelled; equal times are found of each half; the power of the Sun also is the same and perpetual, both that which attracts and that which expels; and its proportion to the inertia of the planet is always the same, in a body that is perennial: so much therefore it gains through one half by attracting as through the other by expelling. Why then should we distrust the restoration of the planetary body to its former interval within one period of time? Do not even in these earthly and violent motions movable things become separated from that which was the cause of motion, as in scorpions, ballistae, catapults, bombards, slings? and projected missiles pass through the free air, and yet strike the destined place no less, and there are marksmen and slingers who are a marvel, with an inimitable certainty of aim? If here the form of that motion which was for a moment directed in the impelling thing to a certain region, impressed on the movable for a short time and vanishing, can do so much that the movable, as long as it is carried by a form not yet wholly languished, does not cease to tend to the destined region: with how much firmer defences will the certainty of the heavenly returns be fortified, which the inner and wholly grown-together and therefore perennial fibres of the movable thing govern, since there the air disturbs the motions by impact and resistance, while here the density of the ether to be traversed is plainly nothing to the effect, even the slightest?
Why are the librations of the different planets not in the same proportion to their mean distances, that is, why is the eccentricity of Mercury greatest, after it that of Mars, after this those of Saturn, Jupiter and the Earth, and the least that of Venus? The instrumental cause is the different strength of the fibres, made so by nature or by situation, but the final cause is the same as that of the eccentricities themselves, namely that from these eccentricities the swiftest and slowest motions of the planets might arise of so great a measure as would suffice for representing the harmonies by them. To this belongs Book V of my Harmonics.
There remains one doubt concerning the direction of the fibres toward the same region of the world: for since you have said that one region of the fibres has kinship with the Sun and the other is at variance with the Sun, so that the Sun according to the one or the other either attracts the very body of the planet or expels it, the Sun seems to be able to do what is less on the planet, namely to remove these fibres from their parallel situation and turn them toward itself, more quickly than the planet is transported into such a situation from which the fibres can look toward the Sun. It has nothing absurd that something of this sort happen, that the Sun so struggles with the direction of the fibres as it struggles with the inertia of the body toward local motion, provided we hold this, that the Sun gains less in inclining the fibres than in moving the whole body from its place, as it also gains less in attracting the planet; which tempering belongs to the design of the Creator, that the planets should not come to contact with the Sun, if they were not transported in a shorter time into the opposite half of the circuit than the whole interval can be consumed by the direct attraction of the fibre. Since then the circulation of the planet about the Sun anticipates the inclination of the fibres, it will come about that although in one quadrant of the circuit the fibres are somewhat inclined, the friendly region toward the Sun, the discordant away from the Sun; yet because the planet is carried into the other quadrant more quickly than the total inclination of the fibres is made (whence there follows as much the exchange of situation of the regions contrary to one another, turned toward the Sun, as if no inclination of the fibres had been made), in the remaining quadrant therefore the Sun with the same force bends back the planetary fibres, set contrariwise and turned to it with the hostile region, to the other side, and thus, contrary to the former inclination, restores the planetary fibres to the parallel situation again. This inclination and reclination will be the chief aid to the calculation in Book V.
Could you give a familiar example of this direction of the fibres and the mixed inclination? An example is in the magnetic needle, which, although it looks to the north if free, yet turns aside somewhat from it, if a magnet approaches obliquely; for then it nods somewhat to the magnet.
What things does the perfect restoration of the fibres to the parallel situation need? That the Sun spend as much force in inclining, for example through the quadrant PIN, attracting downward the sun-seeking region of the fibre II, from the line IS toward itself, as in restoring, as through the quadrant NER, drawing back the same region G of the fibre upward, toward the line EY nearer to itself. But this cannot be done otherwise than thus, if, PR being the line of the apsides, and PN, NR the quarters of the orbit completed, the fibre NQ of the planet standing at N, on the boundary of the quadrants, is directed exactly at the Sun A itself. For although the Sun A administers this inclination SIH, BNQ in the upper quadrant PN at long intervals AP, AI, etc., and so with feeble force, and in the lower NR at short intervals AE, AR, and so with stronger force: yet in turn in the upper PN the planet, tarrying longer also, experiences those feeble inclining forces longer, in the lower NR the stay of the planet is shorter, a shorter time of experiencing the strong reclining forces, and a perfect compensation is made.
For the same perfect compensation can also bring it about that at the same boundary of the quadrants N the interval AN (in the lawfully drawn orbit) be equal to the semidiameter BP, as will be plain in Book V.
What if the planet did not direct the fibre NQ at the Sun exactly after the upper quadrant PN of the orbit was completed, but somewhat later? Here the opposite is in the addition. For that very point is the end of the quadrants, reckoned from the apsides, where the fibre is directed at the Sun. For that inclination of the fibre IH to the perpendicular IS always grows as long as the fibre H seeks the Sun, and the increase of the libration grows together with it, the effect with the cause.
If therefore in this work of attraction of the planet toward the Sun more than a quadrant of the orbit is consumed, with respect to the fixed stars, more than a quadrant also will have to be consumed by the planet in restoring the right angle between the fibre and the Sun at R, and in its effect or the remaining part of the libration, by which the planet is led from the nearness NA to the nearness RA, through the same degrees of increments, now decreasing in the contrary order.
The excess, therefore, of the quadrants over the semicircle, taken separately, will show the quantity of the changed direction of the fibres under the fixed stars in one half of the period, or of the translation of the centre of the orbit B and of the apsides P, R in the order of the signs; this quantity then being taken from what is more than a half of the orbit reckoned with respect to the fixed stars, there will remain not more than half of the elliptical orbit, counted from the apsis P.
Do the apsides therefore remain, or are they transferred from their place under the fixed stars to another? In Jupiter the observations of the ancients compared with the present testify that the apsides stand nearly under the same fixed stars, or even recede very little. In all the rest they are found to have departed from their former seats, the passage being made in the order of the fixed stars, as in the example of the apogee of the Moon, but there by wholly very slow motions, whereas the apogee of the Moon advances very sensibly.
What is the cause why in the primary planets the fibres are found so perfectly restored after whole returns are completed that the progress of the apsides is insensible? Because it is the same Sun which both librates the body of the planet and inclines and restores its fibres, and because the same fibres in both works are those by which, as instruments, the planet is both librated and inclined: there is therefore no cause why we should not measure the forces for each work by the equality of times. For as the planet, tending its fibre NQ directly to the Sun, if it did not go out of the line NA, would in a certain time be joined to it to contact, so also the same planet, placed at the same N, and tending its fibre, by fiction, at right angles with the line NA, would be turned with its fibre fully in an equal time, so that at the end it would tend the fibre NQ to the Sun. But as a third work supervenes on the libration, namely the removal of the planet from the situation AN, so that the fibre NQ no longer tends to the Sun and so is not drawn toward the Sun with the same strength (by which reason it is provided that no full contact be made by sailing along NA, but that it be anticipated by the translation from N to R and be made no greater than RA): so also the same translation of the planet from N to R anticipates this inclination of the fibre, so that the fibre meets the Sun far sooner than it could have been turned from the Sun in a whole quadrant, and so, in place of a quadrant of turning, the work is no more than the arc QM. The angles of the inclinations performed, or the power spent upon them, are estimated by sines, as will be made clear in Book V by examples from natural things. Wherefore as the whole mean distance PB (or in the ellipse NA) is to the quantity of half the libration BA, the work of one quadrant, which is also the eccentricity, so also will the semidiameter of the planetary globe NQ, taken for the whole sine, be to the sine of the angle of greatest inclination MNQ, which happens so far as the quadrant was exactly completed from P by the translation of the planet, from the place of greatest interval PA. From this supposed proportion it is demonstrated that the fibre NQ then tends to the Sun A when PN is exactly a quadrant under the fixed stars. For let AN be equal to PB itself, as in the ellipse, and B the centre of the eccentric, and ABN a right angle, because its measure NR is a quadrant: let also from Q, the sun-seeking end, a straight line fall on BN, which is QM: two rectangles are formed, ABN and QMN, and because NQ is posited to be to QM as NA to AB, N, Q and A will therefore be in one straight line, that is, Q will be directed at the Sun.
Now it has been demonstrated above that if, a quadrant PN under the fixed stars being completed, the fibre Q of the planet is directed at the Sun, so that it is inflected by the angle BNQ, it follows that in the other quadrant NR under the fixed stars the fibre NQ is restored, this angle of inflection BNQ being consumed, so that, the planet standing at R, the fibre is again parallel to BN itself, as it was in P: which is the perfect restoration of the fibres after a semicircle is completed. Let the same judgment be held of the other semicircle, which being completed, the planet returns to the same place under the fixed stars. *)
In turn, since experience testifies that the apsides are transferred insensibly and do not remain in the same places among the fixed stars, it follows that NQ looks to the Sun not exactly at a quadrant from the former place of the apsis P. What is the cause of this departure from the proportion of equality already established? The unobservable slowness of these motions seems to be sought in material necessity, if in anything else, namely in the departure of the said two motions, libration and inclination, from one another, through the intervention of a third motion. For it spreads itself into a certain infinity of time, which has nothing of beauty, being as it were unbounded. But what the intervening cause is it is difficult to disclose, because neither is the matter itself agreed upon by all, nor is the quantity of the thing certain in most cases. The quantity being taken away, we lack the examination of causes, which one might track by conjectures, such as may be the excursion of the planets to the sides of the ecliptic. For that does not occur without an inclination of those fibres NQ to the ray of the Sun AN, as great indeed as the excursion of each. From a greater or lesser such inclination it is fitting that the work of the fibres is somewhat weakened, and that variously according to the various relation of the excursions to the apsides. In Saturn, Mars, Venus, Mercury, *) At the end of Book IV Kepler warns this: The ancient and best considered text is perverted by a new and hasty correction: 1) The principle is begged. 2) It is not of design, what at fol. 380, line 8, was of necessity: the agreement of libration with inclination. 3) The same, a cause that is no cause. 4) One thing is proposed in line 10, another demonstrated at folio 381, there with respect to the fixed stars, here to the apsides. The true cause of the nearly perfect restoration is of physical necessity. For whether the fibres remain parallel, or are inclined in one half downward from the apsis, as NQ, in the other upward, since on both sides the compensation is perfect, as was said at fol. 379, even so the fibres at each apsis are parallel to each other, and the restoration is therefore perfect.
Therefore at fol. 381 something false and self-contradictory is proposed, the departure of libration from inclination: rather this cause should have been stated, which is hinted at the end of fol. 379. For the Sun in the upper quadrant PN inclines a little less, in the lower NE reclines a little more, if indeed the fixed stars fix the limits of the quadrants. Since therefore at E, a point of the fixed stars, the sun-seeking end G is already above SY, and so still nearer the Sun, the planet is therefore still sailing toward it: wherefore beyond R of the fixed stars R will be the perigee apsis. If the latitude of the planet is the cause of this, it will have to be explained otherwise than at fol. 382, where at line 1, for “apsides” read “mean longitudes,” because in Jupiter P is not a node, but a limit. Nor does it suffice to look to Jupiter and to say this is the cause of the standing apsis, that the apsis is at the limit: this too must be explained, why the progress of the apsides in the other planets, of most unequal periods, is nearly equal under the fixed stars. More rightly, therefore, at fol. 382, from line 3 to line 10, one erasure being made, we hand down to posterity the two letters of the Roman court, N. L.
apsides (mean longitudes) have some latitude, in Jupiter none, and according to this analogy the apsides of those advance, of this stand. Since therefore otherwise the force of inclining the fibre of the planetary body is greatest at the apsides P, R, where the fibre is presented to the Sun at right angles, it is credible that this force is somewhat slacker on account of latitude. The reason why the same loss is not also felt in the libration is that there the libration is of itself nearly nothing. Conversely at N the force of inclination is nearly nothing, that of libration greatest: the loss is therefore felt in this, not in that, in proportion to the measure of the latitude, and it can happen that thus the inclination of the fibre is the more retarded; which being granted, it comes about, as has now been explained, that the fibre looks to the Sun more slowly, namely beyond the bounds of the quadrant. But it has been demonstrated before that then the apsides are transferred in the order of the signs. This, then, may be the cause of the said phenomenon, bound up with physical or geometrical necessities according to the principles laid down.
But I would not meanwhile rigidly deny that this effect was rather in the part of design, so that it is, or is not, a mere appendix of necessity, because its quantity we still do not know. Then there will be room to state the final cause, that the tempering among themselves of the forces of libration, of the inclination of the fibres, and of circulation tends to this, in a certain proportion in each: that since the librations indeed are provided for constituting the harmonies of the motions, each of the harmonies might arise not always in one configuration of two planets, but in the succession of ages range over all configurations altogether, and so all the harmonies of the motions (which are in Book V of the Harmonics) be mingled with all the harmonies of the configurations (the matter of Book IV of the Harmonics).
IV. Of the motion in latitude.
By what laws do the planets run out to the sides of the ecliptic? Again by a law of the simplest sort, that the plane which they circumscribe with the centre of their body is in every period exactly straight and inclined to the plane of the ecliptic by a constant and invariable inclination; except in the Moon.
Level planes, if they are inclined to one another, meet and cut each other in one straight line: I ask what is that common line on which the orbit of the planet is inclined to the ecliptic. It passes through the centre of the Sun in all the planets, and the line of each planet is extended to its own proper places of the ecliptic, opposite one another from the centre of the Sun.
Whence is this established? It is established from this, that when the planet, in two different parts of its return, as in diagram 70 at C and D, is seen on the ecliptic, lacking latitude, these two places of it are found by calculation to be with the Sun A in the same straight line CAD: so that if ACM came to 17° of Taurus, the interval of time until the planet was again seen on the ecliptic, joined with the hypothesis of the eccentric, shows the line ADO, of another place of the eccentric, at 17° of Scorpio, namely in the opposite of 17° of Taurus.
What is gathered from this? The same, namely, as above at folio 354. For since the planes of all six eccentrics meet in one common centre of the Sun, they therefore cannot all meet together anywhere except in this centre of the Sun, because the line of section is not common to all but proper to each, and different lines meet in no more than one point.
Since therefore the Sun is the common knot of all the systems, whether nature moves the planets by corporeal powers or mind by rational nods, the Sun altogether is for the planets the goal to which all the circuits look.
What causes of the motion in latitude do you give? The Sun is not a cause to the planets of this deviation from the plane of the ecliptic, except a remote one, nor is a mind needed by the planets for this work, nor the construction of solid orbs refuted above, in which as in chariots they might be borne along their just orbit, and much less for this than for the librations in height and depth or for the motion in length: but some formation of the planetary bodies themselves alone suffices to twist their orbits away and twist them back to the ecliptic.
Why is the Sun not the cause, since it has already been said that the lines of section go through the very body of the Sun? Because one and the same Sun, by one and the same form of its body, which goes round in a uniform and most direct stream under the circle midway between the poles of the Sun’s rotation, cannot carry different planets along different other ways, unless the planets themselves add from their own the causes of this different departure to the sides.
What sort of formation of the planetary bodies do you intimate? It can be either essential, namely inner rectilinear magnetic fibres, or accidental, namely the rotation of the planetary globe about its axis, so arranged that the fibres or axis of gyration retain a parallel situation through the whole circuit of the body, and are so directed that when the planet is on the ecliptic it touches the orbit and with one end turns aside somewhat toward the north, with the other toward the south.
Have you a popular example of this deflection? The oars of ships supply an example of a sort. For if a ship is driven forward by the winds, and an oar is bound obliquely to the stern, then the ship, against the line which the wind brings, is gradually thrust aside.
An oar, rudder or helm always directs a ship into one region only; how then do the planets now go out to the sides of the ecliptic, now return thence to the ecliptic? If the oar of a ship is turned, the ship also turns aside to the other side. The planets, although they hold their fibres directed in parallel situation and unturned, are yet carried to opposite parts of their circuit, in which the fibres, in the former situation, have an opposite inclination to their orbit; wherefore also the planets through the other semicircle are driven into opposite regions.
That I may understand this motion better, say what surface one such fibre or axis creates in the carrying round of the planet about the Sun. Let us suppose, when the planet is on the ecliptic, as here in C, E, that the fibre of latitude AB is not inclined toward the Sun (although it can also be otherwise with the same effect, if the situation is equivalent), but is so inclined to the plane of the ecliptic that the half EA, CA is to be understood as sunk below the paper, which represents the plane of the ecliptic, and the remaining half EB, CB stands out above the paper; let the angle of inclination be as great as the latitude is wont to be at the limits, F being understood above the paper, D below.
(Suppose the letters D, E, F to be at the centres of the little circles.)
Let the motion of the solar form also be as of a river or a wind, from E toward F, C, D.
Since therefore this motion at E will run against the opposite half AE of the fibre, which is sunk, and at C similarly against the opposite BC, which stands out and is opposite to AC itself: accordingly at E it will drive the planet upward from the paper, whither the leading end B tends, at C downward, beneath the paper, whither tends the end A, which at that place is leading. In the rudder the contrary happens, because it is thrust by the force of the stream, not driven by an inborn aptitude. But since meanwhile the fibre AB remains in a situation parallel to itself through the whole compass, hence it comes about that at F, of the planet most northerly, and at D, of the planet sunk and most southerly, neither end, neither A nor B, leads, but the fibre AB, stretched as it were into the depth of this stream, that is, toward the Sun, and receiving the impetus on its side presented straight, affords no cause of further ejection into any region, until at these points the exchange takes place; so that whereas before the point F the end B had led, now after F the end A leads, and thereby the planet begins again to approach the ecliptic, with a progress at first insensible.
From this it now appears what figure is generated. For since the fibre AB is moved from E toward that very region to which the leading end B tends, the surface which is created by AB is attenuated at the point E into a mere line, which yet little by little becomes a surface, and, arising from the point E, acquires at F its greatest breadth, equal to the length of the fibre AB; thence again this surface is attenuated, as far as the parts of the circuit C, which are opposite to the E first mentioned, where that surface again vanishes into a line. Let the same be understood of the opposite semicircle CDE. Being thus carried, inclined, in F and D and always following its own course, it will create a perfect plane, in so far, namely, as it retains the parallel situation: which plane, if continued, will pass through the centre of the Sun, because the fibre AB looks to the Sun, in F with the end A, in D with the end B.
But this continuation of the plane being set aside, if what is created by the fibre alone is considered, the figure will be such as two little lunes present between two ellipses, the outer CBAE and the inner EACB, touching each other in C, E, so that the same line CE is the diameter, of the lesser EACB the longer or straight, of the greater CBAE the transverse.
The centre also of the body of the planet will go round in the perfect plane, which in this figure is made circular, namely CDEF; although it too, as is plain from what was said above, turns aside a little from the perfection of the circle to an elliptical correction of its sides.
The oar or rudder of a ship is stretched forward from the ship into the waves or into the wind; these fibres lie hidden within the round body of the planet: they have not therefore the same force as rudders. It is not necessary that all things answer in some likeness: but in place of the faculty of oars there comes another force of the fibres, much more suitable, that as above the fibres had a natural inertia against their own inclination, or rather a power of retaining the parallel situation in the transport of the body, so now also there be in the fibres of latitude, besides a like force of retaining the parallel situation, also a natural power of agility, or of guarding the same line and deriving along it the motion brought to them, in so far as the motion tends in the same direction as the other end of the fibre.
Compare this form of the motion in latitude with the old astronomy by a popular example. We here commit the planet to a river with an oblique rudder, by whose help the planet itself, as it floats down, crosses from one bank to the opposite.
The old astronomy builds a solid bridge (solid orbs) over this river (the breadth of the zodiac) and carries the planet across it, as it were lifeless in a chariot. But the whole machine being surveyed, it appears that this bridge has no piles to support it, nor rests on the Earth, which they believed the base of the heavens.
Yet this speculation on the motion of latitude is more difficult than if one imagined solid orbs. But you must remember, reader, that we are engaged here in a physical speculation of the causes on account of which any hypothesis is employed, that we may know what of truth underlies such an astronomical hypothesis or fiction. But below in Books V and VI we shall not reject whole circles and their inclinations to the ecliptic, for the sake of understanding, because they are equivalent to these projections of the fibres toward the sides of the ecliptic.
If both that former libration of the planet in height and this excursion in breadth had the same limits under the ecliptic and were effected by the same fibres of the body, the causes you give would be probable. Nay, what hinders one and the same globe from having two kinds of rectilinear fibres, pervading the whole body, so that by the one it is librated in height and depth, by the other it again rows to and fro? So on the surface of a river a threefold motion of parts is seen, each observing its own direction: the first is the downflow of the water, the second that of the waves, which that downflow in a continuous series casts across toward the banks, the third is from the wind, which, if it blows obliquely contrary, roughens the surface of the waves and stirs another series of smaller waves in its own direction also, which run over the former undisturbed. So above in Book I the substance of the stomach was alleged, which represents a certain threefold web, possessing three kinds of fibres distinct in direction, the seats of three faculties, the attractive, the retentive and the expulsive: though it is the texture not of one but altogether of three coats.