Epitome Astronomiae Copernicanae: Liber IV, part 10
Do the greatest excursions of the wandering stars always happen in the same places of the zodiac, or do these also change their places? The observation of the progression of the limits is still obscurer than that of the apsides; yet they seem gradually to creep backward against the order of the sphere of the fixed stars, and indeed more slowly than the apsides advance, both after the example of the motions of the Moon.
If the limits recede while the apsides advance, the fibres of latitude, which administer the one, will not remain interwoven with the fibres of longitude, both of which you have given to the same globe. Here a refuge lies open to us to the inner substance of the globes, into such straits as we were already driven before in comparing the diurnal revolution of the Earth with its librating fibres. We may therefore also here seek in the outer crust a separate globe, as in the white of an egg the yolk, furnished with its own fibres and turnable according to their laws, with a strength of powers also distinct from the outer crust, if need be; so that both may be bent by the same external cause, with distinct measures of swiftness, if there is need of this also. For so also in the example of the stomach already introduced there are three coats, outermost, innermost and middle, of which one can suffer while the others are unhurt, this act while those are idle, although they differ in this respect, that they are not separated from one another.
The old astronomy lays solid and plainly adamantine orbs one over another, though no body in them is visible to us, and the whole region is as clear as if it were empty. It will not therefore take it ill that we contrive something similar in the globes, which are visible and palpable bodies.
Could not that very axis of which you have already made mention, the axis of turning of the outer crust of the planetary bodies, sustain this office of declining the motion of the planets to the sides? That cause rests indeed on great probability, as will have to be said in Books VI and VII in the explanation of the theory of the Sun and of the eighth sphere: yet nothing certain can be brought forward about all of them, because although we said it credible that the other primary planets also turn about their axes, the regions to which these axes verge or decline are unknown to us: so that in the Earth alone we have an example. And the Moon, a secondary planet, does not turn, though she accomplishes her latitudes.
How can this be effected, that the limits of the excursions recede against the order of the signs? Some part of this appearance will be excused in Book VII as accidental, not physical or real. But what is left of this motion and real is effected by the succeeding nod of the fibres of latitude backward, so that they remain indeed in one and the same plane through the whole of their circuit most exactly, but themselves, over the centre of their globular body (that is, the globe itself), are secretly inclined backward along these fibres.
From what causes does this reclination arise? So far indeed the probability of most of the causes brought forward was evident; in this last array of astronomical matters the causes succeed with difficulty, and not only the mind but most of all the credit of what one may devise labours. Let us say, however, as much as we can find. We said that the nature of the fibres of latitude consists in an aptitude for motion forward toward the region of their parallel direction; we said also that while they are carried from the place that has the planet on the ecliptic, namely from C, E, to the place of greatest excursion to north or south, to D and F, they meanwhile remain parallel, and that on this account it comes about that whereas there at C, E they touched the orbit, here now at D, F they are plunged into the depth toward the Sun, to which that motion is not directed to which they are inclined: rather the moving stream from the Sun, so to say, falls upon the transverse AB at right angles, swifter below (namely at A in situation F and at B in situation D) than above and outside. If then they are inclined to the motion, what wonder if this inclination, the lower part seeking the direction of the motion, detracts somewhat from the parallelism, and that at each limit? Thus the recession of the limits will follow, there being no compensation. For at F, A will be pushed forward along the way EAC, at D, B will be pushed along the same way CBE: thus on both sides B will be inclined downward on the paper.
But if this cause is not admitted, let a moving soul be summoned, which turns the inner kernel in the outer crust by its own laws, with this design of the Maker, that by the mutual interweaving of the orbits of one out of another and their frequent multiplication and condensation, in the succession of ages some orbicular solidity might be traversed by the planet.
Why is the recession of the limits slower than the progress of the apsides? Although the matter itself is in doubt concerning Mercury and somewhat concerning Jupiter also, let us nevertheless follow probability, on account of the evident example of the Moon, and let us state this cause, that the disturbance, if any occurs, from the same foreign cause, necessarily becomes more sensible in a great motion than in a small. Now the transposition of the apsides arises from a great motion, which is the inclination and reclination of the fibres in each semicircle as great as the optical equation, and it would become greater and altogether total, if it were not anticipated by the circumduction of the planetary globe. But the transposition of the limits is made by the small motion of an excursion to the sides of a few degrees, and one which in this its measure is not greater, so that it can blame nothing by which it is hindered. Wherefore the same rays of the Sun, impelling both motions by the laws already explained, have more evident effects there than here. It is added that there the rays of the Sun act with greater difference than here, other things being equal. For there the obliquity of the Sun’s rays to the fibres, which tends in the sides, or the angle of latitude, by which their work is weakened, was sensible; here the difference between the parts of the planetary globe and so between the ends of the fibres of latitude, the nearest to the Sun and the most remote from it, to which difference we ascribed the motion of the limits, is very slight: with right, therefore, this work also is less than that.
V. Of the double inequalities of the Moon and their causes.
Are those things which have been argued so far of the causes by which the true motions of the primary planets are made threefold unequal to be understood also of the Moon, a secondary planet? 1. Altogether, indeed, the Moon about the Earth emulates in general the same form of motion which the primary planets observe about the Sun, and therefore we ought to posit also the same causes in her body, namely magnetic fibres and their rectilinear course and the contrary regions of this course, one region friendly to the Earth, the opposite hostile; finally this course of the fibres nearly parallel to itself through the whole circuit of the Moon; so that, the Moon being transferred to the opposite place, an exchange of the regions takes place and according to the friendly one the Moon is attracted by the form of the body of the Earth, according to the hostile repelled, and by that reason her motion in length also is either quickened or slackened; and likewise it is necessary to conceive other fibres of the body by which the Moon’s digressions from the ecliptic are accomplished.
In the present diagram are expressed some situations of the Earth with the surrounding heaven of the Moon, going round the Sun; and in the heaven of the Moon some situations of the Moon herself, going round the Earth. The image of the little magnetic needle signifies the magnetic fibres in the globe of the Moon, by which the Moon becomes eccentric about the Earth. For the point A, B, C, D, E, F, G signifies the region friendly to the Earth and turns to the second mean longitude. And so the Moon at A, G is placed midway between perigee and apogee, at B, R, D a little before apogee, at C a little after apogee, at E, F a little before perigee.
2. But since this motion of the Moon about the Earth, as was explained above, is derived as it were from two sources, namely from the diurnal rotation of the Earth, which is the middle of the small heaven of the Moon, and from the rotation of the Sun about its axis, which Sun is the middle of the great orb common to the Earth and the heaven of the Moon, it will be reasonable that the true and real motion of the Moon about the Earth (even in so far as that common circulation about the Sun, accidental to the whole heaven of the Moon, is removed from it in mind) should have two sources, as it were, and suffer all those double affections which the motions of each of the primary planets have. And this agrees admirably with experience and the knowledge of the craftsmen and the very names that arose from this duplication. For not only in the foregoing, in place of what was in some one of the primary planets, there was in the Moon a solitary mean motion, and a mean and the semimonthly variation of this mean, but also in this place, when we treat of the periodic inequality of this motion, which is not semimonthly like the variation, but monthly or rather half-yearly, we find, in place of the simple inequality of some primary planet, a double intensification and remission of the mean motion in the contrary moments of the period, and finally, in place of the simple digression of the primary planets to the sides, a double one also.
What cause has the Moon’s eccentricity of its quantity? In the Harmonics I demonstrate that the variety of the Moon’s motions determines exactly the diatessaron, which seems to have an affinity with the quadratures and conjunctions of the Moon; in order therefore that this interval might be represented by the composite motion, so great an eccentricity was made.
What difference is found between those inequalities of the Moon common with the planets and these proper to the Moon? 1. As the motion of the Moon about the Earth in the foregoing was composed of two elements, as it were, the one from the Earth, rotated about its axis, the other from the application of the solar light to this moving form of the Earth, of which the former was free from the phases of the Moon, the latter bound to the phases: so now also of two inequalities, the former, the accident of the former element or mean motion, is found to have its own proper limits, which we call the apogee of the Moon; the first form also of the digression to the sides has its own, and these distinct from the limits of the apogee, which they call limits and nodes; but the latter inequality of the latter element, or accident of the conjunctional stirring, called by Ptolemy the nod of the epicycle, has limits common with the lunar month and the phases, as also has the second form of the digression to the sides.
2. The former, both of longitude and of latitude, is always constant through all periods, of a quantity, namely, perpetually the same; of the latter, each becomes greatest in only one month of each half-year, less in the rest, in some, which divide the year into two parts, nearly nothing, namely where the opposite affections of this second acceleration and retardation, likewise of north and south latitude, begin to pass into the contrary halves of the lunations.
3. And so those former inequalities have both their quantity and the laws of their distribution from their own proper causes, the second take their quantities and affections from the presence of the first in each semicircle of the lunation; only the laws of distribution, separate and accommodated to the circuits of the lunations, they have yet like those of the first.
4. Akin to this also is what we find in the Moon, that the motion of the apsides in the order of the signs and of the limits against it is much swifter than in the primary planets, not only in the proportion of the swifter return of the Moon, but plainly sensibly, and the retrocession of the limits more than twice slower than the progress of the apsides.
The Moon is not seen to turn alternately now this now the opposite part of her body to the Earth: for we always behold the same spots of the face of the Moon. Why then can the causes of the approach and recess of the Moon from the Earth not be sought from this? 1. It is not necessary that the lunar magnetic fibres in two opposite times of the period be directed straight toward the Earth: it suffices that at those moments they be inclined in alternate regions toward the Earth, and that this situation of the fibre remain parallel through the whole circuit of the Moon. For even so it can happen that now one region of the fibre nods nearer to the Earth, now the opposite. But if this inclination is small, our sight is not so accurate that it can observe most exactly on the disc of the Moon whether never on the margins of the lunar globe, which look toward the poles of the ecliptic, some minute particles show themselves to be seen which at another time are not seen. For those parts of the globe are sloping and the appearances most slight, and the illumination of the margin, now of this, now of that, often fails, on account of the inconstancy of the lunar countenance.
2. We have long since left it uncertain whether there is not a globe within a globe, as a kernel within a rind, of different turning from that; which both the examples of the Earth and also the motion of latitude itself suggest. And so such an inner globe might tend to the Earth with its regions turned alternately, notwithstanding that the outer crust always turns the same spots to the Earth. For among these and like things the manner of this motion is altogether uncertain; this alone is most certain, that whatever the manner is, it is adapted to physical and magnetic causes, that is, corporeal and so geometrical, of which we have proposed examples here and there.
Is, then, that second inequality of longitude truly from some second eccentricity or digression of the Moon from the Earth, no less than the first has its cause from a change of interval? No: the observation of the parallaxes of the Moon, joined with the contemplation of eclipses, is against it; and the reasons of the proportion of the bodies, set forth a priori in the first part of this fourth book, conspire. But this too could show that plainly no change of interval bound to the phases occurs, that, while various craftsmen correct this hypothesis in various ways, a less and less quantity of this change has always been produced. Ptolemy set it down as enormous, which Regiomontanus refuted; Copernicus halved it and transposed it from the form of an eccentric into the form of a second epicycle; this again Tycho Brahe corrected, a part being claimed for the equant circle, which he himself with Copernicus was wont to excuse by an epicycle of doubled motion; I exchanged the conjunctional intervals with the quadrature ones, and transposed the circuits from the month into the year; but in later times, relying on these discoveries, I at last found that a change of interval, through the circuits of the phases at least, takes place not at all.
Whence then is that second acceleration and retardation bound to the phases? From the different relation of the Moon’s eccentric to the phases. For while by the simple and perpetually uniform law of eccentricity she goes round the Earth, her mover, as each of the primary planets goes round the Sun, it happens by accident that from the other promoter of her motion, which accelerates her at the conjunctions, she is distant now in one way, now in another. For if her greater interval from the Earth falls at the conjunctions, where the acceleration is greatest, then the form of the Earth, unfolded in a more diffused orb, is weakened in one of the conjunctions, not only in its native and archetypal vigour, but also in that acquired strengthening of itself from the Sun. In turn, if this greater interval of the Moon and Earth falls in the quadratures, where there is no acceleration, then there is no loss of any acquired vigour, and no gain at the short perigee interval.
In diagram 72 there are in the globes of the Earth and the Moon the circles of illumination, dividing the illuminated part from the dark. But since the apogee of the Moon through the whole year, and so through all situations of the heaven of the Moon, stays in the same sign, that is, the fibres WF remain nearly parallel to themselves through the whole circuit, while the Earth with the heaven of the Moon passes from sign to sign, it comes about that the fibres are applied to the circles of illumination (which are extended along the circle concentric with the Sun, representing the density of light in length) in one way and another, as you see in the arcs DT, EY, FW, GX, AO, BP. The same therefore happens also to the points of the Moon’s apogee and perigee, as always verge to places distant by a quadrant from the place or region of the point A, B, R, etc.
What if the greater interval be stretched toward the Sun, will the motion also be weakened thus? But then the Moon runs in a denser light. This is just what we guarded against above. For the light of the Sun does not move of itself, but through the form of the body of the Earth, to which it transmits the laws and modes of its work. As therefore above it was not light that gave the direction of motion, but the form of the body of the Earth, in some places plainly contrary to the direction in which the Sun is moved about its axis: so now also the moving form from the Earth is strengthened according to the proportion of its native strength, thinly, where it is thin, in the greater distance of itself from the Earth as source, strongly, where it is strong, in the lesser distance from the Earth, whatever the variety of the Moon’s distance from the Sun may now be, of whose compensations it was said above in the causes of the variation.
What is the measure of this monthly equation when it is greatest, and what cause is there of such a measure? Tycho Brahe makes it equal to the physical part of the periodic equation resolved, according to my form, because, since the whole periodic is nearly 5°, I claim half for the physical cause, usual to all the planets, namely 2° 30′; so great, then, Brahe also gives for the synodic, as if the moving form of the body of the Earth were made exactly double as strong at a near distance, weaker at a far one, by this corroboration from light, and then is what it is without it. This, if anything is sought, seems to be able to have no cause but this very regard to equality, as of the simplest proportion and therefore the most beautiful.
Ptolemy, however, shows a somewhat greater measure, and altogether as great as we gathered above from the appendix of 132° 45′ to the 12 synods, the variation of one quadrant, namely 2° 41′. But if this measure and quantity is to be held on both sides, then the cause seems to be transferred from design to geometrical necessity: namely that the increase of interval, that is the eccentricity, utterly exhausts what the stirring from the light had given at that conjunction; but in turn at the other conjunction the eccentricity, taken from the interval, adds to the swiftness as much as that stirring of light also effected. Thus in the month which lacks the synodic equation, when, that is, in this diagram the apogee is in EF, the perigee in IK, the parts of the luminous stirring at both conjunctions are equal, because the intervals of the Moon and Earth at both conjunctions, as AC, AH, are equal in that month; in the succeeding months, an inequality of the intervals at the conjunctions arising little by little (as if L were apogee, N perigee, then certainly the distance AC would be greater than AH), some monthly equation also arises, which is always greatest at EF, IK in that whole month: finally in that month in which the synodic equation is full (as if the apogee were at CD), the stirring at the conjunction CD is nothing, at the other GH double the stirring in the month first mentioned: then at the quadratures E, K the equation is greatest of all that can be in the whole year; but at G, nearest the perigee, the parts of the smallest little equation are four: 1) optical, as in the planets, 2) physical, as in the planets, 3) variation from light, 4) the intensification of the same on account of the diminished interval; of which parts the proportion to one another is of geometrical necessity; but the sum compounded of all four is so tempered by design that this perigee motion of the Moon GH at the conjunction would be to the apogee motion of the Moon CD at the quadrature as 4 to 3, and it would be the harmony of the diatessaron.
But for what reason it comes about that, although these two things are bound together by no necessity, 1) the measure of the conjunctional stirring, 2) the measure of the eccentricity, the eccentricity yet exactly cancels the stirring at the apogee conjunction and doubles it at the perigee, this, I say, I have not yet been able to investigate.
What probable cause do you give of so great swiftness of the apsides and limits of the Moon, if the apsides and limits of the primary planets are incomparably slower? Surely here also the effect of the composition of the Moon’s moving powers is made conspicuous. For as above we said that the simple power of the Earth is tempered to harmonic numbers, in the revolving of the body of the Earth about its axis to 360 perfect days, in the carrying round of the Moon about the Earth to exactly 12 months in one year, or the return of the Earth’s centre about the Sun: so now also let us say that in the inclination and restitution of the fibres of the Moon, by which both the libration and the deflections of latitude are performed, the simple forces of them are tempered in the same proportion to the length of the Moon’s periodic time as was observed in the other planets. But as above, on account of the accession of help from the Sun, both for turning the globe of the Earth and for carrying round the Moon, the archetypal numbers were disturbed in the last effect (so that for 360 there came to be 365 and a quarter, and for 12 lunations in the year about 12 and a third), so now also on account of the same addition to the acceleration of the Moon from the solar illumination it comes about that the Moon arrives at the middle sides of her circuit before the fibres are inclined by the just measure, and so the fibre looks to the Sun at a place deeper than a fourth part from the apsis; by which being done, we impressed above that the transposition of the apsides takes place. It is fitting that this transposition of the apsides be quite sensible, because that addition is sensible, namely nearly 11 degrees, yet this is less, that is a little more than three degrees, in a month: 1) because those degrees are accumulated for the most part at the conjunctions, whereas the fibres without regard to the conjunctions are inclined and reclined mostly in the mean longitudes, and so for the most part the affections depart from one another in the reckoning of equal degrees (in which knot it is probable that something still lies hidden, by ignorance of which it comes about that the Moon’s motions are not yet exact to a scruple, not even in Tycho’s calculation); 2) because the inclination of the fibres is not only anticipated in place and time but also, for that very reason, is cut down in quantity. For if the Moon had advanced more slowly, or if the inclination of the fibres in the accelerated Moon were still as great as it would have been in the slow, the apsides would also be transposed farther. But by the acceleration of the Moon it happens that the fibre meets the Sun before it reaches the just measure of inclination originally assigned to it; from the mixture of which it comes about that between nothing or the insensible, which would be without the acceleration of the Moon, and the 11° which the acceleration causes, something between, namely 3¼°, redounds into the motion of the apsides. Let the same be said of the impulsion or reclination of the fibres of latitude: for it ought to have been insensible, as in the primary planets, if the Moon, like the primary planets, were to advance by a simple force. But because the accelerating power, supervening to the Moon, was estimated at nearly 11° by the effect on longitude, which, if it bore upon the fibres of latitude throughout the whole compass, would bend them back by the whole 11°, as fibres not defended against it, it, having found the fibres subject to it only at the limits, bends back by one degree and a half in one period, upon which reclination the precession of the limits follows.
Nevertheless, since the quantities and proportion of those two motions, the one of the apsides, the other of the limits of the Moon, are known exactly from the observation of so many ages now, there still remains room for ingenuity. For he who shall bring such causes of these things that from them this very quantity follows, may drive his chariot around the goal. Philosophers ought to strive for this the more, because the Moon, besides so many other experiments, is also in this our question a teacher for the acquiring of knowledge of the heavenly things, and by her example shines before the nature of all the planets.
By what reason does it come about that the Moon besides her customary periodic latitudes makes also synodic excursions to the north and south? That power of light, strengthening the form of the Earth which moves the Moon, as it borrows the direction of motion and the proportions of its work from that very thing which it strengthens, as it also passes for the cause of longitude into the genius of her very orbit, according to the measure of mutual application, so the same must be held to do also for the cause of latitude. It made the motion in length easier because it is itself extended in length; it will therefore make easier also the motion in breadth, because it has also the other extension of latitude, that is because light is a surface, a partaker of density, as we have often alleged from the optical writers. At the conjunctions, therefore, the fibre of latitude touching the orbit and inclined according to the latitude of the form of the Earth, this latitude of light, applying itself to the form of the Earth, makes the excursions easier, so that they are made at a greater angle than the fibre makes with the plane of the ecliptic, and by that reason the star arrives at the quadratures at limits more remote to north and south than those which the fibre showed from the conjunctions. In turn, in another quarter of the year the fibre of latitude, touching the orbit at the quadratures, does not accommodate itself to the extension of the form of light in breadth, but is stretched almost directly against the Sun, as also the orbit itself of the Moon. As therefore in that place the motion of longitude is not made easier by the light but as it were roughened, so the same redounds also into the excursion of latitude, so that it is not made greater than the angle at which the fibre of latitude is inclined to the ecliptic; thus the Moon does not arrive at the conjunctions at limits more remote than those which the fibre showed from the quadratures. But what befalls the Moon when she is at the limits, or with what countenance the light of the Sun regards her, when, that is, the fibre of latitude is stretched to the Earth, is of no concern to latitude. For there is then no action of the fibres of latitude in the exchange of excursion with return, no strengthening of it therefore from light, since we said that it follows the laws of its object.
Since you refer everything to the corporeal fibres of the globes and to the immaterial forms of the turning bodies of the Sun and Earth, and finally to the light of the Sun as a strengthening cause, and leave nothing to the animal faculties, you seem to philosophize just as if someone contended that for the stomach its threefold fibres suffice for its offices, and there is no need of the animal faculty. Nay, in the body of the Sun I admit a soul, set over the turning of the Sun and dispenser of the whole motion of the world, nor did I simply deny in Book I that even for the bodies of the planets there are single souls, set over the turning of the bodies. But as it is not necessary to introduce a peculiar soul into the fibres of the stomach (for it suffices that one common soul runs out from the heart or the liver by its form or by heat into the stomach and uses the faculties of its fibres), so also in the world there seems to suffice that form (as of light, as of heat, so also, if you like, of the solar soul), sent out together with light and heat and penetrating even to where light and heat are shut out, namely into the inner fibres of bodies: so that as the soul in the body can do nothing without the organ of the stomach, so also this soul of the world can do nothing without these laws and the geometrical disposition of bodies.
Let the state of the controversy, then, be observed: for it is far otherwise to refer the whole method of dispensing the heavenly motion, though involving contradictions and therefore impossible, simply to the occult powers of some soul, rejecting the corporeal instruments and all the modes which the human mind can devise, which is the refuge of all ignorance, the mere death of all philosophy, yet the commonest practice with most who write or speak of astronomical matters, noted in part above in Ptolemy himself; and otherwise first to examine everything in bodies fitted to motion, that the possibility of the motions may appear even by popular examples, and afterwards at last to pour over all these, as over a human body compacted of all its muscles and nerves, a moving soul; which, if it can despatch any offices by corporeal instruments, will not need for them counsel and discursive thought, the proper works of an intelligent soul: as on the contrary, if it accomplished all things by counsel and discursive thought, it would not need these corporeal instruments.
In brief, philosophers have devised intelligences which draw out the motions of the heavens from themselves, as from a notebook, which use consent, will, love, understanding of themselves, and finally command; for me the moving soul or souls are of an inferior kind, which bring only the impetus (as a sort of matter of motion) with uniform exertion of powers, without the help of mind, but find the laws (or form) of the motions in the bodies themselves, which were once fashioned in the very beginning of the world by a mind, yet not their own but the Creator’s, and tempered to effecting such motions.