Epitome Astronomiae Copernicanae: Liber VII, part 1
Pertaining at once to the spherical and the theoretical doctrine.
What will be the matter of Book VII? That dispute of the ancients concerning the motion of the ninth and eighth spheres, or of the fixed stars, and what depends upon it, the differing magnitude of the sidereal year and the tropical year; the precession of the equinoxes; the changeable obliquity of the ecliptic and the connected questions.
How many spheres does the old astronomy have? Ten. The first is that of the Moon, the second of Mercury, the third of Venus, the fourth of the Sun, the fifth of Mars, the sixth of Jupiter, the seventh of Saturn (for the semidiameters of these according to the opinion of the ancients see Book IV, folio 333), the eighth of the fixed stars, the ninth and tenth are starless. The more recent have dared to add an eleventh, and some even a twelfth.
What reason did they have for setting other spheres without stars above the sphere of the fixed stars? There was a time when it was unknown that the fixed stars are advanced in the order of the signs, and then the eighth sphere, of the fixed stars, was held to be the last, as in truth it is; and to it was assigned a motion from east to west about the immovable poles of the world, that is, those ever overhanging the same places of the Earth, which were marked out in Book II, folio 190. In this eighth sphere they imagined to be described the equinoctial and the zodiac with their poles, and under the poles of the zodiac they supposed to be fastened, as it were, one concentric with the system of Saturn, so that, the sphere of the fixed stars being whirled round in the space of 24 hours, the poles and the whole concentric sphere of Saturn must be carried round together with it, less the very small arc which the sphere of Saturn, creeping as it were under the eighth, is bidden to traverse daily in the contrary direction, from west to east.
But after it was discovered that the fixed stars also gradually depart from the signs of the zodiac in the order of the signs, before it was known whether they did this continuously or whether they would one day return by the same footsteps, the observers rather believed the latter, led by this likelihood, that they seemed to themselves to see this motion of the fixed stars become swift from slow. And so they set the ninth sphere around the eighth, and transferred to it the diurnal motion and the equinoctial and the zodiac and its poles, so that under the poles of this the poles of the ecliptic of the eighth sphere of the fixed stars were fastened, which should have its own zodiac as before, and here the zodiac of the eighth should in its turn nod to and fro under that of the ninth through a space of some degrees. This motion they called the motion of access and recess, or the motion of trepidation.
But in succeeding ages experience taught that the fixed stars, having passed all the limits prescribed by the opinion of the ancients, go on continually and do not return upon their former footsteps. They therefore thought it necessary to give this ninth sphere also a motion of its own and to set a tenth around it, which, because they had assigned to the eighth the office of the first motion and to the ninth that of the second, should itself administer the circles, the equator and the zodiac and its poles, movable by the diurnal motion, while the poles of the ecliptic of the ninth sphere should adhere to the poles of the ecliptic of the tenth, as the poles of the eighth to those of the ninth. So the tenth went round from east to west in the space of a day about the immovable poles of the world, turning the whole machine; the ecliptic of the ninth crept under the ecliptic of the tenth in the contrary direction, from west to east, about its own poles, completing by an equable motion one period, according to the Alphonsines, in the space of one jubilee of thousands, that is, 49,000 years; the ecliptic of the eighth trepidated under the ecliptic of the ninth by a reciprocal motion as before, if that can be called trepidation which performs one turn in the space of 7000 years; by which trepidation the motion of the ninth was either accelerated or retarded.
This imagination of three ecliptics in one and the same plane I think more suitable to the numbers of the old astronomy. For those who introduced the deflections of the head of Aries and Libra to the sides of the world in little circles have thrown themselves into confusion, and their readers with them, and have not expressed the intent of the first authors and the force of the numbers. I confess that little circles were in their minds for moving the heads, that is, the beginnings of Aries and Libra, but that along the diameters of the little circles, extended according to the length of the ecliptics, not on the circumferences running out to the sides of the world. For by this hypothesis they did not represent the change of obliquity of the ecliptic, certainly not as to quantity, nor was there anyone who even lightly attempted this before Copernicus.
What becomes of these motions of the eighth, ninth and tenth sphere, and indeed of the spheres themselves, in the astronomy of Copernicus? It casts out the ninth and tenth as superfluous and empty of stars, and leaves the eighth, or sphere of the fixed stars, wholly immovable to the world as its outermost wall; but all three motions, and whatever besides has been newly discovered since, it confers on the single globe of the Earth. See Book IV, folio 356, 358.
For the first or diurnal motion, which we think goes from east to west, belongs not to the whole machine of the world but to the single very slight little globe of the Earth upon its own poles, which are called the poles of the world, because the ancients believed that the tenth sphere, and with it the whole machine of the world, was turned about the same axis continued and its poles, as was copiously disputed in Book the First. The imaginary transposition of the fixed stars into the following degrees of the signs is not a true and physical motion, since those degrees are not another physical place but in truth remain the same, only numbered otherwise; nor do spaces accrue to the fixed stars, but a number accrues to the spaces. See Book III, folio 273.
What is the cause why the numbering of the places of the fixed stars increases with the succession of ages? Because the beginning of the numbering of the parts of the zodiac is made from the section of the ecliptic and the equator, as was indicated in Book II, folio 203. Now the points of section go back from east to west, whereby it comes about that the same fixed star, remaining fixed, in a later year also stands farther from the section, or from the changeable beginning of the zodiac.
What transfers the sections, or what makes the ecliptic cut by the equinoctial at ever different points? The inclination or bending of that axis about which the globe of the Earth is turned by the diurnal motion on the axis set in it, as was explained in the folio 273 already cited.
In the present figure let the circle CLD be understood to represent that part of the surface of the fixed stars which communicates with the frigid zone. Let the north pole of the ecliptic be O for Ptolemy, for us P. Within, in the depth of the sphere near the centre, let there be the globe of the Earth, of which half the axis is extended toward the point E of the fixed stars at the time of Ptolemy, the remaining part toward the opposite point of the sphere in the southern zone. While the Earth remains near the centre of the sphere, the axis of the Earth, as on the centre of the body as though immovable, would have been inclined so that after Ptolemy it looked to other and other points of the circle, E, F, M, and at last in our age looks to the point F. Now it was said at folio 189 that the line of the Earth’s axis continued, where it falls upon the fixed stars, there marks the pole of the world or sphere. The pole of the sphere was therefore of old at the fixed stars E and has passed on to the fixed stars F. But if O is the pole of the ecliptic and E the pole of the world, QE will be the great circle, the colure of the solstices, at the time of Ptolemy, and PF at our time. As therefore the pole has gone back from E to F, so also the solstitial points in the underlying sphere have gone back. But the colure of the equinoxes cuts the colure of the solstices in the poles of the world at right angles. Of old, therefore, EI, at right angles to EO, was the colure of the equinoxes, and today it is FK, at right angles to PF. Therefore the equinoctial points also have gone back, and these are the mutual sections of the ecliptic and the equinoctial, in which the colure EI continued and the circle of latitude OQ meet; so also FK and PH, those in the ancient time, these in our time.
But it is not permitted in the hypothesis of Copernicus to place the centre of the Earth immovable in the centre of the world. True: the line of the Earth’s axis, continued on both sides to the fixed stars, is carried round the Sun in a position parallel to itself, and describes on both sides among the fixed stars little orbs equal to the great orb about the Sun, such little orbs as you see represented at E, F; and when the northern part of the line of the axis is nearest the north pole P, then its southern part is farthest from the south pole, and by compensation, when in the opposite part of the year that part has departed farthest from P, this part has come nearest and approached the south pole. Yet these little orbs at E, F are so slight in respect of the immense amplitude of the fixed stars that they scarcely equal the fifty-thousandth part of the diameter; see Book IV, folio 334. Wherefore all these little orbs of all the years, E, F, chained together, can be taken for a simple circle EFM, and it is just as if the centre of the Earth rested in the very centre of the world.
You posit, then, three motions in the one globe of the Earth? Three altogether, if we examine everything, but those distinguished by their subjects no less than by time. One is the annual motion of the centre about the Sun, of which in Book VI; another is the secular inclination of the axis upon the centre of the body, of which we treat here; the third is the diurnal motion of the body about the centre and axis as if immovable, of which in Book I. Examples of all three mixed together are in the top, Book I, folio 173.
What cause do you assign to this inclination of the axis, or what mover? It can be that animal faculty which turns the globe about its own axis, but which, hitherto content with bodily instruments and the disposition of fibres, now uses mind besides: and of this cause mention was made in Book I, folio 178. We also granted such a cause for the motion of the apsides in Book IV, folio 381, and such for the administration of the motion of latitude, folio 386.
But it can also be a concurrence of physically moving causes, an external one, which is the moving species of the Sun, and an internal, which is the disposition of the fibres, and an impediment arising from the concurrence, so that this very slow motion is not truly a motion but is a kind of aberration of the diurnal body from the annual motion of the centre, of which kind many can be conceived, as appears from the examples of the places cited, folio 381 and folios 385, 386; and this cause also of the inclination of the Earth’s axis I hinted at in Book I, folio 174. So there would be two real motions only.
But that there may be less labour and weariness of the wit in working out the manner of this concurrence, see the reasons in the places mentioned; for they hold here much more strongly than in the business of the translation of the apsides and latitudes.
What new thing do you say has been discovered about the motion of the fixed stars and of the ecliptic? It is established that the fixed stars depart not only in longitude from the vernal section of the zodiac but also in latitude; the northern fixed stars about the summer solstitial point are found today nearer the northern pole of the ecliptic, while the southern about this part of the ecliptic approach the ecliptic itself. About the winter solstitial point the same happens analogously, but about Aries and Libra no sensible difference is found. See Book II, folios 193 and 229, and Tycho Brahe’s Progymnasmata, Volume I, from folio 233 to 246.
So also this new thing befell Albategnius 750 years ago, that he found a smaller declination of the ecliptic than Eratosthenes and Ptolemy had handed down, and this diminution has lasted down to us, so that today it is not new to us.
What hypotheses did our predecessors devise to save these new phenomena? To the diminution of the greatest declination of the ecliptic from the equator our predecessors neglected to assign its cause, down to Copernicus. He made the pole of the Earth librate, for the sake of this phenomenon, on the diameter of a little circle which falls in the plane of the colure of the solstices. Into this libration, as a warp, he wove another libration as a woof, twice as swift, to save the inequality of the precession, so that from both compounded there arose the appearance of a twisted garland; see the Narration of Rheticus, joined to my Cosmographic Mystery (first edition). Others, offended by the absurdity of so many motions of the Earth, preferred to set an eleventh sphere around the world. But Tycho Brahe, who first noticed that the latitudes also of the fixed stars have been changed, not of all together, but only of those near the parts of the ecliptic that decline most, understood from this that it is not the poles of the world or of the equator that must be moved, so that the equator may approach the ecliptic, but on the contrary the poles of the ecliptic, having left their former seats among the fixed stars, move toward the poles of the equator and so the ecliptic approaches the equator; which we followed in Book III, folio 292. But in what manner this would come about Tycho Brahe left unexplained.
What, then, moves the poles of the ecliptic and indeed the ecliptic itself from its seats? Again the same annual motion of the Earth about the Sun, if indeed it is made in all respects like the motions of the other planets, as I advised in Book VI, Part I, that is, if it also has its limits of latitude and its nodes, and those movable, like those of the others. See Book IV, folio 358.
But for the other planets limits and nodes could be assigned with respect to the ecliptic, from which they ran out to their limits and which they cut in the nodes: the centre of the Earth, however, by its own motion about the Sun describes the ecliptic beneath the fixed stars, and so does not run out from it, as it follows the one that goes, like a shadow the body. The Earth indeed neither cuts nor leaves her own path, and therefore, that she may be able to have limits and nodes, it is necessary to establish some other fixed ecliptic, as a royal road, to which both the orbit of the Earth and the temporary ecliptic standing over it and the journeys of all the rest are variously inclined.
Whence arises that royal road and its poles, and what is it? From the axis about which the body of the Sun is turned, continued on both sides to beneath the fixed stars, two poles are marked, between which the middle and greatest circle of parallels is the royal road.
As in the figure given before, if the royal pole is A and about it in the little circle QOP the pole of the temporary ecliptic P is moved, and about this age, between Ptolemy and us, is in the circle AC, nearly at right angles to the usual colure of the solstices PF, the pole of the ecliptic P will approach the fixed stars situated in PF as far as the ecliptic itself, and the ecliptic itself will depart from the same about H, being always a quadrant distant from its movable pole.
Investigate the position of the royal ecliptic with respect to the temporary one, or rather of this with respect to that. First, because the nodes of the Moon and of nearly all the other planets go backward, let the sections of the Earth’s orbit with the royal one go backward too, the pole of the ecliptic being moved, namely, toward the quarters QOP. Let the pole of the world of today be F. Because, then, the fixed stars about and below F are today less distant from the pole of the ecliptic, the pole of the ecliptic P descends toward the fixed stars F. The nodes, therefore, or sections of the ecliptics, will fall about the colure PE, about O, or a little earlier toward the quarter AM, where PF and AF are equal; the limits therefore will be in the quadrate places, about Y> YY; so that if FAD is nearly a right angle, the nodes of the Earth’s orbit will be in the circle CAD produced on both sides, and the northern limit will be toward the quarter AD, because the pole of the temporary ecliptic is in the contrary quarter AC, namely at P.
On account of this disposition, which obtains in this age, namely the rectitude of the angle APF, we said above in Book IV, folio 385, that it is likely that the inclination of the planet is administered through the axis of the turning of the body. For under F is that axis about which the Earth is turned, and PF extends toward the nodes of the Earth. But if the same situation of the points A, P, F does not remain through all ages, this likelihood is disturbed.
The ascending node of the Earth is therefore today about O, the northern limit about Yj, the descending node about O, and conversely the royal road which we seek rises from O toward the north from our ecliptic as far as >, approaching the aphelia: of Mars in 29° of ..., of Jupiter in 7° of YY, of Mercury in 15 of /, of Saturn in 25° of /, all northern. From ^ it is south of our ecliptic, approaching the aphelion of Venus in 2° of ..., southern; for her southern limit is in Jf. So the very places of the aphelia of all the planets with their quarters mark this royal road as nearly as may be, yet they also decline somewhat from it, one in one way, another in another.
How does the variation of the obliquity of the ecliptic follow from this hypothesis? This hypothesis is not yet sufficient, for someone could say that the axis of the Earth retains a constant inclination to the plane of its temporary ecliptic and that the arc of distance of the poles FP always remains the same. It is necessary, therefore, to add these things besides: 1) that the axis of the turning of the Earth is inclined at a constant angle to the axis of the turning of the Sun, or the royal pole A (which is likely in itself, since these are the two first principles of all the motions of the world, as was demonstrated in Book IV), and this we followed in Book III, folio 229, or that the circle LCM, under which the pole of the world advances through the fixed stars, is described from A, the royal pole; 2) that the pole of the temporary ecliptic P is swifter about A in the order contrary to the signs than the pole of the world E.
How great is the latitude or greatest inclination of the limit, or how great the distance of the poles PA of the ecliptic and the royal one? and how great the distance of the pole of the world F from the royal pole A, and whence is it deduced? Various quantities of these arcs could be taken. Let conjectures therefore have their way, such as neither the Alphonsines nor Copernicus abstained from. For they defined the returns of the fixed stars and of the little circles by a thousand jubilees and a thousand weeks of years, but Copernicus made the obliquation of the ecliptic and the precession of the equinoxes commensurable in respect of time: each followed probability, short of demonstrative necessity.
Let us also say, then, that the pole of the ecliptic at the beginning of the world was in Q, when the pole of the world was in C, so that QAC was a right angle, and that then AC or QC was 24° 17′ 40″, so that the surface of the temperate zones might equal the surface of the intemperate (see Book III, folio 272).
Let us say secondly that AQ, AP, AB are 1° 47′ 40″, so that the remainder QL or PC or BD is 22° 30′, the eighth part of the circle of the colure which measures the latitudes of the zones, and, the distance of the poles BD, QL or PC being likewise taken, as much of this colure is intercepted by the temperate zones as by the intemperate, which neatness was explained at folio 271.
For if the inclination of the axis, a mean between nothing and 90°, a mean again between nothing and half of 90°, namely 45°, had this final cause, that the temperate zones should lie between the intemperate, interposed between the frigid on one side and the torrid on the other, which indeed was shown at length in the said Book III from folio 268: nothing indeed seems able to be devised more neat than that the exact quantity of the inclination also should be derived from an exact double equality of the zones, of which one is of surfaces, the other of the mean latitude or diameters, which are arcs of the great circle called the colure of the solstices.
Further, this too seems to belong to the beginning of things, that the poles both of the temporary ecliptic Q and of the royal A should be equally distant from the pole of the world C, so that what the inclination AC of the pole of the world to the royal pole promised, the temporary ecliptic itself, or the inclination QC, should in fact perform in the beginning of the motions.
Let the reader compare those things which I disputed in the Commentaries on Mars, Chapter LXVIII, folio 322, about the situation of these little circles, and whatever difference he finds let him ascribe to the difficulty of the matter itself and the scarcity of suitable observations.
How great, then, is the variation of the latitude of the stars and of the obliquity of the ecliptic, what is the correspondence between the two, and what the period? The least obliquity, as was said, would be 22° 30′, or the arc cut off by the side of the octagon on the colure of the solstices; the greatest would grow up to 26° 5′ 20″, but this more than 36 thousand years after it was least. Nor would the solstitial fixed stars always, as today, change their latitudes; at another age it would be found in the equinoctial, namely when the obliquity of the ecliptic should stand immovable. For today, or between Ptolemy and us, the change of obliquity would have been swiftest. And the greatest difference of latitudes would grow to 3° 35′ 20″.
What is in this hypothesis the mean precession of the equinoxes? It is the arc of the royal road, numbered from the circle drawn through the royal pole and the first of Aries, to the section of the equinoctial and the royal road, and that against the order of the signs. Although for those times, when the first of Aries preceded the section, it can for brevity’s sake also be numbered in the order of the signs.
What is to be observed about this beginning of the fixed stars? Tycho Brahe numbers from the bright star of Aries, which is not the first; Copernicus from a point which is 115° 50′ distant from the heart of ..., or which is also 170° 0′ distant backward from the Spike of Virgo, because he thought that the first of Aries truly is only so far from the fixed stars backward. See Tycho’s Progymnasmata.
What is the anomaly of the obliquity of the ecliptic, or also of the precession of the equinoxes? It is the arc of the little circle which the pole of the Earth or of the world describes about the royal pole, from the circle through the poles, the royal and the mundane, to the circle of latitude through the poles of the ecliptic and the royal, numbered against the order of the signs.
Thus if F is the pole of the world, A the royal, P that of the temporary ecliptic, the angle FAP or the arc NP will be the anomaly of the obliquity or also of the precession.