Epitome Astronomiae Copernicanae: Liber VI, part 2
Second Part of the Third Part of the Theoretical Doctrine. Of the three superior planets, Saturn, Jupiter, Mars, and some things common also to the two inferior.
Why, after the explanation of the apparent motions of the Sun (that is, of the proper motions of the Earth), is there subjoined at once the theory of the three superior, then of the two inferior primary planets? Because, since two inequalities of motions are found in the primary planets, the one which truly inheres in them, fully explained in book V, is very like that of the Earth; the other, which does not truly inhere in them but arises from the deceptions of sight, depends on the motion of the Earth itself, because in this vessel, as it were, our sight is carried about the world. Since, then, in these planets the use of the great orb chiefly appears, the theory of the superior planets, and then of the inferior, so far as concerns what remains of them unexplained above in book V, ought to follow immediately after the theory of the Sun, as a kind of appendix or supplement to the doctrine of the great orb; the theory of the superior planets, however, comes first, because in them the deception of sight arising from the great orb is wider, more evident and more obvious to the sense.
How many orbs are needed for the theory of each of the primary planets? Only two besides the ecliptic, one the eccentric, proper to the planet itself, the other that of the Earth, or the great orb, common to all. Their nature, the laws of their motions and the common astronomical terms were explained in book V. The proper quantities of the great orb also have been already premised in the first part of this book VI; the old astronomy needs here many epicycles and other apparatus.
But why, though there are three superior planets, is only one theory proposed, and below also only one of the two inferior? 1. Because the three equally encircle the same orbit of the Earth with their orbits, being superior, but the two equally have courses enclosed within this orbit of the Earth, being inferior. 2. Because the planets are very like each other in their real motions in eccentric orbits, as has been said, while the deceptions of sight, which befall any one of them, arise from the one great orb alone, common to all, yet in two ways, one common to all three superior planets, the other to the two inferior.
3. There is also a mechanical cause, proper to theories (which are instruments of ocular demonstrations). For the theory of Saturn having been drawn and the great orb adjusted, there is so much space between that in it the eccentrics of Jupiter and Mars can also be described, and the same orb, great in name and use, least in quantity, can serve all three; so, the eccentric of Mercury being drawn about the Sun and the great orb adjusted to it (which in respect of the inferior planets also is in fact great, being greater than the orbits of both), there remains also a convenient place for describing the eccentric of Venus between the two. For this reason one single theory can be made for all six primary planets, consisting of six orbits, less conveniently indeed as regards the mechanical construction, yet more properly to the genuine pattern of the moving world itself.
Review what appearances of motions the three superior planets make. 1. The eccentric motions of the three superior planets in longitude, such as were described in book five, astronomers find to agree with observations, or appearances, only at the points of the true oppositions of each with the Sun, scattered through the zodiac.
In the diagram, the planet being placed in Q, its eccentric place under the fixed stars will be D, and although the planet often comes into Q, yet it never appears from the Earth under D, except only if the Earth is at the same time in T, a point of the line SQ; so that from T the Earth sees the Sun S and the planet Q in opposite places C, D.
2. But the visible or apparent motions of the three superior planets outside the points of opposition are very different from those delivered in book V. For if we take with instruments the distances of the planets from the fixed stars near the zodiac on successive nights, exploring how much each advances at any time from one night to another, then we find them nearly as much swifter to the eye as they come nearer to the Sun toward the evening concealments, or as they have gone less far out from the Sun’s rays after the morning emersions, and conversely as much slower as they are more remote from the Sun, so that at last they become plainly stationary to sight, Saturn indeed between a quadrant and a third of the circle before and after the Sun, Jupiter exactly a third from the Sun, Mars beyond a third, namely by three eighths and two fifths of the circle before and after the Sun. Between those points toward opposition with the Sun they become plainly retrograde, and in this retrograde motion they are swiftest in the middle, about the opposition itself in the acronychal position. And Saturn, the highest and slowest planet, has more stations in its eccentric, Mars, the lowest and swiftest, fewer, Jupiter, in his mediocrity, a middle number. Conversely, the higher each is and the more remote from the Sun than itself or than another lower, the longer it is found to remain retrograde; Saturn prolongs its retrogradation into the fifth month, Jupiter occupies four months, Mars three or two, though on the contrary the highest of the three traverses by its retrogression the least arc of the ecliptic, the lowest the greatest, yet in a most different manner, one and the same planet, when higher, the greatest, when made lower, the least: Saturn a little more than 7°, Jupiter 10°, Mars when high almost 20°, when made low less than 12°. So that this second inequality differs very much from the first of book five and in fact has very much of inequality.
3. As regards motion in height, planets placed in the acronychal position are found always greater; the nearer they are to the Sun, the more sensibly they diminish the apparent magnitude of their bodies.
Here, if the Earth is in T, the planet in Q will appear great, but about A small.
4. Finally, by reason of motion in latitude, the apparent latitude of the planet is always greater when it is in opposition to the Sun than if, a return having been completed, standing in the same place of the eccentric, it is now nearer to the Sun, and it is always so much the less as in the completed returns it is nearer to the Sun; though the true inclination of each place in the eccentric through all the returns of the planet is the same and constant, as was shown in book V.
I. Of the eccentric motions.
That we may begin from the first head, say how the real eccentric motions of the three superior planets, and also of the two inferior, stand. They are exactly like the eccentric motion of the Earth, explained in the first part of this book VI, and observe the laws delivered in book V, so that there is no doubt but that the physical causes confirmed in book IV (from which those laws arise) have place in each; that the Sun, namely, in the middle of all the circuits, moves the planets by the law of the balance. For the single orbits are ordered under single great circles of the fixed stars, inclined to the ecliptic, each in a different place; and they are elliptical figures, whose longer diameters pass in common through the centre of the Sun, so that in it they all intersect; the parts of the orbits are unequally distant from the Sun, and in the parts more remote from the Sun the planets become twice as slow as themselves in the more remote, in those near the Sun twice as swift as themselves in the nearer, and this as regards appearance as from the Sun, or the angles standing at the centre of the Sun, which angles the daily journeys of the planets subtend.
How great are the periods of time of each, and their mean diurnal motions? They complete their courses under the fixed stars Each returns to its own apsis a little more slowly, as will soon appear; but to the same point of the ecliptic a little more swiftly, because these points meet them, as we shall be taught in book VII.
What is the proportion of the three superior orbs and of the fourth, the great orb or that of the Earth, to one another, and likewise of the eccentricities and intervals? Here the old astronomy labours in blind ignorance, as one which discovers only the proportions of the eccentricities to the diameters of their own orbs, but delivers no certain proportion of the eccentrics among themselves, except what arises from the material superstructure of the orbs, on which see book IV, f. 316 and f. 333. For since it gives each its own epicycles, it dissolves the common bond of dimensions. In the form of Copernicus, therefore, the intervals stand thus.
Eccentricities, of which the semidiameter is 100000:
These intervals have been constructed from the observation of the parallaxes of the orb, of which below; but I have explained their archetypal causes in book IV, folio 317, from the Cosmographic Mystery, and folio 323, from the fifth book of the Harmonies.
At what places of the ecliptic do the aphelia of the eccentrics rest, and how great is their motion? Ptolemy, having referred the motions of the eccentrics to the Earth, placed the apogee of Saturn in his age at 23° 11′, of Jupiter at 11° 11′, of Mars at 25° 30′. Supposing, therefore, that the centre of the Earth’s eccentric also of old stood distant 1800 parts from the centre of the Sun, as was said in the theory of the Sun, and that the lines from the centre of the great orb through the centres of the eccentrics must be equidistant from the Ptolemaic lines of apsides (of which equivalence see the Commentaries on Mars, chapter VI), and so fall on the places of the ecliptic already indicated: certainly the lines of the true apsides, namely of the aphelion and perihelion, drawn from the Sun through the centres of the eccentrics, fell on other places of the ecliptic: of Saturn at 23° 34′, of Jupiter at 15° 7′, of Mars at 0° 42′. See the Commentaries on Mars, chapter XVII, and its correction, chapter LIX.
But today the aphelion of Saturn is at 25° 48′, of Jupiter at 7°, of Mars at 29° 20′. If the observations of Ptolemy are true, the aphelion of Saturn would be the swiftest, that of Mars slower, though the period of the former has 30 years, of the latter not two, where no cause appears for so greatly disturbed an analogy. The aphelion of Jupiter would be so slow that it would not overtake the precession of the equinoxes, but would become retrograde under the fixed stars, which seems repugnant to the physical causes explained in book IV; nor, even if it stood immovable under the fixed stars, does it agree with the example of the rest. Therefore the observations of Ptolemy are suspect on this account also, especially about Jupiter.
Did the ancients also observe that the parts of the eccentric become twice as swift when nearer the Sun, twice as slow when more remote from the Sun, and by what argument? They observed it altogether; first by this, that if they had ascribed all the inequality of the eccentric motion to the eccentricity alone, or the diversity of the distances of the epicycle (which they set up, one in each planet, in place of the one great orb), then a greater variety in the apparent magnitude of the epicycles would have resulted than observations bore. Then, that in the supposed epicycles themselves the planets, when made highest, were found to be swifter than in proportion to the distance of the whole epicycle from the centre of the world. And so they were compelled to introduce the centre of the equant, whose office was this, to render those solid orbs (as they believed), the eccentric and the epicycle, in truth inconstant and unequal in motion. For if the epicycle advanced at the highest place of the eccentric, then the whole eccentric was rendered in all its parts truly slow by the equant; and if the planet were at the highest place of the epicycle, the whole epicycle was rendered in all its parts truly swift, just as to us the very centre of the body of a high planet is made slower, of a low one swifter, from physical causes and the principle of the balance; nothing of which kind could be elicited from the old astronomy that would promise the cause of this appearance.
II. Of direct motion, station, retrogradation.
Since, then, such eccentrics save the observations only in the opposition and conjunction of the planets with the Sun, say what are the causes of the remaining irregular appearances mentioned. Here the chief virtue of the Copernican astronomy shines forth, that while the old astronomy is silent and only wonders, it speaks and explains the causes of things; and while the old astronomy multiplies epicycles, the Copernican, more simple, saves all these by the sole and single motion of the Earth about the Sun, added to the eccentrics, which was already introduced above for saving the apparent motion of the Sun, in the first part of this book VI. And on account of this very ample use the orbit of the Earth about the Sun was by the highest right called the great orb in the first part of this book VI.
That we may approach the demonstration, say at the beginning what is the line of the seen or apparent motion of a planet, and what its seen place. Speaking technically and in words adapted to manual theories, in which the proportion of the planetary orb to the orb of the fixed stars, immensely great, cannot be expressed, this line is defined as a straight line drawn from the centre of the Sun to the fixed stars, equidistant from the straight line drawn from the centre of the Earth through the body of the planet. But according to the condition of the thing itself (because the great orb, compared with the fixed stars, is insensible) it is the straight line itself drawn from the centre of the Earth through the centre of the body of the planet to the fixed stars; where this falls on the fixed stars, there it marks the seen place of the star.
In the diagram premised, let the Sun, the centre of the fixed stars, be in S, about which let the great orb BT be, and the Earth in it; and about this let the orbit of one of the superior planets AQR be. Let the place of the Earth be T, of the planet R. T, R being joined, therefore, by the straight line TR continued to the fixed stars at N, and a parallel to it drawn from the Sun S, which shall be SL, this SL will indeed in the manual theories be the line of apparent motion of the planet situated in R, but in truth of the matter TRN itself will be the line of sight of the planet R. And since TS, the interval of the Sun and Earth, is not sensible compared with SL or TN, the distance of the fixed stars, therefore TRN and SL continued fall upon the same place of the sphere of the fixed stars to sense; for the distance of the points N, L, on which these parallels fall, is not sensible on the Earth, but is taken for one point.
By what sign do we know in the hypotheses of Copernicus that a planet is direct, stationary or retrograde? One of the planets superior and slower than the Earth then seems to be direct when the lines of sight, or lines of apparent motion, of the planet on successive days cut one another between the Earth and the planet, or in the centre of the Earth, or behind the Earth, in the direction opposite to the planet; it then seems stationary when the two said lines are parallel; finally retrograde when the intersection falls beyond the planet. Of the inferior planets we shall treat in its place; for they have for the most part contrary reasons.
In the accompanying diagram let E be the Sun, ABC the great orb or orbit of the Earth, divided into diurnal arcs, OPK the orbit of one of the superior planets, enclosing the Earth’s orbit, divided HI, and then the line of sight of the following day will cut the former in the very centre of the Earth. The following lines of sight produced will cut one another in a direction which, from the planet going toward T, beyond the position of the Earth between D, H, tends to the opposite part, as if toward O. In all these conditions I say that the planet appears direct, that is in Q, in Q and about I. But in the diurnal arcs of the planet, more in number than those of the Earth’s orbit, and also smaller according to the doctrine of book IV, folio 346. And let the Earth be at the same time in A and the planet in O, and from there, the diurnal arcs being counted in equal number on both sides, let the Earth finally be at the same time in G and the planet in K. And let the divisions of the diurnal arcs of this be joined with the corresponding divisions of that. It will come about, therefore, at the first line of sight AO and its neighbour, that they cut each other in F, a point intermediate between the Earth A and the planet O; so also, the Earth being in C, the planet in Q, the intersection will be in H, again an intermediate place between C and Q. When the Earth has passed beyond H, there will be one of the lines of sight issuing from the neighbourhood of H itself which touches the Earth’s orbit or is equidistant from a tangent, cutting it and passing through two neighbouring divisions of the Earth’s orbit; let the planet now be coming to T, the Earth to D: the two lines DT begin to become parallel: through these lines of sight, therefore, I say that the planet appears stationary. Finally, beyond D, T, toward G, K, from parallels they become again convergent, but beyond the planet’s course TK, toward N, if they are continued; this, therefore, I say, proves that the planet K appears retrograde by the lines of sight GK.
Prove now that it is necessary, the Earth going about the Sun with its annual motion, that the superior planets, returning more slowly, appear direct in conjunction with the Sun, stationary beyond the quadrature from the Sun, retrograde in opposition to the Sun; and this in respect of the second motions tending from west to east. For retrogradation let there be first this popular demonstration. Euclid demonstrates in the Optics, proposition 54: if some things are carried with unequal speed, and among them the eye also, those which are carried with the same speed as the eye seem to stand, those slower than the eye to be carried in the contrary direction, those swifter to run ahead of the eye. This he says word for word. Which, although adapted to popular matters, for example ships running on the sea (where those which seem to stand to one going along with them seem so because they can be compared with swifter and slower ones going along, of which the former seem to run ahead, the latter to move backward), yet can be not ill adapted also to retrogradations in astronomy. For here also the Earth runs the same way with the superior planets; and since unequal returns are given to it and to them, and the orbits of each return into themselves, it comes about that the Earth with the planet now runs from the same quarter, now from the opposite.
When, therefore, they run from the same quarter, as the Earth in G, the planet in K, the Sun E is in the same quarter for both, and so the Earth G is midway between the Sun E and the planet K, that is the planet K is in opposition to the Sun E, to us, namely, who are in G. Here, then, the eye in G is swifter than the planet in K, not so much because the diurnal motions of the Earth in G are greater than the diurnal motions of the planet in K, as because these diurnal motions at G and K are almost parallel to each other in this place: whence it comes about that the lines of sight, which join the ends of the unequal motions, are inclined to one another, about to meet above the planet. But this is the sign of retrogradation. See the optical part of the Astronomy from folio 324 to 334. (II. 333 ff.)
As to the stations and direct motion, this optical axiom will be applied less conveniently to circular courses. For the Earth being posited in A, the planet in O, the planet and the eye no longer go the same way, as was to be supposed in the axiom, but meet each other from opposite parts of the circles, the Earth by the way AB, the planet by the way OP, nor are the planet and the Earth ever moved equally, but the former always more slowly. Wherefore in this part, dismissing the popular demonstration, let us pass to the astronomical, proper to the Copernican hypotheses. As to the stations, then, it is one of the suppositions of Copernicus, confirmed in book IV, folio 331 ff., that the diameter of the great orb, compared with the diameter of the sphere of the fixed stars, is insensible. Much less sensible, therefore, will be the distance of the parallels about D, T, which include at least a particle of the great orb, namely the diurnal eccentric arcs, of the Earth at D and of the planet at T. Those parallels, therefore, drawn out to the fixed stars, will seem to fall, as it were, on the same point of them, and so the planet through both those lines of sight, that is both at the beginning and at the end of the undertaken time of 24 hours, will seem to stick at the same place of the fixed stars, which is to stand, or to make a station. Since it is necessary, then, that parallel lines of sight occur twice, on account of the conjunction and separation in all ways of the running bodies of the planet and Earth, it will be necessary also that the planet appear stationary in two places, which places will now be demonstrated to be removed beyond the quadrature of the Sun.
Of direct motion and retrogradation thus. First it is necessary, the returns of unequal time of the bodies of the Earth and planet, both going in the order of the signs, being posited, that intersections of lines of sight happen in all manners.
For let there be at the beginning, in opposite parts of the circles, the Earth in A, the planet in O, and let the journeys in contrary directions be AB and OP; the contrary ends of the arcs being joined, for example by the lines AO and BP, it is necessary that their mutual intersection occur at F, a place between the arcs. But the opposite parts of the orbs have the Sun E between them, because the Sun is to Copernicus the heart and as it were the common axis of the orbs, as was confirmed in book IV, folio 318. It follows, therefore, that on some of the days on which the intersection F is between the Earth A and the planet O, the Sun E and the planet O are seen from the Earth A conjoined under the same place of the fixed stars by the line of sight AEO produced to the fixed stars. Because the intersection always approaches the Earth (for it is most remote from A, nearer to B and more to C), and it has been proved that the lines of sight at last become parallel, it is necessary that the intersection in the intermediate place fall at last on the Earth itself and thence run out behind the Earth to infinity, that is, that at last no intersection occurs; as when the Earth comes to D, the planet to T. But if the intersection falls on the Earth (let it be H), the centre of the Earth in its later position falls, therefore, on the line of sight of the former day; that line of sight (let it be HI) cuts off from the great orb one diurnal arc at H, and is therefore equidistant from a straight line touching the great orb in the middle of that diurnal arc. But the line drawn from the centre, to which the Sun is near, as EH, to the point of contact, which shall be H, makes a right angle EHI with the tangent HI. If, then, HE is the line of sight through the Sun, HI, the line of sight drawn through the body of the planet I, will tend to the place in quadrature with the Sun. When, therefore, the intersection occurs in the centre of the Earth, the planet appears about in quadrature with the Sun; but if it is seen more than about 90° elongated from the Sun, the intersection is behind the Earth toward O, until it degenerates into parallels. So it follows that stations occur beyond the quadrature of the Sun. Again, that the lines of sight beyond DT toward GK, drawn in order, meet when produced and are cut by one another above the planet, about the opposition of the Sun, was demonstrated before in the use of the optical axiom.
It has now, therefore, been demonstrated that the intersections of the lines of sight at the time of conjunction of the planet with the Sun occur between the planet and the Earth, at the time of the quadrature of the Sun in the very centre of the Earth, then behind the Earth, and further on none occur; but toward the opposition of the Sun they occur above the planet. But if the lines of sight cut one another above the planet, it is necessary that the planet seem retrograde; if below, toward the Earth, or in the Earth, or behind the Earth, the planet, one of the superior, will necessarily seem to advance direct.
For let OPQ be the order of the signs, in which the planets by their true motions go about the Sun E. Since, then, any intersection also in a manner represents the centre of the fixed stars and is midway between the Earth and the planet, through ABC up to H, any intersection will distinguish the opposite directions of motions, as the Sun does. And since the Earth through ABCH on this side of the intersections moves forward under the fixed stars, in the direction FA, FB and so on, standing above, those parts also of the lines of sight which are beyond the intersections will be disposed in the forward direction. So when the intersection occurs in the centre of the Earth about H, two lines of sight issue from the same place H of the centre, one of the former day, the other of the present, and they issue toward the parts of the planet’s orbit following in the order of the signs, the former to a place nearer to Q itself, the latter to a place I, because the true motion of the planet is from Q toward I. Therefore these lines of sight also advance in the order of the signs. So when the intersection occurs behind the Earth H toward D, the Earth and the planet are in the same quarter in respect of the intersection, and in the same parts of the cutting lines, which again hold the order of the points both of the orbit QIT and of the orbit HD. So far, therefore, the lines of sight advance in the order of the signs. But when the intersection is beyond the planet opposite the Sun, toward R or K, then again the parts TD, KG of the cut lines hold the same quarter in respect of the intersection, the Earth in D, G, and the planet in T, K. But because the intersection is in the quarter of the planet opposite the Sun, they are opposed, on the one side E the Sun, on the other the intersection toward R or N, and the journeys of the bodies DG and IK are interposed. They are direct in respect of the Sun E, because they are bent about the Sun, and they are therefore retrograde in respect of the intersections above R, N. Therefore those parts of the lines of sight DT, GK which are above the intersections will themselves also be disposed against the order of the signs; wherefore it is necessary that the planet opposite the Sun seem retrograde.