Epitome Astronomiae Copernicanae: Liber VI, part 3
This most absurd inequality of the stations and retrogradations the old astronomy ascribed to the planets themselves, each separately, and held it to inhere in their true motions, as in their subjects.
What do you call the points of the stations? Either single points of the ecliptic, under which a planet for some time seems to stick immovable, or two points of the planet’s orbit, or also of the Earth’s orbit, terminating those arcs through which, while the planet and the Earth advance, the apparent station lasts.
How many stations are counted for each planet? Two, one before the opposition with the Sun, which is called the first station; the other, after the opposition, named the second.
How can the lines be made parallel, if the diurnal arcs of the eccentrics of the planet and of the Earth are unequal? By as much as the diurnal arc of the Earth is greater than the diurnal arc of the planet, by so much the more obliquely must the lines of sight cut the Earth’s orbit.
How are determined the more distant points of the great orb which represent the stations? Straight lines being drawn from any assumed place of the planet, touching the great orb on both sides; the points in which the Earth, placed there, represents the planet at that place of the eccentric as stationary, withdraw themselves within the points of contact toward the planet; most in Mars, least in Saturn, moderately in Jupiter.
As if inquiry is made about a place S of the eccentric of the planet, the straight lines drawn from it touching the great orb, SB, SC, so that B, C are the points of contact, the points of the two stations are in the arc BC nearer to the planet in S than B, C themselves.
What do you call the arc of retrogradation? 1. Either the arc of the ecliptic, comprehended between the two neighbouring apparent stations of the planet, 2. or the arc of the great orb, comprehended between the two pairs of parallel lines determining the stations. 3. It could also be used of the arc of the planet’s eccentric, which the planet runs from one station to the next, and this the same two pairs of parallels determine.
What was the opinion of the ancients on the causes why planets become stationary, each at a particular elongation from the Sun? The ancients, among whom Apollonius of Perga and Ptolemy who followed him, derive the causes from the proportion of the motions of the epicycle to the eccentric; but the magnitude of the epicycles, attributed to each in a preposterous order, and the causes of the proportion of their motions, they could not state. The Latin philosophers, holding the mathematical disciplines in contempt, ascribed the power of retarding and plainly inhibiting the motions of the planets to the aspects of the Sun, and said that the quartile of the Sun harms Saturn, the trine Jupiter, something more Mars; ignorant that these aspects, as they may be effective on these lower things, are rather terrestrial than celestial beings. For the aspects are not in the planets themselves, but only here on the Earth. Nor did they consider that aspects are beings of reason formally, in which, except in a faculty in some way partaking of reason, no force and efficacy can inhere.
What, then, is the true cause why Saturn between the quartile and the trine of the Sun, Jupiter nearly in the trine, Mars beyond the trine from the Sun, both before and behind, become stationary? Because, if the planet made no progress at all by the proper motion of its eccentric, the places of the apparent stations in the great orb would be the same as the points of contact, that is, the planet would appear stationary in the very quadrature of the Sun, about. For the Earth being posited in C, the planet in S, standing in truth, and CS touching the Earth’s orbit, the Sun would be seen by CE, the planet by CS, and so about 90° from the Sun, because ECS is nearly a right angle. But because the planet does not rest in S, the greater the true diurnal arc of each planet in the eccentric, the farther from the points of contact C must the two lines of sight depart, that they may become parallel and thereby the planet seem to stand.
Conversely, if the diurnal arc of the planet were as long as the arc of the Earth, no station could appear except in the very opposition of the Sun. Now the diurnal arcs of the superior planets are shorter, and that of Saturn shortest, of Jupiter middling, of Mars longest. Wherefore for representing the station of Saturn a small recession of the Earth from the point of contact toward opposition is needed, for Jupiter a greater, for Mars the greatest. The more the Earth is distant from this point, the greater becomes the angle between the lines of sight of the planet and of the Sun. It is concluded, therefore, that the station of Saturn is a little more than 90 degrees from the place of the Sun, that of Jupiter much more, that is about 120°, that of Mars most.
What cause do you assign why the slowest planet more often becomes stationary and retrograde, the swiftest more rarely and more slowly? Because the Earth in the great orb overtakes the slowest, Saturn, sooner and so more often, the swiftest, Mars, more slowly and so more rarely, and as often as it overtakes each, so often that planet appears retrograde, stationary before and after. Here the old astronomy is mute.
Why in general, the higher a planet is, whether than another planet or than itself, does it remain retrograde the longer? There are two causes. 1. If a planet, one of the superior, stood immovable in one place of the eccentric, as in S, then the lower it were, the smaller the arc of the annual orb would fall between the two tangents from the planet, namely between SC, SB; and the higher it is, the greater. And since we have said that, this being posited, the planet S would be seen stationary from the points of contact B, C, a greater arc BC, claimed for the higher planet, would be traversed by the Earth in a longer time.
What, however, follows, the rest of the planet being supposed, in the arc between the two points of contact, the same now, the planet’s own diurnal motion being granted, follows in the arc between the two pairs of parallel lines of sight. There comes in now a second cause, that the planet, the higher it is, is also the slower, or has the smaller diurnal arc of its orbit. But the smaller this arc of the planet, as in T, the less also must the diurnal arcs of the Earth, as in D, recede from the points of contact toward G, the point of opposition, that the lines of sight determining both may become parallel; which has been established above. If these points of the stations recede little inward from the points of contact, the arc of contact for the station of the superior planet is also little diminished; and so the time between the apparent stations, which the planet itself, being truly immovable, would have had (of itself longer than for the inferiors), will be so much the less diminished.
But the planet, the first station being now completed, while it appears retrograde, proceeding through all that interval in its eccentric, seems to transfer the second station farther in the order of the signs. If so, why does the one who proceeds more swiftly lengthen this portion of the annual orb more? But the one who proceeds more swiftly is the inferior, not the superior. That is indeed true; but this lengthening of the arc of the great orb cannot avail equally against that shortening of the same on account of the nearness of the inferior to the great orb; it only brings about that this prolixity of time of the retrogradations does not maintain the exact proportion which arises from the intervals of the different planets. For Saturn is nearly twice as high as Jupiter, and he three times as high as Mars, and yet the time of the retrogradation of Saturn is only slightly longer, and the Jovian time not longer than about one and a third of the Martial.
Whence is it that Saturn traverses the least arc of the ecliptic by his retrogression, Jupiter a middle, Mars the greatest? The cause of this again Copernicus alone can detect from his hypotheses, the old astronomy muttering and introducing unequal epicycles, with no cause of the magnitude indicated. This happens, therefore, because Saturn is farthest from the annual circle of the Earth, or great orb, Jupiter nearer, Mars goes nearest to it.
For hence it follows that if anyone were to look at the orb of the Earth, as BC, from a planet, as from S, it would seem small from Saturn, greater from Jupiter, greatest from Mars. It would be seen, moreover, by two lines of sight, SC and SB, touching it on both sides; to which we have said that the two pairs of parallels which define the stations approximate. And so if even a wholly immovable planet were to stick, then under the same angle under which the orbit of the Earth would be seen from S, toward one region of the zodiac from which the Sun E stands (namely the angle CSB, whose measure would be the arc of the zodiac comprehended between SC, SB produced), under the same angle, I say, or the one vertical to it, CS, BS being continued upward, there would be seen from the Earth at B, C, in the opposite region of the zodiac, the arc of apparent retrogradation, least for the highest planet, greatest for the lowest. For S can now equally be taken for the centre of the fixed stars, since ES also is an insensible interval. But what would be true, the planet being supposed at rest in S, is not much otherwise after it is established that the planet does not rest but is moved forward. For if this disturbs anything and lessens the appearance of the epicycle, it lessens it certainly in all; for all planets go in the order of the signs by their true motions.
But why are these arcs not in a proportion somewhat nearer to the proportion of the intervals of the Sun and of the single planets? As, because Mars is more than six times nearer to the Sun than Saturn, it seems that on account of the first cause stated he ought to have an arc of retrogradation six times greater than Saturn. Although it is not permitted to gather the proportion of angles from the proportion of intervals, as Euclid demonstrates in the Optics, there are nevertheless here other fit causes which disturb any proportion deduced from the intervals. 1) Because, the planets progressing, and the one that is inferior more swiftly, the two pairs of parallel lines of sight recede farther from the points of contact. Wherefore the arc of the great orb between the points of the stations, made smaller, will also appear smaller from the place S of the planet. 2) Because the more swiftly each progresses, the farther it transfers the ecliptic place of the second station, to which CS produced is directed, toward the ecliptic place of the first station remaining, marked under the fixed stars by BS, since that is in the antecedent, this in the consequent places of the zodiac. The ecliptic places of the stations coming together, therefore, the ecliptic arc of retrogradation is again diminished for the swifter planet and so the lower, which it was about to obtain very great.
But you have said that the contrary happens in some one planet, in respect of itself higher or lower. As if Mars is high from the centre of the world, he completes a longer arc of retrogradation, if low, a shorter. What is the cause of this diversity? Because the two causes just stated, shortening the arc in the low planet, in the comparison of different planets do not equal the force of the first, which gives the long arc to that low one; but in one planet compared with itself they exceed the effect of the first. For since both depend on velocity, this velocity of the lower of two is only in sesquialter proportion greater than the nearness of the interval, as was demonstrated in book IV, folio 350; but in one and the same planet, made lower, the velocity about the centre of the Sun is plainly of a double greater proportion than the nearness; which is demonstrated likewise in book IV, folio 351, 373, and used in book V.
What follows hence for the superior planets in conjunction? Saturn being direct, the others cannot be conjoined to him unless they too are direct, and Mars being retrograde, the others not unless they too are retrograde. But Saturn being retrograde, it happens that Jupiter and Mars also are conjoined to him direct, and so also Mars direct to Jupiter retrograde.
Why do the luminaries, the Sun and Moon, not also become retrograde? Not because they are luminaries, but each for its own reason; the Sun, because the Earth goes round him at rest, all the lines of sight therefore cut one another in the Sun himself and so always in the same quarter of the Sun, never in the contrary; the Moon, because she herself goes round the Earth as it goes, always swifter about the Earth than the Earth about the Sun, by at least eleven times.
Can no other causes of these appearances be rendered from another hypothesis? All these things, one and all, follow indeed also from the hypothesis of Brahe, in which the same disposition of the moving world is left, and in its middle the body of the Sun, in which all the orbs are as if connected by a common nail, but in place of the annual motion of the Earth about the Sun the Sun in turn goes about the Earth, carrying and shifting the whole planetary system; the motion of the shifting being depicted for you in book IV, folio 354. By this hypothesis of Tycho Brahe thus corrected astronomy is satisfied; but what can be opposed from physics, especially concerning the interpenetration of the regions of Mars and the Sun, on this see the said place, especially folio 356.
What is the anomaly of commutation, what the elongation from the Sun, and what the angle of each? They are arcs of the ecliptic numbered from the true place of the Sun in it in the order of the signs, the one to the eccentric place of the planet in the ecliptic, the other to the apparent or seen place of the planet. The angle of each is numbered either in the order of the signs from the place of the Sun, or against it, so that it is always less than 180 degrees. They are named from the things themselves, the former the angle at the Sun, the latter the angle at the Earth. In diagram 85, TC is the line of the true place of the Sun, the Sun being in S, the planet placed in P, SPL is the line of the eccentric place of the planet in the ecliptic, in the old form it would be TN, parallel to SL, and TPV is the line of the seen place of the planet; wherefore CSL is the angle of commutation, CTV the angle of elongation.
How is the angle of the anomaly of commutation found? The seen place of the Sun being subtracted from the eccentric place of the planet reduced to the ecliptic; or this from that, so that what remains is less than a semicircle.
What is the parallax of the orb? It is the difference of the angles of commutation and elongation, and is also called the angle at the planet, in the diagram TPS or PTN. And since parallax in Greek is the same as commutation in Latin, ambiguity is to be guarded against; each is an angle, that of the anomaly of commutation, this of the commutation itself. When orb is said, understand of the great or annual orb of the Earth, because this orb causes that commutation or translation of the apparent place from L to V. Though we do not use this parallax separately in the Rudolphine Tables.
How great is the greatest parallax of the orb in each planet? In Saturn it is least, in Jupiter middling, in Mars greatest; in each less in the aphelion, greater in the perihelion, and (the unequal translation of the apsides being granted) not the same in all ages.
What is the index in the Copernican form of astronomy and the calculus of the motion of the planets? Because it cannot be done that we refer the parallaxes of the orb to the semidiameter of the orb, as is done in the old astronomy, therefore in place of the double proportional minutes and the excess of the parallax of the orb, or of the diversity of diameter, in the old astronomy, there has been introduced a number indicating what is the proportion of the sum of the intervals of the Sun and the planet and the Earth (in diagram 85, TS, TP) to their difference.
How is it found? The difference of the distances of the planet and of the Earth from the Sun, extended by five ciphers, is divided by their sum. But for the quotient its logarithm can conveniently be used, as is done in the Tables of Rudolph.
Teach how to find the angle of elongation from the Sun that is visible and the visible place under the ecliptic. The tangent of half the angle of the anomaly of commutation, multiplied into the index, makes the tangent of an arc to be added to that half in the superior planets, to be subtracted in the inferior, so that the angle of elongation from the Sun is constituted. If for the tangent the mesologarithm of that half is used, by simple addition of this to the logarithm of the index there comes out the mesologarithm of the same arc, to be added in the former, subtracted in the latter.
But if the eccentric place of the planet was subtracted from the true place of the Sun, now also the angle of elongation is to be subtracted from the same; but if the place of the Sun was subtracted from the eccentric place of the planet, this angle of elongation will now be added to the place of the Sun, that the visible place of the planet in the ecliptic may come out.
III. Of the increases of the magnitude of the planets.
Prove that, the Earth going about the Sun, it is necessary that the planets in opposition to the Sun, other things being equal, appear greater, toward conjunction with the Sun smaller. Euclid demonstrates in the Optics, prop. 56: when the eye approaches the thing seen near, it is thought to increase. Now the eye approaches the planet so long as the Earth, the dwelling of the eye, hastens to the place of interposition between the Sun and the planet. Therefore the planet will meanwhile seem to increase and conversely, the Earth having passed the place, again to diminish. In diagram 85, Q the planet, T the Earth, interposed between Q and S the Sun, the interval of the planet and Earth will be TQ. Let the bodies be moved in the order of the signs, the planet from Q to P, the Earth through the greater part of the circle from T to B, so that S the Sun and P the planet seem conjoined, and the interval of the planet and Earth will be BP. As, therefore, TQ to BP, so conversely the diameter of the planet in P to the same in Q. How great the diameters of the superior planets appear through the dioptric tube has been said in book IV, folio 329.
Where are the nodes, where the limits of the superior planets, and what is their motion? Ptolemy found the northern limit of the eccentric of Saturn at 3°, of Jupiter at 1°, of Mars at the end; today the limit of Saturn comes at 21°, of Jupiter at 7°, of Mars at 19°. Therefore although the limits, in respect of the vernal equinox, which is held to be the beginning of the ecliptic, move in the order of the signs, yet in respect of the fixed stars all move against it; as do also the limits of the Moon. The nodes are in places of the ecliptic precisely at quadrature, the ascending in the former quadrant.
How and how much are the eccentrics inclined to the ecliptic? Since the nodes are found in diametrically opposite places of the ecliptic, the planes of the eccentrics and of the ecliptic therefore cut each other in straight lines drawn in common through the centre of the Sun. The sections of all the planets (no less than the lines of apsides above) therefore intersect one another in the centre of the Sun. The angle by which the limits are inclined to the ecliptic is in Saturn 2° 32′, in Jupiter 1° 20′, in Mars 1° 50′ 30″, and this today. Ptolemy also gives in Saturn 2° 30′. Does this angle remain invariable? In one period it remains invariable, but from Ptolemy it cannot have changed much; for we cannot ask of the Ptolemaic numbers that they give the truth to a single minute, since the observations were crude. Nevertheless there is a suspicion that, the ecliptic of the time being changed, these inclinations of the eccentrics, reckoned to it, are changed also, and that the limits are inclined constantly to the royal middle ecliptic alone and describe parallels to the royal one; it becomes probable that so the poles of the orbits do not stand equally distant from the poles of the great orb or orbit of the Earth in all ages, on account of the distinct motions of both. See in book VII the probable cause on account of which the greatest inclination of Mars was of old less.
What is the latitude of a planet? It is the arc of a circle of latitude, described under the fixed stars, intercepted between the ecliptic and the seen place of the planet.
What is the kinship, what the comparison, of latitude and the inclination delivered in book V? 1) In the oppositions and conjunctions of the planet with the Sun the circle, both of latitude and of the inclination of each place, is the same. 2) Latitude is always with some inclination, and conversely; and where there is no inclination, there is no latitude. 3) As often as between the planet M and the Earth B there is the same distance as between the planet M and the Sun E, or the triangle EMB is isosceles (Fig. 86), the inclination is equal to the latitude. This happens when between the seen places of the planet BM and of the Sun BE there is less than the fourth part of a circle; which arc in Mars is least, in Jupiter greater, in Saturn nearest to a quadrant. When, therefore, the elongation of the planet from the Sun is greater than this angle, or the distance of the planet from the Earth less, the latitude exceeds the inclination; when it is less, the latter is greater, and is exceeded by it.
Prove that it is necessary that, the Earth going round and so approaching the planet, the latitude of the planet be increased and exceed the inclination. The demonstration is the same concerning the inclination as above concerning the increase of the apparent magnitude of bodies, from prop. 56 of Euclid’s Optics. The old astronomy has here contrived an infinite apparatus of epicycles, diameters and inclinations, reflexions, obliquations, and that most intricate and most difficult to grasp, and has not even so satisfied the observations.
Where is the latitude greatest? Although for the most part the latitude is once greatest in the middle of the retrogradation, about the opposition with the Sun, or in the inferior planets about the inferior conjunction: yet the very turning-points do not come at the oppositions and conjunctions with the Sun themselves, but the greatest latitude occurs, especially in Mars, sometimes before or after the couplings with the Sun, namely when the distance of the planet and of the Earth increases or decreases in the same proportion in which the inclination does.
How is the latitude of a planet found? As the sine of the angle of elongation stands to the sine of the angle of commutation, so the tangent of the complement of the inclination stands to the tangent of the complement of the latitude. Therefore let the sine of commutation, increased by 5 ciphers, be divided by the sine of elongation, and the quotient be multiplied into the tangent of the complement of the inclination, and, 5 last figures likewise being cast away, the tangent of the complement of the latitude comes out. Or by the most compendious way, let the logarithm of elongation be taken from the logarithm of commutation, and the remainder added to the mesologarithm of the complement of the inclination: the sum is the mesologarithm of the complement of the latitude.
How great can the latitudes become? If all the positions of the limits of the planet are interchanged with all the aphelia, both of the planet itself and of the Earth, many ages will have to be awaited, and yet the greatest of Saturn will not attain 2° and 5/6, nor of Jupiter 1° and 5/6, but the greatest of Mars will be able to exceed 7°, as great as nearly the greatest southern is today. Of the parallax of the planets in respect of the diurnal motion we shall treat below in the doctrine of eclipses.