Epitome Astronomiae Copernicanae: Liber V, part 1
Book V
The second of the theoretical doctrine.
Of eccentric circles, or the theories of the planets.
If you set up no solid orbs in the heaven, and if all the motions of the planets are administered by natural faculties which are inborn in the bodies of the planets themselves: I ask then what will be the method of astronomy? For it seems that it cannot do without the imagination of circles and orbs. Of feigned circles and orbs, that useless furniture, it can easily do without; but of the imagination of the true figures into which the journeys of the planets are ordered we are so far from depriving astronomy that the chief work and labour of the true astronomer is to demonstrate from observations what figures the planetary orbits have, and to devise such hypotheses or physical principles that from them figures can be demonstrated agreeing with what is deduced from observations. The figure of the planetary orbit once established, it will in future be the second and more popular exercise of the astronomer to form and govern the astronomical calculation by this genuine figure, or even, that figure being expressed in material instruments, to use it no otherwise than the solid orbs of the ancients, and as a rule to set the courses by these figures before the eyes.
What matter then do you give to the fifth book, or second of the theoretical doctrine, and by what difference do you separate it from the matters of the preceding fourth and the following sixth? So far in the fourth book the physical principles of the motions (among other things) have been demonstrated by reasons and experiments; the fifth will form from these physical principles the figures of the planetary orbits and explain the powers of those figures; where the most hidden store of geometry will have to be searched. The sixth will teach the use of these figures in the theories of the single planets and bring it into work. The fourth therefore has the theory, the fifth the instrument, the sixth the practice; the fourth was physical, the fifth is geometrical, the sixth will be properly astronomical.
How many parts has Book V? Two: in the first the eccentric is connected with its plane, together with the physical causes; in the second are given the definitions of the astronomical terms which occur commonly in all the planets about this eccentric, and the method of calculation as to this part is explained.
What figure, then, is formed of the planetary orbit from the physical principles of the fourth book? If the body of the planet had no magnetic fibres, so that according to one region of them it would be drawn out to the north, according to the other to the south, according to one region be drawn toward the Sun, according to the other be expelled: then the Sun, carrying round by the turning of its body about its axis the immaterial form of its body through the vast spaces of the world, would carry the planet with it, seized by that form, and 1) if that planet had been placed at the beginning under the ecliptic, it would order its whole journey most exactly in the plane of the ecliptic, 2) and would always restore it to that very point from which the beginning was made, 3) the centre of the body of the Sun and of the planetary orbit would be the same, 4) the figure itself of the orbit would be a most perfect circle, 5) the planet would be carried in all equal portions of this circle with most equal swiftness.
But since we have posited that in the body of every planet there are double fibres, it comes about by the mixture of the faculties of the planetary body and of the moving power of the Sun that 1) the planet describes an orbit oblique to the ecliptic; and since the fibres of latitude remain nearly in a parallel situation through the whole circuit, yet not altogether, but rather after many gyrations are gradually inflected, therefore 2) the plane comprehended by the orbit of the planet is very nearly a perfect plane, yet not altogether; rather, one return being completed, the centre of the planetary globe is not exactly restored to its beginning, but, the former having been run through and finished, connects a new circle, in the manner of the circles of the natural days, of which in Book III, fol. 253, or in the manner of the thread which the silkworm spins, casting round itself and building a little house from the interlacing of very many connected circles, by which reason it is also brought about that the longest excursions to the sides do not occur in all ages under the same places of the zodiac. And since the librating fibres make the planet be drawn from one side by the Sun, and from the opposite be driven off, therefore the planet 3) describes an orbit about the Sun, but not as about its centre, that is, eccentric to the Sun, and by this reason there is made 4) not a perfect circle, but one a little narrower and more compressed at the sides, namely of elliptical figure. 5) For the same cause, and because the form of the solar body, which conciliates the planet’s motion, is thinner and feebler in a wider circle, the planet cannot be of the same swiftness in all parts of its orbit, but slow at a long distance from the Sun, swift at a small one. Finally, because the librating fibres also are moved from their parallel situation by the successions of very many revolutions, therefore also the places under the zodiac in which the planets become highest and slowest do not always remain, but gradually advance in the order of the signs.
You have described a perplexed figure of the planetary journey, and not one fit to be set before the eyes, especially on a plane. Although this is true, it is yet not new in astronomy or peculiar to Copernicus, nor is it necessary that everything be represented at once in the same plane, but those perplexities, arising from the very slow translation of the limits of latitude and height, can be separated with the same dexterity which the ancient astronomers used, though with less apparatus.
How did the ancients separate those translations of latitudes and heights? They devised for latitudes one orb, carrying the nodes, the outermost of the whole planetary theory; for heights two orbs, each of unequal thickness, to which they gave the name of the deferents of the aphelia.
Why do you think they are not to be used? Because they were contrived rather for setting before the imagination the physical reasons of the motions than the astronomical. And so by the use of them those false physical opinions of the solidity of the orbs would be established, and in turn the true opinions of the causes of these inequalities and of their very slow transposition, demonstrated in Book IV, would be obscured by them.
What then do you substitute for these three orbs of the ancients, to set the astronomical reasons before the imagination? It suffices that we draw two straight lines from the centre of the Sun, one through the sections of the orbit of the planet with the ecliptic, the other through the planet’s own centre of the orbit, each extended on both sides to the fixed stars, and that we teach the motion of the former under the ecliptic in the antecedent signs, that of the latter under the circle which stands above the orbit in the sphere of the fixed stars, in the order of the signs, each most equable, the former from the middle equinoctial point, the latter from that line of intersections. Unless something here has to be excepted in Book VII, from the ground that the ecliptic also is shiftable, and does not always extend altogether through the same fixed stars.
This separation made, what remains to our imagination of the figure of the planet’s journey? There remains a perfectly elliptical orbit, a mere plane most regular, inclined to the plane of the ecliptic at constant angles, by which plane of the ecliptic this orbit is cut by a line drawn through the centre of the solar body, as was premised at fol. 382 ff. in Book IV. In this orbit the planet is carried with unequal swiftness by parts, but is restored to the sections, and so also to the equinoctial points, nay even to the fixed stars and to the line through the centres, by most equal measures of the periodic times, so far as in itself lies.
Does this imagination err nothing in the physical causes and measures of the motions of one period? Nothing at all, provided we keep in memory that those things which are taken away by the said two lines from the real interlacing and connection of several orbits are physically effected not by those lines themselves but by the inclination of the real fibres of the planetary body.
By what right do you make this also a part of the Copernican astronomy, when that author remained in the opinion of the ancients concerning perfect circles? I confess that this form of the hypotheses is not Copernican. But because that part about the eccentric serves the universal hypothesis, which uses the annual motion of the Earth and the rest of the Sun, the naming is made from the better part. Add that this little part of the hypothesis is tied by necessary physical arguments to that rest of the Sun and motion of the Earth, the Copernican doctrines, and so by a good title these things also can be referred to Copernicus.
By what method must we proceed, that it may be demonstrated that from the physical causes established in Book IV there arises such a figure of the orbit and so great swiftness of the planet by parts of it? We must begin from the approach and recess of the planet to and from the Sun, and first the geometrical measure of the strength of the forces which is put forth upon the planet to be librated in any situation of the fibres must be established. Second, a compendious geometrical measure of the effect of attraction or expulsion, which has been accumulated through some whole arc of the orbit by all the increments of the forces, must be despatched. Third, it must be demonstrated that from such a libration, performed in the going round, the elliptical figure of the orbit arises. Fourth, it must be shown that the plane of the ellipse exhibits the measures of time and of the delays which the planet spends in any arc of its elliptical figure. Fifth, the equivalence between the plane of the circle and the plane of the ellipse, as to this measuring of time, must be taught. Last, it will have to be demonstrated that on the circumduction of the fibres of latitude, so arranged as is laid down in Book IV, the equability of the plane of the orbit rests. These being demonstrated, the curious astronomer is made secure (for popular things are not needed, neither in Book IV nor in this first part of the fifth) of this part of the calculation of the motions, which the other part of Book V will despatch, and Book VI will teach to bring into use by the application of this elliptical orbit and its plane to the great orb.
I. Of the increment of libration.
Begin with the first and say by what principles the manner of the increment of libration in every situation of the planet is formed or determined. Two causes concur to the formation of this increment, active and passive. The active is the measure of the librating forces, in respect of themselves, such as it is found in any one of the equal particles of the eccentric orbit. The passive is the disposition of the planetary body toward the Sun, now one, now another, which does not receive or admit the whole of that measure of forces, but each situation its own proper portion.
What measures, then, the measure itself of the forces for librating the planet? These three: first, the distance of the arc of the orbit from the Sun; second, the quantity of this arc; third, the time which the planet spends while it is in that particle.
What does the distance of the arc, and in it of the planet, from the Sun contribute to the librating forces? As is the proportion of the distances, the contrary is the proportion of the thinness of the form of the Sun, which, one and the same, both carries round and librates the planet, now attracting it, now repelling, as was said in Book IV, fol. 348. And so the farther the particle is from the Sun, the more feebly at any moment of time the planet that is in it is librated. On this account alone the Sun would spend unequal forces on different and mutually equal arcs of the eccentric.
What does the quantity of the particle or arc of the orbit effect? Because in a long arc much of force is poured out, little in a short: with equal arcs posited, then, in this respect alone equal forces are due.
What does time contribute to the increase of the forces, taken separately, and what do all three causes together? Since the planet, as was shown in Book IV, fol. 349 and 351, the farther it is from the Sun, the longer delays in equal particles of the orbit, the longer also feels the moving power of the Sun, such as it is at the distance of that particle, and it has already been said that the farther any one of the equal particles of the orbit is from the Sun, the more feebly also the planet is librated in it: therefore the more feebly it is librated in one moment of time in any of the equal particles of the orbit, so much the longer also it both tarries and is librated in it. Since then the length of time, in which the planet experiences those forces in itself, compensates the feebleness of the forces, and that in the same proportion on both sides, namely of the same distances from the Sun, it is finally brought about that in equal particles of the eccentric the measure also of the librating forces is put forth by the Sun, and in respect of it as agent, wholly equal. See the diagrams at fol. 349 and 374. (Cf. Kepler’s letter, fol. 74 ff.)
Now therefore say the measure of the portion which the planet admits into itself of the measure of the solar forces in any situation of its own toward the Sun. The angle which the Sun’s rays make with the magnetic fibres of the planetary globe must be attended to. For the sine of the complement of this angle measures this admitted portion of the forces. For since the efficient causes of libration are the Sun’s ray and the magnetic fibres of the planetary body, two physical lines, it is fitting that the measure also of the strength of libration be sought from the angle between these lines and its sine.
As if A be the Sun, I, E the centre of the body of the planet, RP the line drawn through A the Sun and the centre of the orbit B, EG and IH will be the magnetic fibres nearly perpendicular to RP (at least the compensation of the semicircles being considered) and H, G the sun-seeking ends. For it was posited in Book IV, fol. 375, that the fibres in the circulation of the body remain nearly parallel to themselves and at P, R give no occasion of drawing or repulsion, because there in place both ends, the sun-seeking and the sun-fleeing, are equally distant from the Sun A; but in the intermediate places, where the ends, sun-seeking or sun-fleeing, look straight at the Sun, the vigour of libration is greatest of all. AE and AI are the Sun’s rays. Let the lines ED and IO be drawn parallel to RP itself, and the perpendiculars to them from the points F and C, in which the Sun’s rays cut the middle circles of the planetary globe, be CL and FK. Here the angles of the Sun’s rays with the fibres are AEG, AIH, the complements of the angles CED, FIO, or the arcs CD, FO, and the sines of these LC, FK, of which IH or EG is the whole sine 100000. It is therefore laid down that as EG, IH are to LC, KF, so is the whole measure of the forces present from the Sun in I or E to the portion which the planet admits in the situations of the fibres EG and IH. (Transpose in figure 74 the point K to the line IO, the point H to the tip of the arrow.)
Why do you set the sine as the measure rather than the angle or the arc of the complement itself? Because any magnetic fibre, although it is in a globular body, is not a circle but a physical straight line, which works most strongly (or is most strongly disposed to suffer the drawing or to admit into itself the forces of the Sun’s ray) when it is directed straight at the Sun, or, which is the same, when it is perpendicular to the plane of illumination of the circle (by which the part of the globe turned to the Sun is bounded): but when it is oblique to that plane, it is equivalent to the perpendicular drawn from its end to that plane, as being shorter. So the Sun’s ray, considered according to the work of heating, when it strikes the plane at a right angle heats most strongly, when at oblique angles heats less, in that measure by which an oblique ray is less than the perpendicular drawn from the Sun to the same plane (continued).
This consideration will be more beautiful if you weigh that the whole globe consists of mere fibres, of which the longest are those which lie in the greatest circle of the globe, the shorter those in the lateral circles; in this way not only EG or IH will be a fibre, but also the sines we have touched, LC and KF, marked by the Sun’s ray AE and AI at their ends C, F, are lateral fibres. By so much as CL, FK are less than GE, HI, by so much less of force from the solar ray does each single fibre of the whole body admit into itself, on account of this very obliquity of the Sun’s ray to it. Thus the ray itself of the Sun, in designating a lateral fibre, designates the sine, which is the measure of the portion of its power received into them.
Besides, all natural or artificial motion, in which the same or analogous principles concur, is dispensed by the sines of angles, chiefly and most evidently the motion or effort of the arms in the balance and the steelyard. Since therefore this libration also is among natural motions in a wider sense (for the librating power of the solar form partakes of dimensions and in a manner, though without matter, is corporeal; but the disposition of the fibres in the planet again is corporeal), it is not absurd that this libration also take the same laws with the balance and the steelyard. This is the more probable for libration toward the Sun, because the very progress of the planet along the length of its orbit, the cause of intensification and remission, of swiftness namely and slowness, imitates the laws of the same balance or steelyard, as was said in Book IV, fol. 351 and 374, and will be made evident below at greater length.
Compare this swiftness of libration with the reasons of the balance. The line from the Sun to the librating planet stands as the handle in the balance, the fibre as the arm of the balance, the regions of the fibres as the scales; and what the weights are in the scales, that in the planet are the attraction toward the Sun or the repulsion from it, and both indeed of the same kind of things. For as the Sun draws the planet, so the Earth draws bodies, on account of which drawing bodies are called heavy. The Sun indeed draws the planet from one region and drives it from the other, and that according to more and less, while the Earth draws weights without difference of situation. What therefore in the balance is the inequality of the weights, that in the planet is the diversity of situation of the fibres toward the Sun, where the planet represents both weights of the balance. And as in the balance the heavier weight descends to the Earth, the lighter departs from it, ascending, so in this business the whole globe of the planet follows the affection of the prevailing region. As if the friendly region is drawn the more by the Sun, the whole planet approaches the Sun; but if the hostile region is the more driven, the whole globe of the planet is expelled from the Sun. Therefore also the measure by which the weights of the balance contend with each other will govern in the dispensing of this attraction and expulsion. Now in the balance the victory of the weights is estimated by the sine of the complement of the angle which is between the handle and the arm of the lighter weight, as will be proved: wherefore also in the libration of the body of the planet toward the Sun the suffering of the region of the fibre nearer to the Sun will overcome the suffering of the opposite region in the proportion of the sine of the complement of the angle which is between the Sun’s ray and the fibre. The effect of the victory, in the motion of the planets, is the strength of libration belonging to each place. This strength, therefore, or the increment of libration born from it, will be estimated similarly by the sine of the complement of the angle to the fibres.
Let AD be the handle or beam, and AB, AC equal arms to it in the same straight line BC, H a lighter weight, hanging from B, I a heavier weight, hanging from C. As great as the length of the arms BC is, so great is the height the weights have (which are in power at the points B, C) over which they contend with each other: let it be DE. For if the greater weight overcame the whole as, the arm BA would be joined to the handle DA, and the greater weight C would be at the place of height E and would raise the less to the very top D; but since it does not overcome the whole as, the perpendicular BF drawn from the end of the arm B to the handle DA shows that the weight B is lifted by the part of the height FA, and by as much the weight C is also depressed, namely by AG. As therefore DF is to FE, so is the weight H to the weight I, and as FE to FG, so the weight I to its excess over H; and as DE to FG or DA to FA, so the sum of the weights to the excess. But if BA is set as the whole sine, FA will be the sine of the angle FBA, which is the complement of the angle FAB.
In the same way if EA be the Sun’s ray, BC the magnetic fibre of the planetary body, H or B the lesser vigour of expulsion, I or C the greater vigour of attraction, since C is taken nearer the Sun than B, then if BA represents the strongest attraction, the angle BAD being nothing, AF will represent the traction, the angle BAF or GAC being given.
Apply these things also to the reasons of the steelyard. The reason of the steelyard is the same, with this one difference, that in the balance the beam A is midway between the ends of the arms B, C, and therefore unequal weights made BC not remain parallel to the horizon: in the steelyard the line of the weights remains parallel to the horizon, but the beam divides the length of the arms not in the middle but nearer to the heavier weight, so that the arms have the inverse proportion of the weights.
As if the handle of the balance DA be equal to the arms BA, AC, the steelyard will be so formed that these weights, hanging from B, C, will hang in equilibrium with the horizon. The perpendicular drawn from D to BC, which let be DK, will be the handle, and the arms BK, KC; and as DF before to FE, so here BK to KC. Then as BK the lesser arm to KC the greater, so the lesser weight H to be hung from C to the greater weight I to be hung from B.
The reader must be warned that mechanical experiment is difficult, because mechanically the weight and thickness of the arms themselves cannot be avoided: but they ought geometrically to be constituted a mere line without weight and breadth. How this impediment is in part to be met, let it be seen in Archimedes (On Equiponderants, two books).
I hold that the measure of the strength or increment of libration in any situation of the fibres of the body of the planet is to be sought from the complement of the angle of the fibre with the Sun’s ray; but since this angle seems to be disclosed with difficulty, because not only is the body continually transferred from place to place, but also its fibres are inclined, this measure seems uncertain and therefore unfit for use. Nay, on account of this very inclination of the fibres, that angle can be converted into an arc of the orbit, so that from this arc the same sine arises, the same measure, namely, by which reason it becomes most fit for use.
Teach and demonstrate this conversion of the said angle into the orbit. Remember that at the beginning, when the planet is at the apsides, that is, at the beginning of the orbit, the angle between the Sun’s ray and the fibre is a right angle. Again in Book IV, fol. 379, it was shown that the fibre NQ of that figure is directed at the Sun A itself, or is united with the Sun’s ray NA, this angle being consumed, when the quadrant of the orbit PN is completed from the apsis P, so that thus the arc of the orbit from the apsis measures the complement of this angle. It remains therefore to demonstrate this, that the intermediate angles also of the fibre with the Sun, as HIA, between the right and none, are so completed by the middle arcs of the orbit, as PI, between none and a quadrant, that joined they make 90°.
And it is demonstrated thus: at fol. 380 it was said that as IS is to NB, so is the sine of the angle HIS to the sine of the angle QNB nearly. This was so used for the sake of comprehension of IS and NB, although by force of physical speculation it is true rather of the sines of the angles IAP, NAP. Now also the sine of AIB is to the sine of the angle ANB as the sine of the angle IAP to the sine of the angle NAP. (For as BI to BA, so the sine BAI to the sine BIA, and as the same BI or BN to BA, so the sine BAN to the sine BNA; as therefore the sine BAI or IAP to the sine BAN or NAP, so the sine AIB to the sine ANB.) Therefore, the members premised being compared with one another, HIS will be found equal to the angle AIB, and QNB to the angle ANB, and equals being taken away, SIB will be equal to the angle HIA (as by analogy BNB to the angle ANA). But the measure of SIB itself is IN, because the measure of SBI itself is PI, therefore also the measure of HIA itself will be IN, the complement of the arc PI. The arc of the orbit PI being given, there is straightway given also SI, the sine of that arc, namely the measure of the increment of libration.
II. Of the sum of the libration completed.
I hold the measure of the increment or vigour of libration at any moment: but I should like to know the measure of the part of the libration completed from the beginning up to that moment. This is had from the versed sine of the same arc of the orbit traversed. For as the whole longer diameter of the ellipse is to the whole libration, or, which comes to the same, as the semidiameter of the orbit to the eccentricity, so is the versed sine, and its arc, of the orbit beginning from the apsis, to the part of the libration which is meanwhile completed while the planet runs through that arc.
By what means is this demonstrated? By means of that very measure of the increments of libration, now fortified by its demonstration.
Let there be a perfect circle PD, whose centre is B, and let A be the Sun, the line of apsides PBAR, and P, R the highest and lowest apsis, and AB the eccentricity, and its double PB the whole libration. Let the circle now be divided into the smallest equal parts, the beginning being made from P, and let them be PK, KG, GD, DN, NS, SR, and from these divisions let the perpendiculars to PR itself be drawn, KX, GF, DB, NA, SY.
Therefore by what has been premised, as the sine KX to GF, DB, NA, SY, RR (a point in place of a line), so are the increments of libration to one another, corresponding to the arcs themselves PK, KG etc., namely PM to MI, IF, FQ, QY, YB, which is true in that respect in which the division is understood to be made into infinites, when KX and RR are understood to be equal. Since therefore the points P, M, I, F, Q, Y, B are posited to mark the said increments of libration, let them be transposed to the several distances of the planet from the Sun A. With the centre A, namely, and the intervals AM, AI, AF, AQ, AV let the arcs ML, IH, FE, QO, YT be drawn, so that thus the elliptical orbit of the planet is understood to descend from P through L, H, E, O, T to R; the distances of the planet from the Sun will be AP, AL, AH, AE, AO, AT, AR, but of the said arcs PK, PG etc. the versed sines will be PX, PF, PB, PA, PY, PR. I say that the whole diameter PR, as the sagitta of the arc PDR, is to the whole libration PB as the sagittae of the single arcs are to the single increments of libration, namely PX to PM, so PF to PI, so PB to PF, so PA to PQ, so PY to PV.
For it is posited that the parts of the libration PM, PI etc. are in the proportion of the sines KX, GF etc. Now also the parts PX, PF etc. of the whole sagitta PR are in the same proportion of the sines KX, GF etc. and with the same condition of infinite division: where (no less than before) the point R holds the place of the line RR.
Therefore, alternately, the parts of the libration answer to the parts of the sagitta in the same proportion, and consequently any portion of the libration, taken whole from the beginning P, answers to its whole sagitta in the same proportion.
Whence do we know that the parts PX, XF of the diameter PR, considered as sagittae, are in the proportion of the sines KX, GF which determine them? Pappus demonstrated, in the Mathematical Collections, Book V, prop. 36: if a spherical body, understood under PGZ, is cut by any number of parallel planes, as KW, GZ etc., the surface of the sphere and the axis of the sections, as PR, are cut in always the same proportion: so that as the spherical surface KPW is to the portion of the axis PX, so also is the surface KWZG to the portion XF, and so of the rest.
But if the spherical surface is understood divided into infinite zones of equal breadth, any zone, as KW or GZ, will be like a circle lacking breadth. But the circles KXW, GFZ are to one another, because of their length, as their semidiameters KX, GF etc. wherefore also the portions of the axis PR corresponding, as PX, XF, will keep the proportion of the sines KX, GF by which they are determined.
The demonstration of the same theorem by numbers and the anatomy of the circle, tried, see in the Commentaries on Mars, chapter LVII. In that place this proportion seemed somewhat to fail, because I had not yet read Pappus. But the cause was that I took the first sagitta of an arc not small enough, which is as if in Pappus you divided the spherical surface into parts not smaller than one degree in breadth. For then the breadth of the smallest zone would necessarily come out the double of what would be true.
Although the arcs of the circle PK, KG and the rest are taken equal, yet the arcs of the true orbit PL, LH etc. do not seem to be equal, but greater toward E: does this disturb nothing the certainty of the demonstration? Nothing. For that the arcs toward E are greater is to be ascribed to these very librations, as will appear below; but the same thing cannot be to itself either a solitary or a concurrent cause, to say nothing of the fact that the disturbance, even if any had to be admitted, would plainly be insensible.
III. Of the figure of the orbit.
I see that the measure of libration lies in the versed sines of the arcs of the orbit begun from the apsis, from the principles and causes of motions assumed; it remains for you to prove that by this form of libration the elliptical orbit is constituted, of which you said that observations testify. That the orbit of the planet PLHEOTR and its opposite becomes an ellipse is demonstrated from the identical properties of this figure; which properties the libration so far set forth expresses.