Epitome Astronomiae Copernicanae: Liber IV, part 7
How does the Sun strengthen the moving power of the Earth, increasing the swiftness of the Earth’s diurnal revolution? It is very probable that this happens by means of the light of the Sun, which is poured into the Earth by the illumination of its hemisphere. For since the physical equation of time demands unequal diurnal revolutions of the Earth, as its intervals from the Sun vary, certainly at a short interval the illumination is strong, as being from a denser light, at a long one weak, as being from a thinner and so scantier light, and that (as to the one dimension of length, in which the motion tends) in the very proportion of the intervals. Thus the abundance of light, which exists at any time, becomes fit, in place of the intervals, to dispense this acceleration through the year.
What are the effects of the diurnal revolution of the Earth, and in general of the primary planets about their axes? Two. The first is proper to the Earth: that to us inhabitants of the Earth all the stars of heaven, fixed and wandering, and so also the Sun and Moon, seem day by day to rise from the east and to set in the west, although in respect of this diurnal motion they in truth remain fixed in their places. Of this deceptive appearance we treated in the first three books of the spherical doctrine. The other effect, physical and most true, common to all the primary planets and even to the Sun itself, is this: that the primary planets, by the form of their body issuing forth as it is established in revolution, stir their secondary planets, as the Earth the Moon, and bring it about that the secondary planets follow in the same direction, yet more slowly and as if left behind.
By what arguments is it made probable that the primary planets themselves impart to their secondaries their motions about themselves, and especially the Earth to the Moon? The Moon and the Earth give the first assurance. For as above, from the fact that planets approaching the Sun are carried on more swiftly, we reasoned that the Sun, by the form of its body, that is, established in rotation, stirs the planets about itself in the same direction, so also, since we find that the Moon, 1) the nearer it approaches the Earth (not, however, the Sun), the more swiftly it goes about the Earth, 2) and indeed in the same direction in which the Earth turns about its axis, with the greatest probability we derive that motion of the Moon from this turning of the Earth, and all the more because 3) this also corresponds, that as the Sun’s turning about its axis is shorter than the shortest period of Mercury, so also the Earth is turned almost thirty times until it restores the Moon once. For if the Moon outran the Earth, her motion certainly could not be from the rotation of the Earth. 4) The credibility of this matter is confirmed by comparing the four Jovian satellites and Jupiter with the six planets and the Sun. For although about the body of Jupiter, whether it also is turned about its axis, we do not have such proofs as are available to us in the bodies of the Earth and especially of the Sun, namely from sense itself, yet sense testifies that exactly as it is with the six planets about the Sun, so also it stands with the four Jovian satellites, that about the body of Jupiter each, the farther it can run out from it, the more slowly returns, and that in a proportion not the same but greater, that is, the sesquiplicate of the proportion of the intervals of each from Jupiter, which is exactly the very one which the six planets used above. For the intervals of the four Jovian satellites from Jupiter Marius gives in his Jovian World as these: 3, 5, 8, 13 (or 14 for Galileo), as if their little orbs were separated by three rhombic figures: 1) the rhombic dodecahedron between the innermost, whose intervals are 3 and 5; 2) the rhombic triacontahedron between the middle ones, 5 and 8; and 3) the cube, not truly rhombic, but a kind of origin of rhombi, between the outermost, 8 and 13 (or 14). The periodic times the same Marius gives; the proportion of these is greater than double, therefore greater than that of the intervals 3, 5, 8, 13 or 14, yet less than that of the squares, which double the proportions of the intervals, namely 9, 25, 64, 169 or 196, just as sesquiplicates also are greater than simple proportions but less than double ones (see above, p. 67).
Since then so exact is the agreement of the Jovian satellites with the six primary planets themselves, not only did we rightly conjecture from this above that the body of Jupiter also turns about its axis after the example of the Sun, so that analogy holds in all its members, but here we now further confirm, not unfittingly, in this same kind, that this rotation of the primary planets about their axes is the cause of the circuit of the secondaries about their primaries; 5) and this the more probably, because we see that as the Sun is greater than all the planets which it moves, so also the Earth is much greater than its Moon, and Jupiter than his satellites, and on that account, equally with the Sun, are fitted to move them. The remaining likelihoods again concern the Moon. For 6) the telescope has taught us that the bodies of the Moon and the Earth are akin, since it shows in the Moon signs of mountains and seas such as are on our globe of the Earth. This kinship Aristotle also acknowledged, otherwise the keenest defender of the fifth essence of the heavens, who, as Averroes reports, said that the Moon seems to be a kind of ethereal Earth. I pass over Plutarch and the other philosophers in Macrobius.
As therefore the kinship of bodies brings it about that the magnet draws a magnet or iron, so also it is not incredible, concerning the Moon, that it is moved by the kindred body of the Earth, although neither here nor there does any contact of bodies intervene. And so 7) what wonder that the Moon is moved by the Earth, when we see in turn that the Moon, by her passage over the zenith of places, causes the reciprocal flow of the ocean on the Earth? Is not this a sufficiently evident proof of a communication of motions between these two bodies? Finally 8) the same is confirmed by this remaining part of the analogy: the Sun and the Earth turn about their axes, which is certain by experience for the Sun by itself, for the Earth at least in Copernicus, namely that by this turning they may impart motion to the planets placed about them, the Sun to the six primary planets, the Earth to the Moon; the Moon in turn is not turned about the axis of her body, as the spots show. But why this, unless because about the Moon no further planet is seen to go round? The Moon therefore has no planet to which she might impart motion by the turning of her body; the turning in the Moon, as superfluous, was therefore omitted.
These eight arguments, if they do not avail singly, will help when joined.
But it seems absurd that the Earth, which lacks light, should be put on a level with the Sun, the source of light; for by this quality the moving power of the Sun is made the more probable. Although the light of the Sun performs its part in carrying out the motion, yet the body of the Sun is not strong in moving power on account of light alone; for nothing hinders two subjects, as it were, of the moving power from concurring in the Sun, light and a corporeal magnetic affection, and the latter of these alone being in the Earth: surely because the Earth also moves only one planet, and that the most ignoble (being one of the secondary ones), and the magnetic power of the Earth does not move it alone without help, as we shall hear, nor does the Earth have all this power from itself, although in itself, but seems in part to have drawn it, by the continuation of a line from the Sun into itself as through a kind of channel, and altogether with the illumination of its own body, and to have derived it into a new source, namely its own body, as was said a little before and will be said more clearly below.
The turning of the Earth observes the circle of the equator, the motion of the Moon the zodiac, which declines much from the equator; it is not therefore probable that the motion of the Moon is from the turning of the Earth. This hinders us no more in the Moon than in the other planets, which, although they decline each into their own regions, and hold the helm, so to say, in their hands and turn it at their pleasure and sail out to the sides or banks of the river, are nevertheless carried along meanwhile by the force of the common moving vortex emanating from the Sun, and so they too owe that their distinct motion to the common motion of the river, just as the Moon owes her oblique motion through the zodiac to the straight motion of the Earth along the equator.
Why, then, does the Moon accommodate her whole journey to the zodiac rather than to the equator? Because, besides the Moon’s proper circuit about the globe of the Earth, of which we have spoken so far, the whole heaven of the Moon is also moved with a common motion with the centre of the Earth about the Sun under the zodiac, like the other planets; from which composition it comes that the Moon, with respect to the centre of the Sun, always keeps a direct course in the order of the signs, not only when the Sun and the Earth with their extended spaces urge her, full, in the same direction, but also when, dark or empty, the Sun impels her forward and the Earth (with respect to the centre of the Sun) backward. For this impulse from the Earth is still much less than that from the Sun, wherefore it diminishes that carried on in the order of the signs, but does not wholly absorb it, much less prevail against it. See the diagram of this compound motion of the Moon in the Commentaries on Mars, fol. 149 (III. 387).
Since therefore that flow of the solar form advancing under the zodiac is greater, and the other, of the terrestrial form, which is under the equator, is less, and since besides the Moon, when joined to the Sun, by reason of the swiftness and the direction of rising or setting, in the space of the world obeys the Sun more than the Earth: hence I judge it comes about that, also in respect of the lateral directions, obeying the Solar as the stronger, as with her whole heaven about the Sun, so also with her body about the Earth, she is compelled to advance under the zodiac, that is, to subordinate her orbit about the Earth to the zodiac.
Does no anomaly of the Moon’s motion arise from this, if in the tropical signs she goes according to the guidance of the terrestrial form, because the zodiac and the equator in those parts are parallel, but in the equinoctial signs crosses this form of the terrestrial body by an oblique path? Again, to dispel this objection I answer the same as about the latitudes. Namely, the form of the body of the Earth in its middle under the equator is strongest, at the sides of the equator weaker, because also at the source, that is, in the globe of the Earth, the parallel circles of the equator, being smaller, are stirred more slowly than the equator, the greatest circle. A compensation therefore occurs: that where the Moon experiences the strong moving form, there she does not follow it entire, going off across it, and where she follows the whole, obeying it completely, there she experiences it weak. Yet of an entire compensation I pronounce nothing, since the observations of the Moon still disagree in the smallest things with any calculations, and it is uncertain to what that discrepancy is to be referred.
How can the Moon also be carried about the Sun in the annual motion, and the four satellites about Jupiter in the common twelve-year motion, so that meanwhile they do not desert or let go, the Moon the Earth, the Jovian satellites Jupiter, if they are attached to no orbs, she to the Earth, they to Jupiter? About the Sun the secondary planets are carried by the same power of the solar form by which their primaries, the Earth and Jupiter, are carried; they would be carried round, however, so much faster than their primaries as they are more ready for motion by density, bulk and weight, were they not retained and held fast by the Earth and Jupiter by a magnetic force like that with which the Sun also is endowed. But this force of holding, as was said above also of the planets, is contained by the contrary powers of the approach and recess of the Moon from the Earth, so that, revolved about its axis, it carries the Moon round with itself by this holding, the Moon meanwhile exchanging the regions of her body by which the approach and recess are effected. Look at diagram 62. Suppose the region of the Moon’s globe friendly to the Earth to be turned toward it and not exchanged with the contrary region; suppose also that the Earth is not rotated about its axis but is carried about the Sun: then the Moon will run the same course with the Earth and meanwhile will be drawn by the Earth until she comes to contact with it. Suppose conversely the same of the unfriendly region: then the Moon will flee the Earth until she has come outside the sphere of the Earth’s magnetic power, and then will surely give herself up to the sole rapture of the Sun, and so will wander off wholly from the Earth.
You have said that the middle circle of the Earth is a little less than sixty times narrower than the orbit of the Moon; but the same circle of the Earth is only thirty times swifter than the Moon, because the Moon returns in thirty days less a half. The circle of the Earth is therefore slower than the circle of the Moon’s centre about the Earth in a double ratio. How then will a body which advances more slowly impart to the Moon a motion more than double its own and swifter? This objection is not peculiarly adverse to the lunar motion but is common to all the planets, and has nothing absurd in it. For these bodies, of the Sun and the Earth, do not move by contact, but by their forms, spread out or unfolded into the whole orbit of the movable body. Now the form of the body of the Earth, emanating through space to whatever extent, is turned with the Earth, its source, in the same time of 24 hours, although at the place where it holds the Moon it is of the same breadth as the orbit of the Moon. This form, therefore, sixty times wider than the Earth, sweeps, I say, through the whole orbit of the Moon thirty times in one month, while the Moon returns within the same space only once, following the form of the Earth. There remains therefore the probability that this form of the body of the Earth, being moved, moves the Moon, yet so that the inertia of the lunar body overcomes about 29 parts of space daily, and is overcome by no more than 30.
Why do you hold that the Sun concurs with the moving form of the Earth also in that motion by which the Moon revolves about the Earth? 1) Because Tycho Brahe found that the mean motion of the Moon (that is, stripped of that anomaly which exists in all the planets on account of the eccentricity of the orbit) is still anomalous or unequal. For the Moon is always swifter at the conjunctions, as here at CD, GH, slower at the quadratures EF, IK, than the ratio of the eccentric brings, whether she has been at apogee in both, or at perigee, or any other place of her eccentric; and (if the so-called Tychonic variation hypothesis is to be pressed closely) exactly so much swifter there as slower here. But the form itself of the Earth, established in rotation, to be understood under the circle DFHK, is of uniform swiftness all around, as well in those parts which are turned to the conjunctions D, H, as in those at the quadratures F, K; understand this at one and the same interval of the Moon and the Earth. There must therefore be added to this moving form other causes of motion, fitted to the phases of the Moon; but the phases of the Moon are produced by the Sun: the Sun therefore aids the motion of the Moon about the Earth.
2) The credibility of this concurrence of the Sun is strengthened by the fact that before, at fol. 360, the same Sun was called in, to help the swiftness of the Earth in revolving by the illumination of the globe, of which the middle circle here is AB. For from this we first understood, as in the example of the Earth, that in the light of the Sun also there is a power of quickening motion, and then we also tie the necessary argument for the Moon from the same source. For if DFHK, the form of the body of the Earth AB, as that is established in turning, moves the Moon, and the Sun urges on this turning, then through the Earth and its urged form it will urge the Moon also.
Is this illumination, then, otherwise when the Moon is at the quadratures F, K, otherwise at the conjunctions D, H? By no means: for in both the halves of the globes are illuminated, both of the Earth AB, which imparts the motion, and of the Moon CD or GH, to which the motion is imparted. Nay, that the swiftness of the Earth from this illumination is equal at both times has just now been said.
Whence then comes to this accessory cause that disparity of effect, that it accelerates the motion of the Moon at the conjunctions D, H very much, at the quadratures F, K not at all? And what on the contrary retards the motion of the Moon at the quadratures F, K? No part of celestial physics has been more difficult to explain than this; that we may despatch it as far as is permitted, the diagram No. 66 will have to be used.
Remember, then, that all the circles which bound the illumination of the globe of the Moon, such as CD, GH and the rest, are parts of as many spherical surfaces into which the light, coming from the Sun as centre, is unfolded, and that the circle DFHK represents the form of the body of the Earth AB, set in its centre, moving the Moon. You see that at the conjunctions D, H the form of the light CD and the form of the body of the Earth OCDL are applied to each other by contact, and cut each other at L, M, N, O at oblique angles, so that the application is imperfect; but at the quadratures EF, IK the section is made at right angles, and the application is therefore plainly nothing, since the section of the Moon tends to the centre of the Earth and corresponds to it on the circle NIO as a mere point.
Since therefore no other cause of the acceleration at the conjunctions appears, it must be held that the strengthening faculty of the moving form of the Earth ODL resides in the light CD taken separately, not as yet in so far as its source, that is, the body of the Sun, is rotated (this modification from motion availed above, when we spoke of the forms of the bodies of the Sun and Earth themselves without regard to light), but as light, namely according to the genuine and in a manner essential figuration of the light. If then we hold that this form of the body of the Earth is strengthened by the modes of its application to the orbs of light, the cause and measure of the acceleration at the conjunctions CD, GH will be at hand, very strong there, none at all at the quadratures. But since by DFHK there is represented not only the form of the body of the Earth as agent or mover, but also the orbit itself of the Moon as of the patient or thing moved (although then the Earth will not be placed in the centre of the circle, but on account of), it must further be conceived either that the body of the Moon at the conjunctions CD, GH is disposed to motion along the diffusion or surface of the light better than at EF, IK, where the Moon cuts the transverse circles of diffusion, or that the very path of the Moon is made as it were slippery at D, H, and roughened at F, K, as on a board across the pores of the wood. Nor will that be very absurd. For since there is in light a power of strengthening motion, as is posited, certainly along the one dimension in which the light extends it is fitting that the passage be easier. Further, they say the same as to the effect: the one who says that the Moon is accelerated at D, H and retarded at F, K, both in the simple proportion which these applications produce, the other who says that the Moon is accelerated very much at D, H and not at all at F, K, but that in the double of the proportion which results from the applications here posited.
Unless someone prefers to ascribe this doubled efficacy of light to the two dimensions of the surface of light, so that, although the immaterial forms of bodies whatever, no less than light, are diffused both in length and in breadth, yet they have been efficacious thus far at least in respect of length, but the light in respect both of length and of breadth, for the reason that the form moves as moved, and is moved only in length, but the light strengthens as light, that is, as it holds its density both in length and in breadth.
Why do you ascribe to light the power of strengthening as a moving cause separately and without regard to its rotated source? Because, in so far as the form of the rotated source moves, it always moves in the order of the signs, OCDL, and of this its effect in moving the Moon we have already treated at the beginning of this place; but this power of light also avails to stir the Moon in the contrary direction MHN with respect to the centre of the Sun, namely when she appears to us empty of light or joined to the Sun. Light does not, then, of itself lend direction to the motion, but through the stirred form MHN.
If this power is in light, it will be greater in the denser light about GH, as being in the neighbourhood of the Sun, less in the more scattered, about the full Moon at CD, since she is farther from the Sun by a thirtieth part of the interval: the new Moon will therefore be swifter than the full, other things being equal. The feebleness of the light at CD is compensated by a more perfect application, since the cavity CD is flatter than GH. Since therefore the strengthening comes about by the application of forms, in the full Moon a more scattered light, applied more perfectly, achieves as much as in the silent Moon a denser light applied less perfectly. But the intervals of the Moon and the Sun are the same, and these measure both the density of the light and the curvature of the circles CD, GH; therefore a perfect compensation of density in length is made by the curvatures of CD, GH. The other part of the effect of light is balanced by the other diversity of application. For even if CD and GH were equally curved, yet there the convex OCDL insinuates itself into the hollow CD; here the convex MGHN is turned to the form of light itself GH, convex toward the Earth.
If that appendix of 133 degrees to the 12 synods in the sidereal year is from that conjunctional stirring of the Moon’s motion, the quantity also of that stirring will have to correspond. Indeed in Tycho Brahe the motion of the Moon is stirred by 1′ 26″ in one degree at the conjunctions, and retarded by as much in one degree at the quadratures; wherefore if the retardation is cancelled by a double stirring, the greatest stirring of the conjunctions will be 2′ 52″. Wherefore if the squared sines of all the 90 degrees bring their small portions into one sum, we shall accumulate 2° 9′, in the year therefore But first, the quantity of the greatest variation in Tycho is not most certain; he gives it in the 45th degree as 40½′, and so if it is set at 51′ we make up the prescribed sum, taking the stirring of the first degree as 3′ 34″ 40‴ (or, in Tycho’s form, 1′ 47″ 20‴ and an equal retardation of the ninetieth, that is, at the quadratures), and thus a sum of 2° 41′ is gathered in one quadrant, which below, when we treat of the causes of the inequalities, will acquire great probability. Next, if we most strictly retain Tycho’s small quantity in the 45th degree, the preceding and following degrees, distributed in another form than the Tychonic, could make up the desired sum, or there lie hidden from us small causes which take something from those 133 in the treatment of the variation. (Cf. note 58.)
In what proportion, then, do you think the monthly motion of the Moon about the Earth is distributed between these two causes, namely the form of the body of the Earth and the circle of illumination of the bodies? We see that while the Earth revolves about its axis thirty times, less a half, the Moon meanwhile returns once about the Earth, that is, from Sun to Sun. Thus it comes about that in one year, or 365 days 6 hours 9′ 26″, the Moon returns twelve times, and of the thirteenth revolution adds more than a third, that is 132° and three quarters. It is fitting therefore that the density of matter in the body of the Moon is so tempered to that archetypal degree of strength in the form of the body of the Earth that, unless illumination helped the diurnal revolution of the Earth and through it also the advance of the Moon, the Moon herself by the simple moving power of the Earth would have returned a little more slowly, namely exactly twelve times. This being posited, it follows that those remaining and, as it were, supernumerary 132° and three quarters of the thirteenth revolution begun must be set down to the other moving cause, namely illumination.
You reckon, then, the temperament of density in the body of the Moon at 12 revolutions of the Moon in one year; what archetypal cause will you give of this number? The cause seems to be compounded of geometrical beauty and of the office of this planet in the world, in this manner. The Moon is a secondary planet, tributary to the Earth, and exercises her courses privately about the Earth. Now to the Earth were destined 360 revolutions while the centre of the Earth returns once about the Sun. As therefore the orbit of the Moon had to become, in the passages above, a mean proportional between the body of the Earth and the orbit in which the centre of the Earth truly, the Sun apparently, goes round, so also the revolutions of the Moon had to be more than one but fewer than 360. And the mean proportional between 1 and 361 is 19; but since the number 361 is not 360, and 19 has no beauty, neither geometrical nor harmonic, two numbers nearest to 19 itself, which multiplied into themselves would make 360, and which are also geometrically and harmonically most beautiful, had to be chosen. The nearest which make 360 are 18 and 20, since by unity alone the one is less, the other greater than 19; but the figure of 18 sides is not demonstrable. The nearest next are 15 and 24, which also make 360. These now have their geometrical demonstrations, but meaner, and do not make between themselves an excellent proportion, but that which is between 5 and 8, nor are they in harmonics the most excellent and first of all. But these, 12 and 30 (for no nearer others make 360), excel in every way: both geometrically, as numbers generated from the first figures inscribed in a circle, and harmonically, because all harmonies are represented by these two divisions of a string. Of those, then, which multiplied into each other would make 360, none were more beautiful. Further, the smaller, 12, was due to the revolutions of the Moon, not the greater, 30, because since the orbit of the Moon bears a certain image of the orbit of the Sun, it was fitting also that as the year, which is the periodic time of the Sun, is divided into 360, a great multitude, so also the month, which is the periodic time of the Moon, should receive parts or days more in number than the whole months that are in the year, and that the multitude should increase in the progress, if first the year, a great time, were divided into 12 months, large parts, then the month, a small time, into 30 days, small parts; for multitude is suited to small things. That would not have been of the same beauty if there had been thirty months in the year, each of twelve days.
Whence do you make it probable that that increase of the Earth’s revolutions in the year to the number 360, and this appendix of the Moon’s motion in the year to the twelve monthly revolutions of the Moon, are from the same cause? Testimony is given to this matter both by the very reasons of this philosophy, that since the diurnal turning of the globe of the Earth moves the Moon, more and swifter turnings of the Earth also move the Moon more swiftly and restore her more often, and especially by the number of days of the solar year, 365 days and a little more than 6 hours, compared both with the archetypal 360 and with the number of days of the lunar year, 354 days and a little less than 9 hours.
For since from the archetype there ought to have been 360 days in the year, revolving the Moon twelve times, but through the accession of another cause they have become 365, all the revolutions have therefore been made swifter in the proportion of 360 to 365, and for that reason stronger to move the Moon. At the same time they have also been made more numerous, namely 365: therefore the power of the archetypal 360 is to be estimated by the number 360, but the power of these 365 must now be estimated not by the number 365, since they are swifter, but by the number which is third proportional, namely 370 and 36′ 50″, if we pursue the minutes. But if the power, marked by the number 360, had moved the Moon so that she completed 12 returns to the Sun, and the last of them to her initial place under the fixed stars: then in the same proportion the power estimated by the number 371 will make her overtake the Sun twelve times and in addition 127° 10′ of her initial place, and because the Sun after the completion of 360 days, as many as were in the archetype, is still distant from the initial place under the fixed stars by 5° 10′, by which space the circle, which in the archetype was divided among the 12 places of the lunations, has been made more contracted: these 5° 10′ added therefore to those 127° 10′ make 132° 20′. Behold how near this reasoning comes to the truth in the astronomical tables, which give the overtaking of the Moon in the sidereal year as 132° 45′, only 25′ more. The same we shall gather also from the days of the lunar year thus. The moving power of the 360 revolutions of the Earth would have restored the Moon to the Sun, at his initial place, twelve times: therefore the power of fewer revolutions, but made so much stronger, will achieve as much. As therefore 365 revolutions are to 360, so is the power of the archetypal 360 to the power of the present ones, 354 with 19 hours 33′. So many revolutions of the Earth, now made more intense, would have restored the Moon to the Sun twelve times, if indeed the spaces between pairs of conjunctions had not been contracted by the increase in the number of revolutions. But because you will bring into the year the supernumerary days, the 360th day, as the archetypal, cuts off the measure of contraction from the zodiac, of which are due, proportionally to the length of the lunar year, 5° 6′ 41″; by so many degrees therefore the Moon is also lightened, so that, even when these are not completed, she returns in the space of the world to the Sun the twelfth time. They are worth 10 hours 4′, which taken away from the 19 hours 33′ found leave in the appendix to 354 days 9 hours 29′, for which the astronomical tables give 8 hours 49′, only two thirds of an hour less, which little difference may be put down to other minute circumstances. Meanwhile it has been proved exactly enough, by either way of numbers, that this departure from whole and beautiful numbers arises from the concurrence of the causes of the Moon’s motion, and the cause is plain why 360 is nearly the mean proportional between the lengths of the years, lunar and solar sidereal.