Epitome Astronomiae Copernicanae: Liber VI, part 10
How is each of them computed? From the known greatest parallax of altitude in the horizon, the definition of which was given beforehand in the explanation of the lunar eclipses. For then, according to the doctrine of the primum mobile delivered in Book III, folio 240, one finds the rising point of the ecliptic and the nonagesimal degree from it, that is, the highest point of the whole ecliptic at the moment proposed, and the distance of the given true place of the star from it; one seeks also the angle of that rising point, or the altitude of this nonagesimal. If, these being given, three logarithms are gathered into one sum, namely 1) of the angle of the rising point, 2) of the elongation of the star from the nonagesimal, 3) of the horizontal parallax, the logarithm of the parallax of longitude will be heaped up. So if you gather into one sum 1) the antilogarithm of the angle of the rising point and 2) the logarithm of the horizontal parallax, you will obtain the logarithm of the parallax of latitude. The foundations of this computation are in the Optical Part of the Astronomy, from folio 312 to 320.
Where is there no parallax of longitude, and where in turn is it greatest? At the nonagesimal the parallax of longitude is indeed nil, but it is not always greatest in the horizon. For if Sagittarius or Capricorn is rising, as the angle of the rising point decreases its logarithm increases, and that sensibly; but if at that time the Moon is also moving toward the west, increasing her elongation from the nonagesimal, the logarithm of this elongation diminishes almost to nothing, and so it overcomes the former increase and overflows into the logarithm of the parallax of longitude, and thus this parallax is diminished. So when the Moon descends toward the horizon in the northern signs. Conversely, therefore, when the Moon is in the same signs and rising from the horizon, the parallax is at first increased, from opposite causes. The greatest parallax of longitude, therefore, is below the horizon in the southern signs, in the equinoctial points in the horizon itself, but in the northern signs above the horizon, and then only when the difference of the logarithms of the star’s distance from the nonagesimal has become equal to the difference of the logarithms of the angle of the rising point.
What must be observed concerning the parallaxes of the Sun? Although it ought not to be neglected, notwithstanding that it does not exceed one minute, it need not be treated separately; for it suffices to subtract the Sun’s greatest horizontal parallax from the Moon’s greatest horizontal parallax, so as to establish the lunar excess, which we call the parallax of the Moon from the Sun. For it matters nothing to the knowledge of eclipses of the Sun in what place of the heavens, within one minute, the Sun is seen, but how much of the Sun the Moon covers.
Does the consideration of diurnal parallaxes also pertain to the planets? Hardly at all. For Saturn, being almost ten times higher than the Sun, makes a parallax of scarcely 8 seconds, Jupiter perhaps 15 seconds at the most. Mercury and Venus, when they approach the Earth most closely, are near the Sun’s rays and therefore not observable; and Mercury does not add another full unit to the Sun’s parallax, while Venus, when she comes nearest, acquires 4 minutes, that is, about four times the solar parallax, and that in the very conjunction with the Sun. There remains only the planet Mars, which when sometimes in opposition to the Sun is distant from the Earth about two fifths of the solar interval; then, therefore, it can make a parallax greater than the Sun’s, namely about 2′ 30″, scarcely observable on account of the spreading of the rays.
What parts does the nonagesimal make in the apparent semicircle of the ecliptic?
It distinguishes in it two quadrants from one another, the eastern, namely, from the western.
How do we use the parallaxes of the Moon from the Sun in longitude and latitude? The parallax of longitude, when the Moon is in the eastern quadrant, is added to the true place of the Moon reduced to the ecliptic, in the western it is subtracted, and so the apparent or seen longitude is established. The parallax of latitude, in our hemisphere, is subtracted from the northern latitude of the Moon, if this is the greater, and the remainder is the seen northern latitude; but if the true northern latitude is the smaller, then it is itself subtracted in turn from the parallax of latitude and the remainder will be the seen southern latitude. But if the true latitude was also southern, the parallax is added to it and again yields a seen southern latitude, understood, as above, with respect to the Sun, as if he made no parallax at all.
How great is the visible or apparent hourly motion of the Moon from the Sun made in this way? Almost always the hourly motion in visible longitude is rendered shorter than the true one, more so, however, about the nonagesimal; for near the horizon they do not differ sensibly, and so sometimes the contrary happens, though this also insensibly. Again, this difference is more evident at a great altitude of the nonagesimal, whether on account of the sign or of the climate, less evident at a small one.
In what order do the true and the visible conjunction follow one another? In the western quadrant the visible conjunction follows, in the eastern it precedes. The quantity of this difference is to be sought from the variety of the parallaxes.
What does parallax effect in the visible motion in latitude? In whatever sign the Moon may be going, when the northern signs are rising she seems to be raised toward the north; when the southern signs are rising, she, whether rising or setting, seems to be depressed toward the south; and each the more swiftly, the nearer to the tropic points the degrees rise.
What shape does the visible path of the Moon’s centre take on the Sun’s disc, and what is its inclination to the ecliptic? It does not appear as a plainly straight line, nor is it inclined to the ecliptic at the constant angle of 5° 18′, but sometimes runs parallel to it, or even seems to be inclined contrary to the true path; which happens when the Moon, before or after the ascending node, is rising by her true motion toward the north while the signs near the winter solstice are rising; or, conversely, when the Moon is before or after the descending node while the signs near the summer solstice are rising.
In figure 97 let S be the Sun, ESQ the ecliptic, RN the true path of the Moon, CLD the visible path inclined in the contrary direction.
Sometimes, however, the angle of inclination is very great, more than double, triple or quadruple the ordinary one of 5° 18′, as if, the Moon being about the ascending node, the parts near the summer solstice should rise, or if, she going about the descending node, Sagittarius or Capricorn and the like should be at their rising. See the Optical Part of the Astronomy from folio 408, and at folio 410 the figure with examples.
What, then, is the middle of the eclipse, and what the greatest obscuration? These do not always coincide, but the greatest obscuration is when the centres of the luminaries are seen nearest to each other. This happens sometimes close to the circle of latitude drawn through the centre of the Sun, namely when either the obscuration is great, or the visible path of the Moon CD is nearly parallel to the ecliptic EQ. But if the angle of inclination is very great and the eclipse small, then the nearest approach of the centres is to be sought on the circle which is inclined from the centre of the Sun toward the visible node. Nor does this variety well bear several rules.
In the figure, if from S, the centre of the Sun, a perpendicular SL be drawn to CD, the visible path of the Moon, then at L there will be the greatest obscuration.
What is to be observed about the beginning and end of eclipses, or the digits of incidence and of emersion? If the visible conjunction is to be central, the beginning and the end require nearly the sum of the semidiameters as the visible distance of the places of the Sun and Moon in longitude; but if the visible latitude of the Moon in the middle is great, the places in longitude at the beginning and end will be distant by less than the sum of the semidiameters. Let the visible latitude therefore be computed for a certain interval of time taken, before or after the visible conjunction. The antilogarithm of this latitude, subtracted from the antilogarithm of the sum of the semidiameters, leaves the antilogarithm of the difference of the places in longitude, about, which the beginning or end of the eclipse requires. But the visible motion of the Moon, gathered through the interval of time taken, ought to be equal to this difference, or if it is not equal, the interval is to be corrected until it is equal.
In the figure let CH be the visible latitude at the beginning, about, and CS the sum of the semidiameters; there will then be given HS, the visible distance of the centres for the beginning, or the digits of incidence; so from DF, the visible latitude at the end, and from SD as before, SF is found, the digits of emersion.
How long does an eclipse of the Sun last at a given place of the Earth? The exact time of duration cannot easily be comprehended by rules, on account of the great variety of the parallaxes through the different altitudes both of the pole and of the Sun. The first element of the duration, however, is the sum of the semidiameters of the Sun and Moon, to be traversed by the visible motion, both for the times of incidence and for those of emersion. Since, then, this sum nearly equals the Moon’s hourly motion from the Sun, a little greater when the Moon is in apogee, less in perigee, the Moon by her true motion would spend two hours in the passage across the Sun’s disc. But because the visible motion is less and slower than the true, half of a third hour nearly makes up the variety of the durations, if we are speaking of central eclipses, the longest of all.
Whoever wishes to proceed exactly, let him compute the digits of incidence HS and of emersion SF by the method given before; for it has already been said that the interval of time must be computed and corrected together with them.
What else is looked for in an eclipse of the Sun? The quarter from which the defect stands on the Sun’s edge, both at the beginning and at the end, and likewise in the middle, and when the horns look upward, when downward, and finally when to one side or the other; this not less for astronomical and astrological uses than to the credit of the prediction.
By what means do we learn these quarters? By investigating the angles RSE, CSE, which two circles make with the ecliptic EQ, one the vertical circle RV through the centre of the Sun S, the other the circle CS, LS, DS drawn through the visible centres of the Sun and Moon: for thus the angle CSR, LSR, DSR is also made up, of the mutual section of these in the centre of the Sun; and the vertical circle, being drawn through the Sun’s centre, shows the highest and lowest edge of the Sun. Further, what variety of addition or subtraction of these angles or their complements is to be observed according to the different positions of the Sun in the eastern or western quarter, as distinguished by the meridian circle, is better learned from figures, or from the inspection of a sphere or globe, than from rules. And a suitable figure exists in Tycho Brahe’s Progymnasmata.
Teach how to compute the angle ESE of the ecliptic with the vertical through a certain point of it, as, for instance, through the centre of the Sun S. There are four ways of this computation. 1. If the altitude of the Sun and the altitude of the nonagesimal, or the angle of the rising point, are had, the angle will be computed from them according to the doctrine of folio 241 of Book III, or by adding the antilogarithms of both, so that the logarithm of the angle sought is made. 2. If the azimuth of the Sun and the altitude of the nonagesimal, or the angle of the rising point, and the Sun’s distance from the nonagesimal are had: subtract the logarithm of this from the logarithm of that, and add the remainder to the antilogarithm of the altitude of the nonagesimal; the logarithm of the angle sought will result. 3. If the angle RVA, of the vertical RV with the equinoctial AV, is had (for various ways of computing which see Book III, folio 224), take with it the angle of the circle of declination with the ecliptic, and proceed either by adding or by subtracting, as the inspection of the sphere or globe shall teach you. Although this way has more requisites and is more laborious, it has nevertheless its own use where several moments are to be computed together. 4. If the angle of the rising point, or the altitude of the nonagesimal, and the Sun’s distance from it are had, which two are for the most part given beforehand, then from the mesologarithm of the complement of the former subtract the logarithm of the latter, and the mesologarithm of the angle sought remains. But here there is very great caution to be used about the signs of excess or defect, as the algebraists commonly do.
Teach also how to compute the angle CSE of the ecliptic with the circle through the centres. Subtract the logistic logarithm of the distance of the centres CS, LS, DS, which is found at any moment, from the logistic logarithm of the visible latitude of that moment, as CH, LS, DF; there remains the logarithm of the angle CSH, LSL (or a right angle) and DSF.
How many kinds of solar eclipses are there? Three: for either the whole Sun is covered, or an entire circle of him remains shining, or the remaining part comes out into horns, which is properly called a partial eclipse.
If the Moon is so much smaller than the Sun, as was said in Book IV, folio 328, how can she cover the whole Sun? By as much as her diameter is smaller than the Sun’s diameter, by so much is she nearer to the Earth than the Sun himself, as is shown in the same place; and things near appear under a greater diameter than if they were the same things remote.
When can there be a remaining circle? When the Moon is in apogee, and the Sun below his apogee, most of all when in perigee. For then the diameter of the Sun Does this never happen unless the Moon is placed in apogee? Indeed, in the year 1567, on the 7th of April, such an eclipse with a remaining circle of the Sun was seen at Rome, when both the Sun and the Moon had their mean distances, and so the Moon’s diameter appeared greater than the Sun’s.
What, then, was the cause of this phenomenon? Not an astronomical one, surely; for it would hold perpetually in a like disposition of the luminaries. But in the year 1605, on the 22nd of September (2nd of October), the Moon, higher or nearer her apogee, covered the whole Sun, placed at the same distance from his own apogee, at Naples in Italy.
We must therefore seek a physical cause, in the lofty ether however; of which two present themselves, also seen above in the eclipses of the Moon. The first is, if the ethereal substance, poured about the body of the Sun in the very seat of the solar body, is at times somewhat thick and is set aflame by contact with the solar body: then, the Sun being covered, a flaming splendour stands out, not allowing a deep and excessive shadow to be made, as Plutarch says. This cause was touched on also in Book I, folio 147, and confirmed by many experiments.
But because such a splendour does not at once appear as a slender circle, let a second cause be called in: a transparent air poured about the globe of the Moon, in whose entrance from above as well as in its exit below the Sun’s rays are broken, and, being refracted, fall upon our sight in the same manner in which the same rays of the Sun are refracted in our air spread about the Earth. In this way it will come about that the Sun’s circle indeed is slender, but is seen not directly, but through the Moon’s air interposed, which breaks the rays; and therefore rather the circle of the Moon, lit by the Sun’s light, is seen.
From what quarter does an eclipse of the Sun begin? Ordinarily from the quarter of the west, in our hemisphere on the right, and the eclipse ends, or the Sun is filled up, from the quarter of the east on the left; contrary to what ordinarily happens in an eclipse of the Moon.
Does this not always happen, then, and from what causes, and in what order? In small eclipses of the Sun, on account of the great inclination of the Moon’s visible path to the ecliptic, it frequently happens, especially about the eastern horizon, in the signs of right ascension, the Moon falling toward the tail of the dragon, that the Sun begins to be eclipsed either at the very top or even a little toward the left, and is filled up from the same quarter lower down; or, on the contrary, about the western horizon in the signs of right descension, the Moon fleeing from the head of the dragon toward the north, that he begins either at the lowest edge or a little higher on the left and ends from the same quarter higher up.
Nay more, things akin to these can happen even about the nonagesimal itself, on account of the great variety of the parallaxes of latitude. For when Gemini or Cancer is rising, the parallaxes of latitude decrease very swiftly, when Sagittarius or Capricorn is rising they increase; if therefore at the same time the Moon there ascends toward the north about the node that lifts her, and here descends toward the south, by the accumulation of the true and the seen variation the angle of the visible path becomes very great, and a fairly large eclipse can both begin and end from the same quarter. So in the year 1598 a very great eclipse was in Pisces and near the ascending node, and began at Graz in Styria on the right, exactly at what was almost the nonagesimal, and ended on the left indeed, but almost at the very vertex, so that if it had been even a little smaller it would have ended on the right, without touching the vertex. On the other hand, in the year 1614 an eclipse in Libra near the descending node began about noon at Munich on the right and ended below. See the Optical Part of the Astronomy, folio 411 and following.
What is the cause why not all total eclipses of the Sun are the cause of such deep darkness that night takes the place of day? This is confirmed beyond doubt by experience, of the two causes just now stated the second, namely, a thick substance about the Sun, not here in our air, but poured at times around the very seat of the Sun, which shines with the Sun’s rays and appears even when the Sun is covered, like a flame darting out in a circle and showing so much light that it cannot be mere night. This matter is not always poured around the Sun; when therefore the ether about the Sun is limpid and pure, then, this splendour also being absent, it is necessary that with the Sun extinguished the day also is utterly extinguished.
Some report that the Moon in a total eclipse is seen with a fairly clear redness: whence this redness of the Moon? Not from the Moon’s own perennial light, for the Moon could not then be lost to sight in her total eclipses; nor from the Sun’s rays transmitted through the body of the Moon, as if it were transparent, as some think, since it is in truth opaque, like the Earth; but from the brightness of the Earth lit by the Sun, reflected to the Moon, of which brightness the Moon’s shadow covers only the smallest part.
Is there any apocatastasis, or return, of eclipses? Because of the many principles concurring to any eclipse, there can be no exact order of their returning. For each later series of them brings something new, which, since it is not very conspicuous in the first return, is yet doubled in the second. This return in lunar eclipses, as being the simpler, is especially notable, and is defined by two hundred and twenty-three months, that is, 18 lunar years completed, so that to one Metonic cycle of nineteen years one simple lunar year remains over, and it often happens that the same eclipses, not differing much, recur and appear after this completed time, yet ten degrees of the zodiac earlier in place. Sometimes, however, two eclipses of the Sun by a beautiful conspiracy also define the nineteen-year cycle, the Sun losing his light again after 19 years completed in the same place of the zodiac. So it happened in the years 1579 and 1598, on the 25th of February in both; for on both days the eclipse of the Sun was great. But this happens by a singular concurrence of causes, nor was anything like it or near it to be expected from the following cycle.
Is the Moon alone the cause of the prodigious darkness by day? By no means; for frequent examples occur to us in histories of darkness more than Cimmerian not at the time of new moon; the causes of which must be physical, so much higher above the Earth as such gloom is observed more widely. The first cause, nearest to us and therefore serving only particular places, is clouds, especially those heavy with a downpour, or pouring forth a copious cloud of snow; to which add a cloud of birds or of cicadas, rarer events yet not so unknown; for all these cover the Sun in certain places and bring night upon day.
A second cause, somewhat higher, though arising from the Earth, is a huge force of ashes, thrown out from the craters of burning mountains; whence this cause for the most part has an earthquake joined with it, or a shower of stones, which are cast up on high together with the ashes from the narrow passages of the Earth as if through channels. Cornelius Gemma writes that something of the kind happened in Portugal in the year 1562, and Dio presents such an obscuration of the Sun by ashes under Domitian: for at Rome this prodigious gloom was seen when Mount Vesuvius near Naples had belched forth ashes and they were scattered as they fell even as far as Palestine. Compare also that darkness which, at the time of the passion of our Lord, overshadowed the Sun, for that too was with an earthquake, although Dionysius the Areopagite in Suidas ascribes this darkness to a prodigious motion of the Moon.
A third cause is akin to that now brought three times upon the stage, a thick substance standing about the Sun in his very seat, of which in Book I, folio 155. If this is a little denser, it will darken the Sun to the whole world, yet will not entirely extinguish him.
This happened throughout the whole year of Caesar’s murder, and in the year 1547 for four days over almost the whole of Europe. Nor does it matter whether such a thing exists close to the Sun, or on the very surface of the Sun.
Do other heavenly bodies also run beneath the Sun, as the Moon does, obscuring him? Venus and Mercury can indeed run under the Sun and be seen in him as very small spots, since they are opaque bodies, as was proved above by the argument of the phases of Venus. These incursions are nevertheless very rare, though for different causes. For the nodes of Venus, as was said above, are in Gemini and Sagittarius. Now the two conjunctions of Venus with the Sun in her lower part are 72° apart. If, then, one conjunction happens in a node, about 248 years are to be waited until, after the returns of eight years, it strays 72 degrees from the node and the next conjunction succeeds to its place. And so in our age Venus cannot eclipse the Sun or pass beneath him. Mercury, though he is often joined to the Sun in his lower part about his nodes, yet seldom comes to so great a nearness as to run under the Sun, on account of the inequality of his motion and the greatness of the inclination of his orbit.
What do you think those spots on the Sun are, some of which have been wrongly passed off for Mercury in the Sun? Here again we have so much the less to assert about solar things than about lunar, as the Sun is higher than the Moon and farther from us. There are nevertheless some things these spots have in common with our clouds. First, just as if one could look at the Earth from some fixed place of the ether toward the Sun, he would see the clouds on its disc turned within half a natural day from the eastern edge to the western, yet so that some of the clouds would run a little ahead, others, whose motion over the Earth is toward the east, would in this universal motion be left behind somewhat; so also it is with the spots of the Sun. For they mostly remain on the Sun’s disc for 12 or 13 days, some however somewhat longer, others somewhat shorter, and they enter from the east and cease to be seen at the west. Then, as one who watched the rotation of the Earth would see, whether clouds or spotted parts of the surface, slow about the extreme edges and their shapes compressed, on account of the convexity of the globe drawn upward away from the sight, but in the middle of the disc swift and their shapes spread out, so the same is seen to happen in the spots of the Sun. This experiment refutes those who devise I know not what spherical surface, woven in the manner of a transparent breastplate with opaque little bodies, and moving beneath the Sun like a planetary sphere with a very slow motion. For if that were so, the spots would always have equal daily motions on the Sun’s disc, as well about the edges as in the middle. Third, as clouds arise, are divided, scattered, vanish, so also we see some of the spots begin to appear very small in the middle of the Sun’s disc and grow larger with the succession of days, others thin out and vanish before they come to the extreme edge, some also divided in two or three. Fourth, as the surface of the Earth, moistened by rains, is of a darker colour even after the clouds are dispersed and then turns green, so on the other side the parts of the Sun in which spots are seen to be dispersed, after they are dispersed, grow bright and appear like little flames scattered here and there on the disc, if the image of the Sun is painted on paper through a good telescope. See also Book IV, folio 343.
Are other stars also hidden by the Moon as the Sun is, or even by themselves? That stars, fixed as well as wandering, are hidden by the Moon is almost daily. That fixed stars are covered by the planets, or the superior planets by the inferior passing beneath them, we observe from time to time, yet not so often as by the Moon, because their bodies are small, their motions slow, and the light even of some so clear that they easily overpower neighbouring stars of duller light. See examples of both kinds in the Optical Part of the Astronomy, folio 304.
What is the method of computing an occultation of some star made by the Moon? One must use the true place of the Moon, copied from the ephemerides or computed from the tables, and also the whole parallax of the Moon, not diminished by the subtraction of the parallax of the Sun as before in eclipses of the Sun. The rest as in an eclipse of the Sun.
What is the chief use both of eclipses of the Moon and of the occultations of the Sun and stars by the interposition of the body of the Moon? 1. This is almost the only aid for inquiring into the longitudes of places, or the differences of meridians, concerning which method see Book III, folio 300. And therefore this observation of eclipses in various places, and the computation from the observations of the true place of the Moon for the time of each place, is exceedingly useful for hydrography and navigation and indeed the whole of geography.
2. In general the eclipses of the luminaries are tutors for learning the motion and the altitude of the Moon from the Earth. In turn, in the theory of the Moon astronomers put forth the rudiments of the whole of astronomical knowledge. For the motions of the Moon everywhere shine before us toward the truth of the doctrine on the motions of the other planets also.
3. The Moon is also as it were the first step on the ladder of the world, because the orb of the Moon is the mean proportional between the globe of the Earth and the orb of the Sun, or great orb, as was demonstrated in Book IV of the Epitome, folio 327. The great orb is the mean proportional between the nearest surfaces of the inferior and the superior moving bodies. On which see the end of Book V of the Harmony of the World. Lastly, the outermost orb of the moving bodies, that is, of Saturn, is the mean proportional between the body of the Sun and the sphere of the fixed stars; see Epitome, Book IV, folio 332 and following.
4. Finally, without the knowledge of astronomy, or of the machine of the world in all its parts, all physical knowledge is maimed and there is no metaphysical knowledge. So, having made a beginning of the ascent through these dreadful toils and labours of the luminaries, we at last arrive at the most sublime things, knowing God the Creator from the book of nature and the works of his hands.
IX. Of the harmonic proportions of the motions.
How is the swiftness or slowness of the apparent motion known? From the quantity of the arc which the planet traverses in one day or in an hour of the time proposed, that is, from the comparison of the daily or hourly motions.
In how many ways can the proportions of the daily motions be considered? In two, as there are two inequalities of the planets. Either, then, the daily motions are considered according to the first inequality alone, how great, namely, they would appear to an eye by a fiction placed in the centre of the Sun, or they are considered according to both inequalities, how great they appear here on Earth.
What is the difference between these two considerations with respect to harmonic tempering? In the former, certain harmonic proportions are attributed to certain planets; in the latter, since the planets become stationary, losing all apparent motion, all the harmonies therefore belong to all without distinction. Yet that former consideration either does not concern us earth-dwellers at all, since it is solar, or at least is only intellectual; the latter, viewed from the Earth, can also be sensual on Earth, understand it thus, that astrologers can attend to the time at which it happens that the motions of the planets are tempered by harmony, and whether any effect of that tempering appears in the meteors, as they have been diligent in this, so as to know that at the time when the planets are configured harmonically by reason of position, then nature is especially stirred up.
How many harmonic proportions are there? The compound ones are infinite, but the simple are eight: the unison between equal arcs, the minor third between arcs which have to each other the proportion that is between the numbers 6 and 5, the major third between 5 and 4, the diatessaron between 4 and 3, the diapente between 3 and 2, the minor sixth between 8 and 5, the major sixth between 5 and 3, the diapason between 2 and 1. For their causes and origin see Book III of the Harmony.
What is the manner of composition of the harmonies? Although four of these eight can also be called compound (since the diapason is resolved into a third and a sixth, or into a diapente and a diatessaron, the sixth into a third and a diatessaron, the diapente into thirds), yet they are properly called compound when with one of the simple ones one or more diapasons are joined: as the diapason with a diapente, between 3 and 1.
For the diapente between 3 and 2 is compounded with the diapason between 2 and 1. So the triple diapason between 1 and 8 is compounded of the diapason between 1 and 2 and two other diapasons, between 2 and 4 and between 4 and 8.
In how many ways do the harmonies belong to the planets in the first consideration, which has place in the Sun? In three. 1) Either to each singly are attributed their own harmonies with respect to the different motions of one planet. These are merely intellectual and archetypal, that is, the two terms of one harmony are not joined at the same time. So the variety of Saturn’s motions attains the terms of the major third, Jupiter’s of the minor third, Mars’s very nearly the diapente, Mercury’s the diapason with nearly a minor third, the Moon’s the diatessaron, and this in relation to the Earth. But the Earth herself is here an exile together with Venus; for the Earth’s motions contain themselves within the semitone 16/15, Venus’s within the comma 80/81, which are not harmonic intervals.
2) Or the harmonies are considered as attributed to two neighbouring planets together, where we attend to how many harmonies can belong to each pair, from diverging motions (that is, of the higher at its highest and the lower at its lowest) up to converging (that is, the higher at its lowest and the lower at its highest). And these harmonies are temporal, that is, they can sometimes happen between motions existing at the same time, and therefore we can call them sensible, in that kind of sense in which aspects are perceived here on Earth. Thus to Saturn and Jupiter belong all the harmonies from the diapente with the double diapason up to the simple diapason; to Jupiter and Mars all from the triple diapason up to that compounded of the double diapason and a minor third; to Mars and the Earth all from that compounded of the diapason and a minor third up to the diapente. To the Earth and Venus no more than two sixths, major and minor, the difference of which two, being a diesis, is the same as the difference between the hard and the soft song; with these two planets, therefore, lies the distinction of the harmonic genera. To Venus and Mercury belong all, from the double diapason up to the major sixth.
3) Or to any two whatever, by a reasoning formed from the two former, which finally ends in the inquiry into the universal harmony of all six, conspiring into one concord, of which, through a sufficiently long stretch of time, four forms can happen. For these and their causes and defects see Book IV of the Harmony.
Teach how to compute what the harmony is between the daily motions, whether eccentric from the Sun or apparent on the Earth. Multiply the smaller of the apparent daily motions, whether from the Sun (there) or from the Earth (here), by 240, divide the product by the greater, and compare what results with this little table.
Let the daily motion of the Sun be 60, of Mars 48; multiply 48 by 240, divide the product 11520 by 60: the quotient 192 is the argument of the major third.
The quotient 30 (a little more) shows that there is nearly the harmony compounded of the diapason and the double diapason, that is, the triple diapason. For if the motion of Venus were 104, this harmony would be perfect.
Let the rest be supplied from the reading of the work of the Harmony of the World.