# Summary of Astronomical Theory Necessary for the Study of the Heavenly Harmonies

##### 26 minutes • 5433 words

In What Features Relating to the Motions of the Planets Have the Harmonic Proportions been Expressed by the Creator, and How?

When therefore the fantasy of retrogressions and stations has disappeared, and the planets’ proper motions, in their own true eccentric orbits, have been stripped to essentials, there still remain in the planets the following distinct features: 1. their distances from the Sun; 2. their periodic times; 3. their daily eccentric arcs; 4. the daily times expended on their arcs; 5. their angles at the Sun, or apparent daily arcs to ob servers, so to speak, on the Sun.

And again, all of these (except for the periodic times) are variable right around their orbit, most indeed at the mean longitudes, and least in fact at the extremities, when they have just turned away from one of them and are returning towards the opposite one.

Hence when the planet is lowest and closest to the Sun, and therefore expends as little time as possible on one degree of its eccentric, and on the other hand completes its greatest daily arc of the eccentric in a single day, and appears fastest from the Sun, then its motion persists for a while in this vigorous state, without sensible variation, until when the perihelion has been passed the planet has begun to increase its linear distance from the Sun. Then at the same time it also expends a longer time on the degrees of its eccentric, or if you consider the motion of a single day, it makes less progress on each following day, and also appears much slower from the Sun, until it approaches its upper apsis, making its distance from the Sun the greatest.

For then it also expends the longest time of all on one degree of its eccentric, or on the other hand completes its smallest arc in one day, and also makes its appearance much smaller and the smallest in its whole circuit.

Lastly, all these features belong either to any one planet at different times, or to different planets; so that if we suppose an infinity of time, all the states of the orbit of one planet can coincide at the same moment of time with all the states of the orbit of another planet, and can be compared; and then the complete eccentrics indeed, compared with each other, have the same proportion as their semidiameters, or their average distances, whereas the arcs of the two eccentrics, designated as equal or by the same number, nevertheless have unequal true lengths in the proportion of the whole eccentrics.

For example, one degree on the sphere of Saturn is almost twice as large as one degree on the sphere of Jupiter. And on the other hand, the daily arcs of the eccentrics, expressed in astronomical numbers, do not show the same proportion as the true paths, which the globes complete through the aethereal air in one day because single degrees each represent on the wider circle of the superior planet a section of its path which is larger, but on the narrower circle of the inferior planet a section which is smaller. Hence a sixth aspect for consideration is now added, concern ing the daily paths of the two planets.

First, therefore, let us take the second of the features listed, that is to say the periodic times of the planets, which comprise the assembled totals of all the times expended on all the degrees of the whole circuit, long, average, and small.

And it has been observed from antiquity up to our own time that the planets complete their journeys around the Sun as follows in the table.

Sixtieths of a day Days Saturn Jupiter Mars Earth with Moon Venus Mercury 10759 4332 686 365 224 87 Therefore mean daily motions. Minutes Seconds Third min 12 2 0 27 37 59 15 42 58 4 31 59 96 245 59 26 31 8 8 11 7 32 39 25 In these periodic times there are therefore no harmonic propor tions, which is readily apparent if the larger periods are continually divided by two, and the smaller ones are continually doubled, so that with the intervals of a diapason suppressed we can look for those which are within a single diapason.^^ Saturn Halves 10759. 5379. 2689. 1344. 672. 12 36 48 54 27 Jupiter 4332. 2166. 1083. 541. 37 19 10 35 Mars 686 . 59 Earth Venus Mercury 365. 15 224. 42 449. 24 87. 58 175. 56 351. 52 Doubles All the last numbers, as you see, are repugnant to harmonic pro portions, and seem similar to inexpressibles. For let the number of

days for Mars, 687, be measured in units in which it represents 120, which stand for a division of a string. In these units Saturn will be represented by a little more than 117, taking a sixteenth part; Jupiter by less than 95, taking an eighth; the Earth by less than 64; Venus by more than 78, taking double; Mercury by over 61, taking quadruple. Yet these numbers do not make any harmonic proportion with 120; but the neighboring numbers 60, 75, 80, and 96 do. Similarly in units in which Saturn comes to 120, Jupiter comes to about 97, the Earth over 65, Venus more than 80, Mercury less than 63. And in units in which Jupiter comes to 120, the Earth comes to less than 81, Venus less than 100, Mercury less than 78. Also in units in which Venus comes to 120, the Earth comes to less than 98, Mercury to more than 94. Lastly in units in which the Earth comes to 120, Mercury comes to less than 116. But if this free selection of proportions had been valid, they would have been absolutely perfect harmonies, without excesses or defi ciencies. God the Creator is therefore not discovered to have intended to introduce harmonic proportions among these sums of times ex pended added together into periodic times. And since it is a very probable conjecture (inasmuch as it depends on geometrical proofs, and on the theory of the causes of the plane tary motions set out in the Commentaries on Mars) that the bulk of the bodies of the planets are in the proportion of their periodic times,^’^ so that the globe of Saturn is about thirty times greater than the globe of the Earth, Jupiter twelve times. Mars less than twice, the Earth greater than one and a half times the globe of Venus, and four times greater than the globe of Mercury, then these proportions of the bodies will not be harmonic either. Since, however, God has established nothing without geometrical beauty unless it is bound up with some other, prior law of necessity, we readily infer that the periodic times get their durations, and there fore the moving bodies also their bulks, from something which has prior existence in the Archetype. It is to express it that these, as they appear, disproportionate bulks and periods are fitted to this measure. But I have said that the periods are the sum of the times expended, very long, medium, and very slow. The geometrical harmonizations must therefore be found either in these times, or in something prior to them in the mind of the Maker, perhaps.

The proportions of the expended times are bound up with the proportions of the daily arcs, because the arcs are in the inverse proportion o f the times. Again, we have stated that the proportions of the times expended and the distances of any one planet are the same. As far as individual planets are concerned, therefore, discussion of these three, the arcs, the times expended on equal arcs, and the remoteness of the arcs from the Sun, or the distances, will be one and the same. And because all these are as it happens variable in the case of the planets, there can be no doubt that if they have been assigned any geometrical beauty, by the sure de sign of the Maker, they acquire it at their extremes, as at their distances in aphelion and perihelion, and not so much at the mean distances in between. For given the proportions of the extreme distances, the design does not need to fit the intermediate proportions to a definite number; for they follow automatically, by the necessity of the plane tary motion from one extreme, through all the intermediate points, to the other extreme. Therefore, the extreme distances are as follows, worked out from the very accurate observations of Tycho Brahe, by the method explained in the Commentaries on Mars, by the most persistent exertions of seven teen years.

Then there is no single planet of which the extreme distances hint at harmonies, except for Mars and Mercury.

But if you compare the extreme distances of different planets with each other, some light of harmony now begins to shine forth. For the divergent extremes of Saturn and Jupiter make a little more than a diapason; their convergent extremes the mean between major and minor sixths. Similarly the divergent extremes of Jupiter and Mars embrace about a double diapason, and their convergent extremes about a diapason and a diapente. However, the divergent extremes of the Earth and Mars have embraced rather more than a major sixth, and their convergent extremes an oversize diatessaron. In the following couple of the Earth and Venus again there is the same oversize dia tessaron between their convergent extremes, but between their diver gent extremes we are deserted by harmonic proportion; for it is less than half a diapason (if we may use the phrase), that is, less than the semiduplicate proportion,’*^ Lastly, between the divergent extremes of Venus and Mercury the proportion is a little less than the combi nation of a diapason and a minor third; between their convergent ex tremes is an oversize diapente, and a little over. Therefore, although one interval departs a little too far from the harmonic proportions, yet this good result was an invitation to pro ceed further. Now the following was my reasoning. First, these distances, insofar as they are lengths without motion, are not appropriate to be examined for harmonies, because the harmonies are more intimately connected with motion, on account of its swiftness and slowness.

Second, in the case of the same distances, insofar as they are diameters of spheres, it is easy to believe that the ratio of the five regular solids should be taken in preference, by analogy. For the ratio of the solid geometrical bodies to the celestial spheres, either enclosed on all sides by celestial matter, as antiquity would have it, or to be enclosed by the accumulation of a great many successive rotations, is also the same as that of the plane figures which are inscribed in a circle (and which are the figures which generate the harmonies) to the celestial circles of the motions, and to the other spaces in which the motions occur.

Therefore, if we are seeking harmonies, let us seek for them not in these latter distances, as they are the semidiameters of spheres, but in the former distances, as they are the measures of the motions, that is, rather in the actual motions. Certainly no other distances can be taken as the semidiameters of the spheres, but the average distances; whereas we are dealing with the extreme distances.

Therefore, we are not dealing with the distances in respect of their spheres, but in respect of the motions.

For these reasons, then, since I had gone over to comparison of the extreme motions, at first the proportions of the motions remained the same in magnitude as those of the distances were previously, except that they were inverted. Hence some proportions were also found between the motions, as previously, to be unmelodic, and foreign to the harmonies. However, again I thought that I deserved that result, inasmuch as I was comparing arcs of the eccentric with each other, which are not expressed or counted by a measure of the same size, but are counted in degrees and minutes which are different in size for different planets. Also they do not anywhere show the apparent size which the numerical value of each indicates, except only at the center of each eccentric, which is not supported by any body; and similarly also it is incredible that there should be any sensation or natural instinct in that position in the world which could grasp this apparent size, or rather it is even impossible, if I was comparing the eccentric arcs of different planets, with respect to their apparent sizes at their own centers, which are different in different cases. However, if the different apparent magnitudes were compared, they ought to be apparent at a single position in the world, in such a way that that which has the opportunity of comparing them would be situated at that position of their common appearance. Therefore, I judged that the apparent sizes of these eccentric arcs should either be put out of my mind or represented in a different way. But iff were to put the apparent sizes out of my mind, and turn my attention to the actual daily paths of the planets, I saw that I should have to apply the precept which I stated in Number IX of the previous chapter.^® Therefore, on multiplying the daily arcs of the eccentrics by the mean distances of the orbits, the following paths resulted.

Thus Saturn completes scarcely a seventh part of the path of Mer cury; and the result is what Aristotle in Book II of his De Caelo {On the Heaven)^^ judged to be in agreement with reason, that the planet which is nearer to the Sun always completes a greater distance than the one which is further away, which is impossible to attain in the ancient astronomy. Therefore, as far as the daily paths of individuals are concerned, the proportions which they comprise ought to be the same in mag nitude as those which were previously in the distances, but inverted in kind, because the eccentric arcs, as has been stated, have the in verse proportion of their own distances from the Sun. However, if we consider the extreme paths of the pairs, either di vergent or convergent, there is much less appearance of anything har monic than previously when we had considered the actual arcs. And indeed if we should ponder the matter more carefully, it will be apparent that it is not very likely that the most wise Creator should have taken thought most of all for harmonies between the actual plane tary paths. For if the proportions of the paths are harmonic, all the other features of the planets will be constrained, and linked to the paths, so that there will be no room for taking thought for harmonies elsewhere. But who will benefit from harmonies between the paths, or who will perceive these harmonies? There are two things which reveal to us harmonies in natural occurrences, either light or sound. The former is received through the eyes, or hidden senses analogous to eyes, the latter through the ears; and the mind seizing on these emanations distinguishes either by instinct (on which plenty has been said in Book IV) or by astronomical or harmonic reasoning between melodic and unmelodic. In fact, no sounds exist in the heaven, and the motion is not so turbulent that a whistling is produced by friction with the heavenly air. There remains light. If it is to teach us anything about the paths of the planets, it will teach us that either the eyes, or some sensory organ analogous to them, are located in a certain position; and for the light to inform us immediately of its own accord, it seems that the sensory organ must be there in its presence. There fore, there will be an organ of sensation all over the world, that is to say in such a way that one and the same is present to the motions of all the planets. For that way which was traversed by dint of obser vations, by way of long drawn out wanderings in geometry and arith metic, of the proportions of the spheres and the rest which had to be learnt beforehand, to reach these actual paths, is too long for some natural instinct, to influence which it seems to be fitting that the har monies were introduced. Therefore, assembling all these points into a single review I have rightly concluded that we should dismiss the true paths of the planets through the aethereal air, and turn our eyes to the apparent daily arcs, all indeed apparent to one definite and prominent position in the world, that is to say to the actual solar body, the source of the motion of all the planets. Also we should look not how high any par ticular planet is from the Sun, nor what space it traverses in a single day —for that is rational and astronomical, not instinctive —but how large an angle the daily motion of each planet subtends at the actual body of the Sun, or how large an arc on one common circle drawn about the Sun, such as the ecliptic, it seems to complete on any particular day. Thus this appearance, brought by the agency of light to the body of the Sun, can along with the light itself flow straight to living creatures, who share in this instinct, just as in the fourth book we have stated that the pattern of the heaven flows to a foetus by the agency of the rays.^^ Therefore, the Tychonic astronomy teaches us (abstracting from the proper motion of the planets the parallaxes of the annual orbit, which impart to them the semblance of stations and retrogressions) that the daily motions of the planets in their own orbits (as they ap pear, so to speak, to those watching on the Sun) are as follows:

Notice that the great eccentricity of Mercury makes the propor tion of the motions differ considerably from the square of the pro portion of the separations.®*^ For if you make the proportion of the motion at aphelion to the mean motion, 245 minutes 32 seconds, that is the square of the proportion of the mean separation, taken as 100, to the separation at aphelion, 121, then the resulting motion at aphelion is 167; and if you make the proportion of the motion at perihelion to the same mean motion, that is the square of the proportion of 100 to the distance at perihelion, 79, the motion at perihelion will be made 393. In both cases it is greater than I have supposed here, naturally because the mean motion at the mean anomaly being viewed very obliquely does not appear as great, that is to say not 245 minutes 32 seconds but smaller by about 5 minutes. Therefore, the motions at aphelion and at perihelion will also be found to be smaller. However, it will be less so for the motion at aphelion, and more so for the motion at perihelion, on account of the theorem in Euclid’s Optics, in accor dance with my warning in the preceding chapter, under Number VI. Therefore, I could assume mentally that between these apparent extreme motions of individual planets there would be harmonies, and their distances would be melodic, and that indeed from the propor tions of their daily eccentric arcs, set out above, since I there saw that square roots of harmonic proportions reigned everywhere, whereas I knew that the proportion of the apparent motions was the square of that of the eccentric motions. But we may verify what is stated by actual observation, indeed without reasoning, as you see in the next table. For the proportions of the apparent motions of individual planets come very close to harmonies. Thus Saturn and Jupiter embrace a very little more than thirds, major and minor: there is an excess in the former case of 53:54, in the latter of 54:55 or less, that is to say about one and a half commas; the Earth embraces a very little more, that is to say 137:138 more, scarcely half a comma, than a semitone; Mars somewhat (that is to say 29:30, which is close to 35:35 or 35:36) less than a diapente; Mercury occupies, over the diapason, nearer a minor third than a tone, that is to say it has less by about 38:39, which is about two commas, in other words about 34:35 or 35:36. Venus alone occupies something smaller than any of the melodic intervals, and is itself just a diesis; for its proportion is between two and three com mas, and exceeds two thirds of a diesis, being about 34:35, almost 35:36, a Diesis diminished by a comma. The Moon also enters into consideration here.®’ For it is found that its hourly motion at apogee in quadrature, that is to say when it is slowest of all, is 26 minutes 26 seconds. At perigee at the syzygies, that is to say when it is fastest of all, it is 35 minutes 12 seconds. By this ratio a diatessaron is formed with great exactness. For a third part of 26'26" is 8'49"—four times which is 35'15". And notice that the harmony of diatessaron is found nowhere else among the appar ent motions. Notice also the analogy of the Fourth in the harmonies with quadrature in the phases.®^ These, then, are found in the mo tions of individual planets. But among the extreme motions of the pairs of planets compared with each other, the clearest light is thrown at once as soon as we look at the heavenly harmonies, whether you compare the receding extreme motions with each other, or the approaching. For between the approaching motions of Saturn and Jupiter the proportion is ex actly double, or a diapason; between their receding motions, it is a very little more than triple, or a diapason with a diapente. For of 5 minutes 30 seconds, a third part is 1 minute 50 seconds, whereas Saturn has instead of that 1 minute 46 seconds. Therefore, the planetary pro portion has one diesis over, or something a little less, that is 26:27 or 27:28; and when Saturn is approaching to within less than a single second from aphelion, the excess will be 34:35, the size of the pro portion between the extreme motions of Venus. Between the diverg ing and converging motions of Jupiter and Mars reign the triple di apason, and the third a double diapason above, though not perfectly. For an eighth part of 38 minutes 1 second is 4 minutes 45 seconds, whereas Jupiter has 4 minutes 30 seconds. Between those numbers there is still a difference of 18:19, which is the mean between 15:16 and 24:25, a semitone and a diesis, that is to say very nearly a perfect limma, 128:135. Similarly, a fifth part of 26 minutes 14 seconds is 5 minutes 15 seconds, whereas Jupiter has 5 minutes 30 seconds. There fore, the deficiency from the fivefold proportion here is about 21:22, the amount of the excess in the other proportion previously, that is about a diesis, 24:25. The harmony 5:24 which takes in a minor in stead of a major third over the second octave comes rather near. For of 5'30" a fifth part is 1'6", and taking twenty four times that pro duces 26'24", with which 26'14" makes no more than half a comma.®’^ Mars has been allotted a very small proportion with the Earth, very exactly the sesquialterate, or a diapente; for a third part of 57 minutes 3 seconds is 19 minutes 1 second, and double that is 38 minutes 2 seconds, the very number which Mars has, that is 38 minutes 1 second. As their greater proportion they have been allotted a diapason with a minor third, 5:12, a little less nearly perfect. For a twelfth part of 61 minutes 18 seconds is 5 minutes 6? seconds, and taking five times that gives 25 minutes 33 seconds, whereas instead of that Mars has 26 minutes 14 seconds. The deficiency is therefore about a narrow diesis, that is 35:36. However, the Earth and Venus have been allotted harmonies in common, the greatest 3:5 and the least 5:8, which are sixths, major and minor, again not quite perfect. For a fifth part of 97 minutes 37 seconds is 19 minutes 31 seconds, and three times that comes to 58 minutes 34 seconds which is more than the motion of the Earth at aphelion by 34:35, which is almost 35:36, the amount by which the planetary proportion exceeds the harmonic. Similarly, an eighth part of 94 minutes 50 seconds is 11 minutes 51 seconds + , and five times that is 59 minutes 16 seconds which is as nearly as possible equal to the mean motion of the Earth. Hence in this case the plane tary proportion is less than the harmonic by 29:30, or 30:31, which again is nearly 35:36, a narrow diesis; and to that extent this smallest of their proportions approaches the harmony of diapente. For a third part of 94 minutes 50 seconds is 31 minutes 37 seconds, and twice that is 63 minutes 14 seconds, from which the motion of the Earth at perihelion, 61 minutes 18 seconds, is deficient by the tiny amount of 31:32, so that the planetary proportion occupies exactly the mean between the neighboring harmonic proportions. Lastly, the propor tions allotted to Venus and Mercury are as the greatest a double diapason, and as the least a hard sixth, though these are not absolutely perfect. For a fourth part of 384 is 96 minutes 0 seconds, whereas Venus has 94 minutes 50 seconds. Therefore, it approaches the four fold within about one comma. Similarly, a fifth part of 164 minutes is 32 minutes 48 seconds, and taking three times that makes 98 min utes 24 seconds, whereas Venus has 97 minutes 37 seconds. There fore, the planetary proportion is in excess by about two thirds of a comma, that is 126:127. These, then, are the harmonies with each other allocated to the planets; and there is none of the direct comparisons (that is to say between convergent and divergent extreme motions) which does not come very close to some harmony, so that if strings were tuned in that way, the ears would not easily be able to detect the imper fection, except for the excess of the single one between Jupiter and Mars.*

Now it follows that if we compare the motions on the same side®^
we shall not be likely to stray far from the harmonies in that case either.
For on multiplying the 4-5 times 53-’54 of Saturn by the intermediate
proportion 1:2 the combined product is 2:5 times 53:54, which is the
proportion between the motions at aphelion of Saturn and Jupiter.®®
Multiply by the 5:6 times 54:55 of Jupiter: the product is 5:12 times
54:55 which is the proportion between the motions at perihelion of
Saturn and Jupiter. Similarly multiply the 5:6 times 54:55 of Jupiter
by the following intermediate proportion, 5:24 divided by 157:158;
the result is 1:6 divided by 35:36, the proportion between the motions
at aphelion.®"^ Multiply the same, 5:24 divided by 157:158, by the 2:3
divided by 29:30 of Mars; the result is 5:36 divided by 24:25, about,
that is 125:864 or nearly 1:7, the proportion between the motions at
perihelion: in fact this alone so far is unmelodic.®® Multiply the third
of the intermediate proportions, 2:3,®® by the 2:3 divided by 29:30 of
Mars: it comes out as 4:9 divided by 29:30, that is 40:87, another un-
melodic interval between the motions at aphelion. If instead of the
proportion for Mars you multiply by the Earth’s 15:16 times 137:138
you will obtain 5:8 times 137:138, the proportion between them at
perihelion.’^® And if you multiply the fourth of the intermediate pro
portions, 5*8 divided by 30:31, or 2*3 times 31:32, by the Earth’s 15:16
times 137:138, you will find the product is very nearly 3:5, the pro
portion between the motions at aphelion of the Earth and Venus.

For a fifth part of 94 minutes 50 seconds is 18 minutes 58 seconds, and three times that is 56 minutes 54 seconds, whereas the Earth has 57 minutes 3 seconds.^’ If you multiply the same proportion by the 34:35 of Venus ,’ 2 you obtain a product of 5:8, the proportion between the motions at perihelion. For an eighth part of 97 minutes 37 sec onds is 12 minutes 12 seconds + , and taking five times that gives a return of 61 minutes 1 second, whereas the Earth has 61 minutes 18 seconds. Lastly, if you multiply the last of the intermediate proportions, 3:5 times 126:127, by Venus’ 34:35, the combined product will be 24:25 times 3:5, and the result is a dissonant interval, made of the two com bined, between the motions at aphelion. Nevertheless, if you multiply by Mercury’s proportion, 5:12 divided by 38:39, now it will fall short of 1:4, or the double diapason, by as nearly as possible a complete diesis, for the proportion between the motions at perihelion. Therefore, perfect harmonies are found between the convergent extreme motions of Saturn and Jupiter, a diapason; between the con verging extremes of Jupiter and Mars, a double diapason together with nearly a soft third; between the converging extremes of Mars and the Earth, a diapente, and between their motions at perihelion a soft sixth; between the motions of the Earth and Venus at aphelion, a hard sixth, and at perihelion a soft sixth; between the converging extremes of Venus and Mercury a hard sixth, and between their di vergent extremes or even between their motions at perihelion, a double diapason.’^ Hence without detriment to the astronomy developed most subtly of all from the observations of Brahe, it seems that the residual very tiny discrepancy can be absorbed, especially in the mo tions of Venus and Mercury. However, you will notice that where there is not a perfect major harmony, as between Jupiter and Mars, there alone I have detected a very nearly perfect intermediate placing of the solid figure, since the separation of Jupiter at perihelion is very nearly three times that of Mars at aphelion, so that this pair aspires in its distances to the perfect harmony which it has not got in its motions."^^ You will no tice further that the greater planetary proportion of Saturn and Jupiter exceeds the harmonic proportion, that is to say the threefold, by al most the same amount as is Venus’ own proportion; and the defi ciency in the common greater proportion of Mars and the Earth is also almost the same as that in the two common proportions of the extremes of the Earth and Venus, convergent and divergent. You will notice thirdly that among the superior planets there are almost fixed harmonies between the convergent motions, but among the inferior planets between motions in the same direction.^^ And notice fourthly that between the motions at aphelion of Saturn and the Earth there are very nearly five diapasons; for a thirty second part of 57 minutes 3 seconds is 1 minute 47 seconds, whereas the motion of Saturn at aphelion amounts to 1 minute 46 seconds. Further, there is a great distinction between the harmonies which have been set out between individual planets, and between planets combined. For the former cannot indeed exist at the same moment of time, whereas the latter can absolutely; because the same planet when it is situated at its aphelion cannot at the same time also be at its perihelion which is opposite, but of two planets one can be at its aphelion and the other at its perihelion at the same moment of time.^® Then the proportion of simple melody or monody, which we call choral music and which was the only kind known to the ancients, to the melody of several voices, called figured and the invention of recent centuries, is the same as the proportion of the harmonies which are indicated by individual planets to the harmonies which they in dicate in combination. Further, then, in Chapters V and VI the indi vidual planets will now be compared with the choral music of the ancients, and its properties will be demonstrated in the motions of the planets; but in the chapters which follow it will be demonstrated that the planets in combination match modern figured music.