Summary of Astronomical Theory Necessary for the Study of the Heavenly Harmonies
26 minutes • 5450 words
I have totally avoided the ancient astronomical hypotheses of:
 Ptolemy in the Theoricae of Peurbach
 the other writers of Epitomes
They do not convey truthfully either the arrangement of the bodies in the world or the commonwealth of the motions.
I replace them solely with the opinion of Copernicus and persuade everyone to believe it.
The common herd of scholars will find it absurd that the Earth is one of the planets, and is carried among the stars around an unmoving Sun.
My speculations about harmonies are consistent with the hypotheses of Tycho Brahe who arranged the bodies and the combination of their motions in common with Copernicus.
It is only the Copernican annual motion of the Earth which he transfers to the whole system of the planetary spheres, and to the Sun, which occupies the middle by the agreement of both authors.
For by this transference of the motion it nevertheless comes about that the Earth, if not in that vast and immense space within the sphere of the fixed stars, yet at least in the system of the planetary world, takes the same position at any given time according to Brahe as Copernicus gives it.
In fact, just as someone who draws a circle on paper moves the writing leg of his corripasses round, whereas someone who fastens his paper or tablet to a revolving wheel describes the same circle, without moving the leg of his compasses or his pen, on the tablet as it moves round; in the same way in this case for Copernicus indeed the Earth traces out a circle by a real motion of its own body, passing in between the circles of Mars on the outside and Venus on the inside; but for Tycho Brahe the whole planetary system (in which among the others are also the circles of Mars and Venus) turns, like the tablet on the wheel, applying to the motionless Earth, as if to the pen of the man who turns the wheel, the blank space between the circles of Mars and Venus.
The effect of this motion of the system is that the Earth marks on it the same circle round the Sun, intermediate between those of Mars and Venus, while itself it remains motionless, as according to Copernicus it marks out by a true motion of its own body, with the system at rest.
Harmonic study considers the motions of the planets as eccentric, as if viewed from the Sun. If an observer were on the Sun, even though it were in motion, to him the Earth, although it were at rest (to make a concession already to Brahe), would nevertheless appear to be going around an annual circuit, placed in between the planets, and also in an intermediate period of time.
Hence if there is a man whose confidence is too weak for him to be able to accept the motion of the Earth among the stars, nevertheless he will be able to rejoice in the marvelous study of this absolutely divine mechanism, if he applies whatever he is told about the daily motions of the Earth on its eccentric to their appear ance from the Sun, the same appearance as Tycho Brahe shows, with the Earth at rest.
However, true enthusiasts for the Samian philosophy’*’’ have no just cause to grudge such people this share in a most delightful speculation.
inasmuch as their joy will be many times more perfect, that is from the complete perfection of the speculation, if they do also accept the immobility of the Sun and furthermore the movement of the Earth.
First, therefore, readers should take it as absolutely settled today among all astronomers that all the planets go round the Sun, with the exception of the moon, which alone has the Earth as its center; and its orbit or course is not large enough to be capable of being drawn in its correct proportion to the rest in the following plan.
Therefore the Earth is added to the other five as a sixth, which either by its own motion, with the Sun at rest, or without moving itself while the whole system of the planets revolves, itself also marks out a sixth circle round the Sun.
Second, it is also settled that all the planets are eccentric, that is, they change their distances from the Sun, in such a way that on one side their orbits are furthest from the J Me 1 Sun, in the other they come closest to the Sun.
In the attached diagram 3 circles have been constructed for each of the planets.
None of them indicates the actual eccentric path of the planet, but the middle one in fact, for instance BE in the case of Mars, is equivalent to the eccentric orbit, with respect to its longer diameter; but the actual orbit, for instance AD, touches the upper of the three, AF, on one side A, and the lower CD on the oppo site side D. The circle GH which is sketched out by dots, and drawn through the center of the Sun, in dicates the path of the Sun accord ing to Tycho Brahe. If it moves on this path absolutely all the points in the whole planetary system here depicted proceed on an equivalent path, each on its own. And if one point on it, that is to say the center of the Sun, stops in one part of its circle, as here at the lowest point, then absolutely all points of the sys tem will stop, each at the lowest parts of their own circles. Also the three circles of Venus on account of the restricted space have merged into one, contrary to my intention.
Third, the reader should remember what I published in The Secret of the Universe, 22 years ago, that the number of the planets, or of courses round the Sun, was taken by the most wise Creator from the five regular solid figures, about which Euclid so many centuries ago wrote a book which is called the Elements, on account of its being made up of a series of Propositions. However, the fact that there cannot be more regular solids, that is, that regular plane figures cannot be congruent in a solid in more than five ways, has been made clear in Book II of this work.
Fourth, as far as the proportion of the planetary orbits is concerned, between pairs of neighboring orbits indeed it is always such as to make it readily apparent that in each case the proportion is close to the unique proportion of the spheres of one of the solid figures, that is to say the proportion of the circumscribed sphere of the figures to the inscribed sphere. However, it is not definitely equal, as I once dared to promise for eventually perfected astronomy.
For after the final verification of the intervals, from the observations of Brahe, I discovered the following facts: if the vertices of the cube are indeed applied to the inside circle of Saturn, the centers of the faces almost touch the middle circle of Jupiter; and if the vertices of the tetrahedron rest on the inside circle of Jupiter, the centers of the faces of the tetra hedron almost touch the outside circle of Mars.
In the same way the vertices of the octahedron, which rise from any of the circles of Venus (as they are all three compressed into a very narrow gap) are pene trated by the centers of the faces of the octahedron, which go down more deeply below the outside circle of Mercury, yet do not reach as far as the middle circle of Mercury.
Finally, the closest of all to the proportions of the dodecahedric and icosahedric spheres, which are equal to each other, are the proportions or intervals between the circles of Mars and the Earth, and between those of the earth and Venus, which are similarly equal to each other, if we reckon from the inside circle of Mars to the middle circle of the Earth, but from the middle circle of the Earth to the middle circle of Venus. For the mean dis tance of the Earth is the mean proportional between the smallest circle of Mars and the middle one of Venus.
However, these two proportions between the circles of the planets are still greater than are those of the pairs of circles in the figures belonging to the spheres, to the ex tent that the centers of the faces of the dodecahedron do not touch the outer circle of the Earth, nor the centers of the faces of the icosahedron the outer circle of Venus. Yet this gap is not filled up by
adding the semidiameter of the moon’s orbit above the greatest interval of the Earth and subtracting it from below the smallest interval.
However, I discover another proportion in the figures: namely that if the augmented dodecahedron to which I have given the name of Echinus (Hedgehog), that is to say the one formed from twelve quinquagonal stars, and therefore very close to the five regular solids, if, I say, it places its twelve points on the inner circle of Mars, then the sides of the pentagons which are individually the bases of the rays or points touch the middle circle of Venus.®’ In brief, the cube and octahedron which are spouses do penetrate their planetary spheres somewhat; the dodecahedron and icosahedron which are spouses do not altogether follow theirs, whereas the tetrahedron exactly touches both.
In the first case there is a deficiency, in the second an excess, and in the last an equality in the intervals of the planets.
From that fact it is evident that the actual proportions of the planetary distances from the Sun have not been taken from the regular figures alone; for the Creator, the actual fount of geometry, who, as Plato wrote, practices eternal geometry, does not stray from his own archetype.®® And that could certainly be inferred from the very fact that all the planets change their intervals over definite periods of time, in such a way that each one of them has two distinctive distances from the Sun, its greatest and its least; and comparison of distances from the Sun between pairs of planets is possible in four ways, either of the greatest distances, or of the least, or of the distances on opposite sides when they are furthest from each other, or when they are closest. Thus the comparisons between pair and pair of neighboring planets are twenty in number, whereas on the other hand there are only five solid figures. However, it is fitting that the Creator, if He paid attention to the proportion of the orbits in general, also paid attention to the pro portion between the varying distances of the individual orbits in par ticular, and that that attention should be the same in each case, and that one should be linked with another. On careful consideration, we shall plainly reach the following conclusion, that for establishing both the diameters and the eccentricities of the orbits in conjunction, more basic principles are needed in addition to the five regular solids. Fifth, to come to the motions, between which harmonies are es tablished, I again impress on the reader that it was shown by me in my Commentaries on Mars, from the thoroughly reliable observations f Brahe, that equal daily arcs on one and the same eccentric are not traversed at the same speed
 but that these differing times expended on equal parts of the eccentric observe the proportion of their own distances from the Sun, the fount of motion;^® and in turn, that supposing equal times, say one natural day in each case,
 the true daily arcs of a single eccentric orbit corresponding wi th them have a proportion to each other which is the inverse of the proportion of the two distances from the Sun.
 At the same time, however, it was shown by me that the orbit of a planet is elliptical,
 and the Sun, the fount of motion, is at one of thefocuses of that ellipse',
 and thus it comes about that the planet, when it has completed out of its whole circuit a quadrant from its aphelion, is at precisely its mean distance from the Sun, between its greatest at aphelion and its least at perihelion.
 From these two axioms the conclusion is drawn that the daily mean motion of the planet on its eccentric is the same as the true daily arc of its eccentric, at those moments at which the planet is at the end of the quadrant of its eccentric as reckonedfrom the aphelion, even though that true quadrant still appears smaller than a proper quadrant.
 Furthermore, it follows that any two really true daily arcs of the eccentric, which are truly at equal distances, one from the aphelion and the other from the perihelion, added together are equal to two mean daily arcs’,’^^
 and in consequence, since the proportion of the circles is the same as that of their diameters, that the proportion of one mean daily arc to the sum of all the mean daily arcs making up the whole circuit, which are equal to each other, is the same as that of a mean daily arc to the sum of all the true eccentric arcs, which are the same in number but unequal to each other. We need to have this knowledge of the true eccentric daily arcs, and of the true . motions, beforehand so that we can now grasp through them the apparent motions, supposing the eye to be at the Sun.
Sixth, as far indeed as concerns the apparent arcs as seen from the Sun, it has been known even from the time of ancient astronomy that of the true motions, even those which are equal to each other, one which has moved further away from the center of the world (such as one which is at aphelion) seems to be smaller, to an observer at that center; and one which is nearer, such as one which is at perihelion, also seems to be greater.
Since therefore in addition the true daily arcs are also greater when close, on account of their motion’s being faster, but lesser when at a distance at aphelion on account of the slowness of the motion,

hence I have shown in the Commentaries on Mars that the propor tion of the apparent daily arcs on a given eccentric is fairly precisely the square of the inverse proportion of their distances from the Sun.’^^ Thus if a planet on a particular one of its days, when it is at aphelion, were at a distance of 10 parts from the Sun, in any units, and on the opposite day, when it is at perihelion, at a distance of 9 parts, in similar units, it is certain that at aphelion its apparent forward motion as seen from the Sun will be to its apparent motion at perihelion as 81 to 100.

However, that is true with the following reservations; first, that the eccentric arcs are not large, so that they do not participate in different distances which vary considerably, that is so that they do not produce an appreciable difference in the distance of their ends from the apsides

second, that the eccentricity is not very large, for the larger the eccentricity, that is to say the larger the arc is, the more the angle which it appears to subtend is increased, beyond the bound set by its closeness to the Sun, by Theorem 8 of Euclid’s Optics."^^

However, it is of no importance in small arcs and at a great distance, as I have commented in my Optics, Chapter XI. But there is another reason for me to comment on this point.

For arcs of the eccentric near the mean anomalies are viewed obliquely from the center of the Sun, and this obliquity diminishes the apparent size,

whereas on the contrary arcs near the apsides present themselves normally to an observer, so to speak, placed on the Sun.’^^ Therefore, when the eccentricity is very large, the sensible damage is done to the proportion of the motions, if we apply the mean diurnal motion undiminished to the average distance, as if it appeared from the average distance to be the size which it ac tually is, as will be apparent below in the case of Mercury. All this is related at greater length in the Epitome of Copernican Astronomy, Book V. Nevertheless it had to be recalled here as well, because it concerns the actual terms of the heavenly har monies, considered separately on their own. VII. Rejection of the motions which are apparent to observers on Earth.
Seventh, if anyone should bring to mind the daily motions not as they appear to observers from the Sun, so to speak, but from the Earth, with which Book VI of the Epitome of Copernican Astronomy deals, he should know that no account of them whatever is taken in this proceeding, and definitely none should be. For the Earth is not the fount of their motions, and cannot be, for those motions degenerate not only into mere rest or apparent standstill, but into definite retro gression, as far as the deceptive appearance is concerned. On that basis all the infinity of proportions is attributable to all the planets simultaneously and equally. Therefore, to make certain what propor tions are established as their own by the daily motions of the true eccentric orbits individually (even though they are themselves still apparent, to an observer, so to speak, on the Sun, the fount of motion), this fantasy, common to all five, of an adventitious annual motion must be removed from those proper motions, whether it arises from the motion of the Earth itself, according to Copernicus, or from the annual motion of the whole system, according to Tycho Brahe; and the motions proper to each planet, stripped of inessentials, must be brought into view.
Eighth, up till now we have dealt with the various elapsed times or arcs of one and the same planet. Now we must also deal with the motions of pairs of planets compared with each other. Here note the definitions of the terms which we are going to need. We shall call the nearest apsides of two planets the perihelion of the upper one and the aphelion of the lower one, notwithstanding the fact that they are tending not to the same side of the world, but to different, and per haps opposite sides.
VIII. What is the proportion of the periodic times to the distances from the Sun of any pair of planets?
 By extreme motions understand the slowest and the fastest of the whole planetary circuit.
 By converging or approaching motions, those which are at the nearest apsides of the two, that is at the perihelion of the upper planet and the aphelion of the lower;
 by diverging or receding motions, those which are at opposite ap sides, that is at the aphelion of the upper planet and the peri helion of the lower. Again, therefore, a part of my Secret of the Universe, put in suspense 22 years ago because it was not yet clear, is to be completed here, and brought in at this point. For when the true distances between the spheres were found, through the observations of Brahe, by continuous toil for a very long time, at last, at last, the genuine proportion of the periodic times to the proportion of the spheres — only at long last did she look back at him as he lay motionless. But she looked back and after a long time she cameA^ and if you want the exact moment in time, it was conceived men tally on the 8th March in this year one thousand six hundred and eighteen, but submitted to calculation in an unlucky way, and therefore rejected as false, and finally returning on the 15th of May and adopting a new line of attack, stormed the darkness of my mind. So strong was the support from the combination of my labor of seventeen years on the observations of Brahe and the present study, which conspired together, that at first I believed I was dreaming, and assuming my conclusion among my basic premises. But it is absolutely certain and exact that the proportion between the periodic times of any two planets is precisely the sesquialterate proportion of their mean distances, that is, of the ac tual spheres,’^^ though with this in mind, that the arithmetic mean
between the two diameters of the elliptical orbit is a little less than the longer diameter. Thus if one takes one third of the proportion from the period, for example, of the Earth, which is one year, and the same from the period of Saturn, thirty years, that is, the cube roots, and one doubles that proportion, by squaring the roots, he has in the resulting numbers the exactly correct proportion of the mean dis tances of the Earth and Saturn from the Sun. For the cube root of 1 is 1, and the square of that is 1. Also the cube root of 30 is greater than 3, and therefore the square of that is greater than 9. And Saturn at its average distance from the Sun is a little higher than nine times the average distance of the Earth from the Sun. The use of this theorem will be necessary in Chapter IX for the derivation of the eccentricities. IX. How large a space any planet traverses rela tively to another in any given time. X. How from the true paths, and the true distances of the planets from the Sun, is found the apparent motion as from the Sun, the subject of celestial harmony. XI. How from the apparent diurnal motions (seen, so to speak, from the Sun), are elicited the distances of the planets from the Sun. Ninth, if you want to measure the actual completely true daily paths of each planet through the aethereal air, with, so to speak, a ten foot rule, two proportions will have to be combined, one that of the true (not the apparent) daily arcs of the eccentric, the other that of the average distances of each planet from the Sun, because it is the same as that of the width of the orbits. That is, the true daily arc of each planet must be multiplied by the semidiameter of its orbit. When that has been done, the resulting figures will be convenient for investi gating whether those paths make harmonic proportions. Tenth, to find definitely the apparent size of any such daily path, to an eye placed, so to speak, on the Sun, although the same thing can immediately be sought through astronomy, yet it will also be re vealed if you multiply the proportion of the paths by the inverse pro portion not of the mean, but of the true distances as they are at any point of the eccentrics: by multiplying the path of the upper by the distance of the lower from the Sun, and correspondingly the path of the lower by the distance of the upper from the Sun. Eleventh, in the same way also from given apparent motions, at aphelion for one planet and at perihelion for the second, or the other way round, the proportions of the distances are elicited, of one at aphelion to that of the second at perihelion. In this case, however, the mean motions must be known in advance, that is the inverse proportion of the periodic times, from which the proportion of the orbits is elicited, by Number VIII stated above: then by taking the mean pro portional between one or the other apparent motion and their mean, it turns out that this mean proportional is to the semidiameter, which has already been revealed, of the orbit, as is the mean motion to the separation or dis tance, which is required. Let the periodic times of two planets be 27 and 8 . Then the proportion of the mean daily motion of the former to the latter is as 8 to 27. Hence the semidiameters of the orbits will be as 9 to 4. For the cube root of 27 is 3; that of 8 is 2; and the squares of these roots are 9 and 4. Now let the apparent motion at aphelion of one be 2, and at perihelion of the other 33 and a third. The mean proportionals be tween the mean motions 8 and 27, and these apparent motions, will be 4 and 30. Therefore, if the mean 4 gives an ______________ 331 average distance for the planet of 9, then a mean motion of 8 yields a distance at is. 9 . 4 . 3f aphelion of 18, corresponding with an apparent motion of 2. And if the other mean, 30, gives an average distance for the other planet of 4, then its mean motion of 27 gives its distance at perihelion as 3. Therefore, I say that its distance at aphelion is to its distance at perihelion as 18 to 3. From that it is evident that the harmonies dictated between the extreme motions of the two, and the periodic times prescribed in each case, entail the extreme and average distances, and so also the eccentricities.^’ Twelfth, from the receding extreme motions of one and the same planet it is possible to find the mean motion. For in this case it is not pre cisely the arithmetic mean between the extreme motions, nor precisely the geo metric mean; but it is less than the geometric mean by the same amount as the geometric mean is less than the mean between the two.
Let there be two extreme motions, 8 and 10. The mean motion will be less than 9 and also less than the square root of 80 by half the difference between the two, that is between 9 and the square root of 80. Thus if the mo tion at aphelion is 20 and at perihelion 24, the mean motion will be less than 22, and also less than the square root o f 480 by half the differ ence between that root and 22. The application of this theorem is in what follows.
Thirteenth, from what has already been stated is proved the propo sition, which will be very necessary to us, that as the proportion of the mean motions in the two planets, so is the inverse of the square root of the cube of the proportion of the orbits. Thus the proportion of two apparent converging extreme motions is always less than the sesqui alterate of the proportion‘d^ of the distances corresponding with those extreme motions; and by the same amount as, multiplied together, the two proportions of two corresponding distances to the two mean distances or to the semidiameters of the two orbits come to less than the square root of the proportion of the orbits, the proportion of the two extreme converging motions is greater than the proportion of the corresponding distances; whereas if that product exceeded the square root of the proportion of the orbits, then the proportion of the converging motions would be less than the proportion of their distances?^ Let the proportion of the orbits be DH’.AE, and the proportion of the mean motions H I’EM, the sesquialterate of the inverse of the former. Let the distance of the orbit, that is CG, be at its smallest in the former case, and of the orbit in the latter case, that is BE, at its greatest; and let the product of the proportions DH’.CG and BFA E be in the first instance less than the square root of DG’AE. Also let the apparent motion of the upper planet at perihelion be GK, and of the lower at aphelion EL, so that they are extreme converging mo tions. I say that the proportion GK’.FL is greater than the inverse of the proportion CG’BF, but less than its sesquialterate. For the pro portion of HI to GK is the square of the proportion of CG to DH\ and the proportion of EL to EM is the square of the proportion of AE to BE. Therefore, the two proportions multiplied together, that of HI to GK and of FL to EM, come to the square of the proportions of CG to DH and of A£ to BT multiplied together. But the proportions of CG to DH and of AE to BE multiplied together are less than the square root of the proportion of AE to DH by a definite amount, as in the assumptions. Therefore, the proportions of H I to GK a b c n and of j FL to EM multiplied together are also less than the square of the square root, that is, less than the whole proportion of AE to DH, by a factor which is the square of the pre vious deficiency. But HI to EM is the sesqui 5400 alterate of the proportion of AE to DH, by M L 3456 F VIII previously stated. Then less than the square of the deficiency divided into the sesquialterate of the proportion, or in other words the proportions of HI to GK and of K 2025 G FL to EM divided into the proportion of HI I 1 6 0 0 // to EM leave as quotient more than the square root of the proportion of AE to DH, by the square of the amount in excess. But they yield as quotient the proportion of GK to FL. There fore, the proportion of GK to FL is more than the square root of the proportion of AE to DH by the square of the factor in excess. But the proportion of AE to DH is made up of three proportions, those of AE to BE, of BF to CG, and of CG to DH. Also the proportion of CG to DH together with that of AE to BF is less than the square root of that of AE to DH, by a deficiency of the simple factor. Therefore, the proportion of BF to CG is more than the square root of that of AE to DH, by the simple factor. But the proportion of GK to FL was also more than the square root of that of AE to DH, in fact by the square of the excess factor. However, the square of the excess is greater than the simple factor. Therefore, the proportion of the motions GK to FL is greater than the proportion of the corresponding distances, BF to CG.
It is shown in the same way that in the opposite case, if the planets come close to each other at G and F, beyond the mean separations at H and E, in such a way that the proportion of the mean separations DH, AE loses more than its square root, then the proporrion of the motions GK to FL becomes less than the proportion of their distances, B F to CG. For nothing more needs to be done than to change the words greater to lesser, more to less, excess factor to deficiency, and the other way round.
In the numbers quoted, the square root of 4:9 is 2‘3, and 5:8 is still greater than 2:3 by a factor of 15:16 in excess.
Also the proportion 8:9 squared is the proportion 1600:2025, that is 64:81; and the proportion 4:5 squared is the proportion 3456:5400, that is 16:25; and lastly the sesquialterate of the proportion 4:9 is the proportion 1600:5400, that is 8:27.
Therefore, also the proportion 2025 to 3456, that is 75:128, is still greater than 5:8, that is 75:120, by an excess factor of the same amount (120:128, that is), 15:16. Hence the proportion of the converging motions, 2025:3456, exceeds the inverse proportion of the corre sponding distances, 5:8, by the same factor as the latter exceeds the square root of the proportion of the orbits, 4:9. Or, which comes to the same thing, the proportion of the two converging distances is the mean between the square root of the proportion of the orbits and the inverse proportion of the corresponding motions.
From that, however, we may infer that the proportion of the diverging motions is much greater than the sesquialterate of the proportion of the orbits, since the sesquialterate is multiplied by the square of the proportions of the distance at aphelion to the mean distance, and of the mean distance to that at perihelion.