Spira Solaris Archytas-Mirabilis Part V


In considering the astronomical elements in ancients works it seemed advisable to examine the origins of later scientific advances associated in one way or another with these earlier sources. This was especially so with respect to the contributions made by both Galileo and Kepler in light of their acknowledged use of materials provided in Plato's Timæus. But this was only one facet of a much wider investigation. Another concerned technical details long-buried in the Babylonian astronomical cuneiform texts of the Seleucid Era [310 BC - 75 A.D.] - information that only surfaced during the latter part of the previous century and has yet to see the light of day even now. Irrespective of how much or how little this neglected corpus of technical knowledge is regarded, there still remains the leading question why Babylonian astronomy was so obviously concerned with synodic motion and varying orbital velocity. It was this question that generated sufficient interest in astronomy and mathematics to apply the general synodic relationship in Parts I and II of the present work. As shown in these earlier sections, this was the vital step that provided the necessary understanding to the "Golden Section" and the Phi-Series in the present astronomical context. From this viewpoint it was a logical step to consider next the contents of the Chaldean Oracles, held in such high esteem by Proclus and others. Here one finds simple statements that are not necessarily simplistic at all, especially from our present point of view:1

Where the Paternal Monad is.
The Monad is enlarged, which generates Two.
For the Dyad sits by him, and glitters with Intellectual Sections
And to govern all things, and to Order all things not Ordered.
For in the whole World shineth the Triad, over which the Monad Rules.
This Order is the beginning of all Section.
For the Mind of the Father said, that all things can be cut into three
Governing all things by mind.
And there appeared in it (the Triad) Virtue and Wisdom.
And Multiscient Verity
This way floweth the Shape of the Triad, being præ-existent.
Not the first (essence) but where they are measured.
For thou must conceive that all things serve three principles.
The first course is Sacred, but in the middle.
Another the third, ærial; which cherished the Earth in Fire.
And fountain of Fountains, and of all Fountains.
The Matrix containing all things
Thence abundantly springs forth the Generation of multi various Matter.
Thence extracted a prester the flower of glowing Fire,
Flashing into the cavities of the World: for all things from thence.
Begin to extend downwards their admirable Beams

A personal opinion, perhaps, and also that of a continuing (albeit slightly optimistic) agnostic, yet there nevertheless appears to be more than a little power and wisdom reflected in such poetic utterances, not only in the above, but elsewhere in the Oracles; witness:2

Subject not thy Mind to the vast measures of the Earth;
For the Plant of Truth is not upon the Earth.
Nor measure the Measures of the Sun, gathering together Canons:
He is moved by the Eternal will of the Father, not for thy sake.
Let alone the swift course of the Moon: she runs ever by the impulse of Necessity.
The Progression of the Stars was nor brought forth for thy sake.
The Æthereal broad footed flight of Birds is not voracious.
And the Dissections of Entrails and Victims, all these are Toys.
The supports of gainful Cheats. Fly thou those,
If thou intend to open the sacred Paradise of Piety.
Where Virtue, Wisdom, and Equity are assembled.
Stoop not down; for a precipice lies below the Earth.
Drawing though the Ladder which hath seven steps,
beneath which Is the Throne of Necessity.

Or, concerning TIME itself:3

The Mundane god; Æternal, Infinite.
Young, and Old, of a Spiral Form.
And another fountainous, who guides the Empyreal Heaven.
Again, indefatigable Nature commands the Worlds and Works.
The Heaven drawing an eternal course may run.
And the swift Sun shall come about the Center as he useth.
Look not into the fatal name of this Nature.
Stirring himself up with the Goal of resounding Light.
Another fountainous, which guides the Empyreal World.
The Center from which all (lines) which way soever are equal.
For the paternal Mind sowed Symbols through the World.
For the Center of every one is carried betwixt the Fathers.
For it is in imitation of the Mind, but that which is
born hath something of the Body.

It is understandable that if the Oracles pertained to the matter at hand why they would have been held in such high esteem by later platonists such as Proclus, who is reputed to have stated: "If I had it in my power, out of all the ancient books I would suffer to be current only the Oracles and the Timæus." But Proclus also mentions in his own Commentaries that according to the divine Iamblicus, "the whole theory of Plato is comprehended in the Timæus and the Parmenides."  It is at this juncture that we return to the "One and the Many," which is the central theme that runs confusingly though the latter dialogue. Here, as perhaps in much of the early material there is a mixture of condensed wisdom and obfuscation, but even so, Plato presents the reader with necessary materials elsewhere, e.g., he provides more than educational requirements when he states in the Epinomis (991):4
The first and most important of them is likewise that which treats of pure numbers - not numbers concreted in bodies, but the whole generation of the series of odd and even, and the effects which it contributes to the nature of things. When all this has been mastered, next in order comes what is called by the very ludicrous name mensuration, but is really a manifest assimilation to one another of numbers which are naturally dissimilar, effected with reference to areas. Now to the man who can comprehend this, it will be plain that this is no mere feat of human skill, but a miracle of God's contrivance. Next numbers raised to the third power and thus representing an analogy with three dimensional things. Here again he assimilates the dissimilar by a second science, which those who hit on the discovery have named sterometry [the gauging of solids], a device of God's contrivance which breeds amazement to those who fix their gaze on it and consider how universal nature molds from and type by the constant of revolution of potency and its converse about the double in the various progressions. The first example of this ratio of the double in the advancing numbers series is that of 1:2; double of this is the ratio of their second powers [1:4], and double of this again the advance to the solid and tangible, as we proceed from 1 to 8 [ 1,2, 22, 23 ]; the advance to a mean of the double, that mean which is equidistant from lesser and greater term [the arithmetical], or the other mean [the harmonic] which exceeds the one term and is itself exceeded by the other by the same fraction of the respective terms-these ratios of 3:2 and 4:3 will be found as mean between 6 and 12 - why, in the potency of the mean between these two terms [6, 12]. with its double sense, we have a gift from the blessed choir of the Muses to which mankind owes the boon of the play of consonance and measure, w ith all they contribute to rhythm and melody." [emphasis supplied]
The mathematical details and concepts given above are clearly recognizable as a fractional and condensed version of the much larger discourse on the same subject in the Timæus. The triple interval [1,3,9,27] is omitted here, perhaps for good reason, since the double interval [ 1,2,4,8 ] is itself sufficient to illustrate the harmonic law of planetary motion, e.g., 4 3 = 8 2 = 64 where the frame of reference (1) is provided by the first number and the second (2) is the mean inverse velocity. Perhaps it is a matter of semantics and/or linguistics, but translators (A.E. Taylor and others) may have confused the golden mean with the harmonic mean with respect to the statement "the other mean which exceeds the one term and is itself exceeded by the same fraction of the respective terms" - a statement that is also applicable to any three successive values of the phi-series in alchemical terms, i.e., "As above, so below," etc. Moreover, Plato states in detail how the various means are to be combined in the Timæus (31b-32c):5
Now that which is created is of necessity corporeal, and also visible and tangible. And nothing is visible where there is no fire, or tangible which has no solidity, and nothing is solid without earth. Wherefore also God in the beginning of creation made the body of the universe to consist of fire and earth. But two things cannot be rightly put together without a third; there must be some bond of union between them. And the fairest bond is that which makes the most complete fusion of itself and the things which it combines; and proportion is best adapted to effect such a union. For whenever in any three numbers, whether cube or square, there is a mean, which is to the last term what the first term is to it; and again, when the mean is to the first term as the last term is to the mean,- then the mean becoming first and last, and the first and last both becoming means, they will all of them of necessity come to be the same, and having become the same with one another will all be one. If the universal frame had been created a surface only and having no depth, a single mean would have sufficed to bind together itself and the other terms; but now, as the world must be solid, and solid bodies are always compacted not by one but two, God placed water and air in the mean between fire and earth, and made them to have the same proportion so far as was possible (as fire is to air so is air to water, and as air is to water so is water to earth); and thus he bound and put together a visible and tangible heaven. And for these reasons, and out of such elements which are in number four, the body of the world was created, and it was harmonized by proportion, and therefore has the spirit of friendship; and having been reconciled to itself, it was indissoluble by the hand of any other than the framer.
As for the expressed significance of the ratios of 3:2 and 4:3 in the first passage - these are the exact and fundamental exponents for the Phi-series that generate the mean distances - for the Phi-series itself (synodics and planets) in the first case, and the mean distances of the planets in the second (see below for Kepler's use of the ratio 3:2 in a similar astronomical context).

Returning to the Epinomis, Plato concludes by saying:

To the man who pursues his studies in the proper way, all geometric constructions, all systems of numbers, all duly constituted melodic progressions, the single ordered scheme of all celestial revolutions, should disclose themselves, and disclose themselves they will, if, as I say, a man pursues his studies aright with his mind's eye fixed on their single end. As such a man reflects, he will receive the revelation of a single bond of natural interconnection between all these problems. If such matter are handled in any other spirit, a man, as I am saying, will need to invoke his luck. We may rest assured that without these qualifications the happy will not make their appearance in any society; this is the method, this is the pabulum, these the studies demanded; hard or easy, this is the road we must tread.
It is possible that once understood, "the whole theory of Plato" (better stated: "The Doctrine of the Timæus") might, perhaps, be comprehended in Plato's Timæus and Parmenides, but it need not automatically follow that it could be apprehended from these two or three sources alone. Moreover, even with such restrictions the quest for understanding would still be complicated by the poetic/theological contents of the Oracles, the largely unintelligible nature of the Parmenides, and uncertainties concerning the context of the "doctrine" in the first place. One might, however, take initial cues from the title of the Timaeus itself and the statement by Critias (Timæus 35c):6
Our intention is, that Timaeus, who is the most of an astronomer amongst us, and has made the nature of the universe his special study, should speak first, beginning with the generation of the world and going down to the creation of man.
but this is clearly not the whole issue, for Timæus responds by stating:7
"All men, Socrates, who have any degree of right feeling, at the beginning of every enterprise, whether small or great, always call upon God. And we, too, who are going to discourse of the nature of the universe, how created or how existing without creation, if we be not altogether out of our wits, must invoke the aid of Gods and Goddesses and pray that our words may be acceptable to them and consistent with themselves. Let this, then, be our invocation of the Gods, to which I add an exhortation of myself to speak in such manner as will be most intelligible to you, and will most accord with my own intent. First then, in my judgment, we must make a distinction and ask, What is that which always is and has no becoming; and what is that which is always becoming and never is?
The last part brings us back to the issue of the "one and the many" which circles endlessly through the Parmenides. Then there is also the apparent significance of the Chaldean Oracles in this particular context, while matters are further complicated when Egyptian and Pythagorean knowledge are also factored into the equation, as would appear necessary if the contents of Timæus are to be fully addressed. It is relevant to remark here that nothing of value results from witless dichotomies concerning the supposed pre-eminence of Greek versus Egyptian science, or Greek Science versus Babylonian, etc., especially when it is realized that a common element pervades the central theme of the present discussion. But just what was the "Doctrine of the Timæus," and why was so much intellectual effort expended on it? There seems little doubt that a large percentage of the Dialogues and the work of many others (Aristotle especially) are, in one way or another, intimately related to it. But even so, it is difficult to know how much is pertinent and how much is not, for if, as Plato states (Laws VII, 991), "ciphering and arithmetic make one subject," then there is in all likelihood padding and disinformation to contend with also. Indeed, as Proclus took pains to point out:8
Plato, for the sake of concealment, employed mathematical names, as veils of the truth of things, in the same manner as theologists employed fables, and the Pythagoreans symbols. For it is possible in images to survey paradigms, and through the former to pass to the latter.
However, Plato expands the methodology still further in the Protagoras (316d-317B), e.g., we are informed that to conceal their activities and the resentments it apparently spawned, the Sophists:9
... fearing the odium which it brings, adopted a disguise and worked under cover. Some used poetry as a screen, for instance Homer and Hesiod and Simonides; others religious rites and prophecy, like Orpheus and Musæus and their school; some even ... physical training, ... [while] Music was used as a cover by ... many others [emphases supplied]
Intimations of secrecy followed by open disclosure serve to both alert and direct the perceptive reader, while the expansion itself suggests that multiple redundancy may even have been an intended feature of the scheme. But although the material in question may have been hidden from direct view, it would seem from all this that it was never intended to be subverted or lost. One might add that from time to time, it appears that commentators (e.g., Proclus and Thomas Taylor) have followed an age-old procedure and added further clarifications in similar cryptic manner (for more on Proclus and Neoplatonism see: Proclus Diadochus by M.Alan Kazlev).
   An underlying theme that also runs through all this is the tacit understanding that the nature and complexities of the "The Doctrine of the Timæus" provides its own inherent filter. This increasingly complex matter is undoubtedly bound up in and by Homer's "Golden Chain" (Theaetetus, 153e-d), for as Proclus observes in the Commentaries on the Timæus:10
"the bond which proceeds from soul and intellect is strong, as Orpheus also says, but the union of the golden chain [i.e., of the deific series] is still greater..."
Elsewhere Proclus also provides the following explanation:11
"Let us therefore, if you think fit, discuss the theory of the proposed words physically. The first analogy then according to which nature inserts harmony in her works, and according to which the Demiurgus adorns and arranges the universe, is one certain life, and one reason, proceeding through all things; which first, indeed, connects itself, but afterwards the natures in which it exists; and according to which sympathy is ingenerated in all mundane essences, as existing in one animal, and governed by nature. This life, therefore, which is the bond of the wholes, total nature [or nature which ranks as a whole] and the one soul of the world constitute. The one intellect likewise generates it; and always more excellent beings, insert in mundane natures, a greater and more perfect union. Let it be said, therefore that the habit which predominates in material subjects, that material form, and the powers of the middle elements, are bonds. All these, however, have a relation of things without which the primary bond not  participated, and are analogous to the middle in mathematical entities, though which the habitude subsists in the extremes.  But the life of which we are speaking, which collects and unites all things, and is suspended indeed from its proper causes, but binds the things in which it is inherent, is truly analogy, and preserves both its own union and the union of its participants. Again, therefore, a bond is threefold. For the common powers of the elements are one bond; the one cause of bodies is another; and a third is that which is in the middle of both of the others, which proceeds indeed from the cause of bodies, but employs the powers that are divided about body. And this is the strong bond, as the theologist says, which is extended through all things, and is connected by the golden chain. For Jupiter after this, constitutes the golden chain.1 according to the admonitions of Night. 
But when your pow'r around the whole has spread
A strong coercive bond,
a golden chain
Suspend from the æther.
1 This golden chain may be said to be the series of unities proceeding from the one, or the ineffable principle of things, and extending as far as matter itself. And of this chain, the light immediately proceeding from the sun is an image."
(n.b., The footnote is Thomas Taylor's; the emphases and italics are my own)

Returning to the previous quotation, it would appear that the allusions to Homer and Hesiod are valid enough, as are the references to Orpheus, the Muses and gymnastics, for as Proclus explains somewhat cryptically:12

...With respect to music and gymnastic, ... the former causes the lives in the universe to be harmonious, and the latter renders divine motion rhythmical and elegant, so as to preserve the same form, and the same immutable habit of the divine vehicles. For through these things Plato elsewhere calls divine souls Sirens, and shows that the celestial motions is harmoniously elegant; for gymnastic is indeed in them.
while Timæus states towards the end of the dialogue:13
".. the mathematician or any one else whose thoughts are much absorbed in some intellectual pursuit, must allow his body also to have due exercise, and practise gymnastic; and he who is careful to fashion the body, should in turn impart to the soul its proper motions, and should cultivate music and all philosophy, if he would deserve to be called truly fair and truly good. And the separate parts should be treated in the same manner, in imitation of the pattern of the universe,.."


In terms of importance Proclus lists only the Oracles and the Timæus; Iamblicus for his part confines his selection to the Parmenides and the Timæus, but there seems little doubt that much of Plato's Dialogues and other works have relevance here also - Aristotle's On the Heavens, On Generation and Corruption, to mention just a few, and even Kepler's Harmonies of the World appears to have it own particular role in this matter. It is clear from the last section of this 1618 publication - the work which contained the third, or Harmonic Law of planetary motion - that Kepler made it quite clear how he came to finally write the Harmonies and where his allegiances supposedly lay. The introduction to Book Five is as follows:14
Concerning the very perfect harmony of the celestial movements, and the genesis of eccentricities and semidiameters, and the periodic times from the same.
[268] As regards that which I prophesied two and twenty years ago (especially that the five regular solids are found between the celestial spheres), as regards that of which I was firmly persuaded in my own mind before I has seen Ptolemy's Harmonies, as regards that which I promised my friends in the title of this fifth book before I was sure of the thing itself, that which, sixteen years ago, in a published statement, I insisted must be investigated, for the sake of which I spent the best part of my life in astronomical speculations, visited Tycho Brahe, [269] and took up residence at Prague: finally, as God the Best and Greatest, Who had inspired my mind and aroused my great desire, prolonged my life and strength of mind and furnished the other means through liberality of the two Emperors and the nobles of this province of Austria-on-the Anisana: after I had discharged my astronomical duties as much as sufficed, finally, I say, I brought to light and found it to be truer than I had even hoped, and I discovered among the celestial movements the full nature of harmony, in its due measure, together with all its parts unfolded in Book III-not in that mode wherein I had conceived it in my mind (this is not last in my joy) but in a very different mode which is also very excellent and very perfect. There took place in this intervening time, wherein the very laborious reconstruction of the movements held me in suspense, and extraordinary augmentation of my desire and incentive for the job, a reading of the Harmonies of Ptolemy, which had been sent me in manuscript by John George Herward, Chancellor of Bavaria, a very distinguished man and of a nature to advance philosophy and every type of learning. There, beyond my expectations and with the greatest wonder, I found approximately the whole third book given over to the same consideration of celestial harmony, fifteen hundred years ago. But indeed astronomy was far from being of age as yet; and Ptolemy, in an unfortunate attempt, could make others subject to despair, as being one who, like Scipio in cicero, seemed to have recited a pleasant Pythagorean dream rather than to have aided philosophy. But both the crudeness of the ancient philosophy and this exact agreement in our meditations, down to the last hair, over an interval of fifteen centuries, greatly strengthened me in getting on with the job. For what need is there of many men? The very nature of things, in order to reveal herself to mankind, was at work in the different interpreters of different ages, and the finger of God-to use the Hebrew expression; and here, in the minds of two men, who had wholly given themselves up to the contemplation of nature, there was the same conception as to the configuration of the world, although neither had been the others guide in taking this route. But now since the first light eight months ago, since broad day three months ago, and since the sun of my wonderful speculation has shone fully a very few days ago: nothing holds me back. I am free to give myself up to the sacred madness, I am free to taunt mortals with the frank confession that I am stealing the golden vessels of the Egyptians, in order to build a temple for my God, far from the territory of Egypt. If you pardon me, I shall rejoice; if you are enraged, I shall bear up. The die is cast, and I am writing the book-whether to be read by my contemporaries or by posterity matters not. Let it await its reader for a hundred years, if God Himself has been ready for His contemplator for six thousand years. [emphasis and italics supplied]
Kepler next provides the contents of the Harmonies and then continues with the following:15
Before taking up these questions, it is my wish to impress upon my readers the very exhortation of Timæus, a pagan philosopher, who was going to speak on the same things: it should be learned by Christians with the greatest admiration, and shame too, if they do not imitate him: [English translation of the Greek; emphasis and italics supplied]:
For truly, Socrates, since all who have the least particle of intelligence always invoke God whenever they enter upon any business, whether light or arduous; so too, unless we have clearly strayed away from all sound reason, we who intend to have a discussion concerning the universe must of necessity make our sacred wishes and pray to the Gods and Goddesses with one mind that we may say such things as will please and be acceptable to them in especial and, secondly, to you too.
That the "Platonic Solids"and related matters treated by Kepler owe their origins to Plato and the Timæus is beyond dispute, and in this context Kepler's long-standing obsession with them is both intriguing and interesting, as Gingerich [1969] recounts.16 But is also apparent from the last part of the introduction that Kepler wishes to make it clear what his intentions are, albeit stated somewhat cryptically. But even so, his "frank confession that I am stealing the golden vessels of the Egyptians, in order to build a temple for my God, far from the territory of Egypt" may surprise many, for it is not a confession that is generally associated with Kepler, the historical development of his ideas, or their place in general scheme of things.


With respect to the third law of planetary motion - the relationshipship between square of the mean period of revolution T and the cube of the mean heliocentric distance A , i.e., T 2 = A 3 or: T = A3/2 - Kepler recounts in the Eighth Article of the Harmonies that:17
... a certain part of my Mysterium Cosmographicum, which was suspended twenty-two years ago, because it was not yet clear, is to be completed and herein inserted. For after finding the true intervals of the spheres by the observations of Tycho Brahe and continuous labour and much time, at last, at last, the right ratio of the periodic times to the spheres.
though it was late, look to the unskilled man,
yet look to him, and, after much time, came.
and, if you want the exact time, was conceived mentally on the 8th of March in this year One Thousand Six Hundred and Eighteen but unfeliciously submitted to calculation and rejected as false, and finally, summoned back on the 15th of May, with fresh assault undertaken, outfought the darkness of my mind by the great proof afforded by my labor of seventeen years on Brahe's observations and meditation upon it uniting in one accord, in such a fashion that I first believed I was dreaming and was presupposing the object of my search among the principles. But it is absolutely certain and exact that the ratio which exists between the periodic times of any two planets is precisely the ratio of the 3/2th power of the mean distances, i.e., of the spheres themselves; provided, however, that the arithmetic mean between both diameters of the elliptical orbit be slightly less than the longer diameter. And so if anyone take the period, say, of Earth, which is one year, and the period of Saturn, which is thirty years, and extract the cube roots of this ratio and then square the ensuing ratio by squaring the cube roots, he will have as his numerical products the most just ratio of the distances of the Earth and Saturn from the Sun. For the cube root of 1 is 1, and the square of it is 1; and the cube root of 30 is greater than 3, and therefore the square of it is greater than 9. And Saturn, at its mean distance from the sun, is slightly higher [280] than nine times the mean distance of the Earth from the sun.
The linking of dreams with scientific revelation is interesting in its own right since it is not entirely uncommon, but on a more technical note, later in the Ninth Article Kepler next gives examples to support the proposition that: "where the mean movements must be known first, viz., the inverse ratio of the periodic times, wherefrom the ratio of the spheres elicited by Article VIII above: then if the mean proportional between the apparent movement of either one of its mean movement be taken, this mean proportional is to the semi-diameter of its sphere (which is already known) as the mean movement is to the distance or interval sought."37 The complete description need not be given here; it is sufficient for our present purposes to simply note that it commences with: "Let the periodic times of two planets be 8 and 27." 18 These are again key numbers from Plato's Timæus, i.e., they are readily recognized as the end values of Plato's double and triple intervals: 1,2,4,8, and 1,3,9,27 respectively. In our own context they represent the frame of reference (unity) followed by the inverse velocities, the mean distances and the mean planetary periods in both instances. Lastly, although it is obvious enough, it is relevant to note that the 4:8 and 9:27 pairings also represent the first and second integer expressions of the harmonic law and Kepler has already mentioned 9 in the specific context of the distance of Saturn from the Sun.

This latter topic may be compared with Galileo's references to Plato (the Timæus again) in a number of asides and comments in The Dialogues Concerning Two New Sciences [1638]. For example, following the dialectic method, Salviati ostensively explains unity and number in terms of what are again essentially the Double [ 1,2,4,8 ] and the Triple [ 1,3,9,27 ] interval:19

SALVIATI. ... unity is at once a square, a cube, a square of a square and all other powers [dignita]; nor is there any essential peculiarity in squares or cubes which does not belong to unity; as, for example, the property of two square numbers that they have between them a mean proportional; take any square number you please as the first term and unity for the other, then you will always find a number which is a mean proportional. Consider two square numbers, 9 and 4; then 3 is the mean proportional between 9 and 1; while 2 is a mean proportional between 4 and 1; between 9 and 4 we have 6 as a mean proportional. A property of cubes is that they must have between them the two mean proportional numbers' take 8 and 27; between them lie 12 and 18; while between 1 and 8 we have 2 and 4 intervening; and between 1 and 27 lie 3 and 9. (First Day, [83], pp.37-38)
Here again, the matter is not quite as straightforward as one might suppose, for Galileo also makes further cryptic allusions to Plato with respect to parabolas and planetary motion.

A further point of interest arises from Kepler's use of the term kinship, not only with respect to his major concerns with the Platonic Solids, but also, it would seem, with respect to both the Fibonacci Series and the Golden (or Divine) Ratio, i.e., he notes in Part 2. On the Kinship between the Harmonic Ratios and the Five Regular Solids: 20

The second degree of kinship, which is genetic, is to be conceived as follows: First, some harmonic ratios of numbers are akin to one wedding or family, namely perfect ratios to the single family of the cube; conversely, there is the ratio which is never fully expressed in numbers and cannot be demonstrated by numbers in any other way, except by a long series of number gradually approaching it; this ratio is called divine, when it is perfect, and rules in various ways throughout the dodecahedral wedding. Accordingly, the following consonances begin to shadow forth that ratio: 1 : 2 and 2 : 3 and 2 : 3 and 5 : 8. For it exists most imperfectly in 1 : 2, more perfectly in 5 : 8, and still more perfectly if we add 5 and 8 to make 13 and take 8 as the numerator, if this ratio has not stopped being harmonic.[emphases supplied]
Lastly, in addition to his obvious familiarity with the Timæus and related works there are also hints of a revived spiritual motivation in Kepler's Harmonies. Partly motivated, perhaps, by a need to avoid the kind of censure meted out to Galileo, and yet still wishing to acknowledge his sources, Kepler provides an odd mixture of contemporary religious enthusiasm (real or otherwise) and homage to both Plato and Proclus in his closing epilogue which is worth giving here in its entirety:21
10. Epilogue Concerning the Sun by Way of Conjecture.
From the celestial music to the hearer, from the Muses to Apollo the leader of the Dance, from the six planets revolving and making consonances to the Sun at the centre of all the circuits, immovable in place but rotating into itself. For although the harmony is absolute between the extreme planetary movements, not with respect to the true speeds through the ether but with respect to the angles which are formed by joining with the centre of the sun the termini of the diurnal arcs of the planetary orbits; while harmony does not adorn the termini, i.e., the single movements, in so far as they are considered in themselves but only in so far as by being taken together and compared with one another, they become the object of some mind; and although no object is ordained in vain, without the existence of some thing which may be moved by it, while those angles seem to presuppose some action similar to our eyesight or at least to that sense perception whereby, in Book VI, the sublunary nature perceived the angles of rays formed by the planets on the Earth: still it is not easy for the dwellers on the Earth to conjecture what sort of sight is present in the sun, what eyes there are, or what other instinct there is for perceiving those angles [p.1081] even without eyes and for evaluating the harmonies of the movements entering into the antechamber of the mind by whatever doorway, and finally, what mind there is in the sun. None the less, however those things may be, this composition of the six primary spheres around the sun, cherishing it with their perpetual revolutions and as it were adoring it (just as, separately, four moons accompany the globe of Jupiter, two Saturn, but a single moon by its circuit encompasses, cherishes, fosters the Earth and us its inhabitants, and ministers to us) and this special business of harmonies, which is the most clear footprint of the highest providence over solar affairs, now being added to that consideration, [324] wrings from me the following confession: not only does light go out from the sun into the whole world, as from the focus or eye of the world, as life and heat from the heart, as every movement from the King and mover, but conversely by royal law these returns, so to speak, of every lovely harmony are collected by the sun from every province in the world, nay, the forms of movement by twos flow together and are bound in into one harmony by the work of some mind, and are as it were coined from silver and gold bullion; finally, the curia, palace, and the prætorium or throne room of the whole realm of nature are in the sun, whatsoever chancellors, palatines, prefects the Creator has given to nature: for them, whether created immediately from the beginning or to be transported hither at some time, has He made ready those seats. For even this terrestial adornment, with respect to its principal part, for quite a long time lacked the comtemplators and enjoyers, for whom however it had been appointed; and those seats were empty. Accordingly the reflection struck my mind, what did the ancient Pythagoreans in Aristotle mean, who used to call the centre of the world (which they referred to as the "fire" but understood by that the sun) "the watchtower of Jupiter," what, likewise, was the ancient interpreter pondering in his mind when he rendered the verse of the Psalm as: "He has placed His tabernacle in the sun."
But also I have recently fallen upon the hymn of Proclus the Platonic philosopher (of whom there has been much mention in the preceding books), which was composed to the Sun and filled full with venerable mysteries; [emphasis supplied] if you excise that one (here me) from it; although the ancient interpreter already cited has explained this to some extent, viz., in invoking the sun, he understands Him Who placed His tabernacle in the sun. For Proclus lived at a time in which it was a crime, for which the rulers of the world and the peoples itself inflicted all punishments, to profess Jesus of Nazareth, God Our Saviour, and to condemn the gods of pagan poets (under Constantine, Maxentius, and Julian the Apostate). Accordingly, Proclus, who from his Platonic philosophy indeed, by the natural light of the mind, had caught a distance glimpse of Son of God, that true light which lightest every man coming into this world, and who already knew that divinity must never be sought with a superstitious mode in sensible things, nevertheless preferred to seem to look to God in the sun rather than in Christ a sensible man, in order that at the same time he might both deceive the pagans by honouring verbally the Tans of the poets and devote himself to his philosophy, by drawing away both the pagans and the Christians from sensible things, the pagans from the visible sun, the Christians from the Son of Mary, because, trusting too much to the natural light of reason, he spit out the mystery of the Incarnation; and finally that at the same time he might take over from them and adopt into his own philosophy whatever the Christians [p. 1082] had which most divine and especially consonant with Platonic philosophy. And so the accusation of the teaching of the Gospel concerning Christ is laid against this hymn of Proclus, in its own matters: let that Titan keep as his private possession [Golden Reins] and [a treasury of light, a seat at the midpart of the ether, a radiant circle at the heart of the world], which visible aspect Copernicus too bestows upon him; let him even keep his [cyclical chariot-drivings], although according to the ancient Pythagoreans he does not possess them but in their place [the centre, the watchtower of Zeus] - which doctrine, misshapen by the forgetfulness of ages, as by a flood, was not recognized by their follower Proclus; let him keep his [offspring born] of himself, and whatever else is of nature; in turn, let the philosophy of Proclus yield to Christian doctrines, [325] let the sensible sun yield to the Son of Mary, the Son of God, Whom Proclus addresses under the name of the Titan, [O lord, who dost hold the key of the life-supporting spring], and that [thou didst fulfill all things with thy mind-awakening foresight], and that immense power over the [fates],and things which were read of in no philosophy before the promulgation of the Gospel2, the demons dreading him as their threatening scourge, the demons lying in ambush for souls, [in order that they might escape the notice of the light-filled hall of the lofty father]; and who except the Word of the Father is that [image of the all-begetting father, upon whose manifestation from an ineffable mother the sin of the elements changing into one another ceased], according to the following: The Earth was unwrought and a chaotic mass, and darkness was upon the face of the abyss, and God divided the light from the darkness, the waters from the waters, the seas from the dry land; and: all things were made by the very Word. Who, except Jesus of Nazareth and the Son of God, [the shepherd of souls], to whom [the prayer of a tearful suppliant] is to be offered, in order that He cleanse us from sins and wash us of the filth [of generation] - as if Proclus acknowledged the forms of original sin - and guard us from punishment and evil, [by making mild the quick eye of justice], namely, the wrath of the Father? And the other things we read of, which are as it were taken from the hymn of Zacharias (or, accordingly, was that hymn a part of the Metroace?) [dispersing the poisonous, man-destroying mist], viz., in order that He may give souls living in darkness and the shadows of death the [holy light] and [unshaken happiness from (p. 1083) lovely piety]; for that is to serve God in holiness and justice all our days.
Accordingly, let us separate out these and similar things and restore them to the doctrine of the Catholic Church to which they belong. But let us see what the principal reason is why there has been mention made of the hymn. For this same sun which [sluices the rich flow of harmony from on high] - so too Orpheus [making move the harmonious course of the world] - the same, concerning those stock Phoebus about to rise [signs marvellous things on his lyre and lulls to sleep the heavy-sounding surge of generation] and in whose dance Paean is the partness [striking the wide sweep of innocent harmony] - him, I say, does Proclus at once salute in the first verse of the hymn as [king of intellectual fire]. By that commencement, at the same time, he indicates what the Pythagoreans understood by the word of fire (so that it is surprising that the pupil should disagree with the masters in the position of the centre) and at the same time he transfers his whole hymn from the body of the sun and its quality and light, which are sensibles, to the intelligibles, and he has assigned to that [intellectual fire] of his - perhaps the artisan fire of the Stoics - to that created God of Plato, that chief or self-ruling mind, a royal throne in the solar body, confounding into one the creature and Him through Whom all things have been created. But we Christians, who have been taught to make better distinctions, know that this eternal and uncreated "Word," Which was "with God" and Which is contained by no abode, although He is within all things, excluded but none, although he is outside of all things, took up into unity of person flesh out of the womb of the most glorious Virgin Mary, and, when the ministry of his flesh was finished, occupied as His royal abode the heavens, wherein by a certain excellence over and above the other parts of the world, viz., through His glory and majesty, His celestial Father too is recognized to dwell, and has also promised to His faithful, mansion in that house of His Father: as for the remainder concerning that abode, we believe it superfluous to inquire into it too curiously or to forbid the senses or natural reasons to investigate that which the eye has not seen nor the ear heard and into which the heart of man has not ascended; but we duly subordinate the created mind - of whatsoever excellence it may be - to its creator, and we introduce neither God-intelligences with Aristotle and the pagan philosophers nor armies of innumerable planetary spirits with the Magi, nor do we propose that they are either to be adored or summoned to intercourse with us by theurgic superstitions, for we have a careful fear of that; but we freely inquire by natural reasons what sort of thing each mind is, especially if in the heart of the world [326] there is any mind bound rather closely to the nature of things and performing the function of the soul of the world - or if also some intelligent creatures, of a nature different from human, perchance do inhabit or will inhabit the globe thus animated (see my book on the New Star, chapter 24, "On the Soul of the World and Some of Its Functions"). But if it is permissible, using the thread of analogy as a guide, to traverse the labyrinths of the mysteries of nature, not ineptly, I think, will someone have argued as follows: The relation of the six spheres to their common centre, thereby the centre of the whole world, is also the same as that of [discussive intellection] to [intuitive intellection], according as these faculties (p.1084) are distinguished by Aristotle, Plato, Proclus, and the rest; and the relation of the single planets' revolutions in place around the sun to the [unvarying] rotation of the sun in the central place of the whole system (concerning which the sun-spots are evidence; this has been demonstrated in the Commentaries on the Movement of Mars) is the same as the relation of , that of the manifold discourses of ratiocination to the most simple intellection of the mind. For as the sun rotating into itself moves all the planets by means of the form emitted from itself, so too - as the philosophers teach - mind, by understanding itself and in itself all things, sirs up racinations, and by dispersing and unrolling its simplicity into them, makes everything to be understood. And the movements of the planets around the sun at their centre and the discourses of ratiocination are so interwoven and bound together that, unless the Earth, our domicile, measured out the annual circle, midway between the other spheres - changing from place to place, from station to station - never would human ratiocination have worked its way to the true intervals of the planets and to the other things dependent from them, never would it have constituted astronomy. (See the Optical Part of Astronomy,Chapter 9.)
On the other hand, in a beautiful correspondence, simplicity of intellection follows upon the stillness of the sun at the centre of the world, in that hitherto we have always worked under the assumption that those solar harmonies of the movements are defined neither by the diversity of regions nor by the amplitude of the expanses of the world. As a matter of fact, if any mind observes from the sun those harmonies, that mind is without assistance afforded by the movement and diverse stations of the abode, by means of which it may string together ratiocinations and discourse necessary for measuring out the planetary intervals. Accordingly, it compares the diurnal movements of each planet, not as they are in their own orbits but as they pass through the angles at the centre of the sun. And so if it has knowledge of the magnitude of the spheres, this knowledge must be present in it a priori, without any toil of ratiocination: but to what extent that is true of human minds and the sublunary nature has been made clear above, from Plato to Proclus. [p.1084]
Under these circumstances, it will not have been surprising if anyone who has been thoroughly warmed by taking a fairly liberal draft from that bowl of Pythagoras which Proclus gives to drink from in the very first verse of the hymn, and who has been made drowsy by the sweet harmony of the dance of the planets begins to dream (by telling a story he may imitate Plato's Atlantis and, by dreaming, Cicero's Scipio): throughout the remaining globes, which follow after from place to place, there have been disseminated discursive or ratiocinative faculties, whereof that one ought assuredly to be judged the most excellent and absolute which is in the middle position among those globes, viz., in man's earth, while there dwells in the sun simple intellect, the source, whatsoever it may be, of every harmony.
For if it was Tycho Brahe's opinion concerning that bare wilderness of globes that it does not exist fruitlessly in the world but is filled with inhabitants: with how much greater probability shall we conjecture as to God's works and designs even for the other globes, from that variety which we discern in this globe of the Earth. For He Who created the species which should inhabit the waters, beneath which however there is no room for the air [327] which living things draw in; Who sent birds supported on wings into the wilderness of the air; Who gave white bears and white wolves to the snowy regions of the North, and a food for the bears the whale, and for the wolves, birds' eggs; Who gave lions to the deserts of Libya and camels to the wide-spread plains of Syria; and to the lions the endurance of hunger, and to the camels an endurance of thirst: did He use up every art in the globe of the Earth so that He was unable, every goodness so that he did not wish, to adorn the other globes too with their fitting creatures, as either the long or short revolutions, or the nearness or removal of the sun, or the variety of the figures wherewith any region is supported persuaded?
Behold, as the generations of animals in this terrestrial globe have an image of the male in the dodecahedron, of the female in the icosahedron - whereof the dodecahedron rests on the terrestrial sphere from the outside and the icosahedron from the inside: what will we suppose the remaining globes to have, from the remaining figures? For whose good do those four moons encircle Jupiter, two Saturn, as our moon this our domicile? but in the same way we shall ratiocinate concerning the globe of the sun also, and we shall as it were incorporate conjectures drawn from the harmonies, et cetera - which are weighty of themselves - with other conjectures which are more on the side of the bodily, more suited for the apprehension of the vulgar. Is that globe empty and the others full, if everything else is in due correspondence? If as the Earth breathes forth clouds, so the sun black smoke? If as the Earth is moistened and grows under showers, so the sun shines with those combusted spots, while clear flamelets sparkle in its all fiery body. For whose use is all this equipment, if the globe is empty? Indeed, do not the senses themselves cry out that fiery bodies dwell here which are receptive of simple intellects, and that truly the sun is, if not the king, at least the queen [of intellectual fire]?
Purposely I break off the dream and the very last speculation, merely crying out with the royal Psalmist:
Great is our Lord and great His virtue and of his wisdom there is no number: praise Him, ye heavens, praise Him, ye sun, moon, and planets, use every sense for perceiving, every tongue for declaring your Creator. Praise Him, ye celestial Harmonies, praise Him, ye judges of the Harmonies uncovered (and you before all, old happy Mastlin, for you used to animate these cares with words of hope): and thou my soul, praise the Lord thy Creator, as long as I shall be: for out of Him and through Him and in Him are all things, both the sensible and the intelligible]; for both whose whereof we are utterly ignorant and those which we know are the least part of them; because there is still more beyond. To him be praise, honour, and glory, world without end. Amen.

The sections enclosed in brackets represent English translations of the numerous Greek texts and quotations carefully included in the Epilogue by Kepler.
But either way, in this respect it cannot be said that Kepler has not honoured his sources, or disguised his inspiration, the latter religious part notwithstanding.

It has been necessary to take this somewhat lengthy side trip to emphasize the obvious influence of the works of Plato and the Platonists on Kepler himself. That he should have been so absorbed with the Five Platonic Solids, should perhaps, be measured against the results that he ultimately obtained. Did Kepler actually discover the Harmonic Law of planetary motion? Did he really claim that he did? He lived in dangerous times and did what was necessary and expedient to survive, to continue with his work, and to ultimately pass on what he had come to understand. Perhaps it is safe to say that he was the first of the moderns to quantify and confirm the law, and that in view of his stated sources, that this was perhaps closer to his own assessment.


  1. The Chaldean Oracles as Set Down By Julianus,{Latin: Francesco Patrizzi; English: Thomas Stanley} Heptangle Books, Gillette, New Jersey, 1939:3.
  2. Op.cit., p.21.
  3. Op.cit., p.17.
  4. Epinomis (990) Translated by A.E. Taylor, The Collected Dialogues of Plato, Eds. Edith Hamilton and Huntington Cairns, Princeton University Press, Princeton, New Jersey, 1982:1531.
  5. Epinomis (991), op. cit., p.1532.
  6. Timæus (27b) Translated by Benjamin Jowett, The Collected Dialogues of Plato, Eds. Edith Hamilton and Huntington Cairns, Princeton University Press, Princeton, New Jersey, 1982: 1161.
  7. Timæus (27c), op. cit., p. 1161.
  8. Taylor, Thomas, T. The Commentaries of Plato on the Timæus of Plato, in Five Books (Two Volumes), Kessinger Publishing, Kila, Montana. p.117.
  9. Protagoras (316b-317d) Translated by W.K.C. Guthrie, The Collected Dialogues of Plato, Eds. Edith Hamilton and Huntington Cairns, Princeton University Press, Princeton, New Jersey, 1982:315.
  10. Taylor, Thomas, T. The Commentaries of Plato on the Timæus of Plato, Kessinger Publishing, Kila, Montana. Book III, p.406.
  11. Op.cit., p. 406.
  12. Taylor, Thomas, T. PLATO: The Timæus and The Critias, [replete with copious notes by Thomas Taylor on the Commentary of Proclus], Bollingen Series, Pantheon Books, Washington, D.C. 1944:118.
  13. Op. cit., pp.124-125.
  14. Kepler, Johannes The Harmonies of the World, Great Books of the Modern World 16, Robert Maynard Hutchins, Editor in Chief. William Benton, Publisher, Chicago, 1952:1009-1011.
  15. Op.cit., p. 1011.
  16. Gingerich, O. "Kepler and the Resonant Structure of the Solar System," ICARUS, Vol. 11,1969:111-113.
  17. Kepler, Johannes, op.cit., p.1020.
  18. Op.cit., p.1021.
  19. Galileo, Galilei. First Day, [83], The Dialogues Concerning Two New Sciences [1638].  First Day, [83]
  20. Kepler, Johannes, op.cit., p.1013.
  21. Kepler, Johannes, op.cit., pp.1080-1085.

Copyright © 1997. John N. Harris, M.A. (Cmns). Last Updated of February 22, 2004.