Man or Matter Part 34

It is an established conviction of the mathematical scientist that, once an observed regularity in nature has been expressed as a mathematical equation, this equation may be transformed in any mathematically valid way, and the resulting formula will still apply to some existing fact in the world. On innumerable occasions this principle has been used in the expectation of providing further insight into the secrets of nature. We came across a typical instance of this in discussing the basic theorem of kinematics and dynamics (Chapter VIII). Another example is Newton's treatment of Kepler's third law, or - more precisely - the way in which Newton's law of gravitation has been held to confirm Kepler's observations, and vice versa,

It will be our task to a.n.a.lyse the Kepler-Newton case on the very lines of our treatment of the two parallelogram theorems. This a.n.a.lysis will give us insight into a truth which we have to regard as one of the basic maxims of the new science. It says that whether a given formula, derived mathematically from one that was first read from nature, still expresses some fact of nature, cannot be decided by pure mathematical logic, but only by testing it against truly observable phenomena.

Through Kepler's third law a certain relation is expressed between the spatial dimensions of the different planetary spheres and the time needed by the relevant planet to circle once round the circ.u.mference of its own sphere. It says: 'The squares of the periodic times of the planets are always in the same proportion as the cubes of their mean distances from the sun.' In mathematical symbols this reads: t12 / t22 = r13 / r23 We shall see later how Kepler arrived at this law. The point is that there is nothing in it which is not accessible to pure observation.

Spatial distances and lengths of time are measured and the results compared. Nothing, for instance, is said about the dynamic cause of the movements. The a.s.sertion is restricted - and this is true also of the first and second law - to a purely kinematic content, and so precisely to what the earthly onlooker can apprehend. Now it is said that Kepler's third law is a necessary consequence of Newton's law of gravitation, and that - since it is based on pure observation - it therefore establishes the truth of Newton's conception. In this a.s.sertion we encounter a misconception exactly like the one in the statement that the theorem of the parallelogram of forces follows by logical necessity from the theorem of the parallelogram of velocities.

For:

(a) The law of gravitation itself derives from Newton's formula for the centripetal force acting at a point which moves along a circle, this formula being itself the result of an amplification of the formula for centripetal acceleration by the factor 'ma.s.s' (as if the latter were a pure number):

Centripetal acceleration: a = 4(?^2)r / t2

Centripetal force: P = am = 4(?^2)mr / t2

(b) The formula for centripetal acceleration - and the concept of such acceleration itself - is the result of splitting circular movement into two rectilinear movements, one in the direction of the tangent, the other in the direction of the radius, and of regarding it - by a mode of reasoning typical of spectator-thinking - as composed of the two.

This procedure, however, useful as it may be for the purpose of calculation, is contrary to observation. For, as we have pointed out earlier, observation tells us that all original movement - and what can be more original than the movements of the planetary bodies - is curvilinear. No insight into the dynamic reality of cosmic movement, therefore, can ever be gained by handling it mathematically in this way.

(c) The transformation of Kepler's formula which is necessary in order to give it a form representing the nucleus of Newton's formula, is one which, though mathematically justified, deprives Kepler's formula of any significance as expression of an observed fact. The following a.n.a.lysis will show this.

Kepler's formula- r1^3 / r2^3 = t1^2 / t2^2 may be written also r1^3 / t1^2 = r2^3 / t2^2 and this again in the generalized form: r3 / t2 = c.

Obviously, by each of these steps we diminish the reality-value of the formula. In its original form, we find spatial extension compared with spatial extension, and temporal extension with temporal extension. Each of the two comparisons is a fully concrete one, because we compare ent.i.ties of like nature, and only then test the ratios of the two - that is, two pure numbers against each other - to find that they are identical. To compare a spatial and a temporal magnitude, as is done by the formula in its second form, requires already a certain degree of abstraction. Still, it is all spectator's work, and for the spectator time is conceivable and measurable only as a rate of spatial displacement. Hence the constant number c, by representing the ratio between the spatial extension of the realm inside a planet's...o...b..t and the time needed by it to perform one round on this...o...b..t - a ratio which is the same for all planets - represents a definite structural element of our cosmic system.

By this last operation our equation has now achieved a form which requires only one more transformation to bring it into line with Newton's formula. Instead of writing: r3 / t2 = c we write: r / t2 = c (1 / r2) All that now remains to be done amounts to an amplification of this equation by the factor 4(?^2)m, and a gathering of the constant product 4(?^2)c under a new symbol, for which we choose the letter f. In this way we arrive at: 4(?^2)mr / t2 = 4(?^2)cm / r2 and finally: P = ... = fm / r2 which is the expression of the gravitational pull believed to be exerted by the sun on the various planetary bodies. Nothing can be said against this procedure from the point of view

of mathematical logic. For the latter the equation r / t2 = c (1 / r2) is still an expression of Kepler's observation. Not so for a logic which tries to keep in touch with concrete reality. For what meaning, relevant to the phenomenal universe as it manifests in s.p.a.ce and time to physical perception, is there in stating - as the equation in this form does - that: the ratio between a planet's distance from the sun and the square of its period is always proportional to the reciprocal value of the area lying inside its...o...b..t?

Once we have rid ourselves of the false conception that Kepler's law implies Newton's interpretation of the physical universe as a dynamic ent.i.ty ruled by gravity, and gravity alone, we are free to ask what this law can tell us about the nature of the universe if in examining it we try to remain true to Kepler's own approach.

To behave in a Keplerian (and thus in a Goethean) fas.h.i.+on regarding a mathematical formula which expresses an observed fact of nature, does not mean that to submit such a formula to algebraic transformation is altogether impermissible. All we have to make sure of is that the transformation is required by the observed facts themselves: for instance, by the need for an even clearer manifestation of their ideal content. Such is indeed the case with the equation which embodies Kepler's third law. We said that in its original form this equation contains a concrete statement because it expresses comparisons between spatial extensions, on the one hand, and between temporal extensions, on the other. Now, in the form in which the spatial magnitudes occur, they express something which is directly conceivable. The third power of a spatial distance (r^3) represents the measure of a volume in three-dimensional s.p.a.ce. The same cannot be said of the temporal magnitudes on the other side of the equation (t^2). For our conception of time forbids us to connect any concrete idea with 'squared time'. We are therefore called upon to find out what form we can give this side of the equation so as to express the time-factor in a manner which is in accord with our conception of time, that is, in linear form.13 This form readily suggests itself if we consider that we have here to do with a ratio of squares. For such a ratio may be resolved into a ratio of two simple ratios.

In this way the equation - r1^3 / r2^3 = t1^2 / t2^2 a.s.sumes the form- r1^3 / r2^3 = (t1 / t2) / (t2 / t1) The right-hand side of the equation is now const.i.tuted by the double ratio of the linear values of the periods of two planets, and this is something with which we can connect a quite concrete idea.

To see this, let us choose the periods of two definite planets - say, Earth and Jupiter. For these the equation a.s.sumes the following form ('J' and 'E' indicating 'Jupiter' and 'Earth' respectively): rJ^3 / rE^3 = (tJ / tE) / (tE / tJ) Let us now see what meaning we can attach to the two expressions tJ / tE and tE / tJ.

During one rotation of Jupiter round the sun the earth circles 12 times round it. This we are wont to express by saying that Jupiter needs 12 earth-years for one rotation; in symbols: tJ / tE = 12 / 1 To find the a.n.a.logous expression for the reciprocal ratio: tE / tJ = 1 / 12 we must obviously form the concept 'Jupiter-year', which covers one rotation of Jupiter, just as the concept 'earth-year' covers one rotation of the earth (always round the sun). Measured in this time-scale, the earth needs for one of her rotations 1 / 12 of a Jupiter-year.

With the help of these concepts we are now able to express the double ratio of the planetary periods in the following simplified way. If we suppose the measuring of the two planetary periods to be carried out not by the same time-scale, but each by the time-scale of the other, the formula becomes: rJ3 / rE3 = (tJ / tE) / (tE / tJ) = period of Jupiter measured in Earth-years / period of Earth measured in Jupiter-years.

Interpreted in this manner, Kepler's third law discloses an intimate interrelatedness of each planet to all the others as co-members of the same cosmic whole. For the equation now tells us that the solar times of the various planets are regulated in such a way that for any two of them the ratio of these times, measured in their mutual time-units, is the same as the ratio of the s.p.a.ces swept out by their (solar) orbits.

Further, by having the various times of its members thus tuned to one another, our cosmic system shows itself to be ordered on a principle which is essentially musical. To see this, we need only recall that the musical value of a given tone is determined by its relation to other tones, whether they sound together in a chord, or in succession as melody. A 'C' alone is musically undefined. It receives its character from its interval-relation to some other tone, say, 'G', together with which it forms a Fifth. As the lower tone of this interval, 'C' bears a definite character; and so does 'G' as the upper tone.

Now we know that each interval represents a definite ratio between the periodicities of its two tones. In the case of the Fifth the ratio is 2:3 (in the natural scale). This means that the lower tone receives its character from being related to the upper tone by the ratio 2:3.

Similarly, the upper tone receives its character from the ratio 3:2.

The specific character of an interval arising out of the merging of its two tones, therefore, is determined by the ratio of their ratios. In the case of the Fifth this is 4:9. It is this ratio, therefore, which underlies our experience of a Fifth.

The cosmic factor corresponding to the periodicity of the single tone in music is the orbital period of the single planet. To the musical interval formed by two tones corresponds the double ratio of the periods of any two planets. Regarded thus, Kepler's law can be expressed as follows: The spatial ordering of our planetary system is determined by the interval-relation in which the different planets stand to each other.

By thus unlocking the ideal content hidden in Kepler's third law, we are at the same time enabled to do justice to the way in which he himself announced his discovery. In textbooks and encyclopaedias it is usually said that the discovery of the third law was the surprising result of Kepler's fantastic attempt to prove by external observation what was once taught in the school of Pythagoras, namely, that (in Wordsworth's language):

'By one pervading spirit Of tones and numbers all things are controlled.'

Actually, Kepler's great work, Harmonices Mundi, in the last part of which he announces his third law, is entirely devoted to proving the truth of the Pythagorean doctrine that the universe is ordered according to the laws of music. This doctrine sprang from the gift of spiritual hearing still possessed by Pythagoras, by which he could perceive the harmonies of the spheres. It was the aim of his school to keep this faculty alive as long as possible, and with its aid to establish a communicable world-conception. The Pythagorean teaching became the foundation of all later cosmological thinking, right up to the age which was destined to bring to birth the spectator-relations.h.i.+p of man's consciousness with the world. Thus it was left to Copernicus to give mankind the first truly non-Pythagorean picture of the universe.

When Kepler declared himself in favour of the heliocentric aspect, as indicated by Copernicus, he acknowledged that the universe had grown dumb for man's inner ear. Yet, besides his strong impulse to meet the true needs of his time, there were inner voices telling him of secrets that were hidden behind the veil woven by man's physical perceptions.

One of these secrets was the musical order of the world. Such knowledge, however, could not induce him to turn to older world-conceptions in his search for truth. He had no need of them, because there was yet another voice in him which told him that the spiritual order of the world must somehow manifest itself in the body of the world as it lay open to physical perception. Just as a musical instrument, if it is to be a perfect means of bringing forth music, must bear in its build the very laws of music, so must the body of the universe, as the instrument on which the harmonies of the spheres play their spiritual music, bear in its proportions a reflexion of these harmonies. Kepler was sure that investigation of the world's body, provided it was carried out by means of pure observation, must needs lead to a re-establishment of the ancient truth in a form appropriate to the modern mind. Thus Kepler, guided by an ancient spiritual conception of the world, could devote himself to confirming its truth by the most up-to-date methods of research. That his search was not in vain, our examination of the third law has shown.

One thing, however, remains surprising - that Kepler announced his discovery in the form in which it has henceforth engraved itself in the modern mind, while refraining from that a.n.a.lysis of it which we have applied to it here. Yet, in this respect also Kepler proves to have remained true to himself. There is, on the one hand, the form in which Kepler p.r.o.nounced his discovery; there is, on the other, the context in which he made this p.r.o.nouncement. We have already pointed out that the third law forms part of Kepler's comprehensive work, Harmonices Mundi.

To the modern critic's understanding it appears there like an erratic block. For Kepler this was different. While publis.h.i.+ng his discovery in precisely the form in which it is conceived by a mind bent on pure observation, he gave it a setting by which he left no doubt as to his own conception of its ideal content. And as a warning to the future reader not to overlook the message conveyed by this arrangement, he introduced the section of his book which contains the announcement of the law, with the mysterious words about himself: 'I have stolen the golden vessels of the Egyptians from which to furnish for my G.o.d a holy shrine far from Egypt's confines.'

1 We must here distinguish sensation from feeling proper, in which sensation and motion merge in mercurial balance.

2 Note how for Ruskin the gulf which for the onlooker-consciousness lies between subject and object is bridged here - as it was for Goethe in his representation of the physico-moral effect of colour.

3 De motu animalium and Theoria mediceorum planetarum ex causis physicis deducta.

4 Knowledge of this biological rhythm is still preserved among native peoples to-day and leads them to take account of the phases of the moon in their treatment of plants. A cosmic nature-wisdom of this kind has been reopened for us in modern form by Rudolf Steiner, and has since found widespread practical application in agriculture. See L. Kolisko, The Moon and Plant Growth.

5 In the order of names given above we follow the ancient usage for the two planets nearest to the sun, not the reversed order in which they are used to-day. This is necessary in a cosmology which aspires at a qualitative understanding of the universe, in view of the qualities represented by these names. Note also the absence of the three most distant planets, Ura.n.u.s, Neptune and Pluto. They are not to be considered as parts of the indigenous astral structure of our cosmic system - any more than radioactivity is an original feature of the earth.

6 Note the 'Venus' character of Ruskin's description of the plant's state of florescence quoted above (p. 336).

7 As to the time-scale of the processes brought about by Mercury and Venus respectively, experience shows that they reveal the cosmic rhythms less clearly than those for which the Moon-activity is responsible. The same is found at the opposite pole. There it is the Saturn - generated processes which show the cosmic rhythm more conspicuously than those engendered by Jupiter and Mars. To learn to recognize rhythmic events in nature and man as reflexions of corresponding planetary rhythms is one of the tasks which future scientific research has to tackle. A practical example of this kind will appear in the further course of this chapter.

8 See L. Kolisko: Working of the Stars in Earthly Substances, and other publications by the same author.

9 The close connexion between the ear and the motor system of the body is shown in another way by the fact that part of the ear serves as an organ for the sense of balance.

10 The muscle-tone can be made audible by the following means. In a room guarded against noise, press the thumbs lightly upon the ears and tense the muscles of the hands and arms - say by pressure of the fingers against the palms or by contracting the muscle of the upper arms. If this is done repeatedly, the muscle-tone will be heard after some practice with increasing distinctness. It is easily distinguished from the sound of the circulating blood as it is much higher. (As an example: the author's muscular pitch, not a particularly high one, has a frequency of approx. 630 per sec., which puts it between Treble D sharp and E.)

11 Compare also the beginning of Traherne's poem Wonder, quoted in Chapter VI (p. 110), where he says that everything he saw 'did with me talk'.

12 For the particular reasons by which Goethe justifies his a.s.sertion, see his essay Leben und Verdienste des Doktor Joachim Jungius.

13 The natural question why Kepler himself did not take this step, will be answered later on.

CHAPTER XXI

Know Thyself

Our inquiries have led us to a picture of man as a sensible-supersensible organism composed of three dynamic aggregates - physical, etheric, astral. As three rungs of a spiritual ladder they point to a fourth, which represents that particular power in man by which he distinguishes himself from all other beings in nature. For what makes man differ from all these is that he is not only fitted, as they are, with a once-for-all given mode of spiritual-physical existence peculiar to himself, but that he is endowed with the possibility of transforming his existence by dint of his free will - that indeed his manhood is based on this capacity for self-willed Becoming.

To this fourth principle in man we can give no better name than that which every human being can apply to himself alone and to no other, and which no other can apply to him. This is the name, I. In truth, we describe man in his entirety only if we ascribe to him, in addition to a physical, etheric and astral body, the possession of an I (Ego).

Naturally, our previous studies have afforded many opportunities for observing the nature and mode of activity of the I. Still, at the conclusion of these studies it is not redundant to form a concise picture of this part of man's being, with particular regard to how it works within the three other principles as its sheaths. For in modern psychology, not excluding the branch of it where efforts are made to penetrate into deeper regions of man's being, nothing is less well understood than the true nature of man's egoity.