Man or Matter / Introduction to a Spiritual Understanding of Nature on the Basis of Goethe's Method of Training Observation and Thought

by Ernst Lehrs

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 4ϲm, and a gathering of the constant product 4ϲc under a new symbol, for which we choose the letter f. In this way we arrive at:
4ϲmr / t² = 4ϲcm / 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 space 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 orbit?

*

Once we have rid ourselves of the false conception that Kepler's law implies Newton's interpretation of the physical universe as a dynamic entity 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) fashion 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 (r³) represents the measure of a volume in three-dimensional space. The same cannot be said of the temporal magnitudes on the other side of the equation (t²). 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 -
r₁³ / r₂³ = t₁² / t₂²
assumes the form-
r₁³ / r₂³ = (t₁ / t₂) / (t₂ / t₁)
The right-hand side of the equation is now constituted 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 assumes the following form ('J' and 'E' indicating 'Jupiter' and 'Earth' respectively):
r_J³ / r_E³ = (t_J / t_E) / (t_E / t_J)
Let us now see what meaning we can attach to the two expressions
t_J / t_E and t_E / t_J.

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:
t_J / t_E = 12 / 1
To find the analogous expression for the reciprocal ratio:
t_E / t_J = 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:
r_J³ / r_E³ = (t_J / t_E) / (t_E / t_J) = 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 spaces 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.