chapter_20_text

c20p96

(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.

chapter_20_text

c20p97

(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 analysis will show this.

chapter_20_text

c20p98

Kepler’s formula-
r13 / r23 = t12 / t22
may be written also
r13 / t12 = r23 / t22
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 entities 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 orbit and the time needed by it to perform one round on this orbit – a ratio which is the same for all planets – represents a definite structural element of our cosmic system.

chapter_20_text

c20p99

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 4Ï2m, and a gathering of the constant product 4Ï2c under a new symbol, for which we choose the letter f. In this way we arrive at:
4Ï2mr / t2 = 4Ï2cm / 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

chapter_20_merge_container

c20p100-both

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?
*
chapter_20_merge_container

c20p101-both

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.
chapter_20_text

c20p102

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.

chapter_20_merge_container

c20p102-both

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.