Centripetal force:
P = am = 4ϲmr / 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 analysis will show this.

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.

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

That it is not in the nature of the I to leave its sheaths in the condition in which it finds them when entering them at the beginning of life, can be seen from the activities it performs in them during the first period after birth. Indeed, in man's early childhood we meet a number of events in which we can perceive something like ur-deeds of the I. They are the acquisition of the faculties of walking, speaking and thinking. What we shall here say about them has, in essentials, already been touched upon in earlier pages. Here, however, we are putting it forward in a new light.

Once again we find our attention directed to the threefold structure of man's physical organism. For the faculty of upright walking is a result of the I's activity in the limb-system of the body; the acquisition of speech takes place in the rhythmic system; and thinking is a faculty based on the nerve-system. Consequently, each of the three achievements comes to pass at a different level of consciousness-sleeping, dreaming, waking. All through the struggle of erecting the body against the pull of gravity, the child is entirely unaware of the activities of his own I. In the course of acquiring speech he gains a dim awareness, as though in dream, of his efforts. Some capacity of thinking has to unfold before the first glimmer of true self-consciousness is kindled. (Note that the word 'I' is the only one that is not added to the child's vocabulary by way of imitation. Otherwise he would, as some mentally inhibited children do, call all other people 'I' and himself 'you'.)

This picture of the three ur-deeds of the I can now be amplified in the following way. We know that the region of the bodily limbs is that in which physical, etheric and astral forces interpenetrate most deeply. Consequently, the I can here press forward most powerfully into the physical body and on into the dynamic sphere to which the body is subject. Here the I is active in a way that is 'magic' in the highest degree. Moreover, there is no other action for which the I receives so little stimulus from outside. For, in comparison, the activity that leads to the acquisition of speech is much more of the nature of a reaction to stimuli coming from outside - the sounds reaching the child from his environment. And it is also with the first words of the language that the first thoughts enter the child's mind. Nothing of the kind happens at the first stage. On the contrary: everything that confronts the I here is of the nature of an obstacle that is to be overcome.

There is no learning to speak without the hearing of uttered sounds. As these sounds approach the human being they set the astral body in movement, as we have seen. The movements of the astral body flow towards the larynx, where they are seized by the I; through their help the I imbues the larynx with the faculty of producing these sounds itself. Here, therefore, the I is active essentially within the astral body which has received its stimulus from outside. In order to understand what impels the I to such action, we must remember the role played by speech in human life: without speech there would be no community among human individuals on earth.

An illustration of what the I accomplishes as it enters upon the third stage is provided by the following episode, actually observed. Whilst all the members of a family were sitting at table taking their soup, the youngest member suddenly cried out: 'Daddy spoon ... mummy spoon. ... ' (everyone in turn spoon) ' ... all spoon!' At this moment, from merely designating single objects by names learnt through imitation, the child's consciousness had awakened to connective thinking. That this achievement was a cause of inner satisfaction could be heard in the joyful crescendo with which these ejaculations were made.