c20p55 chapter_20_text

c20p55

Among the discoveries of the last century in the realm of acoustics, there is one which especially helped to establish a purely kinematic conception of sound. Helmholtz showed that tones which to our ears seem to have a clear and definite pitch may be split up by a series of resonators into a number of different tones, each of them sounding at a different pitch. The lowest of these has the pitch which our ears attach to the entire tone. Thus in any ordinary tone there may be distinguished a ‘fundamental’ tone and a series of ‘overtones’. Helmholtz further showed that the particular series of overtones into which a tone can be resolved is responsible for the colour of that tone as a whole. Naturally, this meant for the prevailing mode of thinking that the experience of the colour of a tone had to be interpreted as the effect of a kind of acoustical adding together of a number of single tone perceptions (very much as Newton had interpreted ‘white’ light as the outcome of an optical adding together of a certain number of single colour perceptions).

c20p74 chapter_20_text

c20p74

What we are thus dimly aware of in physical sense activity, becomes definite experience when the supersensible part of the senses concerned can work unfettered by the bodily organ. Clear testimony of this is again given to us by Traherne in a poem entitled Dumnesse. This poem contains an account of Traherne’s recollection of the significant fact that the transition from the cosmic to the earthly condition of his consciousness was caused by his learning to speak. The following is a passage from the description of the impressions which were his before his soul was overcome by this change:

c20p92 chapter_20_text

c20p92

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 circumference 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 assertion 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 assertion 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:

c20p110 chapter_20_text

c20p110

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):