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An Introduction To Non-aristotelian Systems And General Semantics.

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702            X. ON THE STRUCTURE OF 'MATTER'
The advent of such a crop of geniuses and of several theories expressed quite differently, yet nearly equivalent, is an event of deep human semantic significance. It helps to understand the working of the human nervous system, and is in accord with the present general theory.
From the point of view of the physicist, these new theories are a marked structural improvement over the classical theory, a fact which can best be illustrated by a diagram of a special case. In Fig. 1, the crosses indicate the experimental data, the curves indicate the results as predicted by the classical theory, by the Compton theory, and by the new quantum mechanics. It should be noticed that the new quantum theory appears much more in accordance with the experimental data than the older theories. This fact is of great structural and semantic importance to us as well as to the physicist.
\ Classical
Theory
Section B. The nature of the problem.
At this point we may explain briefly the nature of the problem that was demanding solution.
We have become familiar with the use of co-ordinates. This procedure has been generalized and has given rise to 'generalized co-ordinates'. These are defined as arbitrary variables which represent not merely lengths but may also represent angles, surfaces, volumes . , though they must be capable of representing the 3n orthogonal co-ordinates. If, in a special case, we make the number of generalized co-ordinates equal to the number of degrees of freedom which the system has, these s generalized co-ordinates can be regarded as independent of each other. If we denote by $, the generalized co-ordinate,, then the orthogonal co-ordinates of any of the n particles can be represented as definite functions of the generalized co-ordinates, so that
                                       (1)
We know that kinetic energy is represented bywhere m represents
the mass and v the velocity, or the 'time' derivative of the 'space' travelled.
If we want to find the value for the energy we must differentiate each of the
3n equations (1) with respect to 'time', which gives the components of the
velocity, square them, multiply them by the corresponding masses and add
them together to find the double value for the energy.