History and root of the principle of the conservation of energy . orms are perfectly analogous tospace—for example, the tone-series for hearing, which CONSERVATION OF ENERGY SI corresponds to a space of one dimension—and we donot allow ourselves a like liberty with them. We do notthink of all things as sounding and do not figure toourselves molecular events musically, in relations ofheights of tones, although we are as justified in doingthis as in thinking of them spatially. This, therefore, teaches us what an unnecessaryrestriction we here impose upon ourselves. There is nomore necessity to t
History and root of the principle of the conservation of energy . orms are perfectly analogous tospace—for example, the tone-series for hearing, which CONSERVATION OF ENERGY SI corresponds to a space of one dimension—and we donot allow ourselves a like liberty with them. We do notthink of all things as sounding and do not figure toourselves molecular events musically, in relations ofheights of tones, although we are as justified in doingthis as in thinking of them spatially. This, therefore, teaches us what an unnecessaryrestriction we here impose upon ourselves. There is nomore necessity to think of what is merely a product ofthought spatially, that is to say, with the relations ofthe visible and tangible, than there is to think of thesethings in a definite position in the scale of tones. And I will immediately show the sort of drawbackthat this limitation has. A system of n points is inform and magnitude determined in a space of r dimen-sions, if e distances between pairs of points are given,where e is given by the following table: i 2 3 4 5 r. 2ft — 3 3ft —6 4ft — io 6ft —21 (r+i)ft- (r + i) (r + 2) In this table, the column marked by eT is to be usedfor e if we have made conditions about the sense of thegiven distances, for example, that in the straight line allpoints are reckoned according to one direction; in theplane all towards one side of the straight line through thefirst two points; in space all towards one side of the plane 52 CONSERVATION OF ENERGY through the first three points; and so on. The columnmarked by e2 is to be used if merely the absolute magni-tude of the distance is given. Between n points, combining them in pairs, — distances are thinkable, and therefore in general morethan a space of a given number of dimensions cansatisfy. If, for example, we suppose the ^-column tobe the one to be used, we find in a space of r dimen-sions the difference between the number of thinkabledistances and those possible in this space to be n(n—i
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