# Page:Popular Science Monthly Volume 70.djvu/84

agreement, the aggregate of distinct series ${\displaystyle \Sigma }$ will still form a physical continuum and the number of dimensions will be less but still very great.
To each of these series ${\displaystyle \Sigma }$ corresponds a point of space; to two series ${\displaystyle \Sigma }$ and ${\displaystyle \Sigma '}$ thus correspond two points ${\displaystyle \mathrm {M} }$ and ${\displaystyle \mathrm {M} '}$. The means we have hitherto used enable us to recognize that ${\displaystyle \mathrm {M} }$ and ${\displaystyle \mathrm {M} '}$ are not distinct in two cases: (1) if ${\displaystyle \Sigma }$ is identical with ${\displaystyle \Sigma '}$ (2) if ${\displaystyle \Sigma =\Sigma +s+s'}$, ${\displaystyle s}$ and ${\displaystyle s'}$ being inverses one of the other. If in all the other cases we should regard ${\displaystyle \mathrm {M} }$ and ${\displaystyle Mu'}$ as distinct, the manifold of points would have as many dimensions as the aggregate of distinct series ${\displaystyle \Sigma '}$, that is, much more than three.
For those who already know geometry, the following explanation would be easily comprehensible. Among the imaginable series of muscular sensations, there are those which correspond to series of movements where the finger does not budge. I say that if one does not consider as distinct the series ${\displaystyle \Sigma }$ and ${\displaystyle \Sigma +\sigma }$, where the series a corresponds to movements where the finger does not budge, the aggregate of series will constitute a continuum of three dimensions, but that if one regards as distinct two series ${\displaystyle \Sigma }$ and ${\displaystyle \Sigma '}$ unless ${\displaystyle \Sigma '}$ ${\displaystyle =}$ ${\displaystyle \Sigma +s+s'}$, ${\displaystyle s}$ and ${\displaystyle s'}$ being inverses, the aggregate of series will constitute a continuum of more than three dimensions.
In fact, let there be in space a surface ${\displaystyle \mathrm {A} }$, on this surface a line ${\displaystyle \mathrm {B} }$, on this line a point ${\displaystyle \mathrm {M} }$. Let ${\displaystyle C_{0}}$ be the aggregate of all series ${\displaystyle \Sigma }$. Let ${\displaystyle C_{1}}$ be the aggregate of all the series ${\displaystyle \Sigma }$, such that at the end of corresponding movements the finger is found upon the surface A, and ${\displaystyle C_{2}}$ or ${\displaystyle C_{3}}$ the aggregate of series ${\displaystyle \Sigma }$ such that at the end the finger is found on ${\displaystyle \mathrm {B} }$, or at ${\displaystyle \mathrm {M} }$. It is clear, first that ${\displaystyle C_{1}}$ will constitute a cut which will divide ${\displaystyle C_{0}}$, that ${\displaystyle C_{2}}$ will be a cut which will divide ${\displaystyle C_{1}}$, and ${\displaystyle C_{3}}$ a cut which will divide ${\displaystyle C_{2}}$. Thence it results, in accordance with our definitions, that if ${\displaystyle C_{3}}$ is a continuum of ${\displaystyle n}$ dimensions, ${\displaystyle C_{0}}$ will be a physical continuum of ${\displaystyle n+3}$ dimensions.
Therefore, let ${\displaystyle \Sigma }$ and ${\displaystyle \Sigma +\sigma }$ a be two series forming part of ${\displaystyle C_{3}}$; for both, at the end of the movements, the finger is found at M; thence results that at the beginning and at the end of the series σ, the finger is at the same point M. This series a is therefore one of those which correspond to movements where the finger does not budge. If ${\displaystyle \Sigma }$ and ${\displaystyle \Sigma +\sigma }$ are not regarded as distinct, all the series of ${\displaystyle C_{3}}$ blend into one; therefore ${\displaystyle C_{3}}$ will have dimension, and ${\displaystyle C_{0}}$ will have 3, as I wished to prove. If, on the contrary, I do not regard ${\displaystyle \Sigma }$ and ${\displaystyle \Sigma +\sigma }$ as blending (unless ${\displaystyle \sigma =s+s'}$, ${\displaystyle s}$ and ${\displaystyle s'}$ being inverses), it is clear that ${\displaystyle C_{3}}$ will contain a great number of series of distinct sensations; because, without the finger budging, the body may take a multitude of different attitudes. Then ${\displaystyle C_{3}}$ will form a continuum and ${\displaystyle C_{0}}$ will have more than three dimensions, and this also I wished to prove.