*THE POPULAR SCIENCE MONTHLY*

agreement, the aggregate of distinct series will still form a physical continuum and the number of dimensions will be less but still very great.

To each of these series corresponds a point of space; to two series and thus correspond two points and . The means we have hitherto used enable us to recognize that and are not distinct in two cases: (1) if is identical with (2) if , and being inverses one of the other. If in all the other cases we should regard and as distinct, the manifold of points would have as many dimensions as the aggregate of distinct series , 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 and , 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 and unless , and being inverses, the aggregate of series will constitute a continuum of more than three dimensions.

In fact, let there be in space a surface , on this surface a line , on this line a point . Let be the aggregate of all series . Let be the aggregate of all the series , such that at the end of corresponding movements the finger is found upon the surface A, and or the aggregate of series such that at the end the finger is found on , or at . It is clear, first that will constitute a cut which will divide , that will be a cut which will divide , and a cut which will divide . Thence it results, in accordance with our definitions, that if is a continuum of dimensions, will be a physical continuum of dimensions.

Therefore, let and a be two series forming part of ; 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 and are not regarded as distinct, all the series of blend into one; therefore will have dimension, and will have 3, as I wished to prove. If, on the contrary, I do not regard and as blending (unless , and being inverses), it is clear that 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 will form a continuum and will have more than three dimensions, and this also I wished to prove.