correspond to series of movements where the finger does not budge. I say that if one does not consider as distinct the series Σ and Σ + σ, where the series σ 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 Σ and Σ´ unless Σ´ = Σ + S + S´, S and 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 A, on this surface a line B, on this line a point M. Let C0 be the aggregate of all series Σ. Let C1 be the aggregate of all the series Σ, such that at the end of corresponding movements the finger is found upon the surface A, and C2 or C3 the aggregate of series Σ such that at the end the finger is found on B, or at M. It is clear, first that C1 will constitute a cut which will divide C0, that C2 will be a cut which will divide C1, and C3 a cut which will divide C2. Thence it results, in accordance with our definitions, that if C3 is a continuum of n dimensions, C0 will be a physical continuum of n + 3 dimensions.
Therefore, let Σ and Σ´ = Σ + σ be two series forming part of C3; 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 σ is therefore one of those which correspond to movements where the finger does not budge. If Σ and Σ + σ are not regarded as distinct, all the series of C3 blend into one; therefore C3 will have 0 dimension, and C0 will have 3, as I wished to prove. If, on the contrary, I do not regard Σ and Σ + σ as blending (unless σ = S + S´, S and S´ being inverses), it is clear that C3 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 C3 will form a continuum and C0 will have more than three dimensions, and this also I wished to prove.
We who do not yet know geometry can not reason in this way; we can only verify. But then a question arises; how, before knowing geometry, have we been led to distinguish from the others these series σ where the finger does not budge? It is, in fact, only after having made this distinction that we could be led to regard Σ and Σ + σ as identical, and it is on this condition alone, as we have just seen, that we can arrive at space of three dimensions.
We are led to distinguish the series σ, because it often happens that when we have executed the movements which correspond to these series σ of muscular sensations, the tactile sensations which are transmitted to us by the nerve of the finger that we have called the first finger, persist and are not altered by these movements. Experience alone tells us that and it alone could tell us.
If we have distinguished the series of muscular sensations S + S´ formed by the union of two inverse series, it is because they preserve the totality of our impressions; if now we distinguish the series σ, it is because