That fixed, in order that [Sigma] and [Sigma]' should correspond to the same point _M_ = _M'_ of the first space, it is necessary and sufficient for us to have [Sigma]' = [Sigma] + [sigma]. We shall then have: _S_ + [Sigma]' + _S'_ = _S_ + [Sigma] + [sigma] + _S'_ = _S_ + [Sigma] + _S'_ + _S_ + [sigma] + _S'_. But we have just ascertained that _S_ + [sigma] + _S'_ was one of the series [sigma]'. We shall therefore have: _S_ + [Sigma]' + _S'_ = _S_ + [Sigma] + _S'_ + [sigma]', which means that the series _S_ + [Sigma]' + _S'_ and _S_ + [Sigma] + _S'_ correspond to the same point _N_ = _N'_ of the second space. Q.E.D. Our two spaces therefore correspond point for point; they can be 'transformed' one into the other; they are isomorphic. How are we led to conclude thence that they are identical? Consider the two series [sigma] and _S_ + [sigma] + _S'_ = [sigma]'. I have said that often, but not always, the series [sigma] preserves the tactile impression _A_ felt by the finger _D_; and similarly it often happens, but not always, that the series [sigma]' preserves the tactile impression _A'_ felt by the finger _D'_. Now I ascertain that it happens _very often_ (that is, much more often than what I have just called 'often') that when the series [sigma] has preserved the impression _A_ of the finger _D_, the series [sigma]' preserves at the same time the impression _A'_ of the finger _D'_; and, inversely, that if the first impression is altered, the second is likewise. That happens _very often_, but not always. We interpret this experimental fact by saying that the unknown object _a_ which gives the impression _A_ to the finger _D_ is identical with the unknown object _a'_ which gives the impression _A'_ to the finger _D'_. And in fact when the first object moves, which the disappearance of the impression _A_ tells us, the second likewise moves, since the impression _A'_ disappears likewise. When the first object remains motionless, the second remains motionless. If these two objects are identical, as the first is at the point _M_ of the first space and the second at the point _N_ of the second space, these two points are identical. This is how we are led to regard these two spaces as identical; or better, this is what we mean when we say that they are identical. What we have just said of the identity of the two tactile spaces makes unnecessary our discussing the question of the identity of tactile space and visual space, which could be treated in the same way. 5. _Space and Empiricism_