I have just enunciated two conditions for these points being identical; one is relative to sight, the other to touch. Let us consider them separately. The first is necessary, but is not sufficient. The second is at once necessary and sufficient. A person knowing geometry could easily explain this in the following manner: Let _O_ be the point of the retina where is formed at the instant [alpha] the image of the body _A_; let _M_ be the point of space occupied at the instant [alpha] by this body _A_; let _M'_ be the point of space occupied at the instant [beta] by the body _B_. For this body _B_ to form its image in _O_, it is not necessary that the points _M_ and _M'_ coincide; since vision acts at a distance, it suffices for the three points _O_ _M_ _M'_ to be in a straight line. This condition that the two objects form their image on _O_ is therefore necessary, but not sufficient for the points _M_ and _M'_ to coincide. Let now _P_ be the point occupied by my finger and where it remains, since it does not budge. As touch does not act at a distance, if the body _A_ touches my finger at the instant [alpha], it is because _M_ and _P_ coincide; if _B_ touches my finger at the instant [beta], it is because _M'_ and _P_ coincide. Therefore _M_ and _M'_ coincide. Thus this condition that if _A_ touches my finger at the instant [alpha], _B_ touches it at the instant [beta], is at once necessary and sufficient for _M_ and _M'_ to coincide. But we who, as yet, do not know geometry can not reason thus; all that we can do is to ascertain experimentally that the first condition relative to sight may be fulfilled without the second, which is relative to touch, but that the second can not be fulfilled without the first. Suppose experience had taught us the contrary, as might well be; this hypothesis contains nothing absurd. Suppose, therefore, that we had ascertained experimentally that the condition relative to touch may be fulfilled without that of sight being fulfilled and that, on the contrary, that of sight can not be fulfilled without that of touch being also. It is clear that if this were so we should conclude that it is touch which may be exercised at a distance, and that sight does not operate at a distance. But this is not all; up to this time I have supposed that to determine the place of an object I have made use only of my eye and a single finger; but I could just as well have employed other means, for example, all my other fingers.