Experiment teaches us relations between bodies; this is the fact in the rough; these relations are extremely complicated. Instead of envisaging directly the relation of the body _A_ and the body _B_, we introduce between them an intermediary, which is space, and we envisage three distinct relations: that of the body _A_ with the figure _A'_ of space, that of the body _B_ with the figure _B'_ of space, that of the two figures _A'_ and _B'_ to each other. Why is this detour advantageous? Because the relation of _A_ and _B_ was complicated, but differed little from that of _A'_ and _B'_, which is simple; so that this complicated relation may be replaced by the simple relation between _A'_ and _B'_ and by two other relations which tell us that the differences between _A_ and _A'_, on the one hand, between _B_ and _B'_, on the other hand, are _very small_. For example, if _A_ and _B_ are two natural solid bodies which are displaced with slight deformation, we envisage two movable _rigid_ figures _A'_ and _B'_. The laws of the relative displacement of these figures _A'_ and _B'_ will be very simple; they will be those of geometry. And we shall afterward add that the body _A_, which always differs very little from _A'_, dilates from the effect of heat and bends from the effect of elasticity. These dilatations and flexions, just because they are very small, will be for our mind relatively easy to study. Just imagine to what complexities of language it would have been necessary to be resigned if we had wished to comprehend in the same enunciation the displacement of the solid, its dilatation and its flexure? The relation between _A_ and _B_ was a rough law, and was broken up; we now have two laws which express the relations of _A_ and _A'_, of _B_ and _B'_, and a principle which expresses that of _A'_ with _B'_. It is the aggregate of these principles that is called geometry. Two other remarks. We have a relation between two bodies _A_ and _B_, which we have replaced by a relation between two figures _A'_ and _B'_; but this same relation between the same two figures _A'_ and _B'_ could just as well have replaced advantageously a relation between two other bodies _A''_ and _B''_, entirely different from _A_ and _B_. And that in many ways. If the principles of geometry had not been invented, after having studied the relation of _A_ and _B_, it would be necessary to begin again _ab ovo_ the study of the relation of _A''_ and _B''_. That is why geometry is so precious. A geometrical relation can advantageously replace a relation which, considered in the rough state, should be regarded as mechanical, it can replace another which should be regarded as optical, etc.