They are _conventions_; our choice among all possible conventions is _guided_ by experimental facts; but it remains _free_ and is limited only by the necessity of avoiding all contradiction. Thus it is that the postulates can remain _rigorously_ true even though the experimental laws which have determined their adoption are only approximative. In other words, _the axioms of geometry_ (I do not speak of those of arithmetic) _are merely disguised definitions_. Then what are we to think of that question: Is the Euclidean geometry true? It has no meaning. As well ask whether the metric system is true and the old measures false; whether Cartesian coordinates are true and polar coordinates false. One geometry can not be more true than another; it can only be _more convenient_. Now, Euclidean geometry is, and will remain, the most convenient: 1º Because it is the simplest; and it is so not only in consequence of our mental habits, or of I know not what direct intuition that we may have of Euclidean space; it is the simplest in itself, just as a polynomial of the first degree is simpler than one of the second; the formulas of spherical trigonometry are more complicated than those of plane trigonometry, and they would still appear so to an analyst ignorant of their geometric signification. 2º Because it accords sufficiently well with the properties of natural solids, those bodies which our hands and our eyes compare and with which we make our instruments of measure. CHAPTER IV SPACE AND GEOMETRY Let us begin by a little paradox. Beings with minds like ours, and having the same senses as we, but without previous education, would receive from a suitably chosen external world impressions such that they would be led to construct a geometry other than that of Euclid and to localize the phenomena of that external world in a non-Euclidean space, or even in a space of four dimensions. As for us, whose education has been accomplished by our actual world, if we were suddenly transported into this new world, we should have no difficulty in referring its phenomena to our Euclidean space. Conversely, if these beings were transported into our environment, they would be led to relate our phenomena to non-Euclidean space. Nay more; with a little effort we likewise could do it. A person who should devote his existence to it might perhaps attain to a realization of the fourth dimension. GEOMETRIC SPACE AND PERCEPTUAL SPACE.--It is often said the images of external objects are localized in space, even that they can not be formed except on this condition. It is also said that this space, which serves thus as a ready prepared _frame_ for our sensations and our representations, is identical with that of the geometers, of which it possesses all the properties.