THE MATHEMATICAL CONTINUUM OF SEVERAL DIMENSIONS.--Thence the notion of the mathematical continuum of _n_ dimensions has sprung quite naturally by a process very like that we discussed at the beginning of this chapter. A point of such a continuum, you know, appears to us as defined by a system of _n_ distinct magnitudes called its coordinates. These magnitudes need not always be measurable; there is, for instance, a branch of geometry independent of the measurement of these magnitudes, in which it is only a question of knowing, for example, whether on a curve _ABC_, the point _B_ is between the points _A_ and _C_, and not of knowing whether the arc _AB_ is equal to the arc _BC_ or twice as great. This is what is called _Analysis Situs_. This is a whole body of doctrine which has attracted the attention of the greatest geometers and where we see flow one from another a series of remarkable theorems. What distinguishes these theorems from those of ordinary geometry is that they are purely qualitative and that they would remain true if the figures were copied by a draughtsman so awkward as to grossly distort the proportions and replace straights by strokes more or less curved. Through the wish to introduce measure next into the continuum just defined this continuum becomes space, and geometry is born. But the discussion of this is reserved for Part Second. PART II SPACE CHAPTER III THE NON-EUCLIDEAN GEOMETRIES Every conclusion supposes premises; these premises themselves either are self-evident and need no demonstration, or can be established only by relying upon other propositions, and since we can not go back thus to infinity, every deductive science, and in particular geometry, must rest on a certain number of undemonstrable axioms. All treatises on geometry begin, therefore, by the enunciation of these axioms. But among these there is a distinction to be made: Some, for example, 'Things which are equal to the same thing are equal to one another,' are not propositions of geometry, but propositions of analysis. I regard them as analytic judgments _a priori_, and shall not concern myself with them. But I must lay stress upon other axioms which are peculiar to geometry. Most treatises enunciate three of these explicitly: 1º Through two points can pass only one straight; 2º The straight line is the shortest path from one point to another; 3º Through a given point there is not more than one parallel to a given straight. Although generally a proof of the second of these axioms is omitted, it would be possible to deduce it from the other two and from those, much more numerous, which are implicitly admitted without enunciating them, as I shall explain further on.