Such is the origin of the continuum of the second order, which is the mathematical continuum properly so called.

_Resume._--In recapitulation, the mind has the faculty of creating symbols, and it is thus that it has constructed the mathematical continuum, which is only a particular system of symbols. Its power is limited only by the necessity of avoiding all contradiction; but the mind only makes use of this faculty if experience furnishes it a stimulus thereto.

In the case considered, this stimulus was the notion of the physical continuum, drawn from the rough data of the senses. But this notion leads to a series of contradictions from which it is necessary successively to free ourselves. So we are forced to imagine a more and more complicated system of symbols. That at which we stop is not only exempt from internal contradiction (it was so already at all the stages we have traversed), but neither is it in contradiction with various propositions called intuitive, which are derived from empirical notions more or less elaborated.

MEASURABLE MAGNITUDE.--The magnitudes we have studied hitherto are not _measurable_; we can indeed say whether a given one of these magnitudes is greater than another, but not whether it is twice or thrice as great.

So far, I have only considered the order in which our terms are ranged.

But for most applications that does not suffice. We must learn to compare the interval which separates any two terms. Only on this condition does the continuum become a measurable magnitude and the operations of arithmetic applicable.

This can only be done by the aid of a new and special _convention_. We will _agree_ that in such and such a case the interval comprised between the terms _A_ and _B_ is equal to the interval which separates _C_ and _D_. For example, at the beginning of our work we have set out from the scale of the whole numbers and we have supposed intercalated between two consecutive steps _n_ intermediary steps; well, these new steps will be by convention regarded as equidistant.

This is a way of defining the addition of two magnitudes, because if the interval _AB_ is by definition equal to the interval _CD_, the interval _AD_ will be by definition the sum of the intervals _AB_ and _AC_.

This definition is arbitrary in a very large measure. It is not completely so, however. It is subjected to certain conditions and, for example, to the rules of commutativity and a.s.sociativity of addition.

But provided the definition chosen satisfies these rules, the choice is indifferent, and it is useless to particularize it.

VARIOUS REMARKS.--We can now discuss several important questions:

1 Is the creative power of the mind exhausted by the creation of the mathematical continuum?

No: the works of Du Bois-Reymond demonstrate it in a striking way.

We know that mathematicians distinguish between infinitesimals of different orders and that those of the second order are infinitesimal, not only in an absolute way, but also in relation to those of the first order. It is not difficult to imagine infinitesimals of fractional or even of irrational order, and thus we find again that scale of the mathematical continuum which has been dealt with in the preceding pages.

Further, there are infinitesimals which are infinitely small in relation to those of the first order, and, on the contrary, infinitely great in relation to those of order 1 + [epsilon], and that however small [epsilon] may be. Here, then, are new terms intercalated in our series, and if I may be permitted to revert to the phraseology lately employed which is very convenient though not consecrated by usage, I shall say that thus has been created a sort of continuum of the third order.

It would be easy to go further, but that would be idle; one would only be imagining symbols without possible application, and no one will think of doing that. The continuum of the third order, to which the consideration of the different orders of infinitesimals leads, is itself not useful enough to have won citizenship, and geometers regard it only as a mere curiosity. The mind uses its creative faculty only when experience requires it.

2 Once in possession of the concept of the mathematical continuum, is one safe from contradictions a.n.a.logous to those which gave birth to it?

No, and I will give an example.

One must be very wise not to regard it as evident that every curve has a tangent; and in fact if we picture this curve and a straight as two narrow bands we can always so dispose them that they have a part in common without crossing. If we imagine then the breadth of these two bands to diminish indefinitely, this common part will always subsist and, at the limit, so to speak, the two lines will have a point in common without crossing, that is to say, they will be tangent.

The geometer who reasons in this way, consciously or not, is only doing what we have done above to prove two lines which cut have a point in common, and his intuition might seem just as legitimate.

It would deceive him however. We can demonstrate that there are curves which have no tangent, if such a curve is defined as an a.n.a.lytic continuum of the second order.

Without doubt some artifice a.n.a.logous to those we have discussed above would have sufficed to remove the contradiction; but, as this is met with only in very exceptional cases, it has received no further attention.

Instead of seeking to reconcile intuition with a.n.a.lysis, we have been content to sacrifice one of the two, and as a.n.a.lysis must remain impeccable, we have decided against intuition.

THE PHYSICAL CONTINUUM OF SEVERAL DIMENSIONS.--We have discussed above the physical continuum as derived from the immediate data of our senses, or, if you wish, from the rough results of Fechner's experiments; I have shown that these results are summed up in the contradictory formulas

_A_ = _B_, _B_ = _C_, _A_ <>

Let us now see how this notion has been generalized and how from it has come the concept of many-dimensional continua.

Consider any two aggregates of sensations. Either we can discriminate them one from another, or we can not, just as in Fechner's experiments a weight of 10 grams can be distinguished from a weight of 12 grams, but not from a weight of 11 grams. This is all that is required to construct the continuum of several dimensions.

Let us call one of these aggregates of sensations an _element_. That will be something a.n.a.logous to the _point_ of the mathematicians; it will not be altogether the same thing however. We can not say our element is without extension, since we can not distinguish it from neighboring elements and it is thus surrounded by a sort of haze. If the astronomical comparison may be allowed, our 'elements' would be like nebulae, whereas the mathematical points would be like stars.

That being granted, a system of elements will form a _continuum_ if we can pa.s.s from any one of them to any other, by a series of consecutive elements such that each is indistinguishable from the preceding. This _linear_ series is to the _line_ of the mathematician what an isolated _element_ was to the point.

Before going farther, I must explain what is meant by a _cut_. Consider a continuum _C_ and remove from it certain of its elements which for an instant we shall regard as no longer belonging to this continuum. The aggregate of the elements so removed will be called a cut. It may happen that, thanks to this cut, _C_ may be _subdivided_ into several distinct continua, the aggregate of the remaining elements ceasing to form a unique continuum.

There will then be on _C_ two elements, _A_ and _B_, that must be regarded as belonging to two distinct continua, and this will be recognized because it will be impossible to find a linear series of consecutive elements of _C_, each of these elements indistinguishable from the preceding, the first being _A_ and the last _B_, _without one of the elements of this series being indistinguishable from one of the elements of the cut_.

On the contrary, it may happen that the cut made is insufficient to subdivide the continuum _C_. To cla.s.sify the physical continua, we will examine precisely what are the cuts which must be made to subdivide them.

If a physical continuum _C_ can be subdivided by a cut reducing to a finite number of elements all distinguishable from one another (and consequently forming neither a continuum, nor several continua), we shall say _C_ is a _one-dimensional_ continuum.

If, on the contrary, _C_ can be subdivided only by cuts which are themselves continua, we shall say _C_ has several dimensions. If cuts which are continua of one dimension suffice, we shall say _C_ has two dimensions; if cuts of two dimensions suffice, we shall say _C_ has three dimensions, and so on.

Thus is defined the notion of the physical continuum of several dimensions, thanks to this very simple fact that two aggregates of sensations are distinguishable or indistinguishable.

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 _a.n.a.lysis 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 s.p.a.ce, and geometry is born. But the discussion of this is reserved for Part Second.

PART II

s.p.a.cE

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 a.n.a.lysis. I regard them as a.n.a.lytic 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:

1 Through two points can pa.s.s only one straight;