NUMBER AND MAGNITUDE 641 



Second Stage. We have only taken our first step. We have ex- 

 plained the origin of continuums of the first order; we must now see 

 why this is not sufficient, and why the incommensurable numbers had 

 to be invented. 



If we try to imagine a line, it must have the characters of the phy- 

 sical continuum that is to say, our representation must have a 

 certain breadth. Two lines will therefore appear to us under the 

 form of two narrow bands, and if we are content with this rough 

 image, it is clear that where two lines cross they must have some 

 common part. But the pure geometer makes one further effort ; with- 

 out entirely renouncing the aid of his senses, he tries to imagine a 

 line without breadth and a point without size. This he can do only 

 by imagining a line as the limit towards which tends a band that is 

 growing thinner and thinner, and the point as the limit towards 

 which is tending an area that is growing smaller and smaller. Our 

 two bands, however narrow they may be, will always have a common 

 area; the smaller they are the smaller it will be, and its limit is what 

 the geometer calls a point. This is why it is said that the two lines 

 which cross must have a common point, and this truth seems intuitive. 



But a contradiction would be implied if we conceived of lines as 

 continuums of the first order i.e., the lines traced by the geometer 

 should only give us points, the co-ordinates of which are rational num- 

 bers. The contradiction would be manifest if we were, for instance, to 

 assert the existence of lines and circles. It is clear, in fact, that if 

 the points whose co-ordinates are commensurable were alone regarded 

 as real, the in-circle of a square and the diagonal of the square would 

 not intersect, since the co-ordinates of the point of intersection are 

 incommensurable. 



Even then we should have only certain incommensurable numbers, 

 and not all these numbers. 



But let us imagine a line divided into two half-rays (demi-droites) . 

 Each of these half -rays will appear to our minds as a band of a 

 certain breadth; these bands will fit close together, because there must 

 be no interval between them. The common part will appear to us to 

 be a point which will still remain as we imagine the bands to become 

 thinner and thinner, so that we admit as an intuitive truth that if a 

 line be divided into two half-rays the common frontier of these half- 

 rays is a point. Here we recognize the conception of Kronecker, in 

 which an incommensurable number was regarded as the common 

 frontier of two classes of rational numbers. Such is the origin of 

 the continuum of the second order, which is the mathematical con- 

 tinuum properly so called. 



Summary. To sum up, the mind has the faculty of creating sym- 

 bols, and it is thus that it has constructed the mathematical con- 

 tinuum, which is only a particular system of symbols. The only limit 



