640 SCIENCE AND HYPOTHESIS 



Such terms as A and B, which before were indistinguishable from 

 one another, appear now to be distinct : but between A and B, which 

 are distinct, is intercalated another new term D, which we can distin- 

 guish neither from A nor from B. Although we may use the most 

 delicate methods, the rough results of our experiments will always 

 present the characters of the physical continuum with the contradic- 

 tion which is inherent in it. We only escape from it by incessantly 

 intercalating new terms between the terms already distinguished, and 

 this operation must be pursued indefinitely. We might conceive that 

 it would be possible to stop if we could imagine an instrument power- 

 ful enough to decompose the physical continuum into discrete ele- 

 ments, just as the telescope resolves the Milky Way into stars. But 

 this we cannot imagine; it is always with our senses that we use our 

 instruments; it is with the eye that we observe the image magnified 

 by the microscope, and this image must therefore always retain the 

 characters of visual sensation, and therefore those of the physical 

 continuum. 



Nothing distinguishes a length directly observed from half that 

 length doubled by the microscope. The whole is homogeneous to the 

 part ; and there is a fresh contradiction or rather there would be one 

 if the number of the terms were supposed to be finite; it is clear 

 that the part containing less terms than the whole cannot be similar to 

 the whole. The contradiction ceases as soon as the number of terms is 

 regarded as infinite. There is nothing, for example, to prevent us 

 from regarding the aggregate of integers as similar to the aggregate 

 of even numbers, which is however only a part of it ; in fact, to each 

 integer corresponds another even number which is its double. But 

 it is not only to escape this contradiction contained in the empiric 

 data that the mind is led to create the concept of a continuum formed 

 of an indefinite number of terms. 



Here everything takes place just as in the series of the integers. 

 We have the faculty of conceiving that a unit may be added to a col- 

 lection of units. Thanks to experiment, we have had the opportunity 

 of exercising this faculty and are conscious of it; but from this fact 

 we feel that our power is unlimited, and that we can count indefinitely, 

 although we have never had to count more than a finite number of 

 objects. In the same way, as soon as we have intercalated terms between 

 two consecutive terms of a series, we feel that this operation may be 

 continued without limit, and that, so to speak, there is no intrinsic 

 reason for stopping. As an abbreviation, I may give the name of a 

 mathematical continuum of the first order to every aggregate of 

 terms formed after the same law as the scale of commensurable num- 

 bers. If, then, we intercalate new sets according to the laws of in- 

 commensurable numbers, we obtain what may be called a continuum 

 of the second order. 



