4Q2 POPULAR SCIENCE MONTHLY 



characteristics of a form imposed either upon our sensibility or upon 

 our understanding? 



I wish simply to observe that the last two solutions exclude each 

 other. We can not admit at the same time that it is impossible to 

 imagine space of four dimensions and that experience proves to us that 

 space has three dimensions. The experimenter puts to nature a ques- 

 tion: Is it this or that? and he can not put it without imagining the 

 two terms of the alternative. If it were impossible to imagine one of 

 these terms, it would be futile and besides impossible to consult ex- 

 perience. There is no need of observation to know that the hand of a 

 watch is not marking the hour 15 on the dial, because we know before- 

 hand that there are only 12, and we could not look at the mark 15 to 

 see if the hand is there, because this mark does not exist. 



Note likewise that in analysis situs the empiricists are disembar- 

 rassed of one of the gravest objections that can be leveled against them, 

 of that which renders absolutely vain in advance all their efforts to 

 apply their thesis to the verities of Euclidean geometry. These verities 

 are rigorous and all experimentation can only be approximate. In 

 analysis situs approximate experiments may suffice to give a rigorous 

 theorem and, for instance, if it is seen that space can not have either 

 two or less than two dimensions, nor four or more than four, we are 

 certain that it has exactly three, since it could not have two and a half 

 or three and a half. 



Of all the theorems of analysis situs, the most important is that 

 which is expressed in saying that space has three dimensions. This it 

 is that we are about to consider, and we shall put the question in these 

 terms : When we say that space has three dimensions, what do we mean ? 



3. The Physical Continuum of Several Dimensions 



I have explained in ' Science and Hypothesis ' whence we derive 

 the notion of physical continuity and how that of mathematical con- 

 tinuity has arisen from it. It happens that we are capable of dis- 

 tinguishing two impressions one from the other, while each is indis- 

 tinguishable from a third. Thus we can readily distinguish a weight 

 of 12 grams from a weight of 10 grams, while a weight of 11 grams 

 could neither be distinguished from the one nor the other. Such a 

 statement, translated into symbols, may be written: 



A = B, B = C, A <C. 



This would be the formula of the physical continuum, as crude 

 experience gives it to us, whence arises an intolerable contradiction 

 that has been obviated by the introduction of the mathematical con- 

 tinuum. This is a scale of which the steps (commensurable or incom- 

 mensurable numbers) are infinite in number, but are exterior to one 



