456 INTRODUCTION TO PHILOSOPHY OF SCIENCE 



things perceived has for its border the realm of things con- 

 ceived." For example, "perfect solids, like perfect gases, are 

 nothing but limits, sheer ideals without existence in the 

 world of sense, pure concepts in the domain of reason." 

 These forms "are never actually realized in the subrational 

 world of sense: they are there but indicated as limits be- 

 yond." 1 



On the other hand there is the method of limits, applied 

 within the realm of the rational. If there be inscribed in a 

 circle an equilateral triangle, then a regular hexagon, then a 

 polygon of a dozen sides, and so on, forever, there is created a 

 series of a certain type. ' In respect of size, these approach 

 nearer and nearer, as close as we please, to the size of the 

 circle's area, yet they remain inferior to it forever. And we say, 

 in technical language, that the circle's area is the sequence's 

 limit. It is important to note that the sequence's limit is not 

 a term in the sequence, for all these terms are polygonal 

 areas — shapes bounded by polygons — but that of the circle 

 is not, for the circle is not a polygon." 2 If the series of poly- 

 gons be called the Domain of Polygonal Areas, one may say 

 that the circle does not lie in this realm but upon its border. 

 "Here, then, we have a clear presentation, within a given 

 domain, of something that is not within: we have a clear 

 presentation, by the law of an inner sequence, of a limit on 

 the rim — of an ideal, if you please, which, so long as we 

 operate within the domain, may be aspired unto, approached 

 and pursued forever, but can never be attained." 3 



These, then, are the data : two types of serial order exhibit- 

 ing limits. In the one case the limit is outside of the realm in 

 which the series itself resides; in the other the limit is within 

 the realm though outside of the series itself. What is com- 

 mon to the two cases is the fact that a serial principle of a 

 certain kind apparently affords a basis for an inference be- 

 yond the series itself to an entity which is demanded by the 

 series yet exhibits properties essentially different from the 

 members. Keyser's next step is to generalize this principle 



1 Ibid., pp. 61-63. 2 Ibid., p. 55. 3 Ibid., p. 56. 



