CORRESPONDENCE, 1856-00. 313 



and oo 3 , &c. But quantity which changes sign through infinity 1862. 

 passes though ^. This will become a very important distinction. 



The ^ of common algebra is high up above the of the differ- 

 ential calculus. 



I am rid of all fear about oo 2 . I believe in oo, oo 2 , oo 3 , &c., &c.; 

 and I intend to write a paper against the skim-milky, fast-and- 

 loosish mealy-mouthedness of the English mathematical world 

 on this point. My assertion is that the infinitely great and 

 small have subjective reality. They have objective impossibility 

 if you please ; or not, just as you please. 



I have first to remove an ambiguity, which has played a large 

 part in causing confusion. To imagine is originally to form an 

 image in the mind. But it has been transformed into a synonym 

 of to conceive, to form a concept. The distance from here to the 

 sun is a concept. I have no image of it. But of six feet I have 

 both image and concept when I shut my eyes. Now many per- 

 sons, when they cannot image, speak as if they could not conceive, 

 and use the ambiguous word imagine. We cannot, they say, 

 imagine infinite space. I grant they can't image it, but I am sure 

 by their modes of denial that they have a conception of it. Locke 

 and others affirm that we arrive at the notion of infinity by finding 

 out that when, say, we add number to number, we find the 

 succession incapable of termination, and so fashion interminability 

 in our minds. I say the process is precisely the reverse. If it 

 were not for our conception of infinity we should not know the 

 interminability. 



Who ever tried up to 10,000,000,000,000,000 ? It is certainly 

 not experience. If any one were to affirm that 10 16 is only a 

 symbol, and that any one who should try would find himself 

 brought up by the nature of things, Locke has no answer, 

 unless, as would probably be the case, he should ask permission 

 to bring on the conception of infinity. 



I therefore affirm the concept infinite as a subjective reality 

 of my consciousness of space and time, as real as my conscious- 

 ness of either, because inseparable from my consciousness of 

 either. When, therefore, I think of a finite space say a cubic 

 foot if I compare it with the totality of space, I say infinitely 

 small ; if I compare the totum with it I say infinitely great. 



Now comes a postulate on which there may be a fight. Let 

 A and B be two magnitudes, any whatsoever, arid a third, 

 also any whatsoever. Let these magnitudes be concepts, imagin- 



