Manchester Memoirs, Vol. Ixvi. (1922), No. 2 13 



this fact (3, 73f), and Dr. J. Varendonck has recently recorded 

 his own prooress in this respect (5, 90). 



*' I have even noticed of late that when I happen to read 

 poetry now I am able voluntarily to transform the poet's 

 words into visual imao-es, which adds a hitherto unknown 

 charm to the readino-." 



It iseems, indeed, quite possible that any kind of imagery, 

 even that in which a man may consider himself to be very poor, 

 may be enormously developed during life, so that the facts 

 seem to justify little more than the belief that congenital differ- 

 ances in visualizing power may be significant. 



The progressive complication of the functions of number- 

 forms. 



In books upon the general theory of psychology little 

 attention seems to have been paid to the significance of the 

 number-form as an excellent ' objectifiable ' example of a 

 vehicle of simple or complicated meanings ; an example, more- 

 over, the nature of which can be made plain to other people. 

 An examination of these forms illustrates clearly the different 

 degrees of complexity which the ' image-meaning ' relation 

 may reach. The simplest representation of quantities by the 

 number-form occurs when the actual numbers are seen 

 arranged in fixed spatial relations. Rather more complex is 

 the representation when the numbers are not visualized but 

 only thought of, their positions in space being seen mentally. 

 At a third stage these positions themselves acquire a relative as 

 well as an absolute significance, when the actual quantity 

 attributed to any position mav be merelv a special instance of 

 some more general meaning. 



Such a use of his number-form is often made bv Prof. 

 Tattersall. Any position on the form, though absolutely fixed 

 in space, may represent any one of a large number of related 

 quantities, or even the subject or theme of which a particular 

 quantity is merelv one characteristic.^ Such a form, represent- 

 ing almost any kind of related quantity, has reached another 

 stage in a line of development which, if continued, would 

 lead through infinite gradations to the formation of an image 

 which carries the unquantitative essence of mathematical 

 conceptions. A very high position in this scale of evolution 



1. An example of this variability of meaning of a fixed point on a number- 

 form has been given on p. 5. 



