182 Proceedings of the Royal Society of Edinburgh. [Sess. 
who extended the system were using the signs of the higher values as 
symbols to which no series of actual gestures could possibly correspond — 
just as we use the terms “ five thousand miles ” and “ forty thousand horse- 
power” without implying that the primitive bases for these values (the 
human stride and the drawing power of a horse) have been employed in 
the computation. 
For, just as each new gesture in the series which we have considered 
becomes the basis of, and means to, a further development, so also the 
gesture stage as a whole gives place to the ideographic or symbolic stage ; 
and that again gives us, on the one hand, alphabetic characters, and, on the 
other, numerical figures. A further stage is attained when the written or 
spoken word becomes the basis and expresses some abstract idea that could 
never be represented by any of the preceding symbols ; or, in the case of 
numerical signs, when, as in algebra, the figures themselves are in turn 
superseded by letters or arbitrary symbols, denoting some wholly abstract 
conception. 
( Issued separately February 1 , 1908 .) 
