40 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
It is familiar to all that geometers have adopted the square on the 
unit-line 'as the unit-surface, and the cube on the unit-line as the unit- 
solid. Also, the choice seems from every point of view the best both 
for computation and theoretical investigation. It would be well to re- 
flect, however, that scientific mensuration has discovered that, although 
some fraction of the perigon (e. g., the degree) is the unit-angle best 
adapted to calculation, yet a particular angle incommensurable with the 
perigon (the radian) is the unit-angle positively demanded for theo- 
retical analysis. And again, in like manner, the logarithmic base suit- 
able for calculation must be the same as that adopted for numerical 
notation, while the “natural” logarithmic base, an incommensurable 
number (e=l +T+i+ii+ii . . . ), is discovered to be intrinsically 
the base proper to analysis. 
The particular base, or radix, for the systematic expression of num- 
bers is to be chosen purely for convenience; nevertheless — and here one 
may see precisely the same principle I would apply in teaching men- 
suration — every teacher who has tried the method will testify to the gen- 
eral tonic effect and distinct elucidation resulting from finally teaching 
our decimal notation by using the same system for other bases than ten. 
It is in this particular system of using a basal number that the essence 
and merit of the beautiful algorithm, so familiar to children but so 
imperfectly understood by the majority of their teachers, consists, and 
by no means in the choice of the base ten. Our decimal notation com- 
mands admiration not because it is decimal, but because the orderly 
positions of the figures to the left or right of a point express ascending 
or descending powers of one basal number. A thoughtful consideration 
of the notation would enable anyone to adapt its system to any base; 
and so long as one feels hesitation in doing this, he may be sure he does 
not understand what he has deemed so familiar. Indeed, a really ade- 
quate understanding of the matter requires, still further, examina- 
tion of the system and existence-theorems of scalar notations, with the 
recognition of a radix notation as the special case where the scale merely 
repeats the same number. It is true that other bases (e. g., twelve) 
would afford greater facilities than the base ten; but no one thinks of 
attempting to change the confirmed habits of language. It is not as 
revolutionists that any teachers employ and advocate such comparisons, 
but upon the profound principle that comparison is essential to satis- 
factory knowledge. 
Without going too deeply into the matter, it is not out of place to 
point out how exceedingly profound this principle is. The common 
barrenness of many otherwise widely differing philosophies may be at- 
tributed to their divorcement of immediate apprehension and discursive 
thought. At the root of any nominalist logic lies the opinion that 
