GEORGE BRUCE HALSTED-THE NEW MATHEMATICS. 
91 
A word of the same sort as our word “to score ” is the old Greek word 
for counting pempazo, to finger-fit-by-fives (or hands). 
To five would be the method of races poor in number sense, e. g., the 
Romans; to score would be the method of barefooted or sandaled races, 
for example Mexicans. But most counting used as base the ten digits, 
and so our arithmetic is decimal. It is not that a decimal system, in so 
far as decimal, is a good system. That is one of the widespread errors 
taught to our children. Twelve would have been unquestionably a bet¬ 
ter base. If only our ancestors had had the good fortune to sprout an¬ 
other finger on each hand, there would now be no decimal system, and 
a vast simplification in arithmetic would have occurred. But the ten 
fingers of our ancestors have so grown into our brains, that I think all 
hope of relief now from decimality is chimerical. But this hopelessness 
of our hereditary imperfection is no adequate reason for lauding 
“ decern.’’ Do not attribute to ten the positional notation for number 
made possible by that splendid invention of the Hindoos, the zero, 
nought, cypher, which a digital point empowers to run down as easily 
as up. 
The very ease and facility thus attained for cyphering prepared the 
way for an extension of meaning to number, or rather the attribution to 
it of a second distinct but allowable meaning. 
From the time of Euclid the world has had an exquisite scientific treat¬ 
ment of the exact relation of any two magnitudes comparable by apposi¬ 
tion of their multiples. This relation, their ratio, Euclid always pictured 
as the fingers picture number, by exhibiting another group also possess¬ 
ing the quality; he pictured the ratio of any two magnitudes by the ratio 
of two sects (definite pieces of a straight line). 
The modern era of the world, the scientific, dates from 1637, when 
Descartes published his system of conditions which we now interpret 
as giving to every point in a plane a distinct name consisting of two 
numbers, and to every pair of numbers a point. His conventions, 
though for his use explicable, and by him explained, as a geometric al¬ 
gebra operating with sects, yet get their dual power only when seen as 
setting up a unique one-to-one connection between number-pairs and 
points, so making algebra talk geometry, and inversely, geometry talk 
abgebra. For example, the equation Ax-t-B?/j-C=0, representing each 
pair of numbers which jointly satisfy the equation, pictures now an ag¬ 
gregate of points, which are on a straight line, while number is discrete, 
but which are the straight line if number be continuous. Descartes per¬ 
haps never passed beyond Euclid’s representation of the ratio of two 
magnitudes by two other magnitudes, never reached the conception of 
the systematic representation of the ratio of two magnitudes by one mag¬ 
nitude, a step like that from picturing the numeric quality of a group by 
