GEORGE BRUCE HALSTED-TIIE NEW MATHEMATICS. 
95 
Such an idea is contrary to their preconceived notions, and appears to 
them to contradict the testimony of their senses. 
So their straight line possesses no more continuity than the series of 
rational fractions. What do you gain by holding up your colored scrap 
to the color-blind ? 
No one before Pythagoras, no one since Pythagoras, untaught, ques¬ 
tions the possibility of expressing all size-relations among lines in terms 
of rational number. 
The text-books of America, like those of France and Germany, have 
lost Euclid’s masterful pure geometric theory of proportion for the study 
of size-relations without reference to measure. Following Newton’s 
paradox without knowing it, they make a ratio, not a proportion, the 
primary idea, and define a ratio as a quotient, a number. No wonder 
that Dedekind says this doctrine of ratio can only be clearly developed 
after the introduction of irrational numbers. Newton, leaning on 
Euclid’s Fifth Book, got his irrational number, by an innovation, from 
Euclid’s geometric doctrine of proportion. American books get then- 
proportion from the general irrational number, and then get their irra¬ 
tional from this proportion. To see that this argumentum in circulo fails 
in* both directions around the circle, we have only to meditate on the 
obvious fact that for a great part of the science of space the continuity 
of its forms is not a necessary presupposition. 
In illustration of this, Dedekind gives the following example: 
If we take any three non-collinear points, with only the specification, 
that the ratios of the sects AB, AC, BC, are algebraic numbers, and con¬ 
sider as present in space only those points M, for which the ratios of 
AM, BM, CM, to AB are likewise algebraic numbers, then the space 
consisting of these points is throughout discontinuous [it lacks all points 
D, for which a ratio, as AD to AB, is a transcendent number such as pi 
or e]; yet despite the discontinuity, the perforation, of this space, all 
constructions occurring in Euclid are in it just as achievable as in per¬ 
fectly continuous space. The discontinuity of this space would therefore 
never be noticed, never be discovered, in Euclid’s science. 
“ Um^o schoener erscheint es mir, dass der Menscli ohne jede Vorstell- 
ung von messbaren Groessen, und zwar durch ein endliches System ein- 
facher Denkschritte sich zur Schoepfung des reinen, stetigen Zalilenreiclies 
aufschwingen kann; und erst mit diesem Hiilfsmittel wird* es ihm nach 
meiner Ansicht moglich, die Vorstellung vom stetigen Raume zu einer 
deutlich auszubilden. ’ ’ 
But this is just what has never yet even been attempted in any text¬ 
book in the English language. 
All geometries, the best and the worst, make congruence their basis 
