96 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
for comparison in size. One magnitude is said to be greater than an¬ 
other when this other is congruent to a piece of the first. 
No two magnitudes have a ratio unless one can be congruent to, 
greater than, or less than the other. 
But which American geometry points out that, since no part of a circle, 
no arc, can be congruent to any sect, so no part of a circle can be equiv¬ 
alent to any sect in accordance with the definition of equivalent magni¬ 
tudes as such as can be cut into pieces congruent in pairs? But it follows 
that no circle has any ratio to its diameter from the assumptions in 
Euclid’s geometry. The very attribution to a circle of ratio to its 
radius makes new assumptions. 
High authorities (e. g . Duhamel) maintain that attributing measur¬ 
ability to curves, that the very ascription of length to a curve (that is, 
ratio to the unit sect,) involves the particular idea of a limit which itself 
involves the true conception of an infinitesimal. 
