Statement of Principles. 
241 
was to Ptolemy, that was Lobachevsky to Euclid,” or when 
Prof. Sylvester of Oxford also compares Lobachevsky’s “ release 
of geometry from Euclid’s parallel axiom to Hamilton’s exten¬ 
sion of the power of multiplication. ” 
This assertion of the broadening of geometric truth calls for 
a brief statement of the essential principles of the Non-Euclidian 
geometry, as it was termed by Gauss, although Bolyai gave it 
the name Absolute Geometry, and Lobachevsky the somewhat 
more modest title Imaginary Geometry. This system, or these 
systems, of geometry are based upon a denial of the eleventh 
axiom of Euclid, known as the “ parallel axiom, ” which declares 
“if a straight line, falling upon two straight lines,makes two in¬ 
terior angles on the same side less than two right angles, these 
straight lines continually produced meet upon that side upon 
which the two angles are less than two right angles, ” an axiom 
much longer than the other axioms of Euclid, and so differing from 
them in character that it has always, as it were, rested under a 
suspicion as to its being truly axiomatic. In the second cen¬ 
tury, indeed, Ptolemy denied its axiomatic character, although 
by no means its truth, and attempted to prove it; in which at¬ 
tempt he has been followed by an almost unbroken line of 
seekers. The Italian priest, Saccheri, covered one hundred quarto 
pages with what he considered to be a proof. Even that king 
among mathematicians, Gauss, undertook the same task, and 
about one hundred years ago wrote " if we could prove that a 
rectilineal triangle is possible whose content may be greater 
than any given surface, then I am in condition to prove with 
rigor all geometry. Most ” he said, “ would indeed let that pass 
as an axiom. I not. It might well be possible that, how far 
soever we took the three vertices of the triangle in space, yet 
this content was always below a given limit. ” In 1621 Sir Henry 
Saville spoke of this axiom as “ a blemish on the most beautiful 
body of geometry; ” and this view has been so general that, as 
has before been said, there has been an unbroken succession of 
attempts to remove that blemish, with results mostly of such a 
character that, when printed and distributed, they have been 
consigned by their recipients to some limbo dedicated to the 
productions of Lawrence Sluter Benson who, if still living, is 
16 
