6 
Journal of the Mitchell Society. 
[May 
mathematical world exhausted itself in the effort to prove 
Euclid’s celebrated parallel-postulate. Ptolemy, the great 
astronomer, wrote a treatise purporting to prove it; and 
Nasir Eddin (1201-1274), whose work on Euclid in Arabic 
was printed at Rome in 1594, sought to dispense with the 
problem of parallelism, by taking his stand upon another pos- 
tulate: that two straight lines which cut a third straight 
line, the one at right angles, the other at some other angle, 
will converge on the side where the angle is acute, and diverge 
where it is obtuse. Other mathematicians, notably John 
.Wallis whom I claim as an ancestor, sought to turn the flank 
of the difficulty by identifying the problem of parallels with 
the problem of similitude. In general, we may say that the 
problem was attacked from three sides. 
First, there were those who sought to substitute a new 
definition of parallels for Euclid’s, which reads (I, Def. 35): 
“ Parallel straight lines are such as are in the same plane, 
and which being produced ever so far both ways do not 
meet” 
To cite a few classic definitions, Wolf, Boscovich, and T. 
Simpson use the following: “Straight lines are parallel 
which preserve the same distance from each other.” But 
this is begging the question, asHalsted has remarked, since it 
assumes a definition, viz.: “Two straight lines are parallel 
when there are two points of the one on the same side of the 
other from which the perpendiculars to it are equal;” and at 
the same time assumes a theorem: “All perpendiculars from 
one of these lines to the other are equal.” Those geometers 
who assume that parallel lines have the same direction are 
guilty of a petitio principii, in assuming (Varignon and 
Bezout) the definition that “parallel lines are those that make 
equal angles with a third line,” and also in assuming the 
theorem that “Straight lines that make equal angles with 
one transversal make equal angles with all transversals.” 
The second method of attack, far more logical, was to pro- 
