10 
Journal of the Mitchell Society. [. May 
same side of AB. Join C and D by a straight line; and it 
easily follows that the angle ACD is equal to the angle BDC. 
Now there are three possibilities: (1) The angle ACD is 
acute; (2) the angle ACD is obtuse; (3) the angle ACD is a 
right angle. He undertook to prove the absurdity of the first 
two possibilities so as to leave only the third possibility, viz., 
that the two angles ACD and BDC are each right angles. 
He pursued the lines of argument, following from the first 
two assumptions, at some length — for his book was more 
than a hundred pages long; but was doubtless amazed to dis- 
cover that for quite a time he was unable to involve himself 
in any logical contradiction. In the event, certain of his con- 
clusions were erroneous, and led him to believe that he had 
actually proved the parallel-postulate. What he really did 
do was to identify the assumption of the right angle with the 
parallel-postulate, thus showing the two to be mutually inter- 
changeable postulates. 
In 1766, Johann Heinrich Lambert wrote his theory of par- 
allel lines, in which he starts from the notion of the sum of 
the angles of a triangle being equal to 180 degrees. If the 
sum is equal to 180 degrees, the triangle is a figure in a 
plane; if the sum is greater than 180 degrees, the triangle is 
on a sphere; if the sum is less than 180 degrees, the triangle 
is on the surface of an imaginary sphere (radius equal to the 
square root of minus one)— Lobatchevsky — Bolyai “imaginary 
geometry,” so called because its trigonometric formulas are 
those of the spherical triangle if its sides are imaginary. As 
to the third hypothesis, Lambert naively said: “There is 
something attractive about this which easily suggests the 
wish that the third hypothesis might be true.”* 
France contributed little to the solution of the problem; 
recognition, however, should be given to Legendre, who stud- 
♦Compare The Philosophical Foundations of Mathematics, by Dr. Paul 
Carus; The Monist, vol. 13, pp. 273-294; 370 397; 493-522, to which I am 
indebted. I once had the pleasure of hearing Dr. Carus lecture on this 
subject before the Mathematical Club of the University of Chicago. 
