igo 7] Henderson — Foundations of Geometry. 9 
“Now let us suppose that, while one part of AE, viz., BE, 
revolves into the position BF, another little bit of it, viz., 
AG, revolves, through an equal angle, into the position AH; 
and that, while CF revolves into the position of lying along 
CD, AH revolves — and here comes the fallacy. 
“You must not say ‘revolves through an equal angle, into 
the position of lying along AD,’ for this would be to make AH 
fulfil two conditions at once . 
“If you say that the one condition involves the other, you 
are virtually asserting that the lines CF, AH are equally 
inclined to CD — and this in consequence of AH having been 
so drawn that these same lines are equally inclined to AE. 
“That is, you are asserting, ‘A pair of lines which are 
equally inclined to a certain transversal, are so to any trans- 
versal.’ [Deducible from Euc. I, 27, 28, 29].” 
Thousands of mathematicians have tried in vain to prove 
something that only a genius could see was indemonstrable. 
The history of the evolution and exfoliation of that fertile 
idea is of very great interest to the mathematician of today, 
especially in view of the fact that beyond contradiction the 
most original researches of the last quarter of the nineteenth 
century pertain to the non-Euclidian geometry. 
The most notable attempt to demonstrate Euclid’s parallel- 
postulate that has been preserved to the world is embodied in 
a book entitled Euclid Vindicated from every Blemish, by 
a Jesuit priest named Hieronymus Saccheri (1667-1773).* 
He was in close association with the great Italian geometer 
Giovanni Ceva (through his brother Tommaso), whose name 
a celebrated theorem bears; and by purely geometrical meth- 
ods in Euclidian style, he sought to apply the reductio ad 
absurdum method to the problem of the parallel-postulate. 
His method is essentially as follows: At the end-points of a 
sect AB erect two equal perpendiculars AC and BD on the 
*Euclides ab omni naevo vindicatus ; sive conatus geometricus quo stabili- 
untur prima ipm universae geometriae principia. Auctore Hieronymo 
Saccherio Societatis Jesu, Mediolani. 
