164 On the Theory of 
But, to return to geometry, it may be worth while to inquire, 
what has been the couduct of .the ancients in regard to the dif- 
ficulties that present themselves in that science. They have 
either overcome the obstacles that obstructed their progress ; or, 
when this was impossible, they have fairly laid down what they 
could not demonstrate as principles to be assented to by their 
disciples in the further prosecution of their researches. Of 
the first of these ways of proceeding we have instances in the 
investigations relating to incommensurable quantities, and to the 
proportion between the pyramid and the prism. In both these 
cases too the ancient geometers have succeeded by the same 
means, namely, by employing the indirect mode of investi- 
gation. 
Archimedes furnishes an example of the other way of pro- 
ceeding in the principles prefixed to his treatise on the Sphere 
and Cylinder. These principles, on which are founded the most 
considerable of his discoveries in pure mathematics, are really 
theorems which ought to be demonstrated, but which the an- 
cient geometry affords no means of proving. Another instance 
of the same kind we have in Euclid’s manner of treating parallel 
lines. That geometer has demonstrated, in the 17th of the first 
book, that any two angles of a triangle are together less than 
two right angles. The plan of his work required that the con- 
verse of the same proposition should be proved; and it is the 
want of this proof, which no geometer has been able to invent, 
that constitutes the difficulty in the theory of parallel lines. 
Euclid has therefore, in the 12th Axiom, laid down, as a princi- 
ple to which the assent is demanded, the proposition of which 
no demonstration can be found. It is no doubt inaccurate to 
class a principle of this kind with the axioms to which it has no 
affinity; but this is an objection of no moment, when the inten- 
tion of the author is understood. 
Many attempts have been made by succeeding geometers to 
remove the defect found in Euclid’s doctrine of parallel lines. 
New definitions of the straight line have heen imagined; and. 
new axioms, or rather new principles of reasoning, have been 
proposed; but none of these expedients have been attended with 
complete success. On a deliberate view of the case the prefer- 
ence must, I think, be given to Euclid’s manner of treating the 
subject ; because it places before the student without disguise 
the true nature and origin of the difficulty. 
Legendre, in the first nine editions of his Geometry, has treated 
parallel lines in a manner that is both new and seems to be more 
intimately connected with the real cause of the difficulty than 
any other hitherto proposed, The foundation of it is to ia 
inde- 
