Dr. Day on the Theory of Parallels. 157 



proofs, as Legendre's (by limits), Colonel Thompson's, ]\Jr. 

 Exley's, and lastly Mr. Stevelly's, the first being probably the 

 best in respect of simplicity. No one denies the adequacy of 

 the proof, but there is always some break or chasm in the argu- 

 ment which we must bridge over by an appeal to the ultimate 

 conception of space, and which may therefore as well be done at 

 once without all this din and preparation. It may, after all, be 

 doubted whether Euclid's twelfth axiom is not incomparably the 

 best of all that have been proposed, when the scope and require- 

 ments of elementary geometry are duly considered. I cannot 

 agree with Mr. Hennessy, that the doctrine of parallel lines would 

 be complete without it; for although by his definition parallel lines 

 would never meet, yet it is necessary for many of the demonstra- 

 tions of problems by construction, to know that there are no other 

 but parallel lines which do not meet, to say nothing of the proof 

 of the value of the three internal angles of a triangle. We want 

 an axiom which includes the essential conception of angle as well 

 as parallel straight line, and there is no advantage in separating 

 them. In one sense parallelism is only a particular case of lines 

 which meet : it is the limit of angularity, so that parallel and 

 right angle are only two particular extreme terms when the 

 angle equals 0° and 90°. By parallel lines, therefore, we mean 

 not only two straight lines which never meet, but lines one of 

 which cannot be cut by a straight line without its cutting the 

 other also. Let us imagine that Euclid was endeavouring to define 

 an angle, — how that it was made by the intersection of two lines 

 which continually diverged, and how this divergence necessarily 

 imphes unsymmetrical conditions on the two sides of any other 

 line cutting them transversely. Two straight lines can be con- 

 structed so as to cut a third, "both at right angles, making the 

 angles on each side of the cutting line respectively equal to two 

 right angles. These lines are parallel, and never meet. Any 

 change of position of these lines by which the two ends on one 

 side were made to approach, would be accompanied by a depart- 

 ure of the ends on the other side, by an enlargement of the 

 angles on the latter, and a contraction on the former side. What 

 more self-evident or obvious, than that as the ends have been 

 brought nearer to one another, the lines will meet at some finite 

 distance if continued ? And why should not Euclid make choice 

 of this as the most useful and convenient form of the assumption, 

 seeing that if by ])arallel you moan not meeting, you cannot 

 prove that the angles on one side the cutting line are exactly 

 two right angles, but only that if they make these angles such, 

 they are amongst the number of parallel lines ? 



Mr. llennessy'a definition, that parallels ai'C such that if they 

 meet a third right line the two interior angles on the same side 



