Page:Carroll - Euclid and His Modern Rivals.djvu/293

Rh . II. 8.

Through a given point, without a given Line, a Line may be drawn such that the two Lines are equidistantial from each other.

For, if through the given point there be drawn a transversal, there can also be drawn through it a Line such that the two Lines make equal ∠s with the transversal;

and this Line will be such that the two Lines are equidistantial from each other.

Therefore, &c.

. II. 17.

A Line cannot recede from and then approach another; nor can one approach and then recede from another on the same side of it.



If possible, let ABC first recede from, and then approach, DE; that is, let the perpendicular BG be > each of the two perpendiculars AF, CH.

From GB cut off GK > each of the two, AF, CH.

Now a Line may be drawn, through K, equidistantial from DE;

and the points A, C will lie on the side of it next to DE, and B on the other side;

∴ it will cut AB between A and B, and BC between B and C.

Let L, M be the points of intersection; and join LM;

∴ the 2 Lines LBM, LKM contain a space; which is absurd.

Similarly it may be proved that ABC cannot first approach and then recede from DE on the same side of it.

Therefore a Line &c.