21 4 This form of the 12th Axiom necessitates the introduc tion of the following propositions preliminary to the 20th Pro position of the First Book:- (I) Of all the straight lines that can be drawn from a given point on a given braight line, the perpendicular is the shortest. (2) The hine joining the free extremities of bwo equal per pendiculars to a given straight line makes equal angles with the perpendiculars. (B) The line joining bhe free extremities of two equal per- pendiculars to a given straight line makes right angles with the perpendiculars. (4) The opposite sides of a rectangle are equal. (5) If two perpendiculars be drawn to a line and a straight line be drawn across the perpendiculars, of the four angles made by the line with each perpendicular, the alternate angles shall be equal, the exterior angle shall be equal to the interior and opposite angle upon the same side of the line and the two interior angles upon bhe same side of the line shall be together equal to #wo right angles (6) If the four angles formed by the intersection of two lines be not right angles, then a perpendicular on one of bhe lines shall meet the other line in the direction of the acute angle. (7) If a straight line falls upon two other straight lines and if the interior angles on one side are less than two right angles, then the two straight lines shall meet in that direction only. 5. In the proof of propositions throughout the book, no authorities are given anywhere. For the sake of conciseness a few intermediate steps, which may be understood without being mentioned, are omitted.
