Thompson's Theory of Parallel Lines. 287 



long time before the public without being called in question, (the 

 publication by M. Legendre being believed to have been as early as 

 1815,) it may probably be set down as what there is no reason to 

 dispute. 



What is left then to be determined, is whether the last author has 

 been successful in getting over the remaining step ; or proving that 

 it the equal angles at the base of the quadrilateral figure aveless than 

 right angles, the angles opposite to the base (provision always in- 

 cluded that the sides do not meet) shall be greater than right angles. 



With this view, his proceeding is directed to establish, that if two 

 equal finite straight lines terminated in the same point, make any 

 angle that is less than the sum of two right angles, and this angle 

 be bisected by a straight line of unlimited length which may be 

 called the axis ; and at the outward extremity of each of the two 

 equal finite straight lines be added another straight line equal to the 

 first, having its extremity also terminated in the extremity of the 

 other, and making with it an angle equal to the first-mentioned angle, 

 and on the same side of the line; and at the outward extremity of 

 each of these, be added another straight line as before, and so on 

 continually ; and the extremities of every two equal straight lines so 

 successively added at one and the same time at the two ends of the se- 

 ries, be joined by a straight line or chord; each of these chords shall 

 make the angles at the two cusps, where it meets the equal straight 

 lines, equal to one another ; and (so long as none of the equal straight 

 lines meets the axis) the several chords shall in succession make 

 greater and greater angles at the cusp, each than the preceding. 

 This is done by the help of the inferences previously established 

 with respect to quadrilateral figures, and by joining the extremity of 

 one chord cross-wise with the extremity of the next, so as to make 

 one zigzag line which demonstrates almost by inspection the suc- 

 cessive enlargement of the angles. This is followed by a Scholium, 

 warning against the mistake of supposing that because it has been 

 proved that the angle will continually increase, it can therefore be 

 concluded that it will ever arrive at a certain specified magnitude ; 

 for evidence of which, reference is made in the Notes, to the well 

 known instance of the series ^+^+ \ + &c, in which the magnitude 

 of the sum goes on perpetually increasing, and yet never arrives at 

 being equal to double the first term. But this is stated to form no ob- 

 jection to examining the consequences which must result, if the 

 angle is ever found to arrive at a certain magnitude that may be 

 specified. 



The next Proposition is to prove, that in a series of equal straight 

 lines like the last-mentioned, if the angle at the cusp be ever equal 

 to, or greater than, half the angle made by the two first equal straight 

 lines ; the angular points shall lie in the circumference of a circle, 

 whose centre is in the axis, in the part of it which is cut off by the 

 chord ; and the series, being continued, shall at length meet the 

 axis. The demonstration, though lengthy when given with all the 

 forms, is such as will be easily supplied by any person in the habit 

 of geometrical solutions. 



The 



