Thompson's Theory of Parallel Lines. 289 



ence is made to the preceding demonstrations to show, that the se- 

 ries being continued must meet the axis, and the angular points lie in 

 the circumference of a circle ; whence it is inferred, that the sides 

 opposite to the bases of the quadrilateral figures must also meet the 

 axis, and form an interior polygon to the series formed by the bases. 

 From which it is concluded, that these sides cannot make with each 

 other, angles equal to two right angles, for then they would be in 

 one straight line, and this straight line, with the axis, would inclose 

 a space j and, a fortiori, as before, they cannot make angles less 

 than two right angles; wherefore they must make greater, or the an- 

 gles of the quadrilateral figures which are opposite to their bases must 

 be each greater than a right angle. In a Note, some reasons are 

 advanced for believing, that the third of these cases, or that where 

 the moving line is supposed to quit the series, might be got rid of by 

 demonstrating, that the moving line must always pass through at 

 least one pair of cusps after passing beyond the extremity of the side of 

 the quadrilateral figure which was produced to make the axis j and con- 

 sequently that the other two cases may be made to include all that 

 in reality are possible. 



If all this is considered as established, it is manifestly an easy step 

 to the demonstration that if the equal angles at the base of the quadri- 

 lateral figure are right angles, the angles opposite to the base must 

 be also right angles ; and thence, that the side opposite to the base 

 must be equal to the base. After this it may be shown, that the 

 angles of any right-angled triangle must be equal to two right angles j 

 by completing a quadrilateral figure, which shall have the angles at 

 the base right angles. And the same may be proved in any other 

 triangle, by selecting a side that lies between two acute angles, and 

 drawing a perpendicular to it from the angular point opposite ; which 

 will divide the triangle into two right-angled triangles. 



If all this is supposed to hold good, the Proposition on Parallel Lines 

 commonly known as the Twelth Axiom, may be established in the 

 case where one of the angles is a right angle, by drawing a per- 

 pendicular from any point in the line which makes the angle less 

 than a right angle, to the line which intersects the two; and thence 

 establishing successive ranks of quadrilateral figures, of which all 

 the opposite sides may, from the past Propositions, be proved equal, 

 and all the angles right angles ; and showing that their diagonals 

 taken in an oblique succession shall all lie in one straight line, and if 

 the number of quadrilateral figures is continued, shall of necessity 

 cut the other line. After which, the remaining case, or that where 

 neither of the angles is a right angle, may be solved by the assistance 

 of the first. 



As is intimated in the Preface, it is probable that Ihese demon- 

 strations, if no flagrant fallacy should be detected in them, may be 

 found capable of concentration and abridgement. 



N. S. Vol. 8. No. 46. Oct. 1 830. 2 P XLVI. Pro- 



