404 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



Figure 11. 



characteristic of our rotation, namely that two perpendicular lines 

 approach each other and the fixed line between them scissor-wise, 



as may be seen, in Figure 11, where OC and 

 OD become respectively OC and OD', OC" 

 and OD", • • • • The pseudo-circles traced by 

 OC and OD may be called conjugate pseudo- 

 circles, since the interval OC equals the 

 interval OD, the lines CD, CD', • • • • , being 

 singular, and bisected by a fixed line. 



Since two mutually perpendicular lines ap- 

 proach, during rotation about their point of 

 intersection, the same fixed line, we may 

 extend our definition of perpendicularity by 

 regarding every singular line as perpendicular 

 to itself. This extension is also suggested by 

 the fact that the fixed line may be considered an asymptote of a 

 pseudo-circle. Special caution must be given against the idea that a 

 singular line of one class is perpendicular to a singular line in the 

 other class. The peculiarities of singular lines will become clearer in 

 the work on vector analysis. 



12. A triangle of which two sides are perpendicular will be called 

 a right triangle, and the third side will be called the hypotenuse. A 

 parallelogram of which the two adjacent sides are perpendicular and 

 of equal interval will be called a square. The following theorem is 

 obvious : 



XVm. One diagonal of every square is a singular line and the 

 other diagonal is a singular line of the other class. 



XIX. Pythagorean Theorem. The area of the square on the 

 hypotenuse of a right triangle is equal to the difference of the areas of 

 the squares on the other two sides. 



For by XVIII the diagonals of the squares are lines of fixed direction, 

 and hence parallel each to each. The squares on the two legs are 

 similar. And the proposition is evidently a special case of XI. (In 

 Figure 7 if the dotted lines are singular lines, the lines AC and BC 

 are so drawn as to be approximately perpendicular.) 



XX. Any two squares whose sides are of unit interval are equal in 

 ai'ea. 



For by suitable translation and rotation one ma,y be brought into 

 coincidence with the other. The unit of area will henceforth be taken 

 as the area of a square whose sides are of unit interval. Hence 

 follows: 



