202 PROCEEDINGS OF THE AMERICAN ACADEMY. 



A dual definition for angle area led to, 



Angle area = ABc = AC6 = BCa. 



Coordinate systems. 



The fact that the locus of points equidistant from a given point is a 

 line suggests that one convenient coordinate system would be a 

 point and a line / passing through it. The coordinates of a point A 

 are then the distance 0A and the angle which 0A makes with I. 

 These numbers will fix a point uniquely. The distance between 

 two points (pi, 0i) and (p 2 , 2 ) is then, 



d = pi — p\. 



The angle between the two lines is, 



d = 02 — d\. 



The element of arc is, 



ds = dp. 



This coordinate system is not suitable for a great many problems, 

 for the coordinates of the points of the line OF are indeterminate, 

 each point having the coordinates (0, 00). For this system of 

 coordinates the equation of a straight line is not linear which is 

 another disadvantage. 



A second and more convenient coordinate system is obtained by 

 taking the three points (not on/), F, P (on/), as the vertices of a 

 triangle of reference. Number the points along OP by beginning with 

 and laying off equal distances along OP. Draw lines from F to 

 these points and number them to correspond. Number the lines 

 through by beginning with OP and constructing equal angles about 

 0. Number the points in which these lines cut / to correspond. 

 Finally number the points on OF by taking an assigned interval 0Q 

 as unity and measuring off equal intervals by choosing Qi, to satisfy 

 the relation (/Q | OQO = -1, then choose Q 2 so that (/Qi | 0Q 2 ) = 

 — 1 ecc. If these relations are satisfied we will say that the intervals 

 0Q, QQi, Q1Q2, ecc. are equal. Draw lines from P to these points 

 and number them to correspond. The coordinates of a point will 

 then be the projection of the point on OF and OP from P and F. We 

 will call the numbers along OF, y, and those along OP, x, then it is 

 seen from the fundamental properties of distance that the distance 

 between {x\, y\) and (.r 2 , 2/2) is, 



d = x\ — x 2 , 



