S1MSON. 469 



being ascertained by former discoveries. The thing 

 thus found, the point reached, was the discovery of 

 something which could by known methods be per- 

 formed, or of something which, if not self-evident, was 

 already by former discovery proved to be true ; and in 

 the one case a construction was thus found by which 

 the problem was solved, in the other a proof was ob- 

 tained that the theorem was true, because in both cases 

 the ultimate point had been reached by strictly legiti- 

 mate reasoning, from the assumption that the problem 

 had been solved, or the assumption that the theorem 

 was true. Thus, if it were required from a given point 

 in a straight line given by position, to draw a straight 

 line which should be cut by a given circle in segments, 

 whose rectangle was equal to that of the segments of 

 the diameter perpendicular to the given line the thing 

 is supposed to be done ; and the equality of the rect- 

 angles gives a proportion between the segments of the 

 two lines, such that, joining the point supposed to be 

 found, but not found, with the extremity of the dia- 

 meter, the angle of that line with the line sought but 

 not found, is shown by similar triangles to be a right 

 angle, i. e., the angle in a semicircle. Therefore the 

 point through which the line must be drawn is the 

 point at which the perpendicular cuts the given circle. 

 Then, suppose the point given through which the line 

 is to be drawn, if we find that the curve in which the 

 other points are situate is a circle, we have a local 

 theorem, affirming that, if lines be drawn through any 

 point to a line perpendicular to the diameter, the rect- 

 angle made by the segments of all the lines cutting the 

 perpendicular is constant ; and this theorem would be 



