SIMSON. 125 



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. <?., 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 

 demonstrated by supposing the thing true, and thus 

 reasoning till we find that the angle in a semicircle is 

 a right angle, a known truth. Lastly, suppose we 

 change the hypothesis, and leave out the position of 

 the point as given, and inquire after the point in the 

 given straight line from which a line being drawn 

 through a point to be found in the circle, the segments 

 will contain a rectangle equal to the rectangle under 

 the perpendicular segments we find that one point 

 answers this condition, but also that the problem 

 becomes indeterminate ; for every line drawn through 

 that point to every point in the given straight line has 

 segments, whose rectangle is equal to that under the 

 segments of the perpendicular. The enunciation of 

 this truth, of this possibility of finding such a point 

 in the circle, is a Porism. The Greek geometers of 

 the more modern school, or lower age, defined a 

 Porism to be a proposition differing from a local 

 theorem by a defect or defalcation in the hypothesis ; 



