ANCIENT ANALYSIS. PORISMS. 59 



segments of the two lines, such that, joining the point sup- 

 posed to be found, but not found, with the extremity of the 

 diameter, 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 semircircle. 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 

 rectangle made by the segments of all the lines cutting the 

 perpendicular is constant ; and this theorem would be demon- 

 strated 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 enuncia- 

 tion 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 pro- 

 position differing from a local theorem by a defect or defalca- 

 tion in the hypothesis ; and accordingly we find that this 

 porism is derived from the local theorem formerly given, by 

 leaving out part of the hypothesis. But we shall afterwards 

 have occasion to observe that this is an illogical and imperfect 

 definition, not coextensive with the thing defined ; the above 

 proposition, however, answers every definition of a Porism. 



The demonstration of the theorem or of the construction 

 obtained by investigation in this manner of proceeding, is 



