508 SIMSON. 



point to the same line ; and as these propositions had 

 not been in the elements, (though the one follows 

 from the definition of the circle, and the other from 

 the theorem that the three angles of a triangle are 

 equal to two right angles,) and as it might be supposed 

 that two or more circles, like two or more ellipses, 

 might be drawn on the same axis, therefore the lem- 

 ma is demonstrated by a construction into which the 

 centre does not enter. Again, in applying this lemma 

 to the porism (the proportion of the segments given 

 by similar triangles), a right angle is drawn at the 

 point of the circumference, to which a line is drawn 

 from the extremity of a perpendicular to the given 

 line ; and this, though it proves that perpendicular to 

 pass through the centre, unless two semicircles could 

 stand on the same diameter, is not held sufficient ; but 

 the analysis is continued by help of the lemma to show 

 that the perpendicular to the given line passes through 

 the centre of the given circle, and that therefore the 

 point is found. It is probable that the author began 

 his work with a simple case and gave it a peculiarly 

 rigorous investigation in order to explain, as he im- 

 mediately after does clearly in the scholium already 

 referred to, the nature of the porism, and to illustrate 

 the erroneous definitions of later times (veorepiKoi) of 

 which Pappus complains as illogical. 



Of porisms, examples have been now given both in 

 plain geometry, in solid, and in the higher : that is, 

 in their connexion both with straight lines and 

 circles, with conic sections, and with curves of the 

 third and higher orders. Of an algebraical porism it 

 is easy to give examples from problems becoming inde- 



