82 GREEK GEOMETRY. 



hypotenuse, is equal to half that hypotenuse. Now this 

 follows, if the segment containing the right angle be a 

 semicircle, and it might be thought that this should be 

 assumed only as a manifest corollary from the proposition, 

 or. as the plain converse of the proposition, that the angle in 

 a semicircle is a right angle, but rather as identical with that 

 proposition ; for if we say the semicircle is a right-angled 

 segment, we also say that the right-angled segment is a semi- 

 circle. But then it might be supposed that two semicircles 

 could stand on one base ; or, which is the same thing, that 

 two perpendiculars could be drawn from one 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 

 lemma 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 circum- 

 ference, 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 immediately after 

 does clearly in the scholium already referred to, the nature of 

 the porism, and to illustrate the erroneous definitions of later 

 times (reortpiKoi) 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 con- 

 nexion both with straight lines and circles, with conic sec- 



