78 GREEK GEOMETRY. 



being capable of one or two solutions, to its being capable of 

 an infinite number. Thus it would be no porism to affirm 

 that an ellipse being given, two lines may be found at right 

 angles to each other, cutting the curve, and being in a pro- 

 portion to each other which may be found : the two lines are 

 the perpendiculars at the centre, and are of course the two 

 axes of the ellipse ; and though this enunciation is in the 

 outward form of a porism, the proposition is no more a porism 

 than any ordinary problem ; as that a circle being given, a 

 a point may be found from whence all the lines drawn to the 

 circumference are equal, which is merely the finding of the 

 centre. But suppose there be given the problem to inflect 

 two lines from two given points to the circumference of an 

 ellipse, the sum of which lines shall be equal to a given line, 

 the solution will give four lines, two on each side of the 

 transverse axis. But in one case there will be innumerable 

 lines which answer the conditions, namely, when the two 

 points are in the axis, and so situated that the distance of each 

 of them from the farthest extremity of the axis is equal to the 

 given line, the points being the foci of the ellipse. It is, 

 then, a porism to affirm that an ellipse being given, two points 

 may be found such that if from them be inflected lines to any 

 point whatever of the curve, their sum shall be equal to a 

 straight line which may be found ; and so of the Cassinian 

 curve, in which the rectangle under the inflected lines is 

 given. In like manner if it be sought in an ellipse to inflect 

 from two given points in a given straight line, two lines to a 

 point in the curve, so that the tangent to that point shall, 

 with the two points and the ordinate, cut the given line in 

 harmonical ratio ; this, which is only capable of one solution 

 in ordinary cases, becomes capable of an infinite number when 

 the two points are in the axis, and when the ellipse cuts it ; 

 for in that case every tangent that can be drawn, and every 

 ordinate, cut the given line harmonically with the curve 

 itself.* 



* The ellipse has this curious property, which I do not find noticed by 

 Maclaurin in his Latin Treatise on Curve Lines appended to the Algebra, 



