SIMSON. 153 



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 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 pro- 

 blem 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, 

 and dealing a good deal with Harmonical proportions. If from any 

 point whatever out of the ellipse there be drawn a straight line in any 



