212 On an Instantaneous Demonstration o/Tascal's Theorem. 



angulo ex AB in lineam datum CK (vel ^BM) problema construi 

 nequit." 



Though the determinations are very neatly given here, the 

 circumstance of the double solution has escaped his notice, 

 viz. the point of intersection of CH with the remaining semi- 

 circle, as represented by the accented letters, which I have put 

 in for the purpose of showing it. It will be worth the while 

 of the younger geometrical reader to examine this case, and 

 discover whether this second solution be that of the proposed 

 problem or of a collateral one. It involves no material diffi- 

 culty. 



Shooter's Hill, Aug. 15, 1850. 



XXV. An Instantaneous Demonstration of Pascal's Theo- 

 rem by the method of Indeterminate Coordinates, By J. J. 

 Sylvester, M.A., F.ILS* 



THE new analytical geometry consists essentially of two 

 parts — the one determinate, the other indeterminate. 



The determinate analysis comprehends that class of ques- 

 tions in which it is necessary to assume independent linear 

 coordinates, or else to take cognizance of the equations by 

 which they are connected if they are not independent. The 

 indeterminate analysis assumes at will any number of coordi- 

 nates, and leaves the relations which connect them more or 

 less indefinite, and reasons chiefly through the medium of the 

 general properties of algebraic forms, and their correspond- 

 encies with the objects of geometrical speculation. Pascal's 

 theorem of the mystic hexagon, and the annexed demon- 

 stration of its fundamental property, belong to this branch of 

 the subject, and afford an instructive and striking example of 

 the application of the pure method of indeterminate coor- 

 dinates. 



Let <r, y 9 z 9 t, u, v be the sides of a hexagon inscribed in the 

 conic U. Let the hexagon be divided by a new line <p in any 

 manner into two quadrilaterals, say xyzip, tuvtp. 



Then ay<p + bxz — U = a.u<p + filv ; 



.'. (ay — au)q>z=z(3tv—bxz; 

 .'. ay—cm and <p are the diagonals of the quadrilateral txvz. 



By construction, <p is the diagonal joining x, v (i. e. the in- 

 tersection of x and v) with z, t; and thus we see that ay—uu 

 is the line joining /, x with v 9 z; but this line passes through 

 y 9 u. Therefore x, t; y 9 u; z, v lie in one and the same right 

 line. Q. E. D. 



26 Lincoln's.Inn-Fields, 

 August 1850. 



* Communicated by the Author. 



