296 BEV. T. p. KTRKMAN ON LINEAB CONSTBU0TION8. 



The drawing of 14 additional lines will reduce this to 



k\ OXs. OX^ 0X3 (61) = k^ Ox., Oxj Oxf, (67) ; 

 twelve more lines give us the reduction, 



^=» . OJTe • 0X9 . OXj = k^rn^ OT 

 k* . Oxe . Oxj . Oxg = kW Ot. 

 and the line 



or. (61)— 0^(67)rro = r, 

 is given by drawing other four lines. 



We have thus found a fifth point of the conic (82349) 

 by drawing 32 -}- 30 lines additional to the 67 lines ex- 

 pended in finding a fifth point on (12349): in all, 129 

 lines. 



We can now proceed, after the elegant method of Mr. 

 Weddle, to find the point 9 by five applications of Pascal's 

 theorem, so that the mystic enneagram is completed by at 

 most 150 applications of the ruler, after the drawing of the 

 Cartesian co-ordinates ; a process which will be esteemed 

 simplicity itself by those who have attempted to express x^ 

 and ^9 in terms of ajiyi, &c., or even in terms of their nume- 

 rical values. 



Note. — Since this paper was read, there has appeared in the Cambridge 

 and Dublin Mathematical Journal of this year, 1851, a solution of thia 

 problem of the ninth point, with the ruler only, by the Rev. A. S. Hart, 

 F.T.C.D., which may be pronounced perfect, and which for elegaooe and 

 simplicity leaves nothing to be desired. 



