NOTES. 299 



wonted learning and perspicacity, in his celebrated treatise, 

 Geometric Superieure, Jntrod. p. xxi., and there is good reason 

 to hope that a more full discussion will accompany his work 

 lately announced as in preparation, the restoration of Euclid's 

 three books (Les trois Livres de Porismes d'JEuclide re'taUis pour 

 la premiere /o?s) M. Chasles puts in the front of his title that 

 the restoration is effected after the notice and lemmas of 

 Pappus, and conformably to the view of Simson, touching the 

 form of the enunciation of the propositions. Nevertheless, 

 his theory differs in some particulars from that of Simson. 

 It has been highly gratifying to find that this great geometri- 

 cian refers with approval to a porism in the First Tract 

 (prop, vii.) which he considers to throw light upon one of the 

 kinds of porisms described by Pappus as belonging to Euclid's 

 Third Book. Perhaps he will cast an eye upon an illustra- 

 tion of the views entertained on this subject in Tract III. 



It seems essential to the formation of a porism that there 

 should be a transition from determinate to indeterminate, a 

 change in the data which makes the problem indeterminate, 

 and so capable of innumerable solutions. Take very simple 

 and elementary cases. Suppose the problem is to find in the 

 diameter of a circle produced, a point such that the line drawn 

 from it to a given point in the circumference shall have its 

 square equal to the rectangle under the diameter produced, and 

 the portion of it between the circle and the point to be found. 

 Call the diameter a, the portion without the circle d, and the 

 line cutting the circumference f, then f* = (a + d) d. Let y 

 be the ordinate, and x the abscesse, to the given point in the 

 circumference, then also 



a x 

 f* =. (d + xf + a x - x* = d* + 2 dx + a x ; and d= ^-. 



Ci ~ X 



But if the point in the circumference is such that 



2 (a x x 1 ) ace 2 (a x x 9 ) 



d + x = ; then - + x = , and 



a 2 x a 2 x a 2 x 



2 a x 2 x* = 2 a x 2 x* ; or the point in the diameter pro- 

 duced is found whatever be the point in the circumference ; 



