80 
[178 
178. 
ON THE A POSTERIORI DEMONSTRATION OF THE PORISM OF 
THE IN-AND-CIRCUMSCRIBED TRIANGLE. 
[From the Quarterly Mathematical Journal, vol. n. (1858), pp. 31—38.] 
In my former paper “ On the Porism of the In-and-circumscribed Triangle ” ( Journal, 
t. I. p. 344 [175]) the two porisms (the homographie and the allographic) were established 
à priori, i.e. by means of an investigation of the order of the curve enveloped by the third 
side of the triangle. I propose in the present paper to give the à posteriori demonstration 
of these two porisms ; first according to Poncelet, and then in a form not involving (as 
do his demonstrations) the principle of projections. My objection to the employment 
of the principle may be stated as follows: viz. that in a systematic development of 
the subject, the theorems relating to a particular case and which are by the principle 
in question extended to the general case, are not in anywise more simple or easier 
to demonstrate than are the theorems for the general case ; and, consequently that the 
circuity of the method can and ought to be avoided. 
The porism (homographie) of the in-and-circumscribed triangle, viz. 
If a triangle be inscribed in a conic, and two of the sides envelope conics having 
double contact with the circumscribed conic, then will the third side envelope a conic 
having double contact with the circumscribed conic. 
The following is Poncelet’s demonstration, the Nos. are those of the Traité des 
Propriétés Projectives [Paris, 1822] : 
No. 431. If a triangle be inscribed in a circle and two of the sides are parallel 
to given lines, then the third side envelopes a concentric circle. 
This is evident, for, the angle in the segment subtended by the third side being 
constant, the length of the third side is constant ; hence, the length of the perpen 
dicular from the centre upon the third side is also constant, and the third side envelopes 
a concentric circle.