PHILOSOPHICAL SOCIETY OF WASHINGTON, 111 



anywhere found, and it is the failure by Mr. Talbot to examine 

 this memoir that has led him into error in his history. 



Professor Paucker first analyzes the problem, and then gives 

 a solution. The solution which he gives is almost identical with 

 that given by Steiner. It differs, however, in this : after having 

 inscribed circles in the three partial triangles (Fig. 1) the 

 centres of the three circles are joined, two and two, and then 

 from m a line is drawn through the intersection of K L with 

 O C, and he then shows that this line will be tangent to the 

 two circles K and L ; whereas Steiner draws a line from m tan- 

 gent to the circle L, and he says it will be tangent to the circle 

 K also. This is a slight difference in getting at the same result, 

 but combined with the dates of their papers, which differ by a 

 little more than a year, and with the manner in which Prof. 

 Paucker proves his construction, and also combined with the 

 fact that he makes no allusion to the work of Steiner, while he 

 gives due credit to Malfatti, Gergonne, Crelle, Lehmus, and 

 Tedenat, seems to indicate that he had obtained his results in- 

 dependently of Steiner. 



Prof. Paucker demonstrates the correctness of his construction 

 by Euclidian methods, making no use of the Modern Geometry, 

 so called, and goes into the details with much carefulness. He 

 begins by demonstrating some Lemmas, and in this respect has 

 been followed by all subsequent writers. His first Lemma is as 

 follows: If an angle A is bisected 

 by A M (Fig. 2) and two circles are Fig- 2. 



drawn at pleasure, one tangent to 

 A M and A T, and the other tan- 

 gent to A M and A T', and the 

 points of tangency T and T' are 

 joined, then the cords Ti and T'i' 

 are equal. This he readily proves 

 by similar triangles. 



In paragraph 13 of his memoir 

 he demonstrates that if tangents are drawn (Fig. 2) from T to 

 the circle K, and from T' to the circle M, these tangents are 

 equal. 



These two propositions are also proved by Mr. Hart in 

 the first volume of the Quarterly Journal, without reference to 

 w^hether it had previously been done. 



