46 UNIVEESITY OF VIRGINIA PUBLICATIONS 



Academiae . . . Petropolitanae, XI, 235-8 (1798), by Nicolas Fuss in his 

 memoir De Minimis qiiihusdam geometricis ope principii statici inventis, 

 read to the Petersburg Academy of Sciences on 35th February, 1796. In 

 this memoir Fuss gave the values AP, BP, CP and the minimum value of 

 their sum for the particular problem in terms of algebraic functions of the 

 sides of ABC. 



For the foregoing information the writer is indebted to a paper in 

 the Proceediags of the Edinburgh Mathematical Society, Vol. XV, p. 100 

 (1897), On the Isogonic Centers of a Triangle, by Professor J. S. 

 Mackay, in wliich it is pointed out that the problem of the minimum of the 

 sum of the distances of a point from the vertices of a triangle is closely 

 associated with that of constructing an equilateral triangle of maximum 

 area circumscribing the given triangle A B C. In Gergonne's Annals de 

 Matliematiques, I, 384 (1811), the problem: About any given triangle 

 to circumscribe or inscribe the greatest or least triangle similar to a given 

 triangle, was proposed. It was solved by Eochat, Vecten and others in 

 Vol. II, pp. 88-93 (1811 and 1813). 



The problem of minimizing the sum 



'■i + i\ + ''3 



of the distances of a point from the vertices of a triangle A B C is, & 

 favorite one in all the modern treatises on analysis; it is given especially 

 with the view of pointing out the singularity which occurs when an angle 

 in the triangle is equal to 130°. It is discussed by Humbert, Cours 

 d' Analyse, I, p. 193 (1903) ; by Goursat, Cours d' Analyse, I, p. 144 

 (1903) ; by VaUee Poussin, Cours d' Analyse, I, p. 119 (1903). Joachims- 

 thal in his Bijf. and Int. Calculus, p. 399, devotes a considerable space 

 in the appendix to a geometrical consideration of the problem. In all 

 the treatments of this problem the distances r^ r^, r^ of a point P from 

 the points A, B, C are considered as being absolute values and presenting 

 no ambiguities of sign, a matter to be noticed subsequently. 



3. The writer has been unable to find any systematic and conclusive 

 analytical treatment of the problem of minimizing 



where o, j8, y are arbitrary constants. 



The determination of a critical point in this problem is closely asso- 

 ciated with the problems: (a). To construct a triangle of given form 



