UNIVERSITY OF VIRGINIA PUBLICATIONS 



BULLETIN OF THE PHILOSOPHICAL SOCIETY 



SCIENTIFIC SECTION 



Vol. I, No. 3, pp. 45-84 June, 1910 



ON THE MAXIMUM AND MINIMUM VALUES OF A LINEAR 

 FUNCTION OF THE RADIAL COORDINATES OF A POINT 

 WITH RESPECT TO A SIMPLICISSIMUM IN SPACE OF n 

 DIMENSIONS.* 



BY 



WILLIAM H. ECHOLS. 



1. Tliis problem in the particular form. To find that point the sum 

 of whose (Usiances from the vertices of a triangle is a minimum, is one of 

 much historic interest. Viviani relates that it was proposed by Feiinat to 

 Torricelli and by him handed over as an exercise to Viviani, who gave, 

 in the Appendix to his Treatise Be Maximis et Minimis, pp. 144, 150 

 (1659), the following construction for finding the point: Let ABC 

 be the triangle in which each angle is less than 120°. On A B and A C 

 describe segments of circles containing angles of 120°. The arcs of these 

 segments intersect in the point required. Viviani's proof that this is the 

 point required-is long and tedious. Thomas Simpson in his Doctrine and 

 Application of Fluxions, § 36 (1?50), gives the following construction 

 for determining the same point : Describe on B C a segment of a circle 

 to contain an angle of 120°, and let the whole circle B C Q \i& completed. 

 From A io Q the mid-point of the arc B C Q draw A Q intersecting the 

 circumference of the circle in V, wliich will be the point required. In 

 § 431 Simpson treats the more general problem: Three points being 

 given A, B, C, to find the position of a fourth point P, so that if lines be 

 di-awn from thence to the three former, the sum 



a . AP + 13 . BP + y . CP, 



where a, p. y denote any given numbers, shall be a minimum. Both the 

 particular and the more general problem are discussed in Nova Acta 



*Reafl before the ilatheiiiiitienl and Xatiiral Science Section. December. 100!). 



