GRAPHICAL REPRESENTATION OF FUNCTIONS OF 

 THE A^TH DEGREE. 



By FRANCIS E. NIPHER. 

 (Read April 26, 1919.) 



It has been generally assumed that any algebraic symbol y, the 

 exponent of which is vmity, can only be properly represented graph- 

 ically by a line. When the exponent is 2, the quantity 3'- has been 

 assumed to represent a rectangular area, the four sides of which 

 each have a length 3'. If the exponent is 3, the proper geometrical 

 representation was said to be a rectangular volume, having six faces 

 whose areas were 3'-. If the exponent were 4, the philosophic mind 

 began to speculate on the nature of space having four dimensions. 

 No definite solution of this problem having been reached, it has 

 apparently been considered impracticable to consider the properties 

 of space having six or eight or ten dimensions. 



During this period of uncertainty it has been known to students 

 of elementary algebra that 



I 3;l/3 3,2/3 y y/3 y/3 



^ — etc. 



yi.10 y-l^ y ytlU ^U / U ,,? ' 



,1/3 ^,2/3 ., .,4/3 .,5/3 .,2 



It has also been known that similar right triangles have sides whose 

 lengths are directly proportional. This is illustrated in Fig. i where 

 an arbitrarily chosen length is adopted as a unit of length. Its 

 length in terms of the centimeter is unknown. Another length is 

 similarly chosen to represent ys, where y is to be given a numerical 

 value in terms of the chosen unit, but this value need not be deter- 

 mined. 



These lengths having been laid oft' upon rectangular axes, the 

 lengths y"'^, y, y*''^, 3^^ 3'', etc., can be laid oft' upon the two axes by a 

 well-known geometrical method. By means of three parallel rulers, 

 two of which are linked to the third in the usual manner, the lengths 

 unity and y may be first adopted and laid oft' on both axes, and the 



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