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GRAPHICAL ALGEBRA INVOLVING FUNCTIONS 

 OF THE n*» DEGREE. 



Fkancis E. Niphek.* 



In the study of algebra, there is a strong tendency 

 among the young learners to follow rules, without giving 

 serious attention to the fundamentals upon which rules 

 are based. There are many graduates in algebra who 

 could at once answer the question, What is the area of 

 the square, whose sides are x — a. A diagram showing the 

 original square, with the lengths a cut off from its sides 

 would show that when the two strips 2xa have been sub- 

 tracted from the area x 2 , the area a 2 has been subtracted 

 twice. Therefore the final term a 2 is properly added. 



It often happens that a diagram suggests something 

 which its symbolic representation does not suggest. 

 When they are both presented, the suggestion is still 

 more likely to come. 



As is well known, the difference between the squares 

 of two numbers y and x is 



y 2 — x 2 =(y — x) (y + x) 



= y (y — x) +x (y — x) (1) 



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Fig. 1. 



The two terms of the second member represent the 

 areas of two rectangles marked a and b in Fig. 1. They 

 have equal width y — x. Their lengths are y and x. 



*Presented before The Academy of Science of St. Louis, March 18, 

 1918. 



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