CH. VII] 



MAGIC SQUARES 



149 



following the order of the numbers in the first column of the 

 square : the lines in pencil B must be numbered similarly to 

 correspond to the numbers in the second column of the square, 

 and so on. To prevent confusion in the figure I have not inserted 

 the numbers, but it will be seen that the method of construction 

 ensures that the sum of the 8 numbers which designate the 

 lines in each of these 18 pencils is the same. 



We can proceed a step further, if the resulting figure is cut 

 by two other parallel lines perpendicular to the axis, and if all 



c o t r 



Figure xv. A Magic Pencil, n=8. 



