148 MAGIC SQUARES [CH. VII 



intersecting sides of two equal concentric squares. It is required 

 to place the first 16 natural numbers on the corners and points 

 of intersection of the sides so that the sum of the numbers on 

 the corner of each square and the sum of the numbers on every 

 side of each square is equal to 34. Eighteen fundamental solu- 

 tions exist. One of these is given above *- 



There are magic circles, rectangles, crosses, diamonds, and 

 other figures : also magic cubes, cylinders, and spheres. The 

 theory of the construction of such figures is of no value, and 

 I cannot spare the space to describe rules for forming them. 



Magic Pencils. Hitherto I have concerned myself with 

 numbers arranged in lines. By reciprocating the figures com- 

 posed of the points on which the numbers are placed we obtain 

 a collection of lines forming pencils, and, if these lines be 

 numbered to correspond with the points, the pencils will be 

 magicf. Thus, in a magic square of the nth order, we arrange 

 n 2 consecutive numbers to form 2n + 2 lines, each containing 

 n numbers so that the sum of the numbers in each line is the 

 same. Reciprocally we can arrange v? lines, numbered con- 

 secutively to form 2n + 2 pencils, each containing n lines, so 

 that in each pencil the sum of the numbers designating the 

 lines is the same. 



For instance, figure xii on page 143 represents a magic 

 square of 64 consecutive numbers arranged to form 18 lines, 

 each of 8 numbers. Reciprocally, figure xv represents 64 lines 

 arranged to form 18 pencils, each of 8 lines. The method of 

 construction is fairly obvious. The eight-rayed pencil, vertex 0, 

 is cut by two parallels perpendicular to the axis of the pencil, 

 and all the points of intersection are joined cross- wise. This 

 gives 8 pencils, with vertices A, B,...,H;8 pencils, with vertices 

 A', ..., H'; one pencil with its vertex at 0; and one pencil with 

 its vertex on the axis of the last-named pencil. 



The sum of the numbers in each of the 18 lines in figure xii 

 is the same. To make figure xv correspond to this we must 

 number the lines in the pencil A from left to right, 1, 9, ..., 57, 



* Communicated to me by R. Straohey. 



t See Magic Reciprocals by Ot. Frankenstein, Cincinnati, 1875. 



