52 



KANSAS ACADEMY OF SCIENCE. 



No. 9 is obtained by superimposing the lower half of No. 1 above the upper 

 half. This is called subversion. It places 8 in the key corner. 



No. 10 is obtained by trovertmg No. 9, or by stibverfuirj No. -2. It is called 

 transposing by suhtroversion. It brings 11 into the key corner. 



No. 11 comes by suhverting No. 3 or by diverting No. 9. It is called trans- 

 posing by subdiversion. It brings 13 into the key corner. 



No. 12 is made by reversing No. 9. This is known as resubversion. In this 

 square, 2 comes to the key corner. 



No. 13. 

 By Inversion. 



No. 14. 



By Introversion. 



No. 15. 

 By Indiversion. 



No. 16. 

 By Keiuversion. 



No. 13 is obtained by inverting No. 1; No. 14 by troverting No. 13; No. 15 

 by diverting No. 13; and No. 16 by reversing No. 13. These squares have re- 

 spectively the numbers 10, 5, 3, and 16 in the key corners. 



Each of these fifteen derived squares adds 34 in all the regular ways enumer- 

 ated for No. 1. 



In the sixteen transposed squares above, each of the successive numbers is 

 brought in succession into the key position, and in the exact order in which the 

 same numbers occur in square No. 1. Now, by placing the respective syllables 

 that express the mode of transposition in the square, letting each occupy the 

 position occupied by its key number in square No. 1, we have a sort of key to 

 the name of a square after it is formed, and to the number that should occupy 

 the upper left-hand corner when the square is formed : 



KEY. 



1 



Plan of Construction. 

 The sixteen numbers of the square are considered as consisting of four qua- 

 terniads or "sets" of four consecutive numbers each. 



Additional Definitions. 

 By the term "couplet" is meant the two consecutive numbers, an odd and 

 an even, constituting the anterior or the posterior half of a quaterniad, as 1 

 and 2, 5 and 6, 11 and 12, etc. Numbers falling adjacently in the square, as 1 

 and 15 or 2 and 11 in the squares above, are columnar 'pairs or linear j^airs, 

 according to whether they fall in the same column or same line. A pair in one 

 square is a pair in all sixteen. Indeed that is the key to retaining the harmony 

 of a square throughout all transpositions — the four numbers of any quarter are 



