50 



KANSAS ACADEMY OF SCIENCE. 



The four corners and the four centrals ( fig. 13 ) 2 



The four of each quarter ( fig. 14 ) 4 



Corresponding corners of quarters ( fig. 15 ) 4 



Knight-spring numbers in the margin ( fig. IG ) 2 



3d. — By rectangles — 



^•^^ 



czTvj/r 



cJ^^. /f. 



^if.AXI.^ 



Extremes of medial columns and lines ( fig. 17 ) 2 



Alternate linear pairs ( fig. 18 ) 4 



Alternate columnar pairs ( fig. 19 ) 4 



Transverse diagonals of opposite quarters ( fig. 20 ) 2 



4th. — By rhonihs and rhomboids — ■ 



i: 



c7<;^. XI. , 



S^l-q. Z3 



^£5>.i3. 



=7.>2.^, 



Extremes of one diagonal and means of the other (fig. 12) 2 



Opposite columnar pairs (fig. 21) 4 



Opposite linear pairs (fig. 22) 4 



Alternating numbers in opposite columns (fig. 23) 4 



Alternating numbers in opposite lines (fig. 21) 4 



Total ways of adding 34 52 



Here, then, are 52 regular ways of adding equal sums (in this case .34). There 

 are many other ways, more or less irregular, of obtaining the same result, as 12, 

 15, 4, 3, etc. ; but the only notice taken of them is that, whatever order be taken 

 in obtaining the sum of 34, the sum can be obtained from four numbers in cor- 

 responding positions diametrically opposite. 



We may readily concede, then, that this square is perfectly harmonic in all 

 its parts. All parts of the square are evenly balanced and corresponding with 

 every other part. 



Transpositions. 



Taking this square as a primary, 15 other squares may be derived from this 

 by regular transpositions. All of these derived squares will have the same at- 

 tributes as the primary square; will add equal sums in all the different ways, and 

 in the same order, as the primary square; will, in short, be as perfectly harmonic. 



In order to express the different transpositions, the most expressive words will 

 be used; and, where the language does not furnish a properly expressive word, a 

 word suiting the purpose will be invented from an arbitrary syllable. 



Square No. 1 is here reproduced for the purpose of comparison with the 15 

 derived squares. 



