Miscellaneous Papers. 



243 



lines and opposite column^. This scheme is the same as 

 scheme I turned over on its direct central diagonal. The num- 

 bers of the coupling-arrows in the trial arrangements are to be 

 placed in two columns so that they add 18 vertically and 9 

 horizontally, and are to be applied to the left-hand ends of the 

 arrows, which fall in the first and second columns. That puts 

 15 at the top of the second column adjacent to 1. The result- 

 ing model squares here follow : 



FIRST ARRANGEMENT. 



1 8 



7 2 



6 3 



4 5 



No. 13. 



FOURTH ARRANGEMENT. 



1 8 



6 3 

 4 5 



7 2 



No. 16. 



SECOND ARRANGEMENT. 



1 8 



7 2 



4 5 



6 3 



No. 14. 

 FIFTH ARRANGEMENT. 



1 8 



4 5 



7 2 



6 3 



No. 17. 



THIRD ARRANGEMENT. 



1 8 



6 3 



7 2 

 4 5 



No. 15. 



SIXTH ARRANGEMENT. 



1 8 



4 5 



6 3 



7 2 



No. 18. 



Remark. — It is to be noted that the celebrated square of the 

 Greek Moschopulus, first published, so far as known, m the 

 sixteenth century, translated into Latin by Delahire and read 

 by him before the French Academy of Sciences in 1691, is 

 really a harmonic square and fits nicely into my harmonic sys- 



