Miscellaneous Papers. 247 



three other numbers which are the leaders of the second couplet 

 of the three following sets, as 3 is the leader of the second 

 couplet of the first set. 



The placing of any number immediately determines the po- 

 sition of its complement ; and this complement can and should 

 be laid at once, thus simplifying the construction of the square. 



The first number laid around 16 has a choice of four posi- 

 tions; the second number has a choice of three; the third num- 

 ber has a choice of two positions and may be laid in either; 

 the fourth number has only one place left for it. Multiplying 

 together these factors, 4, 3, 2, and 1, we obtain 24 as the possi- 

 ble ways of arranging the sixteen numbers into a perfect 

 square, with 1 placed in a certain definite place, as, for ex- 

 ample, in the upper left-hand corner. 



VARYING SERIES AND SPECIAL SUMS. 



It is by no means necessary in order to produce a perfect 

 square that a series shall be absolutely uniform; it is only 

 necessary that the common differences shall be harmonic, 

 rhythmic, or concordant. There are four kinds of differences 

 in a series of sixteen terms, namely : 



1. The difference between the antecedent and consequent 

 in each couplet. There are eight couplets in a series and there- 

 fore eight of these differences (=d) . 



2. The differences between the leading couplet and the fol- 

 lowing couplet of each set, thus coming in the middle of each 

 of the four sets. There are four of these in any series of six- 

 teen numbers. For the purpose of distinction these differences 

 will be called notches (=w) . 



3. The differences between adjacent sets of four in each 

 half of the series. There are two such differences, one coming 

 in the middle of each half. These will be called side ga^js 



4. The difference between the two halves of a series. There 

 is but one of this. It is the main gap (=G) . 



Some other terms may be represented by letters for con- 

 venience. For example, let a represent the first term, q the 

 last term, and S the sum of all the numbers in one row of the 

 square, whether columnar, linear or diagonal. 



On arranging the terms of a series of sixteen numbers in a 

 row by number, the differences will appear thus : 



ald2n3rf4i/5d6w7(i8G9c^l0%llrfl2sfl3dl4n.l5dl6a 



