TWENTY-SIXTH ANNUAL MEETING. 



83 



Fig. 76. 



The above circles and squares are all perfectly transposable, without 

 marring the harmony of arrangement of the numbers within the figures. 



Other plane forms and all volumetric forms will be left to a future time. 

 Many of them are already prepared; but there is much to do yet. 



FORMULA. 

 To determine the sum of any line in any square, rectangle, star, or circle, 

 multiply half the sum of the extremes of the series by the number of places 

 in a line. Expressed algebraically, the formula would appear thus: 



—j—'P^S 



In which I represents the last term of the series; f represents the first term; 

 p represents the number of places in a line; and s stands for sum. This is 

 true of any series, whatever the first term and whatever the common differ- 

 ence. In nearly all forms given heretofore, the first term of every series has 

 been 1 and the common difference 1; yet whatever is said of those series 

 is equally true of any other series similarly arranged. 



