Miscellaneous Papers. 



253 



of decimal places might as well be taken, except that the 

 farther the decimal is extended the more time and space it will 

 take. 



First we take half of our number, namely, 1.5708, and divide 

 it up as we please according to the formula, being careful to 

 make our several numbers sum up exactly our required num- 

 ber. Then, if we have taken 2a, 8d, 4n, etc., to make our num- 

 bers correspond with, we should select numbers that are di- 

 visible by 2, 8, etc., in order to avoid fractional numbers when 

 not necessary, as in the case where the sum of a line is odd. 



Here are two examples. In the first one the series begins 

 with the decimal surplus above 3 included in the circumference 

 of the circle; the sum of the eight minor differences (Sd = 



THE CIRCLE SQUARED. 



3.1416 



3.1416 



3.1416 



3.1416 



DIFFERENCES. 



2a = .2832 a = .1416 



SERIES. 



d = 



8d = .7854 



4n = .2618 n = 



2g = .1384 g = 



G=^ .1020 G = 



1.5708 



1416 

 .0982 .4726 

 .0655 .8363 

 .0692 1.1673 

 .1020 



,3053 

 ,6363 



.9345 1.0000 1.0982 

 1.2655 1.3310 1.4292 



.2398 



.5708 



4035 

 .7345 



[3.1416 3.1416 3.1416 3.1416 3.1416 

 No. 39. 



.7854) represents the area of a circle whose diameter is 1 : the 

 sum of the four differences next greater (4?i = .2618) is equal 

 to one-third of that amount; the term in the upper left-hand 

 corner of the square (1.0000) represents the diameter of the 

 circle ; and, as should be expected, the sum of every line in any 

 direction and the sum of the four corners of every quadri- 

 lateral, whether rectangular or rhomboid, equals 3.1416, the 

 circumference of the circle. 



The second example is more complex because it represents 

 more. The sum of the two differences in the middle of each 



Square of the Circle. 



Differences. 



Series. 



3.1416 



3.1416 



3.1416 



3.1416 3.1416 3.1416 3.1416 3.1416 

 No. 40. 



