252 



Kansas Academy of Scieyice. 



quantities used as differences represent any amount advisable 

 or available, add the values of the several quantities, then pro- 

 ceed to lay out the series in strict accordance with the formula. 

 The square may then be built up according to any one of the 

 twenty-four model squares already shown. In the first ex- 

 ample shown above, if we make 2a = 2, Sd = 16, An = 18, 

 2g - 14, and G = 5.5, then will a = 1, d = 2, n = 4.5, g = 1, 

 and G = 5.5 ; total, 55.5. 



111 



111 



111 



Differences. 



Series. 



a =20 

 d= 1 

 n= 1 

 g= 1 

 G= 1.5 



2a =40 

 8d= 8 

 4n= 4 

 2flr= 2 

 G= 1.5 



55.5 



20 

 24 

 28.5 

 32.5 



21 



25 



29.5 



33.5 



22 

 26 



30.5 

 34.5 



23 



27 



31.5 



35.5 



111 111 111 111 111 



No. 38. 



In the second example, in order to differ materially from the 

 other, take 20 for the first term and let each of the other 

 differences be 1 except the difference in the main gap (G), 

 which will be 1.5. From these we obtain 2a = 40, 8fZ = 8, 

 An = 4, 2g = 2, and G = 1.5. A series of sixteen terms pre- 

 pared from these differences give the numbers as above, from 

 which a perfect square adding 111 in all its parts may readily 

 be constructed. 



THE CIRCLE SQUARED. 



Shall we essay the problem that has engaged the most 

 eminent mathematicians for thousands of years, that of squar- 

 ing the circle? But we perform the operation in an entirely 

 new way, by a method that has never before been tried. No 

 claim is here made that the problem is solved. It is entirely 

 a play upon words. But if taking the diameter and circum- 

 ference of a circle and placing the divisions in the form of a 

 square, so that by addition of the parts in any direction the 

 same circumference is obtained, if that is not squaring the 

 circle it certainly is not circling the square. In other words, 

 if it is not a circular square or a square circle it must be a 

 circle squared, so it amounts to the same thing. 



Take the figures that represent the circumference when the 

 diameter is 1.0000, or as near that amount as four decimal 

 places will give us, namely, 3.1416; though any other number 



