254 Kansas Academy of Science. 



half of the series (2g) equals .7854, the area of a circle whose 

 diameter is 1 ; no other difference is significant, except that the 

 first term (a = — .0032) is less than zero, and being a minus 

 quantity must be subtracted instead of added whenever in- 

 cluded in any sum. But in the square no less than three of 

 the terms are significant : The number in the upper left-hand 

 corner (1.0000) represents the diameter of a circle, as before; 

 the opposite number (the one in the lower right-hand corner, 

 .8862) represents an equivalent square, that is to say the side 

 of a square equivalent in area to a circle whose diameter is 1 ; 

 the lower left-hand corner (.7071) represents the side of the 

 greatest square that can be inscribed in that same circle; and 

 finally, the sum of every line and quadrilateral equals 3.1416, 

 the circumference of the circle. 



SMYTH'S THEOREM. 



This paper will be closed with a theorem which, while it 

 may not be new, is not taught in the schools as one of the in- 

 teresting and instructive features of mathematics. It is a 

 principle that upon careful inspection must be acknowledged 

 as a truth ; yet it is not sufficiently self-evident to be called an 

 axiom. The proposition is this : 



Theorem. — The sum of any numbe7' of terms (quantities) 

 is equal to the sum of the products of the several terms dimin- 

 ished each by the preceding term and multiplied by the number 

 of terms folloiving that difference. 



The principle is not only true of any line of any perfect 

 square but of any number of numbers whatever, taken in any 

 order, and the numbers may be above zero or below, or mixed 

 in any manner. Before presentmg a working formula a few 

 illustrations will be presented by way of demonstration. 



The sum of any set of four numbers is equal to four times 

 the first number, plus three times the second minus, the first, 

 plus twice the third minus the second, plus the fourth minus 

 the third. When the number to be subtracted is greater than 

 the minuend, then the product of the difference between the 

 two numbers is to be subtracted in the addition. If the num- 

 bers be taken in numerical order, the smallest first, then the 

 sum of a series of four numbers is equal to four times the first, 

 plus three times the difference between the first and second, 

 plus twice the difference between the second and third, plus 

 the difference betv/een the third and the fourth. 



