256 Kansas Academy of Science. 



40 100 



number of the entire series, and as sum of the products the sum 

 of any line or subsquare in a square made from that series. 

 This is true of any series and of a square of any size. 



One more illustration ought to suffice : Let it be required to 

 add according to the theorem some numbers of which we 

 know the sum, say 15, 11, — 8.5, 19, 26.5, 37, no matter what 

 the numbers are or the order in which they occur. The first 



37 100 



difference (which is the first number, the difference being the 

 difference between itself and 0) is to be multiplied by 6, as 

 there are six numbers to be added ; the next difference, minus 4 

 (11 — 15), is to be multiplied by 5 and subtracted; the next 

 difference is minus 19.5, which is to be multiplied by 4 and 

 subtracted ; the other differences, 27.5, 7.5, and 10.5, are to be 

 multiplied in their order by 3, 2, and 1, and added. The entire 

 sum of the products equals 100, as the numbers do; and the 

 differences sum up 37, which is equal to the last number taken. 



The foregoing principle may be stated as follows : The sum 

 of any number of terms is equal to the lowest or smallest term 

 multiplied by the number of terms, plus the difference between 

 the lowest term and the second in order multiplied by the num- 

 ber of terms less 1, plus the difference between the second and 

 third multiplied by the number of terms less 2, plus the next 

 difference multiplied by the number of terms less 3, and so on 

 to the end or highest term. The sum of the differences will be 

 equal to the highest term and the sum of the products will be 

 equal to the sum of all the terms. 



Again, when a series of numbers is arranged in sets of equal 

 length, the sum of the several differences in the first set, each 

 multiplied in order by the number of terms between that 

 difference and the end, plus the differences between the initials 

 of the several sets multiplied by the number of the difference 

 in order beginning with the last, equals the sum of an average 



