444 



THE POPULAR SCIENCE MONTHLY 



sition that the sum of the odd numbers, beginning with unity, is the 

 square of their number. Thus, the sum of those numbers between 1 

 and 199 is a hundred times 100, or 10,000. 



Suppose we desire to make a table of the squares of all the num- 

 bers up to 1,000, for example, the way that first suggests itself is to 

 make a thousand multiplications, 2x2, 3x3, to 999x999. But this 

 method is of little value ; it is exceedingly long, and there is no way 

 of verifying it. We have a surer and more expeditious method. Fig. 

 6 represents the table for calculating the squares of the first ten num- 

 bers. The column Ds, which need not be written, contains 2's ; col- 

 umn Dj represents the series of odd numbers, and may be written off- 

 hand ; column Q may be formed after the following law, which ap- 

 plies to all the numbers in the table : Each number is equal to the one 

 above it in the same column, augmented by the one that follows it in 

 the same line. Thus, 81 = 64-[-17; and 19==17+2. A thousand ad- 

 ditions of two numbers will then be sufiicient to construct our table up 

 to the square of 1,000. But here, it may be said, the results all depend 

 one upon another ; any error will carry itself to the next computa- 

 tion, and grow, like a snow-ball that at last becomes an avalanche, 

 and overthrow the whole calculation. The remedy for this incon- 

 venience is easy. When we have got the squares of the first ten num- 

 bers, we have only to add two ciphers to have the squares also of the 



Fitt. 6.— Squaees. 



Fig. 7.— TRiANGtTLAK Numbers. 



numbers 20, 30, 40, etc., to 90 ; we write them immediately in the 

 place they should occupy ; and then we must get the same numbers 

 again at the proper places in the course of our operations. 



An arithmetical ^progression of the second order is one in which, if 

 we form a series of the excesses of each number over the preceding 

 one, we obtain numbers in arithmetical progression. Of this order are 

 the series of the squares, and of the triangular numbers (Fig. 7). 



There may be also arithmetical progressions of the third and fourth 

 orders, and so on to infinity. They are all calculated in the same 

 manner ; and we take for a single example the series of the cubes of 

 whole numbers, which form an arithmetical progression of the third 



