*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 numbers up to 1,000, for example, the way that first suggests itself is to make a thousand multiplications, 2 X 2, 3 X 3, to 999 X 999. 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 numbers. The column D_{2}, which need not be written, contains 2's; column D_{1} represents the series of odd numbers, and may be written off-hand; column Q may be formed after the following law, which applies 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 additions of two numbers will then be sufficient 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 computation, and grow, like a snow-ball that at last becomes an avalanche, and overthrow the whole calculation. The remedy for this inconvenience is easy. When we have got the squares of the first ten numbers, we have only to add two ciphers to have the squares also of the

N. | D. | D_{1}. |
D_{2}. |
N. | T. | D_{1}. |
D. | |

1 | 1 | 3 | 2 | 1 | 1 | 2 | 1 | |

2 | 4 | 5 | 2 | 2 | 3 | 3 | 1 | |

3 | 9 | 7 | 2 | 3 | 6 | 4 | 1 | |

4 | 16 | 9 | 2 | 4 | 10 | 5 | 1 | |

5 | 25 | 11 | 2 | 5 | 15 | 6 | 1 | |

6 | 36 | 13 | 2 | 6 | 21 | 7 | 1 | |

7 | 49 | 15 | 2 | 7 | 28 | 8 | 1 | |

8 | 64 | 17 | 2 | 8 | 36 | 9 | 1 | |

9 | 81 | 19 | 9 | 45 | 10 | |||

10 | 100 | 10 | 55 |

Fig. 6. — Squares. | Fig. 7. — Triangular 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