In calculating the preceding tables of circumferences, squares, cubes, areas, &c.; the following simple rules have been adopted.

1. The Circumferences.

The circumferences were obtained by adding of 3.1416, or, .3927 constantly for the first table; of 3.1416, or, .31416 for the second; and or .2618 for the third table; thus,

The circumference of a circle, whose diameter is

2. The Squares.

After the first square of each succeeding series was found by the common rule; twice the root of that square, plus 1, added to the square number obtained, gave the square of the next number required; thus,

In finding the cubes; the first of each series was also found in the usual form, then the root of the number so obtained being multiplied by 3, and by the root of the number required, and the product, plus 1, added to the former cube, gave the cube of the next number required; thus,

After the first area of each succeeding series was obtained by the common rule, the others were found by the following: namely,- -a table of constant numbers was formed by multiplying .7854 by twice the fractional number contained in the diameter; thus,

Then twice the whole numbers of the first circle, plus 1, and multiplied by .7854, produced a sum which, when added to the area and constant number of the fractional part, gave the area of the next in the series; thus,

The area of a circle whose diameter is 1