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To find the Square of a greater Number than is contained in the Table.

RULE 1.-If the number required to be squared exceed, by 2, 3, 4, or any other number of times, any number contained in the table, let the square affixed to the number in the table be multiplied by the square of 2, 3, or 4, &c., and the product will be the answer sought.

EXAMPLE.-Required the square of 2595.

2595 is three times greater than 865; and the square of 865, as per table, is 748225. Then, 748225X32=6734025, Ans.

RULE 2.-If the number required to be squared be an odd number, and do not exceed twice the amount of any number contained in the table, find the two numbers nearest to each other, which, added together, make that sum; then, the sum of the squares of these two numbers, as per table, multiplied by 2, will exceed the square required by 1.

EXAMPLE.-Required the square of 1865.

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To find the Cube of a greater Number than is contained in the Tabię.

RULE.-Proceed, as in squares, to find how many times the number required to be cubed exceeds a number contained in the table. Multiply the cube of that number by the cube of as many times as the number sought exceeds the number in the table, and the product will be the answer required.

EXAMPLE.-Required the cube of 3984.

3984 is 4 times greater than 996; and the cube of 996, as per table, is 988047936.

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To find the Squares of Numbers following each other in arithmetical

progression.

RULE.-Find, in the usual manner, the squares of the first two numbers, and subtract the less from the greater. Bring down, in a separate column, the square of the largest of these two numbers, and add it to the difference, with the addition of 2 as a constant quantity; the product will be the square of the next ensuing number.

EXAMPLE 1. Suppose it be required to extend the foregoing table of squares.

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In a similar manner the squares of any numbers, following each other in arithmetical progression may be found; or the foregoing table may easily be extended to any required length.

To find the Cubes of Numbers following each other in arithmetical

progression.

The cubes of a natural series of numbers may be found by a method very similar to that used for the squares; but as two series of differ ences have to be added, in the cubes, the operation is somewhat more complex.

RULE. Find the cubes of the first two numbers, and subtract the less from the greater. Then, multiply the least of the two numbers cubed by 6; add the product, with the addition of 6 as a constant quan. tity, to the difference; and thus continue the first series of differences.

For the second series of differences, bring down, in a separate column, the cube of the highest of the above numbers, and add the difference to it. The amount will be the cube of the next general number.

EXAMPLE.-Required the cubes of 1001, 1002, 1003, and 1004.

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To facilitate the Mensuration of the Surfaces and Solidities of Bodies.

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