EXAMPLE V.

What is the product of 41 3ii Oiii Oiv 5v by 4ii Siii 5iv?

What is the product of 4f. 3ì 2ii 8iii 9iv 5v 3vi by 31 Oii

When there is a great number of feet in the multiplier, the

best way is by the rule of practice, as follows:

Mul

Multiply the feet together, as in common multiplication; and for the odd inches, &c. take the aliquot parts.

Thus, for li take of a foot, or 1ii take of li, &c.

If the feet in both the multiplicand and multiplier be large numbers,

Multiply the feet only into each other: then, for the inches and seconds in the multiplier, take parts of the multiplicand; and for the inches and seconds of the multiplicand, take aliquot parts of the feet only in the multiplier; and the sum of all will be the product.

A POWER is a number produced by multiplying any

given number continually by itself a certain number of times.

Any number is called the first power of itself; if it be multiplied by itself, the product is called the second power,

VOL. I.

T

and

and sometimes the square; if this be multiplied by the first power again, the product is called the third power, and sometimes the cube; and if this be multiplied by the first power again, the product is called the fourth power, and so on; that is, the power is denominated from the number which exceeds the multiplications by 1.

The number which exceeds the multiplications by 1, is called the index, or exponent of the power; so the index of the first power is 1, that of the second power is 2, that of the third is 3, and so on.

Powers are commonly denoted by writing their indices above the first power: so the second power of three is de

noted thus, 32; the third power thus, 3'; the fourth power thus, 3; and so on: also the sixth power of 503 thus, 5036.

Involution is the finding of powers; to do which, from their definition there evidently comes this

PROBLEM IX.

To raise a given number to any given power required.

I. Multiply the given number, or first power, continually by itself, till the number of multiplications be 1 less than the index of the power to be found, and the last product will be the power required.

II. But because fractions are multiplied by taking the products of their numerators and of their denominators, they will be involved by raising each of their terms to the power required: and if a mixed number be proposed, either reduce it to an improper fraction, or reduce the vulgar fraction to a decimal, and procced by the rule.