POWER

, in Mechanics, denotes some force which, being applied to a machine, tends to produce motion; whether it does actually produce it or not. In the former case, it is called a moving Power; in the latter, a sustaining power.

Power is also used in Mechanics, for any of the six simple machines, viz. the lever, the balance, the screw, the wheel and axle, the wedge, and the pulley.

Power of a Glass, in Optics, is by some used for the distance between the convexity and the solar focus.

, in Arithmetic, the produce of a number, or other quantity, arising by multiplying it by itself, any number of times.

Any number is called the first power of itself. If it be multiplied once by itself, the product is the second power, or square; if this be multiplied by the first power again, the product is the third power, or cube; if this be multiplied by the first power again, the product is the fourth power, or biquadratic; and so on; the Power being always denominated from the number which exceeds the multiplications by one or unity, which number is called the index or exponent of the Power, and is now set at the upper corner towards the right of the given quantity or root, to denote or express the Power. Thus, 3 or 3¹ = 3 is the 1st power of 3, 3 X 3 or 3² = 9 is the 2d power of 3, 3² X 3 or 3³ = 27 is the 3d power of 3, 3³ X 3 or 3⁴ = 81 is the 4th power of 3, &c. &c.

Hence, to raise a quantity to a given Power or dignity, is the same as to find the product arising from its being multiplied by itself a certain number of times; for example, to raise 2 to the 3d power, is the same thing as to find the factum, or product . The operation of raising Powers, is called Involution.

Powers, of the same degree, are to one another in the ratio of the roots as manisold as their common exponent contains units: thus, squares are in a duplicate ratio of the roots; cubes in a triplicate ratio; 4th powers in a quadruplicate ratio.—And the Powers of proportional quantities are also proportional to one another: so, if , then, in any Powers also, .

The particular names of the several Powers, as introduced by the Arabians, were, square, cube, quadratoquadratum or biquadrate, sursolid, cube squared, second sursolid, quadrato-quadrato-quadratum, cube of the| cube, square of the sursolid, third sursolid, and so on, according to the products of the indices.

And the names given by Diophantus, who is followed by Vieta and Oughtred, are, the side or root, square, cube, quadrato-quadratum, quadrato-cubus, cubo-cubus, quadrato-quadrato-cubus, quadrato-cubo-cubus, cubocubo-cubus, &c, according to the sums of the indices.

But the moderns, after Des Cartes, are contented to distinguish most of the Powers by the exponents; as 1st, 2d, 3d, 4th, &c.

The characters by which the several Powers are denoted, both in the Arabic and Cartesian notation are as follow:

Hence, 1st. The Powers of any quantity, form a series of geometrical proportionals, and their exponents a series of arithmetical proportionals, in such sort that the addition of the latter answers to the multiplication of the former, and the subtraction of the latter answers to the division of the former, &c; or in short, that the latter, or exponents, are as the logarithms of the former, or Powers. Thus, , and ; ; also , and ; .

2d. The 0 Power of any quantity, as a⁰, is = 1.

3d. Powers of the same quantity are multiplied, by adding their exponents: Thus,

4th. Powers are divided by subtracting their exponents. Thus,

5th. Powers are also considered as negative ones, or having negative exponents, when they denote a divisor, or the denominator of a fraction. So , and , and , &c. And hence any quantity may be changed from the denominator to the numerator, or from a divisor to a multiplier, or vice versa, by changing the sign of its exponent; and the whole series of Powers proceeds indefinitely both ways from 1 or the 0 Power, positive on the one hand, and negative on the other. Thus, &c a^-4 a^-3 a^-2 a^-1 a⁰ a¹ a² a³ a⁴ &c, or &c 1/a⁴ 1/a³ 1/a² 1/a 1 a a² a³ a⁴ &c.

Powers are also denoted with fractional exponents, or even with surd or irrational ones; and then the numerator denotes the Power raised to, and the denominator the exponent of some root to be extracted: Thus, , and , and , &c. And these are sometimes called imperfect powers, or surds.

When the quantity to be raised to any Power is positive, all its Powers must be positive. And when the radical quantity is negative, yet all its even Powers must be positive: because - X - gives +: the odd Powers only being negative, or when their exponents are odd numbers: Thus, the Powers of - a, are + 1, - a, + a², - a³, + a⁴, - a⁵, + a⁶, &c. where the even Powers a², a⁴, a⁶ are positive, and the odd Powers a, a³, a⁵ are negative.

Hence, if a Power have a negative sign, no even root of it can be assigned; since no quantity multiplied by itself an even number of times, can give a negative product. Thus √- a², or the square or 2d root of - a², cannot be assigned; and is called an impossible root, or an imaginary quantity.—Every Power has as many roots, real and imaginary, as there are units in the exponent.

M. De la Hire gives a very odd property common to all Powers. M. Carre had observed with regard to the number 6, that all the natural cubic numbers, 8, 27, 64, 125, having their roots less than 6, being divided by 6, the remainder of the division is the root itself; and if we go farther, 216, the cube of 6, being divided by 6, leaves no remainder; but the divisor 6 is itself the root. Again, 343, the cube of 7, being divided by 6, leaves 1; which added to the divisor 6, makes the root 7, &c. M. De la Hire, on considering this, has found that all numbers, raised to any Power whatever, have divisors, which have the same effect with regard to them, that 6 has with regard to cubic numbers. For finding these divisors, he discovered the following general rule, viz, If the exponent of the Power of a number be even, i. e. if the number be raised to the 2d, 4th, 6th, &c Power, it must be divided by 2; the remainder of the division, when there <*>s any, added to 2, or to a multiple of 2, gives the root of this number, corresponding to its Power, i. e. the 2d, 6th, &c root.

But if the exponent of the power be an uneven number, i. e. if the number be raised to the 3d, 5th, 7th, &c Power; the double of this exponent will be the divisor, which has the property abovementioned. Thus is it found in 6, the double of 3, the exponent of the Power of the cubes: so also 10, the double of 5, is the divisor of all 5th Powers; &c.

Any Power of the natural numbers 1, 2, 3, 4, 5, 6, &c, as the nth Power, has as many orders of differences as there are units in the common exponent of all the numbers; and the last of those differences is a constant quantity, and equal to the continual product , continued till the last factor, or the number of factors, be n, the exponent of the Powers. Thus,| <*>he <*>st Powers 1, 2, 3, 4, 5, &c, have but one order of dif<*>erences 1 1 1 1 &c, and that difference is 1. The 2d <*>wrs. 1, 4, 9, 16, 25, &c, have two orders of differences 3 5 7 9 2 2 2 and the last of these is . The 3d Pwrs. 1, 8, 27, 64, 125, &c, have three orders of differences 7 19 37 61 12 18 24 6 6 and the last of these is .

In like manner, the 4th or last differences of the 4th Powers, are each ; and the 5th or last differences of the 5th Powers, are each . And so on. Which property was first noticed by Peletarius.

And the same is true of the Powers of any other arithmetical progression , &c, viz, , &c, the number of the orders of differences being still the same exponent n, and the last of those orders each equal to , the same product of factors as before, multiplied by the same Power of the common difference d of the series of roots: as was shewn by Briggs.

And hence arises a very eafy and general way of raising all the Powers of all the natural numbers, viz, by common addition only, beginning at the last differences, and adding them all continually, one after another, up to the Powers themselves. Thus, to generate the series of cubes, or 3d Powers, adding always 6, the common 3d difference gives the 2d differences 12, 18, 24, &c; and these added to the 1st of the 1st differences 7, gives the rest of the said 1st differences; and these again added to the 1st cube 1, gives the rest of the series of cubes, 8, 27, 64, &c, as below.

Commensurable in Power, is said of quantities which, though not commensurable themselves<*> have their squares, or some other Power of them, commensurable. Euclid confines it to squares. Thus, the diagonal and side of a square are commensurable in Power, their squares being as 2 to 1, or commensurable; though they are not commensurable themselves, being as √2 to 1.

Power of an Hyperbola, is the square of the 4th part of the conjugate axis.

PRACTICAL Arithmetic, Geometry, Mathematics, &c, is the part that regards the practice, or ap- plication, as contradistinguished from the theoretical part.