INVOLUTION
, in Arithmetic and Algebra, is the raising of powers from a given root; as opposed to Evolution, which is the extracting, or developing of roots from given powers. So the Involution of the number 3, or its powers, are thus raised: 3 - or 31 or 3 is the 1st power, or root, 3 × 3 or 32 or 9 is the 2d power, or square, 3 × 3 × 3 or 33 or 27 is the 3d power, or cube, and so on.
And hence, to find any power of a given root, or quantity, let the root be multiplied by itself a number of times which is one less than the number of the index; i. e. onc<*> multiplied for the 2d root, twice for the 3d root, thrice for the 4th root, &c.
| Thus, to Involve | .12 | to the 3d power. |
| .12 | ||
| .0144 | square, or 2d power. | |
| .12 | ||
| .001728 | cube, or 3d power. |
So also, in Algebra, to Involve the binomial a + b, or raise its powers.
And in like manner for any other quantities, whatever the number of their terms may be. But compound algebraic quantities are best involved by the Binomial Theorem; which see.
Simple quantities are Involved, by raising the numeral coefficients to the given power, and the literal quantities are raised by multiplying their indices by that of the root; that is, the raising of powers is performed by the multiplication of indices, the same as the multiplication of logarithms. Thus,
The 2d power of a is a2.
The 2d power of 2a2 is 22a2 × 2 or 4a4.
The 3d power of 3a2b3 is 27a6l9.
The 3d power of a1/2b2/3 is a3/2b2.
The nth power of am cp is amncpn or (―(am cp))n.
INWARD Flanking Angle, in Fortification, is that made by the curtin and the razant flanking line of defence.
