Involution and Evolution — Chambers's Encyclopaedia

Involution and Evolution are two operations the converse of each other. The object of the first is to raise a number to any power, which is effected by continually multiplying the number by itself till the number of factors is equal to the number designating the power. Thus, 2 raised to the third power is $2 \times 2 \times 2$ , or 8; 7 raised to the fourth power is $7 \times 7 \times 7 \times 7$ , or 2401, &c. Evolution, on the other hand, is the extraction of a root of any number—that is, it is a method for discovering what number, when raised to a certain power, will give a certain known number. Thus, the square root of 64 is 8—that is, 8 is the number which, raised to the second power, will give 64; 3 is the fourth root of 81—that is, 3 raised to the fourth power is 81; and so on. The symbols expressive of the two operations are as follow: $5^3$ means that 5 is to be raised to the third power; $(7^2)^5$ means that the square or second power of 7 is to be raised to the fifth power; $\sqrt{9}$ or $\sqrt[3]{9}$ or $9^{\frac{1}{3}}$ signifies that the extraction of the second or square root of 9 is required; $\sqrt[4]{256}$ or $256^{\frac{1}{4}}$ , that the fourth root of 256 is to be extracted; and so on. Involution and evolution, like multiplication and division, or differentiation and integration, differ in the extent of their application; the former, or direct operation, can always be completed, while there are numberless cases in which the latter fails to express the result with perfect accuracy.