Elements of Algebra

CHAP. XXI. OF ALGEBRA. 63 $b = 3$ and $a^3 = c$, we know that the cube of $a$ must be equal to the given number $c$, and consequently that $a = \sqrt[3]{c}$. It is therefore easy to conclude, generally, from this, how to determine the letter $a$ by means of the letters $c$ and $b$; for we must necessarily have $a = \sqrt[b]{c}$. 218. We have already remarked also the consequence which follows, when the given number is not a real power; a case which very frequently occurs; namely, that then the required root, $a$, can neither be expressed by integers, nor by fractions; yet since this root must necessarily have a determinate value, the same consideration led us to a new kind of numbers, which, as we observed, are called surds, or irrational numbers; and which we have seen are divisible into an infinite number of different sorts, on account of the great variety of roots. Lastly, by the same inquiry, we were led to the knowledge of another particular kind of numbers, which have been called imaginary numbers. 219. It remains now to consider the second question, which was to determine the exponent, the power $c$, and the root $a$, both being known. On this question, which has not yet occurred, is founded the important theory of Logarithms, the use of which is so extensive through the whole compass of mathematics, that scarcely any long calculation can be carried on without their assistance; and we shall find, in the following chapter, for which we reserve this theory, that it will lead us to another kind of numbers entirely new, as they cannot be ranked among the irrational numbers before mentioned. --- CHAP. XXI Of Logarithms in general. 220. Resuming the equation $a^b = c$, we shall begin by remarking that, in the doctrine of Logarithms, we assume for the root $a$, a certain number taken at pleasure, and suppose this root to preserve invariably its assumed value. This being laid down, we take the exponent $b$ such, that the power $a^b$ becomes equal to a given number $c$; in which case this exponent $b$ is said to be the logarithm of the number $c$. To express this, we shall use the letter L. or the initial letters log. Thus, by $b = L. c$, or $b = log. c$,