arithmetical series formed by the continued addition of unity, commencing with the cypher, then will the numbers in the lower line express the number of ratios of unity to the first term, of which the ratio of unity to all the other terms is made up, and therefore they will be the logarithms of the numbers in the line above them. For example : 1, 3, 9, 27, 81, 243{ of which 0, 1, 2, 3, 4, 5 So, 1, 7, 49, 343, 2401{ of which 0, 1, 2, 3, 4 { : are the numbers forming a geometrical series; are the logarithms forming an arithmetical series. are the numbers forming a geometrical series; are the logarithms forming an arithmetical series. And again, 1, 10, 100, 1000, 10000 are the numbers forming a { geometrical series; are the logarithms forming an arithmetical series. Now, from the very nature of a geometrical series, it follows, that its terms are all powers of the constant number by the multiplication of which they are produced, and therefore, in place of writing the numbers themselves, we might introduce the expression denoting the power, without actually performing the multiplication, and we should thus obtain for the three geometrical series above, writing them vertically instead of in horizontal lines*, In these we perceive immediately that the numbers denoting the powers, or, as they are termed, the indices or exponents of the powers, are the same as the arithmetical series given above, and that they are therefore the logarithms of the numbers in the first columns. The constant number, of which the powers are successively taken, is termed the root or radix, and may have any value that we please assigned to it. Thus, we derive another definition of a logarithm, which may be de It must be borne in mind that ao = 1, and a1 = a, whatever the value of a may be. scribed as the index or exponent, to which a certain root or base must be involved, in order to be equal to the number of which it is the logarithm. It is, therefore, evident that a given number may have any number of logarithms corresponding with it; or that the same logarithm may serve for several different numbers, according to the value assumed for the base or root to be involved, or what is the same thing, the common ratio of the geometrical progression *. Thus, in the examples above, the bases or common ratios are 3, 7, and 10. We have, therefore, three distinctly different definitions which may be given of logarithms, depending upon the particular way in which they are regarded, and we shall recapitulate these definitions, before proceeding farther, in order to insure their being thoroughly understood. 1. The logarithm of a given number is the number of ratios of some assumed constant number to unity, contained in the ratio of the given number to unity. 2. Logarithms are a series of numbers in arithmetical progression, answering to another series of numbers in geometrical progression; so taken that 0 in the first corresponds with 1 in the latter. 3. The logarithm of a number is the index or exponent of the power, to which a given constant base or root must be involved, to be equal to that number. Whichever of these definitions may be adopted, the same general properties may be deduced as belonging to logarithms; we shall, however, in the following pages, consider them under the notion involved in the third definition, as the exponents of the powers of some constant root. And, in order to a more perfect conception of the subject, we shall first consider the properties of the exponents of powers generally. CHAPTER II. On the Exponents of Powers. IN algebra, the powers of a quantity, or the number of times that that quantity has been employed as a factor to produce a given quantity, are denoted by that number being written * See page 13. somewhat to the right and above the number or letter expressing the original quantity or root of the power. Thus, the square of 6 is written 62; the cube of x, x3; and the fifth power of 12, 125. In the first example, as 6 enters twice as à factor, it is called the second power, and is denoted by 2 written over the 6; in the second example, as a enters three times as a factor, it is called the third power, and is denoted by a 3 written above the x; and in the last example, as 12 enters five times as a factor, it is termed the fifth power, and is written 12'. The number thus placed over a number, to denote the power to which it is required to be raised, is termed the index or exponent of that power; as the former of these terms is sometimes employed in a different sense, to avoid ambiguity we shall use only the last. Thus, in the foregoing examples, 2, 3, and 5 are the exponents of the powers, to which the quantities 6, x, and 12 are to be respectively raised or involved. Frequently letters are employed instead of numbers as exponents of powers; thus, a denotes that the quantity represented by is to be raised to the power represented by a; and b", that the quantity b is to be raised to the power of n, or the nth power. The quantities, as a or b, in the foregoing examples, which have to be involved, or the powers of which are to be taken, are termed the roots or bases. When it is desired to multiply any two powers of a quantity, a very little consideration will show that their product will be equal to a power of that same quantity, whose exponent is the sum of the two exponents of the powers to be multiplied. For, let us suppose the powers to be multiplied to be x and x", then xx.x.x, and xx.x, therefore, x3 × 2 = x . x . x . x . x=x', the exponent of which 5 is equal to 3 + 2, the sum of the exponents of the two factors. And the converse of this rule holds good, for if it is required to divide a power of a given quantity by any other power of the same quantity, it is only necessary to subtract the exponent of the divisor from the exponent of the dividend to obtain the exponent of their quotient. Thus, let it be required to divide x" by x', we have ∞ ÷ x2 = x.x.x.x.x.x x.x the exponent of which is equal to 6 2. = x.x.x.x=x', Let us next examine the value of the power of a power; for instance, the square of a3. In this case, we see at once that the square of x is nothing more than a3 multiplied by itself, and by our former rule for the multiplication of powers, we have 3× x3 x; if we had required the cube of 3, it would have been x3. x3. x3 = x3, and for every higher power of 3 we must add another 3 to the exponent; it is therefore obvious, that as the exponent of the original power has to be taken as many times as the exponent of the power to which it has to be raised, that the new exponent will be equal to the product of the other two; thus, in the above examples, 3 × 2: 6, therefore (3)3 = x, and 3 × 3 = 9, therefore (x3)3 a. The converse of this rule also holds good, for if it is required to extract any root of a power, we have only to divide the exponent of the power by the exponent of the root, to obtain the exponent desired. Thus the square root of x is a2, because 4 ÷ 2 = 2; and the square root of a" is a3, because 6 ÷ 2 = 3. = = The four processes which we have here described are those which are of the most frequent occurrence, and as it is essential that they should be perfectly comprehended before entering on the use of logarithms, we shall recapitulate them in the form of rules. 1. The multiplication of the powers of any quantity is performed by the addition of their exponents; that is, an × m x(n+m). 2. The division of the power of a quantity by any other power of the same quantity, is performed by subtracting the exponent of the divisor from the exponent of the dividend; that is, an÷xm x(n − m). 3. The involution of any power of a quantity to some power is performed by multiplying its exponent by the exponent of the power to which it is to be raised; that is, the nth power of am is xn.m 4. The extraction of the root of any power is performed by dividing its exponent by the exponent of the root required; m that is, the nth root of am is xn In the last example we have an exponent differing from any which we have previously met with, namely, a fractional exponent; its use, however, in that example sufficiently explains its meaning, which is, that the quantity to which it is attached is to be raised to the power denoted by the numerator of the fraction, and is then to have the root extracted, which is denoted by the denominator of the same; or, the processes may be reversed, and the root first extracted, and then the power raised, since the order in which these operations are performed makes no difference in the final re sult. For example, let a above equal 4, m equal 3, and n m = 3 equal 2; then an 4, and if we take the cube of 4, which equals 64, and extract its square root, we obtain 8; or, if we first extract its square root we obtain 2, the cube of which is also equal to 8. And therefore we perceive that the final result is the same, whichever process is first performed. In the example to the second rule, namely, (-n) = x* ; if m is less than n, then the exponent r is a positive number, and a is termed a direct power of x; if, however, m exceeds n, then will r be a negative number, and in this case a" is termed an inverse power of x. In order to arrive at a correct idea of the value of an inverse power, we will take a direct power, and successively divide by its root, or subtract unity from its exponent, until we obtain a negative value; thus, let us start with a3, then 1 -n or, in more general terms, æ = ; that is to say, the in verse power of any number is equal to unity divided by the direct power with an equal exponent. This last rule holds equally, when the exponent of the inverse power is a fraction, as it does when an integer; thus, 72 is equal to unity divided by the nth root of x, or to We have then four different forms in which an exponent may be presented. 1. The positive integral exponent, as a", which denotes the |