hence the origin of subtraction; but when this last operation cannot be performed in the order in which it is indicated, the result becomes negative. The repeated addition of the same quantity gives rise to mul tiplication; a representing the multiplier, b the multiplicand, and c the product, we have and hence arises division, and fractions, in which this division terminates, when it cannot be performed without a remainder. The repeated multiplication of a quantity by itself produces the powers of this quantity; if b represent the number of times a is a factor in the power under consideration, we have a = c. This equation differs essentially from the preceding, as the quantities a and b do not both enter into it of the same form, and hence the equation cannot be resolved in the same way with respect to both. If it be required to find a, it may be obtained by simply extracting the root, and this operation gives rise to a new species of quantities, denominated irrational; but b must be determined by peculiar methods, which I shall proceed to illustrate, after having explained the leading properties of the equation ao = c. 240. It is evident, that if we assign a constant value greater than unity to a, and suppose that of b to vary, as may be requisite, we may obtain successively for c all possible numbers. Making b = 0, we have c = 1; then since b increases, the corresponding values of c will exceed unity more and more, and may be rendered as great as we please. The contrary will be the case, if we suppose b negative; the equation a = c being then changed into a = c, or = c, the values of c will go on decreasing, and may be rendered indefinitely small. We may, therefore, obtain from the same equation all possible positive numbers, whether entire or fractional, upon the supposition, that a exceeds unity. The same is true, if we have a <1; only the order, in which the values stand, will be reversed; but if we suppose a = 1, we shall always find c = 1, whatever value be 1 assigned to b; we must, therefore, consider the observations which follow, as applying only to cases, in which a differs essentially from unity. In order to express more clearly, that a has a constant value, and that the two other quantities b and c are indeterminate, I shall represent them by the letters x and y; we then have the equation a*=y, in which each value of y answers to one value of x, so that either of these quantities may be determined by means of the other. 241. This fact, that all numbers may be produced by means of the powers of one, is very interesting, not only when consid ered in relation to algebra, but also on account of the facility with which it enables us to abridge numerical calculations. Indeed, if we take another number y', and designate by x' the corresponding value of x, we shall have a = y', and, consequently, if we multiply y by y', we have y y' = a* × a*' = a*+*'; if we divide the same, the one by the other, we find lastly, if we take the nth power of y, and the nth root, we have = (ax)m = amx for the one, and for the other. ут yn = (a+)n = añ It follows from the first two results, that knowing the exponents and belonging to the numbers y and y', we may, by taking their sum, find the exponent which answers to the product yy', and by taking their difference, that which answers to the quotient From the last two equations it is evident, that y' y the exponent belonging to the mth power of y may be obtained by simple multiplication, and that which answers to the nth root, by simple division. Hence it is obvious, that by means of a table, in which against the several numbers y, are placed the corresponding values of x, y being given, we may find x, and the reverse; and the multiplication of any two numbers is reduced to simple addition, because, instead of employing these numbers in the operation, we may add the corresponding values of x, and then seeking in the table the number, to which this sum answers, we obtain the product required. The quotient of the proposed numbers, may be found, in the same table, opposite the difference between the corresponding values of x, and, therefore, division is performed by means of subtraction. These two examples will be sufficient to enable us to form an idea of the utility of tables of the kind here described, which have been applied to many other purposes since the time of Napier, by whom they were invented. The values of x are termed logarithms, and, consequently, logarithms are the exponents of the powers, to which a constant number must be raised, in order that all possible numbers may be successively deduced from it. The constant number is called the base of the table or system of logarithms. I shall, in future, represent the logarithm of y by ly; we have then xly, and since y = a*, we are furnished with the equa―aly. tion y 242. As the properties of logarithms are independent of any particular value of the number a, or of their base, we may form an infinite variety of different tables by giving to this number all possible values, except unity. Taking, for example, a = 10, we have y = (10)1, and we discover at once, that the numbers 1, 10, 100, 1000, 10000, 100000, &c., which are all powers of 10, have for logarithms, the numbers 0, 1, 2, 3, 4, 5, &c. The properties mentioned in the preceding article may be verified in this series; thus if we add together the logarithms of 10 and 1000, which are 1 and 3, we perceive, that their sum, 4, is found directly under 10000, which is the product of the proposed numbers. 243. The logarithms of the intermediate numbers, between 1 and 10, 10 and 100, 100 and 1000, &c. can be found only by approximation. To obtain, for example, the logarithm of 2, we must resolve the equation (10) = 2, by the method given in art. 221, finding first the entire number approaching nearest to the value of x. It is obvious at once, that is between 0 and 1, since (10) 1, (10)1 10; we make therefore x = • = , the equation then becomes (10) =2, or 10 = 2; now z is found 1 between 3 and 4; we suppose, therefore, z=3+, and hence As the value of z is between 3 and 4, we make we have then whence we obtain (៖) = 2(1)3 = 121, or (123) = { and after a few trials we discover that z' is between 9 and 10. The operation may be continued further; but as I have exhibited this process merely to show the possibility of finding the logarithms of all numbers, I shall confine myself to the supposition of z = 9; we have then, going back through the several steps, 2′ = 2, 2 = } }, x = 2;. 28 This value of x, reduced to decimals, is exact to the fourth figure, as it gives x = 0,30107. By calculations carried to a greater degree of exactness, it is found, that x = 0,3010300, the decimal figures being extended to seven places. Regarding this value of a as an exponent, we must conceive the number 10 to be raised to the power denoted by the number 3010300, and the root of the result to be taken for the degree denoted by 10000000; we thus arrive at a number approaching very nearly to 2; that is (10) =2, very nearly; the first member is a little greater than 2; but (10) is smaller.* 3010300 3010299 The method explained in this article becomes impracticable, when the numbers, the logarithms of which are required, are large; another method however, which may be very useful, is given by Long, an English geometer, in the Philosophical Transactions for the year 1724, No. 339. 244. By multiplying the logarithm of 2, successively, by 2, 3, 4, &c., we obtain logarithms of the numbers, 4, 8, 16, &c., which are the 24, 3d, 4th, &c. powers of 2. By adding to the logarithm of 2 the logarithms of 10, 100, 1000, &c. we obtain those of 20, 200, 2000, &c., it is evident, therefore, that if we have the logarithms of the former numbers, we may find the logarithms of all numbers composed of them, which latter can be only powers or products of the former. The number 210, for example, being equal to its logarithm is equal to 2 X 3 X 5 X 7, 12+13+15+ 17, and since 5 = 1o, we have 15110-12. As the process for determining z in the equation (10) y is very laborous, we may, reversing the order, furnish ourselves with the several expressions for z, then forming a table of the values of y corresponding to those of x, we shall afterwards, as will be perceived, be able, in any particular case, to determine x by means of y. We take first for x the values comprehended between 0,1 and 0,9; we have then only to determine the value of y, which answers to x = 0,1, or (10), because the several other values of y, namely, (10), (10)TM*, &c. are the 2a, 3d, &c. powers of the first. By extracting the square root, we discover at once, that then taking the fifth root of this result, we have (10) TT 1,023292992; and raising the result to the 24, 3d,..... 9th powers, we obtain the values of y, corresponding to those of x comprehended between 0,01 and 0,09. It will be readily seen, that by this method, we may also find the |