0, and that of 10 is 1, and hence the logarithms of all numbers between 1 and 10 are greater than 0 and less than 1, that is, they are fractions. In the same manner, between 10 and 100 they are greater than 1 and less than 2, that is, they are 1 with some fraction annexed, and so on. The whole numbers or integers in the logarithmic series are hence easily obtained, being always a unit less than the number of figures in the integral part of the corresponding natural number. On this account it is customary, in the common printed tables, to put down only the fractional part in the form of a decimal, the computer supplying the whole number or integer under the name of index. (1) 5. In order to generalize, let us assume the two following series: &c. y'", &c. y, y' y", (2) in which r is some given number greater or less than unity, and x, x', x'', x'"', &c. any variable quantities chosen in such a manner that r2=y, 1x′ =y', px''=y'', px"=y'"', &c., then the several exponents, x, x', x', x'"', &c. of the series (1) are called the logarithms of the corresponding terms in the series (2). Thus if y, y', y", y'", &c. be a series of numbers such that ry, put!=y', p2=y'', prz!!. y"", &c., then ax=log. y, x'=log. y', x"=log. y", x""log. y"", &c. 6. For the purpose of adapting the series (1) to the series of natural numbers 1, 2, 3, &c. the given number r must be greater than unity, the first index x must be equal to 0, and the several indices x', x", x'", &c. must continually increase. For, since by the principles of algebra, r°=1, whatever r may be, this series will increase from 1 to infinity; and by properly adjusting the values of x', x'', x"", &c. it is evident that the several quantities r2, px“, r2, &c. may be made to coincide with the numbers 2, 3, 4, &c. For example, let r=10; then since 10°-1, and 10=10, the indices of 10, which would give 10", 10", 10", &c. equal to the numbers 2, 3, 4, &c., must be fractions between 0 and 1. If we take the number 3 we have 10—3.16 nearly, from which we infer that a fraction (x') somewhat less than or 0.5, being made the index of (r) 10, would give 103. This fraction is found by calculation to be 47712; hence 1047712-3; therefore, when r=10, the logarithm of 3 is 47712. In like manner, if we assume the number 5, whose logarithm is to be found in place of that of 3, we have 103-4.64; whence a fraction, x()' somewhat greater than, or 666, being made the index or exponent of 10, would give 10")=5. This fraction more accurately computed is found to be .69897, that is, when r=10, the logarithm of 5 is .69897. 7. From this it appears, that the value of the logarithm of any given number depends upon the value of the number r, and that by assuming it equal to different numbers, as many different systems of logarithms may be formed as we please. In every system, however, since 1, the logarithm of 1 must be 0. This constant quantity r from the powers of which the natural numbers are formed, is called the radix or base of the system to which it belongs. 8. In the general equation ry, (art. 5.), let us make a vary, and observe the correspondent variations of y. If r is greater than 1, on making x=0, we have y=1; when x=1 then yr or the logarithm of the base is=1; in proportion as a increases from 0 to infinity, y will increase from 1 towards r, and afterwards to infinity, so that if we suppose x to pass through all the intermediate values, in following the law of continuity, y will increase also in the same manner, though much more rapidly. If we put for x, negative values, we shall have y=r-3, or 1 y === more y or Here we see, in like manner, that the more x increases the 1 decreases, so that in proportion as x augments negatively, y takes all possible values less than 1, as far as 0, in which case x becomes infinite. This was the proposition which Napier made to Briggs on their celebrated meeting at Edinburgh, when conversing on the propriety of changing the logarithmic scale. If r is less than 1 we shall make r=,6 being greater than 1, and 1 we have y=or y=b3, according as x is positive or negative. We bx fall here upon the same case, with this difference, that x is positive when y is less than 1, and negative when y is greater than 1. This proposal Briggs made to Napier, but immediately abandoned it on Napier suggesting that mentioned above, which was finally adopted. If r=1, we have y=1 whatever x may be. We may then say generally, that provided r is not unity, there can always be found a value for x, which renders equal to any given number y. The constant use that is made of the properties of the equation y=r" requires the denominations of its parts to be fixed in order to avoid circumlocution. Hence, as before remarked, x is called the logarithm of the number y, the invariable number r is called the base, and, finally, the logarithm of a number, the power to which the base must be raised in order to produce that number. With regard to the base r it is arbitrary, and when we write x=log. y to show that x is the logarithm of the number y, or that y=r*, the base r is always understood, because when once chosen it is supposed to remain fixed. If it should be changed the new base ought to be indicated. 9. From these principles are derived several properties. 1°. In every system of logarithms, the logarithm of 1 is 0, and that of the base r is 1. 2o. If the base r is greater than 1, the logarithms of numbers greater than 1 are positive, the others are negative. The contrary takes place if r is less than 1. 3°. The composition of a table of logarithms consists in determining all the values of x when y is made successively equal to 1, 2, 3, &c. in the equation y=r. If we suppose r2=μ, on making x=0, ç, 2ę, 3e, &c. We find y=1, μ, μ2, μ3, &c. ng fen The logarithms therefore increase in progression by differences, while the numbers increase in progression by the product or quotient, according as is an integer or a fraction. The ratios are the arbitrary numbers ę and μ. в We may, therefore, regard the systems of values of x and y which satisfy the equation 0, and that of 10 is 1, and hence the logarithms of all numbers between 1 and 10 are greater than 0 and less than 1, that is, they are fractions. In the same manner, between 10 and 100 they are greater than 1 and less than 2, that is, they are 1 with some fraction annexed, and so on. The whole numbers or integers in the logarithmic series are hence easily obtained, being always a unit less than the number of figures in the integral part of the corresponding natural number. On this account it is customary, in the common printed tables, to put down only the fractional part in the form of a decimal, the computer supplying the whole number or integer under the name of index. 5. In order to generalize, let us assume the two following series: y, &c. , y, y', y", y"", &c. (1) (2) in which r is some given number greater or less than unity, and x, x', x'', x''', &c. any variable quantities chosen in such a manner that r2=y, 12=y', p*"=y", rt"=y"", &c., then the several exponents, x, x', x", x''', &c. of the series (1) are called the logarithms of the corresponding terms in the series (2). Thus if y, y', y', y'", &c. be a series of numbers such that ry, px'=y', px'=y'', p*"=y'", &c., then x=log. y, x'=log. y', x"=log. y", ""log. y", &c. 6. For the purpose of adapting the series (1) to the series of natural numbers 1, 2, 3, &c. the given number r must be greater than unity, the first index x must be equal to 0, and the several indices x', x", "", &c. must continually increase. For, since by the principles of algebra, r°=1, whatever r may be, this series will increase from 1 to infinity; and by properly adjusting the values of x', x'', x"", &c. it is evident that the several quantities *", r", r*", &c. may be made to coincide with the numbers 2, 3, 4, &c. For example, let r=10; then since 10°-1, and 10=10, the indices of 10, which would give 10", 10", 10"", &c. equal to the numbers 2, 3, 4, &c., must be fractions between 0 and 1. If we take the number 3 we have 10—3.16 nearly, from which we infer that a fraction (x') somewhat less than or 0.5, being made the index of (r) 10, would give 103. This fraction is found by calculation to be 47712; hence 1047712-3; therefore, when r=10, the logarithm of 3 is 47712. In like manner, if we assume the number 5, whose logarithm is to be found in place of that of 3, we have 103-4.64; whence a fraction, (n) somewhat greater than §, or 666, being made the index or ex ponent of 10, would give 10(). 5. This fraction more accurately computed is found to be .69897, that is, when r=10, the logarithm of 5 is .69897. 7. From this it appears, that the value of the logarithm of any given number depends upon the value of the number r, and that by assuming it equal to different numbers, as many different systems of logarithms may be formed as we please. In every system, however, since r°1, the logarithm of 1 must be 0. This constant quantity r from the powers of which the natural numbers are formed, is called the radix or base of the system to which it belongs. 8. In the general equation ry, (art. 5.), let us make a vary, and observe the correspondent variations of y. If r is greater than 1, on making x 0, we have y=1; when x=1 then yr or the logarithm of the base is=1; in proportion as x increases from 0 to infinity, y will increase from 1 towards r, and afterwards to infinity, so that if we suppose x to pass through all the intermediate values, in following the law of continuity, y will increase also in the same manner, though much more rapidly. If we put for x, negative values, we shall have y=r, or 1 y=ut Here we see, in like manner, that the more x increases the 1 more y or decreases, so that in proportion as a augments negatively, y takes all possible values less than 1, as far as 0, in which case x becomes infinite. This was the proposition which Napier made to Briggs on their celebrated meeting at Edinburgh, when conversing on the propriety of changing the logarithmic scale. 1 If r is less than 1 we shall make r=1,6 being greater than 1, and 1 we have y=or y=b*, according as a is positive or negative. We fall here upon the same case, with this difference, that x is positive when y is less than 1, and negative when y is greater than 1. This proposal Briggs made to Napier, but immediately abandoned it on Napier suggesting that mentioned above, which was finally adopted. If r=1, we have y=1 whatever x may be. We may then say generally, that provided r is not unity, there can always be found a value for x, which renders r equal to any given number y. The constant use that is made of the properties of the equation y=r requires the denominations of its parts to be fixed in order to avoid circumlocution. Hence, as before remarked, x is called the logarithm of the number y, the invariable number r is called the base, and, finally, the logarithm of a number, the power to which the base must be raised in order to produce that number. With regard to the baser it is arbitrary, and when we write x=log. y to show that x is the logarithm of the number y, or that y=r*, the base r is always understood, because when once chosen it is supposed to remain fixed. If it should be changed the new base ought to be indicated. 9. From these principles are derived several properties. 1o. In every system of logarithms, the logarithm of 1 is 0, and that of the base r is 1. 2o. If the base r is greater than 1, the logarithms of numbers greater than 1 are positive, the others are negative. The contrary takes place if r is less than 1. 3°. The composition of a table of logarithms consists in determining all the values of x when y is made successively equal to 1, 2, 3, &c. in the equation y=r*. If we suppose r2=μ, on making x=0, ę, 2, 3, &c. We find y=1, μ, μ2, μ3, &c. ne fen The logarithms therefore increase in progression by differences, while the numbers increase in progression by the product or quotient, according as is an integer or a fraction. The ratios are the arbitrary numbers g and . We may, therefore, в regard the systems of values of x and y which satisfy the equation y=r, as classed in these two progressions, which coincides with what has already been said in art. (2.) 10. We shall now demonstrate algebraically the various properties of logarithms. Let N and n be any two numbers belonging to the series (1); and for example, let N=r and n=r, then N n=rx *=*+*, but, by art. 5, the logarithm of r*+*" is x+x'=log. r2+log. r2=log. N+log. n. In like manner, if n, n', n" be any set of numbers in the series (1) it might be shown that the logarithm of nxnxn", &c.=log. n+log. n'+log. n", &c., from which we infer that the logarithm of the product of any number of factors is equal to the sum of their logarithms. N 11. Again n N but the logarithm of x-x'; therefore, the logarithm of —x—x'=log. r2-log. r-log. N-log.n; hence n it appears, that the logarithm of the quotient of any two numbers is equal to the difference of their logarithms; and that the logarithm is equal to the logarithm of its numerator minus N of a fraction () the logarithm of its denominator. If N be less than n, then log. N-log. n is negative; therefore, the logarithms of all proper fractions are negative. 12. Let N=r be raised to the mth power, then Nm=rmx; but the logarithm of rmx is=mx, hence the logarithm of Nm =mx=m log. r* 1 =m log. N ; for the same reason, since"/ N=N"=r”, the logarithm of "N___log. N ; from which we infer, that the logarithm of the mth power of any number is found by multiplying its logarithm by m, and that of the mth root of any number, by dividing its logarithm by m. m m SECTION II. Of the Construction of Tables of Logarithms. 13. Let express generally any term of the series, (1), and let N be the corresponding number, then r-N. Hence to find the logarithm of N is merely to solve the equation -N where x is the unknown quantity. In order to accomplish this purpose let r=1+b and N=1+n, then take the yth power of each side of this equation, and we obtain (1+6)=(1+n) which, by expansion, gives 1+xyb + xy(xy—1) e +xy (xy—1) (xy—2) b2 b3 +&c.= 1+yn+ 2.3 y(y-1) y(y-1) (y-2) n2 + 2 2.3 n3 + &c. Rejecting 1 from each side of the equation, and dividing by y, it becomes x { b + * y = b2 + (xy—1) (xy—2) b3 + &c. } = |
