30 along that line until you find the nearest less logarithm to the given one, and you wil have the fourth figure of the given number at the top, which is to be placed to the right of the three other figures; if you wish for greater accuracy, you must take the difference, D, between this tabular logarithm and the next greater, also the difference, d, between that least tabular logarithm and the given one; to the latter difference, d, annex two or more ciphers at the right hand, and divide it by the former difference, D, and place the quotient to the right hand of the four figures already found, and you will have the number sought, expressed in a mixed decimal, the integral part of which will consist of a number of figures (at the left hand) equal to the index of the logarithin increased by unity.t Thus, if the number corresponding to the logarithm 1.52634 was required, we find 52634 in the column marked 0 at the top or bottom, and opposite to it is 336; now, the index being 1, the sought number must consist of two integral places; therefore it is 33.6. If the given logarithm was 2.32838, we find that 32838 stands in the column marked 0 at the top or bottom, directly opposite to 213, which is the number sought, because, the index being 2, the number must consist of three places of figures. If the number corresponding to the logarithm 2.57345 was required, we must look in the column 0; and we find in it, against the number 374, the logarithm 57287; and, guiding the eye along that line, we find the given logarithm, 57345, in the column marked 5; therefore the mixed number sought is 3745; and, since the index is 2, the integral part must consist of 3 places; therefore the number sought is 374.5. If the index be 1, the number will be 37.45; and if the index be 0, the number will be 3.745. If the index be 8, corresponding to a number less than unity, the answer will be 0.03745, &c. Again, if the number corresponding to the logarithn 5.57811 was required, look in the column 0, and find in it, against 378, and under 5, the logarithin 57807, the difference between this and the next greater logarithm, 57818, being 11, and the difference between 57807 and the given number, 57811, being 4; to this 4 affix two ciphers, which make 400, and divide it by 11; the quotient is 36 nearly; this number, being connected with the former four figures, makes 378536, which is the number required, since, the index being 5, the number must consist of six places of figures. To show, at one view, the indices corresponding to mixed and decimal numbers, we have given the following table. Logarithms. Decimal number. Mixed number. 40943.0.. .Log. 4.61218 4094.3. .Log. 3.61218 409.43. .Log. 2.61218 0.0040943. 40.943. Log. 1.61218 4.0943. ...Log. 0.61218 0.00040943. Logarithms. ...... MULTIPLICATION BY LOGARITHMS. RULE. Add the logarithms of the two numbers to be multiplied, and the surn will be the logarithm of their product. *This quotient must consist of as inany places of figures as there were ciphers annexed, conformable to the rules of the division of decimals. Thus, if the divisor was 40, and the number to which two ciphers were annexed was 2, making 2.00, the quotient must not be estimated as 5, but as 05, and thea two figures must be placed to the right of the four figures before found. the index corresponds to a fraction less than unity, you must place as many ciphers to the left of that number as are equal to the index subtracted from 9, the decimal point being placed to the left of inese ciphers; in this manner you will obtain the sought number may find the fifth figure of the required number by means of the marginal tables, by entering the table corresponding at the top to the proposed value of D, and in the right-hand column with d; the corresponding number is the fifth figure of the required natural number. In the last example, the sum of the two indices is 16; but since 10 was borrowed in each number, we have neglected 10 in the sum; and the remainder, 6, being less than the other 10, is evidently the index of the logarithm of a fraction less than unity. DIVISION BY LOGARITHMS. RULE. From the logarithm of the dividend subtract the logarithm of the divisor the remainder will be the logarithm of the quotient. In Example III. both the divisor and dividend are fractions less than unity, and the divisor is the least; consequently the quotient is greater than unity. In Example IV. both fractions are less than unity; and, since the divisor is the greatest, its logarithm is greater than that of the dividend; for this reason it is necessary to borrow 10 in the index before making the subtraction; hence the quotient is less than unity. INVOLUTION BY LOGARITHMS. RULE. Multiply the logarithm of the number given, by the index of the power to which the quantity is to be raised; the product will be the logarithm of the power sought. But in raising the powers of any decimal fraction, it must be observed, that the first significant figure of the power must be put as many places below the place of units as the index of its logarithm wants of 10 multiplied by the index of the power In the last example, the index 28 wants 2 of 30 (the product of 10 by the power 3); therefore the fire significant figure of the answer, viz. 1, is placed two figures distant from the ole of units EVOLUTION BY LOGARITHMS. RULE. Divide the logarithm of the number by the index of the power; the quotient will be the logarithm of the root sought. But if the power whose root is to be extracted is a decimal fraction less than unity, prefix to the index of its logarithin a figure less by one than the index of the power,* and divide the whole by the index of the power: the quotient will be the logarithin of the root sought. When three numbers are given to find a fourth proportional, in arithmetic, we make a statement, and say, As the first number is to the second, so is the third to the fourth; and by multiplying the second and third together, and dividing the product by the first, we obtain the fourth number sought. To obtain the same result by logarithms, we must add the logarithms of the second and third numbers together, and from the sum subtract the logarithm of the first number; the remainder will be the logarithm of the sought fourth number. To 100 dollars add its interest for one year; find the logarithm of this sum, and reject 2 in the index; then multiply it by the number of years and parts of a year for which the interest is to be calculated; to the product add the logarithm of the sum put at interest; the sum of these two logarithms will be the logarithm of the amount of the given sum for the given time. In this rule it is supposed that 10 is borrowed in finding the index to the decimal according to EXAMPLE. Required the amount of the principal and interest of 355 dollars, let at 6 per cent compound interest, for 7 years. Adding 6 to 100 gives 106; whose logarithm, rejecting 2 in the index, is Multiplied by Product.... Principal, 355 dollars. Sum gives the logarithm of 533.83.. Therefore the amount of principal and interest is 533 dollars and 83 cents. To find the logarithm of the sine, tangent, or secant, corresponding to any number of degrees and minutes, by Table XXVII. The given number of degrees must be found at the bottom of the page when between 45° and 135°, otherwise at the top; the minutes being found in the column marked M, which stands on the side of the page on which the degrees are marked; thus, if the degrees are less than 45, the minutes are to be found in the left-hand column, &c.; and it must be noted that if the degrees are found at the top, the names of hour, sine, cosine, tangent, &c., must also be found at the top; and if the degrees are found at the bottom, the names sine, cosine, &c., must also be found at the bottom. Then opposite to the number of the minutes will be found the log. sine, log. secant, &c. in the columns marked sine, secant, &c. respectively. EXAMPLE I. Required the log. sine of 28° 37'. Find 28° at the top of the page, directly below which, in the left-hand column, find 37'; against which, in the column marked sine, is 9.68029, the log, sine of the given number of degrees; and in the same manner the tangents, &c. are found. EXAMPLE II. Required the log. secant of 126° 20′. Find 126° at the bottom of the page, directly above which, in the left-hand column, find 20′; against which, in the column marked secant, is 10.22732 required. To find the logarithm of the sine, cosine, &c. for degrees, minutes, and seconds, by Table XXVII. Find the numbers corresponding to the even minutes next above and below the given degrees and minutes, and take their difference, D; then say, As 60" is to the number of seconds in the proposed number, so is that difference, D, to a correction, d, to be applied to the number corresponding to the least number of degrees and minutes; additive if it is the least of the two numbers taken from the table, otherwise subtractive. If the given seconds be 2, †, †, †, or, or any other even parts of a minute, the like parts may be taken of the difference of the logarithms, and added or subtracted as above, which may be frequently done by inspection. These proportional parts may also be found very nearly by means of the three columns of differences for seconds, given, for the first time, in the ninth edition of this work. The first colunm of differences, which is to be used with the two columns marked A, A, is placed between these columns. The second column of differences, whica is to be used with the two colun.ns B, B, is placed between these two columns. In like manner, the third column of differences, between the columns C, C, is to be used with them. The correction of the tabular logarithms in any of the columns A, B, C, for any number of seconds, is found by entering the left-hand column of the table, marked S' at the top, and finding the number of seconds; opposite to this, in the column of differences, will be found the corresponding correction. Thus, in the table, page 215, which contains the log. sines, tangents, &c., for 30°, the corrections corresponding to 25", are 9 for the columns A, A, 12 for the columns B, B, 3 for the columns C, C; so that, if it were required to find the sine, tangent, or secant of 30° 12′ 25′′ we must add these corrections respectively to the numbers corresponding to 30° 12′; thus, these corrections being all added, because the logarithms increase in proceeding from 30° 12′ to 30° 13'. Instead of taking out the logarithms for 30° 12, and adding the correction for 25", we may take out the logarithms for 30° 13′, and subtract the correction for 60" - 25", or 35", found in the margin S'; thus, The corrections are in this case subtracted, because the logarithms decrease in proceeding backward 35" from 30° 13', to attain 30° 12′ 25. The tangents and secants, in this example, are the same by both methods; the sines differ by one unit, in the last decimal place, and this wili frequently happen, because the difference of the logarithms for I', sometimes differ one or two units from the mean values which are used in the three columns of differences. The error arising from this cause is generally diminished by using the smallest angle✶ S', when the seconds of the proposed angle are smaller than 30"; or the greatest angle G', when the number of seconds are greater than 30". Thus, in the above example, where the angle S30° 12', and the angle G30° 13', it is best to use the angle S' when the given angle is less than 30° 12′ 30′′, but the angle G' when it exceeds 30° 12′ 30′′. Thus, if it be required to find the sine of 30° 12′ 51′′, it is best to use the angle G'30° 13′, and find the correction by entering the margin marked S', with the difference 60′′-51′′-9′, opposite to which, in the column of differences, is 3, to be subtracted from log. sine of 30° 13′ 9.70180, to get the log. sine of 30° 12′ 51′′=9.70177. To save the trouble of subtracting the seconds from 60", we may use the right-hand margin, marked G', and the correction may then be found by the following rules:— RULE 1. When the smallest angle S' is used, find the seconds in the column S', and take out the corresponding correction, which is to be applied to the logarithm corresponding to S'; by adding, if the log. of G' be greater than the log. of S'; otherwise, by subtracting. RULE 2. When the greater angle G' is used, find the seconds in the column G', and take out the corresponding correction, which is to be applied to the logarithm corresponding to G'; by adding, if the log. of S' be greater than the log, of G'; otherwise, by subtracting; so that, in all cases, the required logarithm may fall between the two logarithms corresponding to the angles S' and G'. The correctness of these rules will evidently appear by comparing them with the preceding examples; and by the inverse process we may find the angle corresponding to a given logarithm, as in the next article. We have given at the bottom of the page, in this table, a small table for finding the proportional parts for the odd seconds of time, corresponding to the column of Hours A. M. or P. M.; to facilitate the process of finding the log. sine, cosine, &c., corresponding to the nearest second of time in the column of hours, or, on the contrary, to find the nearest second of time corresponding to any given log. sine, cosine. &c. Thus, in the preceding examples, where the angle S'30° 12′, and the * If we neglect the seconds m any proposed angle whose sine, &e, is required, we get the angle denoted above by S', and this angle increased by V', is represented by G'; so that the proposed angle falls between S'and G'; S' being a smaller, and G'a greater angle than that whose log. sine, &c., is required; the letters S' and G', accented for minutes, being used because they are easily remembered |