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roots, of numbers whose logarithms are already computed, may be obtained by much simpler means. Thus since
= 2, and e1 0986123 = 3, we have e'6931472 x el 09861236 el·7917595, or the log of 6 is obtained by adding together the logarithms of 2 and 3. Hence, instead of computing the logarithms of 4, 6, 8, 9, and 10, by the series above, they may be computed by simple addition, or by doubling, tripling, &c. when the number is the square, cube, &c. of a number whose logarithm has been already computed; or by subtraction or division by two, three, four, &c. in the
3. To compute the logarithms to any other base, as where a = 10.
Hence M10 log, 2 = log10 2 = ·43429448 × 6931472 = ·3010300
M10 log, 10 = log10 10 = 43429448 × 2·3025851 = 1.0000000,
To render logarithms really useful in computation, we must have them registered in tables. To compress them into the smallest possible space, and at the same time render them convenient for use, several contrivances have been adopted. The editors of such tables differ in the minutiae of their arrangements; but the general principles of their construction are alike in all. Those in most general use are Dr. Hutton's, and hence this description will have reference especially to the last edition of that work. The great accuracy, too, of these tables, independent of the convenience of their arrangement, is a strong reason for this choice.
(1.) The significant figure of a number N is the figure which stands highest in the numerical scale. Thus 5 is the significant figure of 54.69, of 5·469, of ·5469, of '05469, of '005469, &c.
(2). The distance of the significant figure is the number of places in the decimal scale which it is distant from the units' figure it is considered positive when to the left, and negative when to the right of the units' place. Thus in 5469, the significant figure is +3, being in the third place to the left of the units' place 9 in 0.05469 it is 2, (or as more conveniently written, 2) being in the
second place to the right of the units' place O..
(3). The integer part of a logarithm is called the characteristic or index of that logarithm.
(4). The decimal part, which is always positive, is called the mantissa of the logarithm.
(5). The arithmetical complement of a logarithm is its defect from 10.
2. Tabular theorems.
(1). The removal of the unit-place, whilst the effective figures composing N remain the same, will alter the characteristic but not the mantissa.
For let log10 N = m + d, m being any integer number whatever, positive or negative, and d the decimal part, always positive. Then the removal of the decimal point in N, p places will be the same as multiplying N by 10", where p is positive if the removal of the unit-place be to the right, and negative if to the left. In this case we have
log10 10'N = log 10 10 + log N = p + log N = p + m + d;
and since p is an integer, p + m is an integer. Whence the decimal d, or mantissa, remains the same, whilst the characteristic is increased or diminished by p, according as p is + or -.
This, of course, is to be understood in reference to the fourth definition; that the mantissa is always to be taken positive.
(2). The characteristic of a logarithm is that number which expresses the distance of the significant figure from the unit-place.
Let the number N lie between 10+1 and 10"; then its logarithm lies between p + 1 and p; or it is p+ decimal. But the number N is composed of p + 1 places, or its significant figure is p places to the left of the unit-place.
If p be negative, then the number lies between 10+1 and 10"; and hence the logarithm is -p + decimal. But the number 10-" commences in the pth decimal place, and hence p places to the right of the unit-place.
(3). When we have to subtract a logarithm from another, we may add its arithmetical complement and subtract 10 from the sum.
(4). When m is very small with respect to N, we shall have, very nearly,
be still smaller, and have their significant figures further and further removed from the unit-place: and if they be so taken that the significant figure fall more
remotely from that place than the extent to which we calculate our table, these terms may be rejected as insensible. In this case we have
and since the second terms are equal, the first are so, or
1. The hyperbolic or napierean logarithms are given for numbers from 1:01 to 10 to a mantissa of seven places with the proper characteristics, in Table V. pp. 219-223, Hutton's Tables. Then in Table VI. are given those from 1 to 1200 for every unit.
The number is given in the column headed "N," and the corresponding logarithm in the adjacent column headed “ Logar."
These logarithms are only used in calculating integrals: those used for all other purposes being to the base a = 10.
2. The common, or Briggs's logarithms, are given, characteristics and mantissæ, for the numbers from 1 to 100 in Table 1, p. 2. These occupy the first two pairs of columns.
The numbers from 100 to 999 occupy to the bottom of p. 5; but the characteristics are not inserted, they being always determinable by inspection, theorems (1), (2).
The mantissæ of the logarithms of all numbers composed of four places follow these, forming the columns headed "0," from p. 6 to 185; and tabulated as the last, without the characteristics.
The three leading figures of the mantissæ are omitted from those of all the logarithms after the first in which they occur, and the places they would occupy left blank. These spaces are, therefore, to be understood as occupied by the three figures which occur above them. Thus, mantissa of log. 1091 (p. 7) is to be read 0378248.
The mantissæ of those for numbers of five places, as far as 10799, are given in the same manner from p. 186 to 201.
The mantissæ of the logarithms of all numbers of five places are given on the same pages as those of four. This method has been adopted on account of the three leading figures of these mantissæ being the same for several succeeding logarithms; and thereby rendering it only requisite to repeat in the table the last four.
Thus mantissa of log. 10500 is '0211893,
10501 is 0212307,
... 10502 is 0212720,
and so on. Hence the first four figures 1050 are given on the left margin, and the mantissa of its logarithm (which is the same with that of 10500, theor. 1) is given in the column under the fifth figure 0 at the head. The first four figures of the mantissa of 10501 being the same, and the mantissa itself being the same in the first three places, these figures 021 are taken from the first column headed."0," and the remaining four, viz. 2307, from that headed "1": thus giving mantissa of log. 10501 equal to 0212307, as above.
Whenever there is a change of the third figure of the mantissa in any of the columns not headed “0”, the circumstance is indicated by a small line drawn
over the fourth figure of that mantissa, and in this case the first three figures will be taken from column 0 in the horizontal line immediately below. Thus mant. log. 10544 is 0230054, and not '0220054. In like manner, mant. log. 10545 is 0230466, and so on to mant. log. 10549 and at 10550 the first three figures take their regular position in the horizontal lines.
The small tables on the right hand of the page, marked "Dif. and pro. pts.," enable us to obtain the mantissæ of logarithms to numbers of six places of figures. They are constructed on the principle of theorem 4, p. 253.
It will be seen, that the differences between the logarithms of two consecutive numbers of five places vary very slowly, or are nearly the same for several numbers together. It amounts to this, that of several consecutive (and therefore in arithmetical progression) numbers of five figures, the logarithms are also nearly in arithmetical progression; and hence also must the logarithms of numbers in arithmetical progression, and lying between any two consecutive numbers, be also in arithmetical progression. But to appeal to theorem 4, we have
The tablets referred to are composed of the values of this expression for different values of N and m, in the manner of the following example.
Let N 10000, and m = 1. Then M1 = '43429448.
- log.10 10000 = '000043429
or to seven places taking the nearest number, it is '0000434, the effective figures of which are put down at the head of the tablet-column at p. 6 of the tables. In the same manner are the headings of all the tablets calculable. They were, however, not found in this manner; but by subtracting the calculated logarithms, that of each number from that of its next higher consecutive number differing by 1 in the unit-place.
The parts against the numbers 1, 2, 3 9, in the tablets are the values of log10 (N + m) — log10 N for the several values 1, ·2, ·3, 9, or for the numbers 1, 2, 3, in the number whose sixth figures are 1, 2, 9 in the unit-place. These are called proportional parts of the logarithm for the sixth figure, and are inserted for the purpose of being taken out by inspection, instead of having to compute them in each individual case. These corrections are additive to the logarithm if taken to the first five figures of N, and subtractive if to the first five figures of N + 1. The former is the most convenient, and most generally adopted. Thus, to recur to the first tablet, and dropping the ciphers, we have
As for the smaller values of N to six places, especially under 107999, the
values of M10 vary more rapidly than in other higher numbers of the same
local extent, the mantissa for six places have been given in the tables from p. 186 to 201, with their tablets of proportional parts for the seventh figure in the unit-place. These are to be used where great accuracy of approximation is sought, (which, however, is rarely necessary) and their structure is the same as already described.
Of the tables to twenty places, it is unnecessary to say anything here, as well
as of those that follow: since they are well described in the introductory matter of the volume.
4. The usage of the Tables.
The direct use, viz. taking out the logarithms of numbers, is mainly implied in what has been said on the structure of the tables. It will, however, be well to recapitulate briefly in a didactic form the processes.
I. The characteristic. Count how many places to the right or left of the unit-place the significant figure stands. This number is the characteristic (theor. 2, p. 253); and is marked minus if to the right, and considered plus if to
II. The mantissa. 1. If the effective figures be not more than four, the mantissæ of their logarithms will be found in juxta-position with them in the tables, and may be taken out at once *.
2. If the effective figures be five, find the first four in the column marked N, and the fifth in the horizontal line at the top. The last four figures of the mantissa are found at the angle formed by the horizontal and vertical lines in which the first four and the fifth figures are situated, meet and the first three figures of the mantissa adjacent to the first four figures in the horizontal line, or in that immediately below, according to the explanation already given.
Thus, to find the mantissa of log. 74695, look for 7469 (p. 135, Tables) in the column N, and for 5 in the horizontal line at the top. We find at the angle of the lines in which 7469 and 5 stand, the last four figures of the mantissa, viz. 2915, and adjacent (the blank expressing the number above) to it, the first three, viz. 873. So that the mantissa is 8732915.
Or again, had we sought the logarithm of 74819, the last four figures are 0119; and the dash over the O signifying that instead of 873 we must take 874 from the line immediately below that which contains 7481, for the first three figures of the mantissa. Hence the mantissa of 74819 is 8740119.
3. If the number be composed of six effective places of figures, find for the first five as just directed. In the marginal tablet marked "pro," look for the sixth figure, and place the adjacent number below the number already found: add them together: then the sum is the mantissa of the six figures. This is obvious from theor. 4, and from what is there said on the subject.
4. If the given number be a vulgar fraction or mixed number, the fractional part may be reduced to a decimal, and the logarithm of the expression then taken. But if the decimal be of many places, it will be better reduced to a vulgar fraction, and the process adapted to division by logarithms followed.
Note. In actual practice, it is better to write down the first three figures before looking out the remaining four; though, for convenience of explanation, they have been spoken of in a reverse order.
The inverse use, that of finding the number when its logarithm is given, will, obviously, be equally simple and easy.
5. If the mantissa appear exactly in the table, we have but to write it down, and assign the unit-place in conformity with the rule already laid down according to the characteristic.
* In the logarithms of the numbers from 1 to 99, the characteristics are given also, on the supposition of the number being entirely integer. When this is not the case, the characteristic must be assigned according to the general rule.