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is 2, since 658 is greater than 100 or 10°, and less than 1000 or 10'; and so on. But the necessity of considering between what two powers of 10, any proposed whole number lies, is superseded in practice by the following simple rule, the reason of which must now be obvious.
Rules for the characteristic.-Io. Where the given number is an integer, or an integer with decimals annexed.-Count the number of digits preceding the decimal point, and write as the characteristic one less than this number. Examples. Characteristic of the logarithm of 78645 is 4 (5 integer digits less 1.)
7864.5 , 34
7-8645 ; 0 (1 In each of these instances, the characteristic evidently expresses the lower of the two powers of 10, to which the given number is intermediate.
II.° Where the number consists entirely of a decimal fraction. Let the number which denotes the place of the first significant figure of the given decimals be written down, and a bar (negative sign) be placed over it. Examples. Characteristic of •78645 is 1, since 1st sig, fig. stands in 1st decimal place.
•07864 , 2,
3rd , &c. &c. With respect to the reason of this rule it will be sufficient to observe, that as 1 = 1 the exponent of the quotient of 1 + 10, or 10o = 10' (see last table) should be found by subtracting the exponent 1 from the exponent 0; but this is impossible arithmetically, and the fact that the operation cannot be performed is indicated by the use of the expression 0-1, or for shortness' sake, — 1, 0 being understood. It is convenient also to place the negative sign, as directed in the rule, over the characteristic instead of before it, for otherwise the computer might suppose that the entire logarithm was affected by the mark, when it is only the characteristic that is to be considered negative.* Again, the characteristic of
100 •01 is on the same principle written 2, because the exponent of or
100 OF 102 sho be 1) - 2; but the impossible subtraction here directed, is indicated by the symbol – 2, and so on. Now that which applies to :1, .01, &c., must, it is obvious, apply also to any number of tenths, hundredths, &c.
It is further to be remarked, that if some such device were not adopted to distinguish the characteristic of the logs. of decimal fractions, the operator would be unable to tell from what kind of number a given characteristic was derived.
Characteristics are rarely, if ever, inserted in tables of logarithms, for no one characteristic can refer to more than one particular value of the number opposite to wbich it is placed, and it is necessary, in the course of calculation, to employ several distinct decimal values of the same succession of figures. There is, besides, no difficulty in supplying the characteristic of any required log. from the proper rules laid down on the subject.
The short specimen of a table of logs. given at page 225 convevs but a very inadequate idea of the great utility of such numbers in lessening the labor of computation, since most of the processes that can be performed by aid of the few logs. included in that table, may be as easily effected by simple arithmetic. Some examples, however, of the use of a more extensive table will clearly demonstrate the advantage to be obtained by operating with logarithms in the case of numbers composed of several figures.
• As an illustration of the propriety of connecting the bar or negative sign with the characteristic only and not with the entire log., suppose it required to express the log. of the decimal 7. Log. of 7 = log. of 7 - log. of 10, that is (800 last table) 0.84510 – 1= 0 + •84510 -1= -1 + 84510, otherwise written, 1.84510.
The tables from which the following examples are worked out, are those contained in the Appendix, No. 17, pages 445 to 448. By the word «Antilogarithm,” the title of the second table, is meant, the natural number opposite or answering to a logarithm. * As the foot-notes appended to the tables in question show how the numbers in the columns of « Proportional parts” are to be applied to the finding of a fourth figure of each log. and antilog. respectively, there is nothing further needed in the way of preliminary explanation. The proper mode of treating the negative characteristic will be better understood from the practical illustrations afforded, than from any formal precepts.
To find the product of two or more numbers by means of logarithms.-
Log. 7.281 = 0.8622
Sum 3.3587 Antilog. 2284. Answer. The decimal figures •3587 correspond to the antilog. 2284; and as the characteristic of the logarithmic result is 3, it indicates that there must be four integer figures in the product. According to actual multiplication of the given numbers, the product should be 2284.0497. As regards this instance, therefore, all the figures obtainable, from the present table of logs, are correct; but the accuracy of the last or fourth figure cannot be depended on in every case, especially towards the end of the tables.
To find by means of logarithms, the quotient of one number divided by another.-
Log. .0649 = 2.8122
Difference 1.3547 Antilog. .2263. Answer. To subtract a negative characteristic, the rule is, -Add an equal positive characteristic; for to take away that which is negative is plainly equivalent to adding or bestowing the same amount of that which is positive. It is immaterial, for instance, whether we remove a debt or present an equal sum in the shape of a gain. Thus, 1 subtracted as above from 2, is tantamount to 1 added to 2, that is, to one positive added to two negatives, the difference or balance of which is, of course, one negative (1), since these signs have exactly opposite values.
If, after deducting the tenths, there was anything to be carried, such positive remainder would have the effect of cancelling a unit in the negative characteristic of the subtrahend; e. g., let it be proposed to subtract 3.4643 from 2.3785.
4-9142 Here we say, in the tenths' column, “4 from 13 and 9, carry 1 to 3; balance 2, and 2 taken from 2 is the same as 2 added to 2; result 4.” Another illustration of this mode of operating with the negative characteristic will be found in the next example.
* Tables of antilogarithms are, in general, more trustworthy, and more facile of reference than tables of logarithms when it becomes necessary to translate logarithmic results back into ordinary numbers,
Fxample (3.) Multiply together 858, 496, 4.7, and .0004508.
Log. 858 = 2.9335
496 = 2.6955 , 4.7 = 0.6721 „.0004508 = 4.6540
2-9551 Antilog. 901.8. Answer. There is 2 to carry from the sum of the tenths; 2 and 4 equal 2; 2 and 2 and 2, equal 2. Or carrying the 2 from the tenths to the top row, we may say, 2 and 2 and 2 = 6, 6 and 7 = 2."
To involve a given number to any power by means of logarithms.RULE.-Multiply the log. of the number by the exponent of the power required ; the antilog. of the product will be that power. Example (4.) Required the square of 53-87.
Log. 53.87 = 1.7314
2 (expt. of power)
3.4628 Antilog. 2902. Answer. Example (5.) Involve -5387 to the 6th power.
Log. -5387 = 1.7314
2.3884 Antilog. .02445. Answer. To arrive at this approximate result, in the ordinary way, two multiplications of 5387 by itself, and one multiplication of the result by itself would, at least, be requisite. The labour of this process, even by the method of contracted multiplication, would be very considerable.
To extract any root of a number by means of logarithms.-RULE.—Divide the log. of the number by the exponent of the required root. The antilog. of the result will be that root. Example (6.) What is the cube root of 1291 ?
Log. 1291 = 3.1109
: 3 1.0370 Antilog. 10.89. Answer. To divide a log, with a negative characteristic by a whole number, where the characteristic will not contain the divisor an exact number of times begin by increasing the characteristic until it is divisible, without remainder, by the whole number; set down the quotient with the negative sign over it, and before dividing the decimal part of the logarithm, conceive that it has prefixed to its first figure the number by which it was found necessary to increase the characteristic, Example (7.) Extract the 5th root of .4759.
Log. 4759 = 1.6775
5 1.9355 Antilog. -8620. Answer. Here, as the characteristic I will not contain 5, we add 4 to it, and then divide by 5; the quotient is written 1, and the increment of the characteristic being carried to the next figure makes it 46, which divided by 5 gives 9 and 1 over, and so on to the end, in the manner of ordinary division.
A few additional examples showing the applicability of logarithms to certain questions occurring in the practice of the Excise will be given in the next two chapters.
With respect to the power of a table of logarithms, that is, the extent to which the results afforded by it can be depended upon as accurate, it may be observed generally, that the natural number answering to any logarithm contains, at least, as many places of figures as the log. has places of decimals ; thus, a table of fourfigure logarithms enables us to obtain results, each consisting of an ordinary number of four figures. But the last of these figures cannot always be regarded as correct to a unit, especially towards the end of the table. It will be safest, therefore, in employing a table for business purposes, to consider its absolute
correctness as limited to one figure less than the number of decimal places of which each logarithm is composed. The numbers furnished by logs. of five figures, for instance, are invariably correct as far as their fourth figure, and so on.*
(15.) DESCRIPTION AND THEORY OF THE SLIDE-RULE.—The slide-rule, or sliding-rule, is a portable calculating instrument, constructed chiefly on logarithmic principles, by means of which certain processes of arithmetic may be executed mechanically in a very simple and expeditious manner, and with a degree of accuracy proportioned to the power or dimensions of the instrument, the minuteness and precision of its graduations, and the skill of the operator.
A series of scales in logarithmic progression,-that is, as distinguished from scales of equal parts,-is laid down once or twice over on a flat surface of wood, ivory, or metal, the successive divisions of these scales being placed at distances apart corresponding to the differences between the logarithms of the natural numbers from 1 up to 10, from 10 up to 100, &c., or within some other range and following some definite order suited to particular calculations. Instead of marking the several points of graduation with appropriate logarithms, the natural numbers represented by such logs. are attached to the larger divisions, and the necessity of translating the results back into ordinary arithmetical values is thus avoided.
Portions of the rule consist of moveable pieces or slides, which fit into m grooves cut for their reception along the edges of the fixed scales. The slides are graduated and numbered similarly to the principal lines on the 73 stock of the rule, and are capable of being used in connexion, that is, one in continuation of the other, when inserted in the same groove ; by this xbe arrangement, the facility of working is greatly enhanced, and the instrument is also invested with double the power it would otherwise possess.
It may be readily conceived in a general way, that the adjustment of 1 the slides to various points on the stock will enable us to add together or subtract from one another the distances included between any two sections of the scales ; and that as the distances are laid down in logarithmic progression, the lineal sums or differences so obtained will bave marked against them respectively, the product or quotient of the WITH corresponding natural numbers inscribed upon the rule.
Let A B be a rule bearing a scale of equal parts, and furnished with a slide CD, divided in a similar manner. It is evident that an instrument of this construction will enable us to perform the operations of Addition and Subtrac-ih tion on all whole numbers between 1 and 10, the limits of its graduation.
If, for instance, we wish to add 2 to 7, we have but to draw out the slider from its initial position in which the divisions 0 0,1 1,2 2, &c. correspond on both scales) two intervals to the right, so that 0 on the slide shall coincide with 2 on the rule. The number 7 on the slide will then stand opposite to 9 on the rule; and 9 is the sum of 2 and 7. The principle of this operation is obvious. To a line of the length of 2 on the rule has been added by means of the slide, a line of the length of 7, and the result, as marked on the rule, must be 9.
Subtraction may, of course, be effected on a like principle. Suppose we wish I tal. to take 7 from 9. Make the 7 on the slide correspond to 9 on the rule; then il TIL the 0 on the slide will correspond to 2 on the rule, which is the difference willel required. In this case, as may be seen by adjusting the slide as directed, a H HIS line equal to 7 is taken away from a line equal to 9.
Now a rule which gives us the power of executing no other processes than those of simple addition and subtraction would be of very little practical utility, and would be hardly worth the trouble of its construction. We are enabled, however, by adopting a different mode of graduation, to render an instrument of this kind available to a certain extent for the performance of multiplication and division, involution and evolution.
* The proper mothods of using the numbers in the column of differences attached to tables of five, six, or seven figure logarithms must be acquired by reference to some of the standard treatises on logarithms. See the popular work in Wealo's Rudimentary Series.
Let it now be supposed that the divisions on the rule instead of being equal to one another are all of unequal length, and that they are proportional to the differences between the logarithms of the natural numbers from 1 to 10 inclusive. (See fig. opposite.) Let it also be supposed, as before, that the slide is graduated in a similar manner, and that the divisions on both scales are marked with the numbers corresponding to the logarithms, and not with the logarit thms themselves. We shall thus have the digits, 2, 3, 4, &c., inscribed at distances proportional to 301, 176, 125, -097, .079, 067, 058, 051, .046, which numbers are the differences respectively of the logarithms, 0, 301, 477, .602, 699, 778, •845, :903, 954, and 1.
With an instrument graduated on this principle, it is plain, that we may still effect the addition or subtraction of any two of the given unequal lengths: but inasmuch as these lengths are arranged in logarithmic progression, the distance obtained by adding together two or more intervals of the similar scales laid down on the rule and slider will have placed opposite to it the product of the numerical values of such distances instead of the sum of such values; and the distance measured by the difference of two intervals of the scale will be marked with a number expressive of the quotient of the value of the greater distance divided by that of the less, and not of the remainder left on subtracting one distance from the other, as would be the case were we to employ a scale of equal parts. Hence it will be apparent, that by a simple adjustment of the divisions on the slide to those on the upper line of the rule, operations of multiplication and division may be performed in respect of any of the numbers comprised within the range of the scales.
In the accompanying figure, 1 on the slide is set in a line with 2 on the rule. Then over the numbers 2, 3, 4, 5, &c., on the slide, appear the numbers 4, 6, 8, 10, &c., on the rule, that is, the products 2 x 2, 2 X 3, 2 X 4, &c. A rule thus set, therefore, has all the power of one of the columns of an ordinary multiplication table. Again, in the present position of the slide, a series of lengths equal respectively to 2, 3, 4, &c., on the slide, may be conceived as taken away from a series of lengths equal respectively to 4, 6, 8, &c., on the rule. The difference of these lengths, as shown at the point B on the rule is replaced by 2, the uniform quotient of 4 = 2, 6 = 3,
8 4, &c.t Thus, then, in order to effect multiplication and division by means of a rule, such as is represented in the margin, we have the following rules :
For products.- Place the origin or division marked 1 on the slide opposite to either of the two given numbers on the rule. The product required will be found on the rule opposite to the other number on the slide.
For quotients. Place the divisor on the slide opposite to the dividend on the rule. Then under 1 on the slide will be found the quotient on the rule.
The Excise Officer's Pocket Slide Rule.-The principal lines upon all slide-rules are laid down in the manner above described ; that is, at intervals corresponding to the differences of certain logarithms. Each variety of rule employed in the arts bears a set of scales adapted to special calculations, and only partly available for general purposes ; in the present work it is unnecessary to enter into an account of any other form of the rule than that which is used by officers of excise to facilitate the casting up of their gauges, or the working out of some of the practical questions arising in their every-day business.
* It would, of course, be useless to extend these logarithms to a greater number of places where the object sought is the laying down of a series of short distances by means of a sector or a diagonal scale.
+ The construction and use of the lower line graduated on the rule will be explained a little further in advance.