Part 2

It will now be evident that for numbers

between 1 and 10 the logs. will be between 0 and 1 „ 10 „ 100 „ „ 1 „ 2 „ 100 „ 1000 „ „ 2 „ 3 „ 1000 „ 10,000 „ „ 3 „ 4

In other words, the logarithms of numbers between 1 and 10 will be wholly fractional (i.e., decimal); the logs. of numbers between 10 and 100 will be 1 followed by a decimal quantity; the logs. of numbers between 100 and 1000 will be 2 followed by a decimal quantity, and so on. These decimal quantities for numbers from 1 to 10 (which are the logarithms of this particular series) are as follows:—

Numbers 1 2 3 4 5 6 7 8 9 10 Logarithms 0 0·301 0·477 0·602 0·699 0·778 0·845 0·903 0·954 1·000

Combining the two tables, we can complete the logarithms. Thus for 3 multiplied successively by 10, we have:—

Numbers 3 30 300 3000 30,000 etc. Logarithms 0·477 1·477 2·477 3·477 4·477 „

We see from this that for numbers having the same significant figure (or figures), 3 in this case, the decimal part or mantissa of the logarithm is the same, but that the integral part or characteristic is always one less than the number of figures before the decimal point.

For numbers less than 1 the same plan is followed. Thus extending our first table downwards, we have:—

Numbers 1 0·1 0·01 0·001 0·0001 etc. Logarithms 0 −1 −2 −3 −4 „

so that for 3 divided successively by 10, we have:—

Numbers 3 0·3 0·03 0·003 0·0003 etc. Logarithms 0·477 ̅1·477 ̅2·477 ̅3·477 ̅4·477 „

Here again we see that with the same significant figures in the numbers, the mantissa of the logarithm has always the same (positive) value, but the characteristic is one more than the number of 0’s immediately following the decimal point, and is negative, as indicated by the minus sign written over it. Only the decimal parts of the logarithms of numbers between 1 and 10 are given in the usual tables, for, as shown above, the logarithms of all tenfold multiples or submultiples of a number can be obtained at once by modifying the characteristic in accordance with the rules given.

An examination of the two rows of figures giving the logarithms of numbers from 1 to 10 will reveal some striking peculiarities, and at the same time serve to illustrate the principle of logarithmic calculation. First, it will be noticed that the addition of any two of the logarithms gives the logarithm of the product of these two numbers. Thus, the addition of log. 2 and log. 4 = 0·301 + 0·602 = 0·903, and this is seen to be the logarithm of 8, that is, of 2 × 4. Conversely, the difference of the logarithms of two numbers gives the logarithm of the quotient resulting from the division of these two numbers. Thus, log. 8 − log. 2 = 0·903 − 0·301 = 0·602, which is the log. of 4, or of 8 ÷ 2.

One other important point is to be noted. If the logarithm of any number is multiplied by 2, 3, or any other quantity, whole or fractional, the result is the logarithm of the original number, raised to the 2nd, 3rd, or other power respectively. Thus, multiplying the log. of 3 by 2, we obtain 0·477 × 2 = 0·954, and this is seen to be the log. of 9, that is, of 3 raised to the 2nd power, or 3 squared. Again, log. 2 multiplied by 3 = 0·903—that is, the log. of 8, or of 2 raised to the 3rd power, or 2 cubed. Conversely, dividing the logarithm of any original number by any number n, we obtain the logarithm of the nth root of the original number. Thus, log. 8 ÷ 3 = 0·903 ÷ 3 = 0·301, and is therefore equal to log. 2 or to the log. of the cube root of 8.

Only simple logs. have been taken in these examples, but the student will understand that the same reasoning applies, whatever the number. Thus for 20^3 we prefix the characteristic (1 in this case) to log. 2, giving 1·301. Multiplying by 3, we have 3·903 as the resulting logarithm, and as its characteristic is 3, we know that it corresponds to the number 8000. Hence 20^3 = 8000.

In this brief explanation is included all that need now be said with regard to the properties of logarithms. The main facts to be borne clearly in mind are:—(1.) That to find the product of two numbers, the logarithms of the numbers are to be added together, the result being the logarithm of the product required, the value of which can then be determined. (2.) That in finding the quotient resulting from the division of one number by another, the difference of the logarithms of the numbers gives the logarithm of the quotient, from which the value of the latter can be ascertained. (3.) That to find the result of raising a number to the nth power, we multiply the logarithm of the number by n, thus obtaining the logarithm, and hence the value, of the desired result. And (4.) That to find the nth root of a number, we divide the logarithm of the number by n, this giving the logarithm of the result, from which its value may be determined.

NOTATION BY POWERS OF 10.

A convenient method of representing an arithmetical quantity is to split it up into two factors, of which the first is the original number, with the decimal point moved so as to immediately follow the first significant figure, and the second, 10^{n} where n is the number of places the decimal point has been moved, this index being positive for numbers greater than 1, and negative for numbers less than 1. In this system, therefore, we regard 3,610,000 as 3·61 × 1,000,000, and write it as 3·61 × 10^6. Similarly 361 = 3·61 x 10^2; 0·0361 (= (3·61)/(100)) = 3·61 × 10^{−2}; 0·0000361 = 3·61 × 10^{−5}, etc. To restore a number to its original form, we have only to move the decimal point through the number of places indicated by the index, moving to the right if the index is positive and to the left (prefixing 0’s) if negative. This method, which should be cultivated for ordinary arithmetical work, is substantially that followed in calculating by the slide rule. Thus with the slide rule the multiplication of 63,200 by 0·0035 virtually resolves itself into 6·32 × 10^4 × 3·5 × 10^{−3} or 6·32 × 3·5 × 10^{4–3} = 22·12 x 10^1 = 221·2. It will be seen later, however, that the result can be arrived at by a more direct, if less systematic, method of working.

THE MECHANICAL PRINCIPLE OF THE SLIDE RULE.

The mechanical principle involved in the slide rule is of a very simple character. In Fig. 1, A and B represent two rules divided into 10 equal parts, the division lines being numbered consecutively as shown. If the rule B is moved to the right until 0 on B is opposite 3 on A, it is seen that any number on A is equal to the coinciding number on B, plus 3. Thus opposite 4 on B is 7 on A. The reason is obvious. By moving B to the right, we add to a length 0·3, another length 0·4, the result read off on A being 7. Evidently, the same result would have been obtained if a length 0·4 had been added, by means of a pair of dividers, to the length 0·3 on the scale A. By means of the slide B, however, the addition is more readily effected, and, what is of much greater importance, the result of adding 3 to any one of the numbers within range, on the lower scale, is immediately seen by reading the adjacent number on A.

Of course, subtraction can be quite as readily performed. Thus, to subtract 4 from 7, we require to deduct from 0·7 on the A scale, a length 0·4 on B. We do this by placing 4 on B under 7 on A, when over 0 on B we find 3, on A. It is here evident that the difference of any pair of coinciding numbers on the scales is constantly equal to 3.

An important modification results if the slide-scale B is inverted as in Fig. 2. In this case, to find the sum of 4 and 3 we require to place the 4 of the A scale to 3 on the B scale, and the result is read on A over 0 on B. Here it will be noted, the sum of any pair of coinciding numbers on the scales is constant and equal to 7. This case, therefore, resembles that of the immediately preceding one, except that the sum, instead of the difference, of any pair of coinciding numbers is constant.

To find the difference of two factors, the converse operation is necessary. Thus, to subtract 4 from 7, 0 on B is placed opposite 7 on A, and over 4 on B is found 3 on A.

From these examples it will be seen that with the slide inverted the methods of operation are the reverse of those used when the slide is in its normal position.

It will be understood that although we have only considered the primary divisions of the scales, the remarks apply equally to any subdivisions into which the primary spaces of the scales might be divided. Further, we note that the length of scale taken to represent a unit is quite arbitrary.

THE PRIMITIVE SLIDE RULE.

The application of the foregoing principles to the slide rule can be shown most conveniently by describing the construction of a simple form of slide rule:—Take a strip of card about 11 in. long and 2 in. wide; draw a line down the centre of its width, and mark off two points, 10 in. apart. Draw cross lines at these points and figure them 1 and 10 on each side, as in Fig. 3. Next mark off lengths of 3·01, 4·77, 6·02, 6·99, 7·78, 8·45, 9·03 and 9·54 inches, from the line marked 1. Draw cross lines as before, and figure these lines, 2, 3, 4, 5, 6, 7, 8 and 9. To fill in the intermediate divisions of the scale, take the logs, of 1·1, 1·2, 1·3, etc. (from a table), multiply each by 10, and thus obtain the distances from 1, at which the several subdivisions are to be placed. Mark these 1·2, 1·3, 1·4, etc., and complete the scale, making the interpolated division marks shorter to facilitate reading, as with an ordinary measuring rule. Cutting the card cleanly down the centre line, we have the essentials of the slide rule.

The fundamental principle of the slide rule is now evident:—Each scale is graduated in such a manner that the distance of any number from 1 is proportional to the logarithm of that number.