If the diff. (100 above) is not found exactly in the little table, take the nearest number to it, and take the difference between these again, and annex to it a cipher, seek this result again in the right hand column of the little table, the figures on the left of the two numbers taken out of the right hand column of the little table will be the sixth and seventh figures of the number required A third remainder might be found in the same manner, and an eighth figure of the required number be found.

The last two figures of the number sought are 93, and the number itself is 3.141593. The same method is applicable to our tables, though not with the same degree of accuracy.

The nearest No. to which in the little table is 34 which subtract; on its left is 9.

The rem is.

The number required is therefore 1.15293.

11

on left of which in the marginal table is 3.

EXAMPLES IN MULTIPLICATION AND DIVISION BY LOGARITHMS.

54. 1. Required the product of 26784 and 7.865.

5.323574 is the log. of 210656, which last number is, therefore, the product required.

In looking for the log. of this number, look first for that of 2678, multiply the tab. diff. by 4, the last figure of the given number, and cut off one figure from the product.

2. Required the product of 3′586, 2∙1046, 8372, and •0294.

Instead of using negative characteristics, a method is sometimes employed of taking the difference between the negative characteristic and 10, which is really adding 10 to the negative characteristic, and writing this difference as a positive characteristic; thus, in the above example,

twice 10 must be rejected from the sum. That is a 10 for each posi

tive characteristic employed in the place of a negative.

The result thus obtained 1.26896 is written 9.26896, 10 being added again to avoid the negative characteristic.

I.35222 is, therefore, the log. of the quotient which from the tables, observing the converse rule for pointing off decimals according to the characteristic (3 Art. 50), is 225019.

55. We shall now demonstrate rules for raising numbers to powers, and for extracting the roots of numbers, by means of logarithms.

Resume the equation,

raising both members to the m" power, we have, observing the rule of Algebra, which is to multiply the exponent by the degree of the power,

blm = nm

From this last equation, it appears that Im is the power to which it

is necessary to raise the base b in order to produce n"; hence the following

RULE. To raise a number to any power, by means of logarithms, multiply the logarithm of the given number by the exponent of the power, and the product will be the logarithm of the power.

5.84816 is the log of 000070494, which last number is the power required.

In multiplying the first decimal place by 10, the product is 80, then 10 times 1 is 10, and 8 to carry is 2.

The same example, with positive characteristics, according to the method pointed out on p. 45 would stand thus

9.80618

10

8.06180

The 10 added to the 1 is repeated 10 times and therefore 100 must be rejected from the product, which leaves 2 for the characteristic to be written 8. The rule for placing the decimal point in the number corresponding to the logarithm will be to place one less cipher after the decimal point than is expressed by the difference between the characteristic and 10. 56. To find a rule for extracting the root of a number by means of logarithms, assume again the equation

Take the m2 root of both members, applying in the first member the rule to divide the exponent by the number expressing the degree of the root, and there results

m

is here plainly the logarithm of ✔n; hence the following

RULE. To extract the root of a number by means of logarithms, divide the logarithm of the given number, by the index of the root, and the quotient will be the logarithm of the root.

EXAMPLES.

1. Required the 4th root of 434296.

log. of ⚫434296

1.63779

of this logarithm is obtained by observing that the index, which alone is negative, must be divided separately, as we should divide a minus term, followed by a plus term in Algebra; the I can be rendered divisible by borrowing 3, and afterwards carrying +3 before the 6, rendering it 36; that is, the proposed logarithm is viewed under the form 4+3.63779.

The quotient is I-90945* which is the logarithm of 8118, the fourth root required.

57. This is Table XXVII. It contains the logarithm of the sine, tangent, cosine and cotangent, secant and cosecant, corresponding to every degree and minute in the quadrant.t

These logarithms are those of the trigonometrical lines in a circle, the radius of which is 10000000000, or the tenth power of 10, the common logarithm of which is 10. As the sine is never greater than radius, its logarithm will always be less than 10, except for the arc 90°, the logarithmic sine of which is equal to 10..

The last quotient figure is nearer 5 than 4.

+ Without this table we should have been obliged to employ the other two tables which have been already described, as follows. sine, tangent, &c., of the given arc or angle in entered Table XXVI. and found its logarithm. the columns of secants and cosecants.

First we must have found the natural
Table XXIV. then with this have
The tables of Callet do not contain

PROBLEM.

58. To find from the table the logarithm of the sine, tangent, or cosine of the number expressing any arc.

CASE I. If the given number be composed of degrees and minutes, seek first for the number of degrees among those which are written at the top or bottom of the pages; at the top and on the left of the page if it be less than 45°; at the bottom and on the right of the page if it be greater. Run the eye down the first column which goes on increasing from top to bottom, if the number of degrees is found at the top of the page; or up the last column, which goes on increasing from the bottom upwards, if the number of degrees is found at the bottom; run the eye, I say, through one or the other of these columns in the direction in which it increases until you have found the number of minutes given; upon the same horizontal line with the minutes thus found you will find the logarithm of the sine, cosine, tangent, or cotangent which you seek. In order not to mistake the column, it is necessary to consult the title at the head of the column, if the number of degrees given is at the top of the page, but if it is at the bottom, the inferior title must be consulted.*

EXAMPLES.

1. Required the logarithmic sine, tangent, secant, cosine, cotangent, and cosecant of 19° 55'.

I find 19° at the top of page 204, I descend the first column at the left marked м, which goes on increasing downwards till I find 55'; upon the same horizontal line, and in the column entitled sine at top, I find 9.53231, in the column entitled cosine 9.97322, in the column of tangents 9.55910, and in that of cotangents 10.44090, that of secants 10.02678, that of cosecants 10.46769; and these numbers are therefore the numbers required.

2. Required the logarithmic sine and tangent of 70° 10'.

I find 70° at the bottom of p. 204†; I ascend the last column marked M at bottom which goes on increasing upwards; I find 10' in that column; upon the same horizontal line I find in the column marked sine at bottom 9.97344, and in the column marked tangent at bottom 10.44288, which are the logarithms sought.

The columns marked Hour A. M. and P. M. are connected with Nautical and Practical Astronomy, and will be explained under the proper head.

+ For the reason mentioned in note p. 33, the paging of the tables will be found to exhibit gaps.