N Again, has n cyphers after the decimal point, before 10n+1

the first significant digit, and its logarithm is - (n + 1) + d. Hence, if there are n cyphers after the decimal point, the characteristic is (n+1)

This last case requires some explanation :

− (n + 1) + d= − n − (1 − d) — — n — d' = − (n + d'),

where d' = 1-d and is some decimal. Here the logarithm is negative, and the integral part -n: also the mantissa d' of N log 10+1 is not the same as d the mantissa of log N, which seems to contradict what was before observed: but it is found more convenient to consider all numbers as having logarithms with positive mantissæ, and it is with this convention that we

say that log N and log 10+1 have the same mantissæ, and that

the characteristic of the latter is (n+1). Without this convention we should be obliged to have two tables, one for numbers greater, and the other for numbers less than unity.

Ex.

and

the sign placed above the 3 in the last case signifying that the characteristic alone is negative.

Thus 3-7399835 and — 3·7399835 are different: in fact

3.7399835 == 3+ 7399835

=

- 2.2600165.

58. In the tables in common use the mantissæ of the logarithms of all numbers from 1 to 100000, calculated to seven decimal places, are arranged in order. Thus when we have any number containing only five significant digits we find the mantissa of its logarithm directly from the tables, and prefix the characteristic by inspection; and if we have a logarithm, we can find the logarithm nearest to it in the

tables, and so find the corresponding number correctly to five significant digits.

Numbers of more than five digits must be taken from the tables by means of what are called proportional parts: for an explanation of which the student must consult more advanced treatises on the subject. We will only observe that in determining the side of a triangle of about a mile in length to five digits, the error cannot exceed one inch; a degree of accuracy which is fully equal to that of the ordinary instruments for surveying.

59. Since the Trigonometrical Ratios of angles are real quantities, they also can have logarithms, just in the same way as other quantities. The logarithms of the sines, cosines, tangents, cotangents, secants, cosecants of all angles are arranged in ordinary tables at intervals of one minute. In some tables they are given for every second. They are calculated to seven decimal places as for common numbers; but since they are mostly negative 10 is added to them, to make them positive: hence the tabular logarithms of the Trigonometrical Ratio of any angle is equal to the real logarithm increased by 10. Or as they are written,

L sin a = log sin a + 10,

and so for the others.

60. Also since sin a= cos (90 — a), therefore log sin a= log cos (90 — a).

Hence if we have the log-sines of all angles from 0° to 45o, we also have the log-cosines of all angles from 90° to 45°: and if we have the log-cosines of all angles from 0° to 45°, we have also the log-sines of all angles from 90° to 45°. Hence a complete table of log-sines and log-cosines from 0° to 45° is also a complete table to 90°. This is taken advantage of in the arrangement of the tables. The column headed sines and counted downwards from the top of the page is headed cosines at the bottom of the page and reckoned upwards, as may be best understood by inspecting the tables. The same arrangement applies to the tangents and cotangents, the secants and cosecants.

1

1. Find the logarithms of

309-017, 0000309017, 309017000.

2. Find the logarithms of all numbers from 1 to 10.

3. Find the logarithms of the Trigonometrical Ratios of 30° and 45o.

6. Find the value of

Ans. 1.69408.

3. Find the value of sin 18°; and shew that the value found above (page 40) is equal to it.

5. Shew that log, x: log, x = log, b: log, a.

into equations between the tabular logarithms of the quantities involved.

7. Assuming that log 250 and log 256 differ by 0103, shew that log 2 = 30103.

8. Find the logarithm of 9 to base 3/3, and of 125 to base √/5.2/5.

Ans. 1.3, 3.6.

77

SECTION VII.

APPLICATION OF LOGARITHMS TO THE SOLUTION OF
TRIANGLES; ILLUSTRATION.

61. WE have seen that when we know the logarithms of any numbers we can at once find the logarithm of their product and quotient, but that we cannot apply logarithms directly to the determination of the sum or difference of numbers. Hence in calculating any parts of a triangle from the others by logarithms we must use the formula which connect the parts by means of factors only, and if we wish to use one of the others we must transform it to the form of a product.

Thus in calculating a from the formulæ

we cannot apply logarithms directly, but if we transform this equation to

logarithms become immediately applicable.

We may observe by the way, that in using the first equation we should have to perform four operations in multiplication and one in division; in using the latter we should have to look out five logarithms. If a, b, c were simple numbers the first operation would probably give the least trouble, but as in practice they are generally quantities of five or six digits, the latter is much the shortest.

62. In most cases of solution of triangles the formulæ to be employed are in the form of products: the case which offers most difficulty is when two sides are given and the included angle; this we accordingly proceed to discuss.