As this part of the subject is rather intricate, it is not further entered on here.

131. Using the tables of proportional parts, the working of the foregoing examples would be as follows:

(2.) To find the number whose logarithm is 4·8026755.

.. required number = 00063485643.

132. Tables of Natural Sines, Cosines, &c. In some of the previous chapters, numerical expressions have been obtained for the trigonometrical ratios of the angles 15°, 18°, 30°, &c. These expressions with but little difficulty may be turned into decimal fractions, which, however, very seldom terminate. For instance,

tan 75° 2+ √3 = 2+1·73205... = 3·73205...

Decimal expressions may be obtained not only for the angles referred to, but for any angles; and are called the natural functions of the angles. Tables of them are compiled for all angles from 0 to 90°, at

M

intervals of 1', or sometimes of 20" or 10′′; and are called tables of natural sines, cosines, &c.

In order to use these tables for angles not registered in them, the principle of proportional parts is applied, and may be stated as follows in this case: The change in any function of an angle is approximately proportional to the change of the angle, if it be small in comparison with the magnitude of the angle.

It will be remembered, that up to 90°, as the angle increases in magnitude, the sine, tangent, and secant also increase; but the cosine, cotangent, and cosecant decrease. The application of this fact will be seen in the following examples, and must be carefully attended to.

133. (1.) Find the value of sin 36° 4′ 13′′; given from the tables, sin 36° 4' 5887262, sin 36° 5′ = 5889613.

=

OBS. 1.-The differences between each pair of consecutive functions are usually given in tables, omitting their decimal points and the ciphers immediately following the points.

OBS. 2.-If in the above example we had been seeking cos 36° 4′ 13′′, the proportional part for 13′′ would have been subtracted from, not added to, cos 36° 4′.

(2.) Find the angle whose natural cosine is 3784241.

134. Tables of Logarithmic Sines, Cosines, &c.

Tables of logarithmic functions simply comprise the

logarithms of all the numbers which form the tables of natural functions; and the method of using them will be no difficulty after what has been said of the other tables. They have one peculiarity, however, which will require to be noticed.

The sines and cosines of all angles less than 90°, the tangents of angles less than 45°, and the cotangents of angles greater than 45°, are all less than unity; and their logarithms are therefore all negative. In order to render the tables of logarithmic functions all positive, which it is found convenient to do, every log function is increased by 10 before being registered. Each log function thus increased is called a "tabular logarithm,' which designation is generally represented by the letter L. Thus L sin A = log sin A+10, L cos A = log cos A+10, L tan A= log tan A+10, &c.

This difference between the real log function and the tabular log function must be carefully remembered in applying the tables; as well as the fact, that as the angle increases, the log sine, log tangent, and log secant increase, but the other log functions decrease.

135. Complementary Logarithms.-All the mantissæ registered in a table of logarithms are positive; so that before consulting one to find a number corresponding to a given negative logarithm, the logarithm must be transformed so as to have its mantissa positive, and its characteristic alone negative. How this may be done is easily seen from the following reasoning:

Let (c+m) be a negative logarithm, of which c is characteristic and m is mantissa. Then

or

-(c+m) = —c—m = -c-1+1-m,

-(c+m) = (c+1)+(1−m);

which gives an equivalent for the given negative logarithms having 1-m as new positive mantissa and (c+1) as new characteristic. The following rule is hence evident: To transform a negative logarithm so that

its mantissa may be positive, subtract the mantissa from 1 for a new positive mantissa, and add 1 numerically to the characteristic.

The operation just described may be made useful in other cases of much more frequent occurrence than that mentioned above-in what manner, will be shown after the following explanations.

When a logarithm preceded by the sign has been transformed so that its mantissa is positive, it is then called a complementary logarithm; and is written briefly "colog." These complementary logarithms are always of service in finding the value of an expression consisting of more than two logarithms, of which one or more are negative, and not otherwise; and the utility of them at all times depends upon the readiness with which a mantissa can be subtracted from 1. When required, this should be done at sight of the mantissa. It is most readily done by going from left-hand to right, and subtracting each digit from 9 except the last, which must be subtracted from 10.

There are various minor points with which the student may most easily become familiar by studying the following examples. He may perhaps require to be reminded that the sign above a characteristic, when it occurs, as much affects the characteristic as if it stood before it, but has no influence whatever over the mantissa. Thus, for instance,

while

1-6824468=-1+6824468,

-13421682 -(-1)+ ·3421682

=

−(−1)+

= +1-3421682.

136. In the first of the Examples which follow, a negative logarithm is transformed, so that it may be looked for in the tables.

Obs. 1.-This last mantissa is best obtained by subtracting successively 8, 0, 6, 1 from 9, and then 8, the last digit except ciphers, from 10.

Obs. 2.-The chief use of complementary logarithms is to change two or more operations of addition and subtraction into one of addition alone. The following example will illustrate :

137. Relation between logarithms of the same numbers to different bases.

Let x, y be the logarithms of the same number N referred to a, b respectively as bases; so that a* and b are each equal to N. Then

Hence the logarithm of a number in one system may be changed to the logarithm of the same number in another system by multiplying it by the reciprocal of the logarithm in the first system of the base of the second.