than that which has been given in connection with the subject of logarithms.

Table II, with the exception of the last two columns, which contain natural sines and cosines, is a table in which are arranged the logarithms of the numerical values of the several trigonometrical lines corresponding to the different angles in a quadrant. The values of these lines are computed to the radius 10,000,000,000, and their logarithms are nothing more than the logarithms, each increased by 10, of the natural sines, cosines, and tangents, of the same angles; because the values of these lines, for arcs of the same number of degrees taken in different circles, are directly proportional

to the radii of the circles.

The natural sines are made to the radius of unity; and, of course, any particular sine is a decimal fraction, expressed by natural numbers. The logarithm of any natural sine, with its index increased by 10, will give the logarithmic sine. Thus, the natural sine of 3° is .052336.

In this manner we may find the logarithmic sine of any other arc, when we have the natural sine of the

same arc.

If the natural sines and logarithmic sines were on the same radius, the logarithm of the natural sine would be the logarithmic sine, at once, without any increase of the index.

The radius for the logarithmic sines is arbitrarily taken so large that the index of its logarithm is 10. It might have been more or less; but, by common consent, it is settled at this value; so that the sines of the smallest arcs ever used shall not have a negative index.

In our preceding equations, sin. a, cos. a, etc., refer to natural sines; and by such equations we determine their values in natural numbers; and these numbers are put in Table II, under the heads of N. sine and N. cos., as before observed.

When we have the sine and cosine of an arc, the tangent and cotangent are found by Eq. (3) and (6); thus,

and the secant is found by equation (4); that is,

sec. =

R2

COS.

For example, the logarithmic sine of 6° is 9.019235, and its cosine 9.997614. From these it is required to find the logarithmic tangent, cotangent, and secant.

The secants and cosecants of arcs are not given in our table, because they are very little used in practice; and if any particular secant is required, it can be determined by subtracting the cosine from 20; and the cosecant can be found by subtracting the sine from 20.

The sine of every degree and minute of the quadrant is given, directly, in the table, commencing at 0°, and extending to 45°, at the head of the table; and from 45° to 90°, at the bottom of the table, increasing backward.

The same column that is marked sine, at the top, is marked cosine at the bottom; and the reason for this is apparent to any one who has examined the definitions of sines.

The difference of two consecutive logarithms is given, corresponding to ten seconds. Removing the decimal point one figure, will give the difference for one second; and if we multiply this difference by any proposed number of seconds, we shall have a difference of logarithm corresponding to that number of seconds, above the preceding degree and minute.

For example, find the sine of 19° 17′ 22′′.

The sine of 19° 17', taken directly from the table, is 9.518829 The difference for 10" is 60.2; for 1", is 6.02; and

6.02 × 22 =

Hence, 19° 17' 22" sine is

132

9.518961

From this it will be perceived that there is no difficulty in obtaining the sine or tangent, cosine or cotangent, of any angle greater than 30'.

Conversely: Given, the logarithmic sine 9.982412, to find its corresponding arc. The sine next less in the table is 9.982404, which gives the arc 73° 48′. The difference between this and the given sine is 8, and the dif ference for 1" is .61; therefore, the number of seconds corresponding to 8, must be discovered by dividing 8 by the decimal .61, which gives 13. Hence, the arc sought 18 73° 48' 13".

These operations are too obvious to require a rule, When the arc is very small,—and such arcs are sometimes required in Astronomy,-it is necessary to be very accurate; for this reason we omitted the difference for seconds for all arcs under 30'. Assuming that the sines and tangents of arcs under 30' vary in the same proportion a the arcs themselves, we can find the sine or tangent o any very small arc, with great exactness, as follo

The sine of 1', as expressed in the table, is

6.463726

Divide this by 60; that is, subtract logarithm

1.778151

The logarithmic sine of 1", therefore, is

4.685575

Now, for the sine of 17", add the logarithm of 17 1.230449

Logarithmic sine of 17", is

5.916024

In the same manner we may find the sine of any other small arc.

For example, find the sine of 14' 211"; that is, 861.5".

Two lines drawn, the one from the surface and the other from the center of the earth, to the center of the sun, make with each other an angle of 8.61". What is the logarithmic sine of this angle?

One figure will be sufficient to represent the triangle in all of the following examples; the right angle being at B.

PRACTICAL PROBLEMS.

1. In a right-angled triangle, ABC, given the base AB, 1214, and the angle A, 51° 40′ 30′′, to find the other parts.

REMARK. When the first term of a logarithmic proportion is radius, the required logarithm is found by adding the second and third logarithms, rejecting 10 in the index, which is dividing by the first term.

In all cases we add the second and third logarithms together; which, in logarithms, is multiplying these terms together; and from that sum we subtract the first logarithm, whatever it may be, which is dividing by the first term.

To find this resulting logarithm, we subtracted the first logarithm from the second, conceiving its index to be 13.

Let ABC represent any plane triangle, right-angled at B.

2. Given, AC 73.26, and the angle A, 49° 12′ 20′′; required the other parts.

Ans. The angle C, 40° 47′ 40′′; BC, 55.46; and AB, 47.86. 3. Given, AB 469.34, and the angle A, 51° 26' 17", to fnd the other parts.

Ans. The angle C, 38° 33′ 43′′; BC, 588.7; and AC, 752.9. 4. Given, AB 493, and the angle C, 20° 14'; required, the remaining parts.

Ans. The angle A, 69° 46'; BC, 1338; and AC, 1425.5. 5. Let AB 331, and the angle A = 49° 14'; what are the other parts?

=

Ans. AC, 506.9; BC, 383.9; and the angle C, 40° 46′. 6. If AC=45, and the angle C-37° 22', what are the remaining parts?

Ans. AB, 27.31; BC, 35.76; and the angle A, 52° 38'