angle in those columns, at the top of which those terms are placed. The headings of the columns run along the top, thus as far as 44°. The degrees from 45° to 90° are placed at the bottom of the page, and the minutes of the degree arranged in a right hand column, so that the angles read off on the right hand side are complementary to those read off at the points exactly opposite on the left hand side, the values of the sines, cosines, tangents, &c., being found in the columns at the bottom of which those terms are found. This arrangement is rendered practicable by the circumstance of the sines, tangents, &c., of angles being respectively equal to the cosines, cotangents, &c., of the complements of the same angle.

Besides the column headed sine, tangent, &c., are three smaller columns headed "Diff." They contain the difference between the values of the sines, cosines, &c., of consecutive logarithms corresponding to a change of 100" in the arc; and it should be kept in mind that the same difference is common to the sine and cosecant, to the tangent and cotangent, and to the secant and cosine. They are inserted for the convenience of finding the values of the sines and cosines, &c., of angles which are expressed in degrees, minutes, and seconds.

In the use of these tables, as in that of the natural sines, two questions present themselves :-First, having given the angle in degrees, minutes, and seconds, required the log. sine, log. cosine, &c. Second, having given the log. sine, log. cosine, &c., required the value of the angle in degrees, minutes, and seconds.

If the angle whose logarithmic sine, tangent, &c., it is required to find, be given in degrees and minutes, look for the degrees, if the angle be less than 45°, at the top of the page, and for the minutes in the left hand column: if the angle be greater than 45°, look for the degrees at the bottom of the page, and for the minutes in the right hand column: the logarithm of the proposed function of the angle will be found opposite the numbers in its proper column.

If the value of the angle be given in degrees, minutes, and seconds, we proceed by

RULE XVII.

I°. Find from the table the sine, cosine, &c., which corresponds to the degrees and minutes.

2°. Multiply the tabular difference by the seconds, and divide by 100.

3°. If the required quantity be a sine, tangent, or secant, add the result to the last figures obtained in 1°; if it be a cosine, cotangent, or cosecant, subtract. The result will be the required sine, cosine, &c.

When the angle is greater than 90°, subtract it from 180°, and look for the remainder, which is called its supplement, in the tables.

Thus, to find the log. sine of 120° 24', subtract it from 180°, and look for the log, sine of the remainder (namely 69° 36′), which is 9'971870; or log. sine 120° 24′ = 9'971870.

3. Find the log. tangent of 128° 55′ 47′′.

Supplement of the given angle = 51° 4′ 13′′.
Log. tangent 51° 4′ = 0·092664

+

56

0'092720

Tab. diff.

431

X 13

1293

431

56,03

or 56

But a readier way will, in general, be to diminish the given angle by 90°, and to look out the remainder according to the following rule:—

RULE XVIII.*

If A denote any angle less than 90°, then

For sine......(90+ A) take out......cosine A

Thus, to find the log, cosine of 110°, or log. cosine (90+20), take out the log. sine 20°, which is 9'534052.

To find the log, secant of 160° 12', take out the cosecant 70° 12', which is 10'026465.

Required the log. sine, tangent, secant, cosine, cotangent, and cosecant corresponding to the following are:-

If the value of the log. sine, log. cosine, &c., be given, and it is re

quired to find the angle, we use

RULE XIX.

1o. Find in the tables (XXV, Norie) the next lower† log. sine, log. cosine, &c., and note the corresponding degrees and minutes.

20. Subtract this from the given log. sine, log. cosine, &c., multiply the difference by 100, i.e., annex two cyphers, divide by the tabular difference, and consider the result as seconds.

* This rule may easily be remembered by observing that to the sine, tangent, and secant, co is prefixed, while from the cosine, cosecant, and cotangent, the co is dropped, and in each case the excess above 90° of the angle is used.

+ If the given log. be a cosine, cosecant, or cotangent, we may seek out the next greater to the given log.; then proceed by 2° to find the seconds, which add to the degrees and minutes as found by 1°.

3°. If the given value be that of a log. sine, log tangent, or log. secant, add these seconds to the degress and minutes found in 1°; if it be that of a log. cosine, log. tangent, or log. cosecant, subtract.

The result will be the required angle.

EXAMPLES.

1. Given log. sine = 9°422195 (or ī‘422195): find the angle.

2,

Given log. sine 9'422195

Tab. log. sine next less 9:421857 = log. sine 15° 19′

Tab. diff, for 100′′ = 768)·33800(44" additional seconds

3072

3080
3072

Therefore 9'422195 = log, sine of 15° 19′ 44′′.

Given log. cosine 9'873242 (or ī·873242): find the angle.

86

357

= *240896, a negative mantissa = 1·759104, adding 10 to the index gives 9759104 for the tabular log. tangent; and very nearly corresponding to this is found the angle 29° 52'.

Required the Angles (to the nearest second), the Log. Sine of which is :

1. 9'741279 5. 8.600700 9. 9'929638 13. I'559234

2. 9'518317 6. 9'926100

IO. 9'500000
9.800000
12. 9706176

16. 9700000 17. 8.846217 18, 8*462167

9. 8.967391

8. 9'998970 IO. 9'000000

7.

8.

0'022716
0'054304

9. 0'315400 IO. 0'497691

10. 0.070362

13. 0630000

5. O'121000 8. 11*467931 II. O'900000
6. 0168600 9. 11711010

12.

14. 0061462 15. 0109761

G

4. 0974476 7. 11000873

Find the Arc to the Log. Cotangent of

7. 8.327691 10. 0000276

I. 9'742691 4. 0060431
2. 0.876432 5. 0710880 8.
3. 9'846175 6. 0'545104

13. 8.460000

9.

NAVIGATION.

DEFINITIONS.

NAVIGATION is a general term denoting that science which treats of the
determination of the place of a ship on the sea, and which furnishes
the knowledge requisite for taking a vessel from one place to another.
The two fundamental problems of navigation are, therefore, the finding
at sea the present position of the ship, and the determining the future

course.

The place of a ship is determined by either of two methods, which
are independent of each other:-1st. By referring it to some other
place, as a fixed point of land, or a previous defined place of the ship
herself. 2nd. By astronomical observations.

It has been customary to employ the term NAVIGATION in a restricted
sense to the first of these methods: the second is usually treated of
under the head of NAUTICAL ASTRONOMY.

Navigation and Nautical Astronomy are the two great co-ordinate divisions of the
Art of Sailing on the Sea," as the old writers quaintly worded it. The first branch
of the art is accomplished by means of the Mariner's Compass, which shows the
direction of the ship's track; the Log, which with the help of sand-glasses for measuring
small intervals of time, gives the velocity or the rate of sailing, and thence the distance
run in any interval, and also a Chart of appropriate construction; in short this
branch of the art relates to the directing the ship's course under the varying forces
of winds and currents, and the estimation of her change of place. The second division
is that branch of practical astronomy by which the situation of the observer on the
globe is ascertained by a comparison of the position of his Zenith with relation to the
heavens with the known position of the Zenith of a known place at the same moment.
principal instruments are the sextant for measuring the altitudes and taking the
distances of heavenly bodies; and a chronometer to tell us the difference in time
between the meridian of the ship and the first meridian; also, a pre-calculated astro-
nomical register, such as the Nautical Almanac, the Connaissance de Temps of France,
&c. The solution of problems in nautical astronomy requires the use of spherical
trigonometry, which is therefore characteristic of this method of navigation.

The