Ex.

x. 3. Required the log. tangent 99° 32′ 58′′.

Log. tangent 99° 32′ 58′′ cotangent 9° 32′ 58′′.

Log. cotangent 9° 32′ 0" 10'774844

Parts for

+

58 =

747

Log. cotangent 9 32 58 10'774097

Tab. diff.

1288

58

10304

6440

747,04

EXAMPLES FOR PRACTICE."

Required the log. sine, tangent, secant, cosine, cotangent, and cosecant corresponding the following arcs :

107. If the value of the log. sine, log. cosine, &c., i.e., the logarithm of a trigonometrical ratio, be given, and it is required to find the corresponding angle in degrees and minutes, we use

RULE XLII.

Look for the logarithm in the several columns of the Table marked at the top or bottom, with the name of the given trigonometrical ratio, which being found exactly or the nearest to it, will give the degrees and minutes answering to the given logarithm, being careful to observe that when the name of the given ratio is found at the top of the Table, then the degrees are to be taken from the top and the minutes from the left hand marginal column; but if the name of the ratio is found at the bottom of the Table, take the degrees from the bottom and the minutes from the right hand side of the page.

NOTE.-In using the Table inversely, for example, in searching for the angle which has 9.611294 for the logarithm of its sine, the student must not distinguish sine from cosine, nor tangent from cotangent, but must consider sines and cosines as one table, tangents and cotangents as one table, and must cast an eye on both, and get to 9'611294 as fast as he can. For want of this caution some beginners will turn over page after page until they come to 45°, and then back again to the very page that was first opened.

Ex. 1. Required the angle corresponding to the log, sine 9.729223.

In page 142, Table XXV, Norie, under the word "Sine," and opposite 25′ in left hand marginal column, are the exact figures, the degree (being sought at the head of the page, because the column in which the figures are found is named at the head) is 32°; therefore, the angle is 32° 25′.

* When the tabular difference is considerable, as in this instance, the log. is easier reduced from the log. of the nearest minute.

If the angle for the cosine of the same logarithm be required, the degrees are found at the bottom, and the minutes in the right hand column, and is 57° 35′ accordingly.

108. If the value of the log. sine, log. cosine, &c., be given, and it is required to find the corresponding angle, in degrees, minutes, and seconds,

we use

RULE XLIII.

1. Find in the Tables (XXV, Norie) the next lower log. sine, log. cosine, c., and note the corresponding degrees and minutes; also, take the number from the corresponding part of the adjoining column of "Diff.”

2o. 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.

3°. If the given value be that of a log. sine, log. tangent, or log. secant, add these seconds to the degrees 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.

NOTE.-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 1o.

EXAMPLES.

1. Given log. sine = 9'422195 (or 7'422195): find the angle.

We take out 9:421857, the log. sine of 15° 19', as it is the logarithm next less than the given one, which we take, as the logarithms in the columns increase with the angle. The difference of these logarithms is 338, and if two cyphers be affixed to the difference, and the number then divided by 768, taken from the column of Diff. in the Table, we have 44 for the number of seconds to be added to the degrees and minutes before taken out. The work will stand thus:

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

2. Given log. cosine 9873242 (or ī·873242): find the angle,

Here we take out 9.873223, the log. cosine of 41° 41′, as it is the log. cosine in the Table next less than 9.873242. The difference between these two logarithms is 19; and if two cyphers be affixed to the difference we get 1900; whence 1900 divided by 187, the number from the column of "Diff." gives 10 for the number of seconds to be subtracted. Hence the required angle is 41° 40′ 50′′. The work will stand thus:

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

I. 9'742961 4. 10.060431 7. 8.327691 10. 9'100100 13. 8.460000 2. 10.876432 5. 10710880 8. ΙΟΟΙΟΙΟΙ 3. 10.287632 6. 11*197568

9.

II. 10.825001 14. 9'374611 8.781464 12. 8.272775 15. 12.069844

If tang. A 3, find log. tang. A.

=

3. Given log. cos. A 9'236713, find nat. cos. A.

4. Given log. tang. 35° 20' 9'850593, find log. cotang. 35° 20' without using any tables at all.

5.

Find the log. cosec. 68° 45′ 24′′ from the table of natural sines only.

6. Given log. sec. A= 11024680, find nat. cos. A.

7.

Given log. cosine A = 9'450981, find A (1) from a table of log. cosines, and (2) from a table of nat. cosines.

8. Given nat. sec. A=2'005263, find A (1) from a table of nat. sines and cosines, and

12.

Find nat. cot. 45° 18′ 17′′ from the table of cotangents. 13. Find to the nearest second the angle whose sine is ; ;

109. It is also necessary to have a distinct conception of the limits to which the Trigonometrical Ratios tend when the angles become right angles. The following are the Trigonometrical Ratios for the angles o° and 90°:

And the following, therefore, are the Logarithms of their Trigonometrical

IIO.

When these values occur amongst others requiring to be added to or subtracted from them, the learned must be careful to remember that the addition to or subtraction from them of finite numbers cannot alter them. Hence the explanation of the results in the following:

may be

III. In the event of a bad or obliterated figure in the table, it convenient to know that the tangents are found by subtracting the cosines from the sines, adding always 10, or the radius; the cotangents are found by subtracting the tangents from 20, or the double radius, and the secants are found by subtracting the cosines from 20, the double radius.

*This mathematical symbol is called infinity.

M

NAVIGATION.

DEFINITIONS.

112. 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.

113. 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.

114.

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 posi tion of his Zenith with relation to the heavens with the known position of the Zenith of a known place at the same moment. The 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 precalculated astronomical 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.

115. A Sphere is a solid body bounded by a surface, every point of which is equally distant from a fixed point within it; this fixed point is called the centre; the constant distance is called the radius.

Every section of a sphere by a plane is a circle.

116. A Great Circle of a sphere is a section of the surface by a plane which passes through its centre. A Small Circle of a sphere is a section of the surface by a plane which does not pass through its centre.

Or, a great circle is a circle of the sphere having for its centre the centre of the sphere, thus dividing the sphere into two equal parts; no greater circle can be traced upon its surface. All other circles are called small circles.

All great circles of a sphere have the same radius. All great circles bisect each other.