Ex. 2. Given log. cosine

9.873242 (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 diff ence 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:

Given log. cosine

9.873242

Tab. log. cosine next greater 9.873335 = 41° 40′ (next less angle).
Tab. diff for 100"187)9300(50ʻ, nearly, additive.

.. angle required = 41° 40′ 50′′.

EXAMPLES FOR PRACTICE.

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

1. If eine A='432651, find log. sine A.

2. 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 = 11°024680, find nat. cos. A.

7. Given log. cosine A 9450981, find A (1) from a table of log. cosines, and (2) from a table of nat. cosines,

8. Given nat. sec. A = 2·005163, find A (1) from a table of nat. sines and cosines, and (2) secant from a table of log. secants.

9. Sine 36° X tang. 54°='654.

10. Find by the tables the angle whose sine is √.

11. Given log. cot. A = 11015627, find nat. cot. A.

12. Find nat. cot. 45° 18′ 17′′ from the table of cotangents.

13. Find to the nearest second the angle whose sine is,,, and 183.

60. 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

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

Hence

Ex. 2. Add together log. cos. 90° and log. tang. 45°.

Log. cos. 90° = 18

Log. tang. 45° = 10'000000

Ans. 00

Ex. 4. From log. tang. 21° 48' 30" take log. cot. 90°.

Log. tang. 21° 48′ 30′′ = 9.602212
Log. cot.

90°

=

62. In the event of a bad or obliterated figure in the table, it may be convenient to know that the tangents are found by subtracting the cosines from the sincs, 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 subtractting the cosines from 20, the diameter of a circle whose radius is 10.

Tais mathematical symbol is called infinity.

NAVIGATION.

SPHERICAL TRIGONOMETRY.

An acquaintance with the properties of the sphere being absolutely necessary for the nautical student we give the enunciations of the most important, but state the definitions at length for convenience of future reference.

Sphere.

Centre.

Radius.

Diameter.

Section,
Great Circle.
Small Circle.

Axis.

Poles.

Adjacent and
Remote.

A Sphere is a surface, every point of which is equally distant from a fixed point within it; this fixed point is called the Centre. A sphere may be conceived to be generated by the revolution of a semicircle about its diameter, which remains fixed throughout the motion.

The straight line which joins any point of the surface with the centre is called a Radius.

A straight line drawn through the centre and terminated both ways by the circumference is called a Diameter. Every Section of a sphere by a plane is a circle.

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.

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

The Axis of any circle of a sphere is that diameter of a sphere which is perpendicular to the plane of the circle.

The extremities of that diameter of a sphere which is perpendicular to the plane of a circle are called the Poles of the circle.

The poles of a great circle are equally distant from the plane of the circle.

The poles of a small circle are not equally distant from the plane of the circle. In the case of a small circle the poles may be distinguished as the adjacent and remote poles.

All parallel circles have the same poles. The distance of every point in the circumference of a circle from either of its poles is the same. The poles of a great circle are 90° distant from every point of the circle.

Co-ordinates.-A set of lines, angles, or planes, or combination of these, which, taken together, define the position of the several points of a given surface, or points in space. The method was invented by DESCARTES, the French geometer, who first expressed algebraically theorems involving the position of lines. To represent the position of a curve on a plane he chose a certain right line, to the different points of which he referred all the points of the given curve; then he chose a certain point in this line from which to commence the reckoning (“ad ordiendum ab es calculum.") Hence the series of lines by which the curve was referred to the chosen line were called ordinates (derived from a word of the same root,

ordinare, to range in order), and the portions of the line "cut off" by this series, from the chosen point, were named abscissæ (L'abscindere, to cut off.) If, however, two lines are taken intersecting each other at a given angle in a fixed point, the several points of the curve in question may be referred to each of them in turn, and thus two sets of ordinates be contemplated which, taken together, define every point of the curve; hence the term co-ordinates.

O is the centre of the sphere; OB and OD are Radii; PP and BD Diameters.

ABQD is a Great Circle.

RST is a Small Circle.

PP is the Axis of the circles ABQD and RST, and PP' are the Poles of those circles; P is the Adjacent to RST, P' the Remote.

PBP and PNP are Secondaries to ABQ D.

BPN is a Spherical Angle measured by BN.

RST is a Parallel to A BQ D.

XN and NS are spherical co-ordinates which determine the position of S with respect to the fixed Primary ABQD and the fixed Secondary PXP. Primary Circles. Regarding any great circle as a Primary circle, all great Secondaries. circles which pass through its poles are called its Secondaries.

All secondaries cut their primary at right-angles.

Spherical
Co-ordinates.

Spherical Angle.

Arc of a Great
Circle-how

measured.

The position of a point on a sphere may be referred to any great circle and a fixed secondary by measuring its distance from them along the great circle and along the secondary which passes through the point. These distances are called the Co-ordinates of the point.

The angle between two great circles is called a Spherical Angle.

The arc of a great circle is measured by the angle subtended by it at the centre of the sphere, which is also the same thing as the angle of inclination, at its pole, of two secondaries drawn through its extremities.

The two following propositions are necessary for the proper understanding therefore, here introduce proof of them.

of our subject, we, A great circle

may be drawn through any two points of the sphere.

A great circle may be drawn through any two points on the surface of the sphere. For if these two points be joined by a straight line, and a plane in which this line lies be made to revolve about it till it meets the centre of the sphere, it is evident that this plane must then cut the sphere in a great circle which passes through the two points.

The shortest distance will be a curve in a plane, or, in other words, a circle; and of the indefinite number of circles which planes passing through the two points make with the surface of the sphere the great circle is the largest, and, therefore, its arc joining the two points has less curvature than the arc of any other circle joining them. Hence it most nearly approaches the chord, which is a straight line, and a straight line is the shortest distance in space between two points.

THE TERRESTRIAL SYSTEM.

The Primary Circle is the Equator.

The Secondary Circles are called Meridians,

Co-ordinates, Latitude and Longitude (see above).

The first step in applying the principles above to the terrestrial sphere is to fix upon the primary great circle for the axis of co-ordinates. The question is decided by a

physical characteristic of the earth, viz., its diurnal rotation, from which we get a definite line, itself immovable, as the primary one on which to base our definitions of the curves and points of the terrestrial sphere. The great circle whose plane is perpendicular to this fixed line is chosen as the primary circle for co-ordinates, on which are measured the abscissm, and on secondaries to it are measured the ordinates of the various points on the earth's surface. The abscissæ are reckoned to the right and left of the initial position of the secondary or origin, and designated respectively cast and west longitudes; and the ordinates are reckoned from the primary circle towards the poles of the primary, and designated respectively north and south latitudes. The initial secondary is variously chosen by different nations, usually their principal observatory, and is commonly called the first meridian. We are now in a position to give the following definitions.

Axis of the
Earth.

The Axis of the earth is that diameter about which it revolves from west to east with a uniform motion.