earth, in precisely the same time, i.e., 291⁄2 days, or a lunar month, with the result that we only see one side; the other side, with the exception of a very slight "edge," and occasionally a small area round the poles, being always turned away from the earth.

The Path of the Moon round the Sun is complicated by the fact that while she is revolving round the earth, the earth itself is constantly moving forward. "We may," says Lockyer, "get an idea of the moon's path round the sun, if we imagine a wheel going along a road to have a pencil fixed to one of its spokes so as to leave a trace on the wall: such a trace would consist of a series of curves with their concave sides downwards.” Such a path would also be traced in the air by a stone swung round by a string held in a boy's hand, the boy running forward at the same time.

The earth, then, is 50 times larger than the moon, while the sun is 1,279,000 times larger than the earth. The moon appears about as large as the sun, because it is 400 times nearer to us than the sun, and 100 times nearer than the nearest planet-its distance varying from 239,000 to 253,000 miles.

Phases of the Moon: The moon, like the earth, is round on globular in shape; and, receiving its light from the sun, only on half of its surface is illumined at the same time, so that the differ ent phases or aspects are due to the different positions from which the illumined half is seen during its monthly journey round the earth. When the earth is between the sun and the moon, the entire lit-up half is seen; it is then full moon; and, on the other hand, when the moon is between the sun and the earth, the illumined side is not seen at all: this is the case at new moon. Between new and full moon, we see a crescent moon gradually increasing to a half moon and a gibbous moon, and finally becoming a full moon, when it as gradually assumes the same forms in an inverse order. In other words, the moon waxes from new to full moon and wanes from full to new noon. The moon does not travel exactly in the same plane as the sun, otherwise, whenever she got between us and the sun there would be an eclipse of the sun; and, similarly, whenever the earth passed between the sun and the moon there would be an eclipse of the moon. But occasionally, when the moon crosses the earth's orbit, its shadow falls on the earth, and then we have an eclipse of the sun. At other times, the earth's shadow falls on the moon, producing an eclipse of the moon. There can be no more than five, nor less than two, solar eclipses, and two lunar eclipses, in a year.

Eclipses The planets and their satellites, being opaque, necessarily cast long conical shadows on the side furthest from the sun; when any of them happens to pass into a shadow thus cast, there is an eclipse.

When the moon passes into the earth's shadow there is an Eclipse of the Moon; and when the earth passes through the moon's shadow there is an Eclipse of the Sun. Eclipses of the moon thus take place when the earth passes directly between the moon and the sun.

The Tides are the regular rising and falling in level of the waters of the sea at intervals of about six hours, so that the tides ebb and flow twice a day.

The explanation of the Tides is astronomical. They are caused by the attraction which the moon exerts over the waters of the earth, causing them to "bulge" out on the side nearest to and on that furthest from her. The sun also exerts a similar attraction on the waters of the globe, but its effects are not so sensible as those of the moon, owing to its vastly greater distance from the earth. The sun, however, is so enormously larger than the moon (28,400,000 times) that it attracts the earth as a whole much more strongly than the moon does. But the tides are produced not by the attractive power of the sun and

the moon on the earth as a whole, but are due to the difference of their attraction on different parts of the earth. The sun is so far away that its attraction on opposite sides of the earth is nearly the same; the moon is so near that her attractive power on the waters nearest to her is greater than that on the solid part, and that again is greater than the force she exerts on the waters on the side furthest from her.

The Moon thus produces a tidal wave on either side of the globe, and as the earth rotates on her axis, every part of the ocean is brought under the moon once a day; hence there are two tides a day, i.e., the waters rise and fall— flow and ebb-twice every 24 hours 50 minutes.

The sun also produces a tidal wave, which is generally imperceptible, except at new and full moon, when its attraction unites with that of the moon, thus producing the highest or Spring Tides. The sun's action makes itself again felt when the moon is in her first or last quarter (i.e., 7 or 21 days old), as his attraction is then diametrically opposed to that of the moon, thus causing the lowest or Neap Tides.

Spring Tides are thus the sum and Neap Tides the difference of the solar and lunar tides.

The earth turns on its axis, with regard to the sun, once in 24 hours, but to bring the same place again under the moon, the earth must perform rather more than an entire rotation, because the moon has in the meantime been travelling eastward on her journey round the earth.

The Lunar Day is thus 50 minutes longer than the solar day, and the tides will not, therefore, occur at the same time every day, but each day about 50 minutes later than on the previous day.

The Moon's Attraction on the waters of the sea is not felt at the exact time of her passage over a given place, but always about two hours later. Similarly, spring tides do not occur exactly at new or full moon, nor neap tides when the moon is in her quadratures, but always a day or two later.

II.

In order to represent the earth's surface accurately by means of globes, maps, or charts, it is absolutely necessary to have some means of indicating the exact positions of places.

How necessary this is may be seen by attempting to describe the exact position of a point within a circle or square without indicating its distance and direction from one or more points already fixed or easily found, such as the corners of the square or the centre of the circle. To fix the position of a point accurately, we must either indicate its exact direction and distance from a given point, or draw straight lines from two given points in given directions, so that they intersect or cut each other at the exact point required. Before, then, we can make any intelligible representation of the whole or any part of the earth's surface, we must find means to fix the actual position of everything we intend to show-the coast-line, rivers, mountains, towns, &c. Hence the necessity for the fixing of certain points, lines and circles, without which no map or plan could be drawn with any degree of accuracy.

Preliminary Definitions: A clear understanding of the following definitions will aid the student to master the essential facts and main principles of Mathematical Geography :

A Point has no parts and no magnitude: i.e. a point indicates position only

A Line is length without breadth. Its extremities are points.

A Straight Line is that which lies evenly between its extreme points.

A Plane is that which has only length and breadth, and is such that any two points being given, a straight line between them lies wholly in that plane.

Parallel Lines are lines in the same plane, which, being produced ever so far both ways, do not meet.

An Angle is the inclination of two lines to each other in a plane which meet together, but are not in the same straight line.

A Right Angle is formed when a straight line standing on another straight line makes the adjacent angles equal to each other. Each of these angles is called a right angle, and the straight line which stands on the other is called a perpendicular to it. An Obtuse Angle is greater, and an Acute Angle is less, than a right angle.

A Term or Boundary is the extremity of anything, and a Figure is that which is enclosed by one or more boundaries. Hence

A Circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another. This point is the centre of the circle.

A Diameter of a circle is a straight line drawn through the centre and terminated both ways by the circumference. A diameter cuts a circle into two equal parts, each of which is a Semi-circle.

A Quadrant is the fourth part of a circle.

A Radius is a straight line drawn from the centre of a circle to the circumference, and is therefore equal to half of the diameter. All the Diameters (and also all the Radii) of the same or equal circles are equal to one another.

An Arc is a part of the circumference of a circle.

Degrees: When two straight lines cut one another at right angles, the space round the point of intersection is divided into four equal or right angles. The circumference of a circle described from this point as centre subtends or extends across these four angles.

For the purpose of measuring angles and distances, every circle, whether great or small, is supposed to be divided into 360 equal parts or Degrees; a semi-circle contains, therefore, 180 equal parts or degrees, and a quarter of a circle, or quadrant, contains go degrees.

The size of an angle, and the length of the arc or part of the circumference which subtends the angle, are both expressed in degrees. And as every circle is supposed to be, irrespective of size, divided into 360 degrees, it follows that the length of a degree on the circumference must vary according to the size of the circle; but the angles formed at the centre are always the same, whatever the size of the circle. Thus, if from the same centre we draw one circle within another, and draw straight lines from the centre to indicate 10, 20, 30, &c. degrees, the arcs of the outer circle are longer than the corresponding arcs of the inner one, although the angles subtended are the same.

Minutes-Seconds: Every Degree is divided into sixty equal parts called Minutes, and every minute is divided into sixty equal parts called Seconds.

If it is

The following signs are used to express degrees, minutes, and seconds-*'". necessary to express distances (either of latitude or longitude) which are less than a degree, we say that a place is in so many degrees, so many minutes, and so many seconds, using these signs. Thus, 24° 16′ 8′′, means twenty-four degrees, sixteen minutes, and eight seconds; just in the same way as, in order to express an amount of money, we might say £24, 16s. 8d., that is, twenty-four pounds sixteen shillings, and eight pence.

A Sphere is a solid body, every point on the surface of which is equally distant from a point within it called the centre.

Or (Euc. xi., Def. 14) a sphere is a solid figure described by the revolution of a semi circle about its diameter, which remains unmoved.

Down-Up: In a sphere, down means the direction from any point on or above the surface towards the centre; and up, the direction from the centre to any point on or outside the surface.

A Great Circle of a sphere divides it into two equal parts, each of which is a hemisphere, i.e., half a sphere. The plane of a great circle thus passes through the centre of the sphere.

A Small Circle of a sphere is a circle whose plane does not pass through the centre of the sphere. A. small circle thus divides a sphere into two

unequal parts.

All circles on a sphere, great and small, are supposed to be divided into the same number of degrees (360°), hence the length of a degree varies, while the corresponding angle is invariable.

A Cone is a solid figure, the base of which is a circle, but which tapers to a point from the base upward.

An Ellipse is the section of a cone produced by a plane which is not parallel to the base. (All sections of a cone which are parallel to the base must be circles, as the base is a circle.)

An ellipse has thus two diameters, unequal in length-the long diameter or major axis, and the short diameter or minor axis. (The diameters of a circle are all of equal length.) Now there are two important fixed points in the major axis called the foci (singular, focus), equally distant from the centre or point of intersection of the two diameters, and these points are such that the sum of two straight lines, drawn from them to any point in the circumference, is equal to the length of the major axis. The shape or eccentricity of the ellipse will thus depend on the distance between the foci-the nearer they are, the closer will the ellipse approach to a circle. An ellipse may be roughly drawn by passing a loop of string loosely over two tacks or pins stuck through a sheet of paper, on which, by tightening the string by means of a pencil, we may trace an elliptical figure. We can then draw the two diameters and find the two foci.

Applying these definitions to the Earth as a mathematical figure, we find that certain

Points, Lines, and Circles are absolutely necessary to indicate the positions of places on its surface, and to enable us to represent the whole or parts of its surface by means of globes or maps.

From two points on the surface of a sphere, or from a line passing through its centre, the positions of all other lines and circles can be determined. The rotation of the earth fixes the line required, and its two extremities the two essential points.

This line is the Axis of the earth, or the imaginary line which passes through its centre, and on which it turns or rotates.

The two essential points are the Poles, the extremities of the earth's axis. And as the carth's axis always points in the same direction in space, namely, to a point close to the star called Polaris (the position of which is easily found by observing the 'Pointers' in the constellation of the Great Bear), we have a fixed

Immovable point in the heavens from which we can determine the positions of all other points. The pole of the earth which is always directed towards this fixed point is called the North Pole; and the pole pointing in exactly the opposite direction in the heavens is the South Pole.

Relative Positions of Places: North, South, East, and West are terms used to express the relative positions of places to one another. These are the Four Cardinal Points of the Compass—an instrument used to determine the respective bearings of places.

There are altogether 32 points of the Compass, 28 of these being intermediate between the 4 Cardinal Points.

The Cardinal Points are generally abbreviated thus: N., S., E., W.

The Four Collateral Points are the points lying midway between the four Cardinal Points. The Collateral Points are generally abbreviated thus: N.E., N. W., S. E., S. W.

The Intermediate Points are named according to their position relative to the 8 principal points.

The 32 points are divided into half-points, and anyone knowing all these is said to be able to " Box the Compass."

In steering by compass, there are further divisions into quarter-points. In steering steamers, the course is now generally given in degrees, that is, the compass card is divided into 4 quadrants of 90° each, i.e., N. to E. 90°; N. to W. 90°; S. to E. 90'; S. to W. 90°; or 360 degrees in all. The course is always given from N. or S., thus N. 50° E. (not E. 40° N.)

We can find our bearings' or determine the direction of any point in several ways.

(1) By the Pole Star: This star, being nearly due north, if we turn towards it our fa will be towards the north, and away from the south-the east will be to our right, and the west to our left.

(2) By the Sun: The sun rises in the east and sets in the west. If we turn towards the sun at noon we face the south, then the east will be to our left, and the west to our right. (3) By the Compass: In ordinary Pocket Compasses a magnetised needle is placed over a fixed card, which must be turned until the needle is at rest over the N. and S. points. In the Mariner's Compass one or more needles are attached to the card, which turns with the needle or needles, and thus the direction in which the ship is moving, and the "bearings" of other ships or objects on shore, are seen at a glance.

Form of the Earth: A sphere (Greek, sphaira, a ball) is a perfectly round body. The earth is round or globular, but it is not a perfect sphere, being slightly compressed at the poles.

That the earth is round or globular in shape is evident from the proofs already stated, but its rotundity has been also proved by actual measurements, from which its exact shape and size have been mathematically calculated. We thus know that the earth is not a perfect sphere, but is slightly compressed at the poles, so that its exact shape is that of an oblate spheroid. The amount of compression is, however, comparatively very small. Were the earth a perfect sphere, its diameter would be 7,9121⁄2 miles; as it is, its Polar Diameter is 7,899 miles, and its Equatorial Diameter 7,926 miles, a difference of only 27 miles. The distance, therefore, from the surface to the centre is 131⁄2 miles less at the poles than at the equator-a difference of only both of the earth's diameter.