axis is perpendicular to the circle when it is perpendicular to any two radii, a point on the surface of a sphere 90° distant from any two points of a great circle, will be the pole.

All angular distances on the surface of a sphere, to an eye at the centre, are measured by arcs of great circles. Hence all triangles formed upon the surface of a sphere, for the solution of spherical problems, must be formed by the arcs of great circles.

Secondaries to a great circle are great circles which pass through its poles, and consequently must be perpendicular to their great circles.

The axis of the earth is that diameter about which it performs its diurnal motion; and the extremities of this diameter are called the poles.

The terrestrial equator is a great circle of the earth perpendicular to its axis. Hence the axis and poles of the earth are the axis and poles of its equator. That half of the earth which lies on the side of the equator in which Europe and the United States of America are situated, is called the northern hemisphere, and the other the southern; and the poles are respectively called the north and south poles.

The latitude of a place upon the earth's surface is its angular distance from the equator, measured upon a secondary to it. These secondaries to the equator are called meridians.

The longitude of a place on the earth's surface is an arc of the equator intercepted between the meridian passing through the place, and another, called the first meridian, passing through that place from which you begin to measure; or it is the angle formed at the pole by these two meridians. The Americans and English generally place the first meridian at Greenwich; the French place it at Paris, the Spaniards at Cadiz ; some geographers place it at Teneriffe, and others at other places. Throughout this work, Greenwich will be reckoned as the first meridian. The longitude is counted from the first meridian, both eastward and westward, till it meets at the same meridian on the opposite point; therefore the longitude (and also the difference of longitude between any two places) can never exceed 180°.

If the plane of the terrestrial equator be produced to the sphere of the fixed stars, it marks out a circle called the celestial equator; and if the axis of the earth be produced in like manner, the points of the heavens, to which it is produced, are called poles, being the poles of the celestial equator. The star nearest to each pole is called the pole star.

Secondaries to the celestial equator are called circles of declination; of these 24, which divide the equator into equal parts, each containing 15°, are called hour circles. Small circles parallel to the celestial equator are called parallels of declination. The sensible horizon is that circle in the heavens whose plane touches the earth at the spectator. The rational horizon is a great circle in the heavens, passing through the earth's centre, parallel to the sensible horizon.

If the radius drawn from the centre of the earth to the place where the spectator stands be produced both ways to the heavens, the point vertical to him is called the zenith, and the point opposite, the nadir. Hence the zenith and nadir are the poles of the rational horizon.

Secondaries to the horizon are called vertical circles, because they are perpendicular to the horizon. On these circles, therefore, the altitude of a heavenly body is measured. The secondary common to the celestial equator, and the horizon of any place, is the celestial meridian of that place. This meridian corresponds with the terrestrial meridian of the same place, which passes through the poles of the earth, the zenith and nadir crossing the equator at right angles, and cutting the horizon in the north and south points; that point being called north which passes through the north pole, and the opposite direction is called south. The vertical circle which cuts the meridian of any place at right angles is called the prime vertical; the points where it cuts the horizon are called the east and west points, and to an observer, with his face directed towards the south, the east point will be to his left hand, and the west to his right hand. Hence the east and west points are 90° distant from the north and south. These four are called the cardinal points. The meridian of any place divides the heavens into two hemispheres, lying to the east and west; that lying to the east is called the eastern hemisphere, and the other the western hemisphere. When the sun is at its greatest altitude on the meridian of any place, it is noon, or mid-day.

The azimuth of a heavenly body is its distance on the horizon, when referred to it by a secondary, from the north or south points. The amplitude is its distance from the east or west points, at the time of rising or setting.

The ecliptic is that great circle in the heavens which the sun appears to describe in the course of a year. The ecliptic and equator, being great circles, must bisect each other, and their angle of inclination is called the obliquity of the ecliptic; and the points

to these points are called the equinoxes. The ecliptic is divided into 12 equal parts, called signs:-viz. Aries 9, Taurus 8, Gemini II, Cancer, Leo Virgo mg, Libra Scorpio m, Sagittarius ↑, Capricornus, Aquarius, Pisces. The order of these is according to the apparent motion of the sun. The first point of Aries coincides with one of the equinoctial points, and the first point of Libra with the other. The first six signs are called northern, lying on the north side of the equator; and the last six are called southern lying on the south side.

The zodiac is a space extending eight degrees on each side the ecliptic, within which the motion of all the planets is contained, except the newly-discovered planets. The right ascension of a body is an arc of the equator intercepted between the first point of Aries, and a circle of declination passing through the body, measured according to the order of the signs.

Right ascension of the meridian, or mid-heaven, is the distance of the meridian from the first point of Aries, and is found by adding the apparent time past noon to the sun's right ascension.

The ascensional difference of any object is the difference between the right ascension of the object and that point of the equator which rises or sets with it.

The declination of a star or any celestial object is its angular distance from the equator, measured upon a secondary to it passing through the object.

The longitude of a star or any celestial object is an arc of the ecliptic intercepted between the first point of Aries, and a secondary to the ecliptic passing through the body, measured according to the order of the signs. If the observer be on the earth, the longitude is called the geocentric longitude; but if seen from the sun, it is called the heliocentric longitude; the body in each case being referred perpendicularly to the ecliptic in a plane passing through the eye.

Nonagesimal degree of the ecliptic is its highest point at any given time, and is 90° from the points where the ecliptic intersects the horizon.

The latitude of a star or any celestial object is its angular distance from the ecliptic, measured upon a secondary to it drawn through the body. If the body be observed from the earth, its angular distance from the ecliptic is called the geocentric latitude; but if observed from the sun, it is called the heliocentric latitude. The secondary circle drawn perpendicular to the ecliptic is called a circle of latitude.

The tropics are two parallels of declination touching the ecliptic. One, touching it at the beginning of Cancer, is called the tropic of Cancer; and the other touching it at the beginning of Capricorn, is called the tropic of Capricorn. The two points where the tropies touch the ecliptic are called the solstitial points.

Colures are two secondaries to the celestial equator, one passing through the equinoctial points, called the equinoctial colure; and the other passing through the solstitial points, called the solstitial colure. The times when the sun comes to the solstitial points are called the solstices.

Aberration of a star, or any heavenly body, is a small apparent motion, occasioned by the progressive velocity of light. This is calculated by means of Tables XXXIX., XLI., or XLII.

Nutation is a small apparent motion of the heavenly bodies, occasioned by a real motion of the earth's axis, arising from the attractions of the sun and moon on the spheroidal form of the earth. The effect of this on the right ascension and declination is given in Table XLIII., and on the longitude in Table XL.; the correction in this last table being generally called the equation of the equinoxes in longitude.

Precession of the equinoctial points is a small motion of about 50" per year, occasioned by the same cause as the nutation. By this motion the equinoctial points are carried backward from east to west; consequently, the heavenly bodies appear to move forward the same quantity from west to east. The annual variations of the places of the stars from precession, and the secular equations arising from the change of the earth's orbit by the attraction of the planets, are given in Tables VIII. and XXXVII.

The arctic and antarctic circles are two parallels of declination, the former about the north, and the latter about the south pole, the distance of which, from the two poles, is equal to the distance of the tropics from the equator, which is about 23° 28'. These are also called polar circles. The two tropics and two polar circles, when referred to the earth, divide it into five parts, called zones; the two parts within the polar circles are called the frigid zones; the two parts between the polar circles and tropics are called the temperate zones; and the part between the tropics is called the torrid zone. Besides the imaginary divisions of the earth, there are various natural divisions of its surface, such as continents, oceans, islands, seas, rivers, &c.

A continent is a large tract of land, wherein are several empires, kingdoms, and countries conjoined; as Europe, Asia, Africa, and America.

An island is a part of the earth that is environed or encompassed round by the sea as Long Island, Block Island, &c.

A peninsula is a portion of land almost surrounded with water, save one narrow neck which joins it to the continent; as the Morea.

An isthmus is a narrow neck of land joining a peninsula to the adjacent land, by which the people may pass from one to the other; as the isthmus of Darien.

A promontory is a high part of land stretching itself into the sea, the extremity of which is called a cape or headland.

A mountain is a rising part of dry land, overtopping the adjacent country.

An ocean is a vast collection of water, separating continents from one another, and washing their borders or shores; as the Atlantic and Pacific Oceans.

A sea is part of the ocean, to which we must sail through some strait, as the Mediterranean and Baltic Seas. This term is sometimes used for the whole body of salt water on the globe.

A strait is a narrow part of the ocean lying between two shores, and opening a way into some sea; as the Straits of Gibraltar, that lead into the Mediterranean Sea.

A creek is a small narrow part of the sea or river, that goes up but a little way into the land.

A bay is a great inlet of the land; as the Bay of Biscay, and the Bay of Mexico; otherwise a bay is a station or road for ships to anchor in.

A river is a considerable stream of water issuing out of one or various springs, and continually gliding along in one or more channels, till it discharges itself into the ocean: the smaller streams are called rivulets.

A lake is a large collection of waters in an inland place; as the Lakes Superior and Huron in America.

A gulf is a part of the ocean or sea, nearly surrounded by the land, except where it communicates with the sea; as the Gulf of Venice.

Thus we have given the most useful definitions of Astronomy and Geography, and to assist the learner there is also given Plate V., in which those terms are explained at one view. We may further observe, that, as the latitude of any place upon the earth is counted from the equator upon an arc of the meridian, the difference of latitude between two places, both north or both south, is found by subtracting the less latitude from the greater; but if one latitude be north, and the other south, the difference is found by adding both latitudes together.

1. Consequently, if a ship in north latitude sails northerly, or in south latitude southerly, she increases her latitude; but in north latitude sailing southerly, or in south latitude sailing northerly, she decreases her latitude, because she sails nearer to the equator, from whence the latitude is reckoned.

2. Wherefore, in north latitude sailing northerly, or in south latitude sailing southerly, the difference of latitude, added to the latitude left, gives the latitude in.

3. "In north latitude sailing southerly, or in south latitude sailing northerly, the difference of latitude, subtracted from the latitude left, gives the latitude in.

4. When the latitude decreases, and the difference of latitude is greater than the latitude sailed from, subtract the latitude left from the difference of latitude, and the remainder will be the latitude in, but of a different name, for it is evident, in this case, that the ship has crossed the equator.

5. The difference of longitude between two places, being both east or west, is found by subtracting the less longitude from the greater; but if one be in east longitude and the other in west, their sum is the difference of longitude, when it does not exceed 180°, but if it exceeds 180°, that sum must be subtracted from 360°, and the remainder will be the difference of longitude.

6. Therefore in east longitude sailing easterly, or in west longitude sailing westerly, the difference of longitude, added to the longitude left, gives the longitude in, when that sum does not exceed 180°; but if it exceeds 180°, the sum, subtracted from 360°, leaves the longitude in, but of a different name from that left.

*

7. In east longitude sailing westerly, or in west longitude sailing easterly, the difference of longitude, subtracted from the longitude left, gives the longitude in; but when the difference of longitude is greatest, the longitude left must be subtracted from that difference, and the remainder will be the longitude in, but of a different name from the longitude left.

* In this rule it is supposed, that the sum of the longitude left, and the difference of longitude, is less than 360, which is always the case when the difference of longitude is less than 180°, which we have