CHAPTER II.

The Circles of the Sphere-Extent of the Visible Horizon- Method of Drawing a Meridian Line-Circles of Position-Equator-LatitudeFirst Meridian-Longitude.

THE modes of precisely fixing the situation of places upon the earth are founded upon the circumstance just now proved of its spherical form, and upon the supposition which, for the purposes to which it is applied, is not a false one, that it is enclosed in a concave or hollow sphere of the heavens, of which it occupies the middle spot or centre.

and p the south pole; now suppose a plane to cut this sphere into two equal Fig. 1.

E

Upon the surface of the earth considered as a globe, various lines are supposed to be drawn for the purposes of geographers, and in order to determine or explain the truths of their science; and as the heavens present to us a concave sphere, having the same centre as the earth, there are also imaginary lines supposed to be traced upon the inner surface of the heavens, which exactly correspond with those traced upon the earth. By this device geography has become allied with astronomy, and has thence derived its most important improvements. We now proceed to the description of the abovementioned lines which are supposed to divide the earth, and which are seen drawn upon the common geographical globe.

parts in a direction perpendicular to the axis, this plane will pass through the centre O, and the circle Qab Q cd, which is the boundary of the cutting plane upon the surface of the globe, will represent the equator, and is every where at an equal distance from both poles. This circle and all other circles, the planes of which pass through the centre of the sphere, are called great circles. All circles such as PapcP Pbpd P, which pass through both poles P, p, of the earth, and which have the axis of the earth Pop for a common diameter, are called meridians, because when the centre of the sun is over or upon that one of these circles which passes through any place, it is mid-day or noon at that place. The plane of every meridian cuts the plane of the equator at right angles, so that the equator divides every meridian (as for instance Papc P) into four equal parts; thus Pa and ap, and pc and cP are equal to one another, and are called quadrants or quarters of a circle. Meridians are also called circles of latitude, because upon them the latitudes of places are measured. The Ecliptic found traced upon common globes (although it is properly an imaginary circle in the concave sphere of the heavens representing the apparent path of the sun in the course of a year) is a great circle upon the globe of the earth, the plane of which is inclined at a certain angle to the plane of the equator, and is represented in the figure by the circle ÉL. All circles upon the sphere which do not pass through its centre are called small circles; those which are parallel to the equator, as mrns, are called

The earth has a daily motion from west to east, about one of its diameters (called the earth's axis), which causes all the heavenly bodies to appear to move daily round the earth in an opposite direction from east to west. The two extremities of this axis are called the poles of the earth, from a Greek word signifying a pivot; one is called the North pole, being that which is opposite or nearly opposite to the star in the heavens called the pole star; the other extremity of the axis is called the South pole; the north pole is also known by the name of the arctic pole, from a Greek word signifying a bear, the 'great bear' being the name of a constellation or collection of stars in the immediate neighbourhood of the pole star, and commonly known as Charles's wain: the South pole has the corresponding term of the antarctic pole, or the pole opposite to the arctic.

Let PEPQP (fig. 1) be a sphere representing the globe of the earth, O the centre, POp the axis, P the north

circles of longitude or parallels of latitude; and as all meridians cut the equator at right angles, they also cut all circles of longitude at right angles, which is evident from these latter circles being parallel to the equator. Every circle traced upon the earth is supposed to have a corresponding circle traced upon the concave or hollow spherical surface of the heavens. All circles, whether great or small, are divided into 360 equal parts called degrees; every degree is again divided into 60 equal parts called minutes, and every minute into 60 seconds these various parts are distinguished by certain signs, thus 15 degrees is written 150, 32 minutes is written 32', and 5 seconds 5"; so that 15° 32′ 5", signifies 15 degrees together with 32 minutes and 5 seconds. The magnitude of degrees is of course different in great and small circles; the amount and variation of this difference in the circles of the globe, will be explained afterwards.

The zenith of any place on the earth is that point in the concave surface of the heavens which is immediately opposite to the extremity of a line drawn from that place to the centre of the earth, or in the direction of a plumb line; it is the point in the heavens directly over our heads. The nadir is the corresponding point in the opposite hemisphere of the heavens. Of all the meridian circles, that which passes through the zenith of a place in the heavens, or through the place itself upon the earth, is the meridian of that place. The horizon of a place is the boundary of view at that place: with respect to the earth it is called the visible, sensible, or apparent horizon; with respect to the heavens it is called the rational, true, or astronomical horizon. The visible horizon is most accurately observed upon the sea where it is distinct and unbroken, and is, therefore, sometimes called the horizon of the sea. The extent of the visible horizon may easily be found if the height of the spectator's eye above the surface of the earth be known, and also the length of the earth's radius or semi-diameter. For if (fig. 2.) BEDF be a great circle, C the centre of the earth, A E the height of the eye, E C the semi-diameter of the earth, and A B be drawn from A, just touching the earth's surface at B, E B will be the extent in one direction of the visible horizon. If B and C be joined, BC will be perpendicular to A B. The

Fig. 2.

F

length of AE and E C or AC and also of B C, which is the semi-diameter, being known, the angle AC B may be found by a very simple mathematical process; and this angle being measured by the arc EB, the required distance is found. AB, the direct distance of the horizon at B from the spectator's eye at A, may also be found in a somewhat similar manner :-if A E be equal to 5 feet, and E C, the semi-diameter of the earth, be taken at 29,949,655 feet, the angle at C, or the arc BE, will be found to be equal to 2' or 12,188 feet, which is nearly equal to 2 miles and 532 yards; D B is of course equal to twice BE, as the spectator sees as far one way as another, therefore D B is equal to 4 miles 1064 yards. This, however, is not quite true in practice, as by the refracting power of air and vapour, the apparent horizon is a little more extensive. The rational horizon is in every part of it 90°, or a quadrant distant from the zenith. When a heavenly body first appears above the horizon of a place, it is said to rise, and to set when it disappears or sinks below the horizon. When a heavenly body is upon the meridian of any place, it has obtained its greatest height or altitude above the plane of the horizon of that place.

The north point of the horizon is that which is nearest to the north pole of the heavens or the pole star, the point 180° distant from it is the south. The meridian line of a place passes through the north and south points. The east point is 90° distant from the north or south in that portion of the heavens where the sun, stars, &c. appear to rise; the west is 180° distant from the east point. Thus all the cardinal or principal points of the compass are determined.

By means of the observed altitudes of heavenly bodies, when at their highest or on the meridian of a place, many

geographical problems are solved; it is, therefore, of great importance to ascertain the direction of the meridian lime at the place of the observer. The operation in its more scientific and correct shape is one of very considerable nicety; but the following method will determine it, if much accuracy be not required. On the 15th of June, or the 24th of December, plant a stick A B in a position perpendicular to the horizon, (fig. 3.) at an hour or two before the Fig. 3.

B

sun has arrived at its greatest altitude in the heavens, that is, be at ten or eleven o'clock in the morning; mark accurately the extremity C of the shadow BC cast by the stick; then from the base B of the stick as a centre, and with the length of the shadow B C as a radius, trace a circle G H upon the ground; as the sun gradually arrives at its greatest altitude, the shadow of the stick will become gradually shorter, and will fall within the circumference of the circle which has been traced. The shadow will be at its shortest B E, when the sun is at its greatest altitude, or when it is on the meridian of the place which is the moment of noon; after this the sun will gradually decline, the shadow of the stick will become longer and longer, until at last it again reaches to the circumference of the circle in the point D, at which time in the afternoon the sun is at the same height in the heavens as it was when the shadow of the stick was of the same length B C before noon. Now it so happens, that, on the abovementioned days, the altitude of the sun above the horizon, at one hour or two hours before noon, is equal to its altitude at the same time after noon; and as the sun has in these equal times before and after noon, described equal

spaces in the heavens (supposing those spaces to be measured from the meridian), the middle point of the whole space described by the sun in the sum of those times will be that point in the heavens which the sun occupies when it is noon; at this time the sun is on the meridian. Hence, if the arc C D of the traced circle be divided into two equal parts C F and FD, and the point F of division and the base of the stick be joined, the line B F, joining these two points, and which will be the direction of the shadow of the stick at noon, will be the meridian line. The longest and most accurate meridian line in the world, is that drawn by Cassini (a celebrated astronomer and mathematician) upon the pavement of the church of St. Petronis, at Bologna, in Italy: it is 120 feet in length.

One of the principal objects in mathematical geography is to ascertain the position of any particular spot upon the earth's surface. This term position is strictly a relative one-applied to a body, it has no meaning unless there be some other body or mark, which is fixed, and to which the first body may be referred. If, in the midst of infinite space, there existed but a single body, it could hardly be said to have position, or at any rate the meaning of the term as applied to such a body would pass our comprehension. This may be illustrated by what is said in some books on mechanics, that motion could not

exist if there were but a single body in the universe; by which is meant that in such a condition of things motion could be neither measured nor perceived: it is not intended by such expressions to assert that motion cannot exist independently of other bodies, because the existence of a foreign body cannot really affect the state or condition of motion in any moving body; it only enables us to ascertain the fact of motion, and its measure. In the idea of position, therefore, there is contained a reference to something which is fixed and which is independent of the body, the position of which is required. The distance of any body from this something which is fixed being known, and also the direction given in which that distance is to be measured, its position may be determined.

In order to ascertain the situation of any spot upon the surface of the globe, it is sufficient to fix upon two great

globe PE p P be required-it may be determined if we have given in position the great circle C Bb Q, and the great circle Pa bp, the planes of which are perpendicular to each other. For we need only make a great circle PABP perpendicular to the circle E B b Q pass through A, and then a small circle pass through A parallel to E B b Q, and the distances of their intersection from the given great circles, viz. the arc A B being the distance from ΠB b Q, and the arc A a, or the corresponding arc B b, being the distance from Pabp, will determine the exact position of the point A.-In applying this to the practical purposes and wants of geography, it is evident that the first object is to fix the position of the two great circles CE BbQ and P a bp, and then to devise some mode for ascertaining the distances A B, A a from each of them. The astronomers and geographers of all countries have concurred in fixing upon the equator or equinoctial line (as it is sometimes called) for the position of the circle CE BbQ. The equator has been already defined as a great circle dividing the globe into two equal parts or hemispheres, and the plane of it as perpendicular to the axis of the earth. The distance A B measured upon the meridian of A, which is a great circle perpendicular to the equator, is called the latitude of A. The latitude of a place is north or south latitude, as it is situated towards the north or south of the equator. It is very evident that astronomers were led to fix upon the equator for one of the great circles of position, by the circumstance of the ap

parent daily motion of heavenly bodies, being performed either in a circle in the heavens corresponding with the equator itself, or in circles which are parallel to it. But as there was nothing in the apparent courses of heavenly bodies, or in any particular spot upon the earth to regulate the choice of astronomers in fixing upon a first meridian or the other great circle of position perpendicular to the equator, and which is represented in the preceding figure by the circle Pabp, the consequence has been, that astronomers and geographers of different ages and countries have assumed different circles for their first meridian, from which they have measured the arc A a or B b.

The ancient geographers took for their first meridian the meridian of the Fortunate Isles, a line passing, as they conceived, through the western extremity of the habitable earth. Many of the moderns have employed the same meridian, or rather that of the island of Ferro, one of the most westerly of the Canaries. In general, however, nations adopt as their first meridian the meridian of their own metropolis, or of their principal observatory, as the English do that either of London or Greenwich, the French that of Paris. The angular distance on the arc A a or B b of any place from the first meridian is called its longitude, and is either east or west longitude as the place is to the east or west of the first

meridian.

The English map-makers frequently adopt the meridian of London instead of that of Greenwich for the first meridian, but as London (taking St. Paul's as the point referred to) is 5' 47" west of Greenwich, longitudes given from London may be easily reduced to longitudes reckoned from Greenwich, by adding to them 5' 47" if they are west longitudes, and subtracting the same quantity if they are east longitudes.

CHAPTER III.

General Description of the Method of finding the Latitude of a Place.

HAVING fixed upon the two circles of position by a reference to which the position of a place is to be determined, it will now be necessary to explain how distances from each of these circles (being the latitudes and longitudes of places) may be ascertained. This depends entirely upon the supposition that

ridian of A passing through p, the pole of the earth, and at right angles to the equator (e q), the plane of the paper is the plane of the meridian, O the centre of the earth, p O its semi-diameter, to which (e q), the equator, is at right angles, HR the rational horizon of A. Then Ae, the arc or angular distance of A from the equator measured upon the meridian of A, is the latitude of A. But ep, or the distance of the pole from the equator, is 90°, or a quadrant, and Ar the distance of A from the point r, where the rational horizon meets the surface of the earth, is also 90°, or a quadrant. Hence e p is equal to Ar; if therefore Ap, which is common to both ep and Ar, be taken away from each, the remaining quantities Ae and pr will be equal; and as Ae is the latitude of A, it follows that (pr) or the height of the pole above the horizon is equal to the latitude of the place.

Again, Ap is the distance of the pole from A and as A e is the latitude of A, and pe is 90°, Ap is the difference between 90° and the latitude, so that if Ap be known, A e, or the latitude, is found, by subtracting Ap from 90°. Apis called the complement of the latitude, or the co-latitude.

Again, Ah is 90°, and, therefore, A e being the latitude, he is the co

latitude,-he, being the height of the equator above the horizon; so that if he be known, the latitude is found by subtracting he from 90°.

It appears, then, that if we can find any one of the above four arcs, viz., Ae, pr, Ap, he, the latitude of A will be known: and the mode of determining these arcs, is by measuring similar arcs of corresponding circles in the heavens. Let ZPRQ HÆ be the circle in the heavens which corresponds with the meridian circle passing through A, and Z, P, R, Q, H, Æ, points in the heavens corresponding with A, p, r, q, h, e. The attention of the geographer is then transferred from the consideration of the several arcs Ae, pr, Ap, he, to the corresponding arcs in the circle in the heavens, ZE, PR, ZP, HÆ: for if any of these be determined in their number of degrees and parts of degrees, the latitude is found directly. Thus it is, that the geographer depends so much upon the science of astronomy for the solution of the most important geographical problems. Persons who are in the slightest degree acquainted with geometry, or with the most simple properties of the circle, will not object to the above-mentioned mode of determining the latitude of places on the earth by means of corresponding arcs in the heavens, that these corresponding arcs are of different magnitudes; for in computing the latitude, we do not so much want the actual admeasurement and linear quantity of the arc of the meridian intercepted between the given place and the equator, as the number of degrees and parts of a degree which it contains, or, in other words, the proportion which this intercepted arc bears to the whole circumference of the meridian circle. And as arcs are the measures of angles, the arcs, Z Æ and A e, are both measures of the same angle at o; and, therefore, although they are unequal in magnitude, yet they mutually bear the same proportion to the circumference of the circles of which they are parts; that is, Z contains the same number of degrees as Ae: and as the latitude of a place is always expressed in degrees and parts of degrees, the number of degrees contained in the arc in the heavens, Z A, which corresponds with the arc of the meridian A e, will be the latitude of A. If, after having ascertained the latitude in this manner,