no component of rotation there, the pendulum would continue to move in one and the same plane. At intermediate stations the rate of rotation is easily calculated; and observations confirm the calculations, and have made the earth's rotation actually visible. The poles of the earth are the points in which the axis of rotation, or of figure, meet the surface; and the equator is the circle in which the surface is intersected by a plane through the earth's centre, perpendicular to the axis of rotation Every point of the equator is therefore equidistant from the poles. To determine the position of a point in space three co-ordinates or measurements are necessary; they may be three lines, or two lines and one angle, or two angles and one line. Thus, to define the precise position of a point on the earth's surface, we express it by latitude, longitude, and altitude; the first two are angular measures, the third a linear magnitude, namely the height above the surface of the sea. The line in which the surface of the earth is intersected by a plane through the axis of rotation is called a meridian, and all meridians are evidently similar curves. A line perpendicular to the surface at any point is called a vertical line; it corresponds with the direction of gravity there; being produced outwards, that is, away from the earth's centre it meets the heavens in the zenith; and produced downwards it intersects the axis of revolution; it would of course pass through the earth's centre were it a sphere; as it is, it passes near the earth's centre. The angle between the meridian planes of two stations as A and B is called the difference of longitude of A and B, or the longitude of B with reference to A. In British maps the longitudes of all places are expressed with reference to the Royal Observatory of Greenwich. meridian. As the radius of curvature of an ellipse is variable, increasing from the extremity of the major axis to the extremity of the minor axis, so on the earth's surface a degree of the meridian is found by geodetic measurement to increase from the equator to the poles. The actual length of a degree of the meridian at the equator is 362746-4 feet; at either pole it is 366479-8 feet. The length of one degree of the equatorial circle is 365231·1 feet. With regard to the figure of the earth as a whole, the polar radius is 3949-79 miles, and the radius of the equator 3963-30 miles; the difference of these, called the ellipticity, is of the mean radius. A spheroid with these semiaxes is equivalent in volume to a sphere having a radius of 3958-79 miles. Without referring further here to the spheroidal figure, we shall now, having given the precise dimensions, regard the earth as a sphere whose radius is 3959 miles. On such a sphere one degree is 69.09 miles. From the definitions given above it appears that the radius of the parallel which corresponds to all points whose latitude is is 3959 cos; and that one degree of this circle, ie., one degree of longitude in the latitude is 69.09 cos expressed in miles. E P In the representation of the spherical earth (fig. 2) P is the pole, QQ the equator, E,F any two points on the surface, PEe, PF/ the meridians of those points intersecting the equator in e and ƒ Join EF by a great circle; then in the spherical triangle PEF the angle at P is the difference of longitude of E and F, PE is the co-latitude of E, and PF the co-latitude of F, the latitudes being eE and fF respectively. The angle at E, being that contained between the Fig. 2. meridian there and a vertical plane passing through F, is the azimuth of F (measured in this case from the north), while the angle at F is the azimuth of E. If, then, there be given the latitudes and longitudes of two places, to find their distance apart, and their relative bearings, it becomes necessary to calculate a spherical triangle (PEF) in which two sides and the included angle are given, the calculation bringing out the third side, which is the required distance, with the adjacent azimuthal angles. The latitude of any point is the angle made by the vertical line there with the plane of the equator, or the co-latitude is the angle between the vertical line and the axis of rotation. The surface of the earth being one of revolution, any intersecting plane parallel to the equator cuts it in a circle. If we imagine the vertical lines drawn at any two points, as P and Q, in such a circle it is evident from the symmetry of the surface that these verticals make the same angle with the equator; in other words, the latitudes of all points on this circle are equal. Such circles The latitudes and longitudes of places on the earth's are called parallels; they intersect meridians at right angles. surface are determined by observations of the stars, of the If we suppose that at any point Q of the surface the sun, and of the moon. As the earth rotates, the zenith of meridian, or a small bit of it, is actually traced on the any place (not being on the equator) traces out among the surface, and also a portion of the parallel through the same stars a small circlé having for centre that point in which point, then these lines, crossing at right angles in Q, mark the axis of rotation meets the heavens. If there were a there the directions which we call north and south, east star at this last point it would be apparently motionless, and west-the meridian lying north and south, the parailel having always the same altitude and azimuth. The pole east and west. Planes containing the vertical line at Qstar, though very conveniently near the north pole of the are vertical planes there. A vertical plane is defined by heavens, and without perceptible motion to the unaided its azimuth, which is the angle it makes with the meridian eye, is in reality moving in a very small circle. The zenith plane; the azimuth at Q of any object (or point) celestial of a point on the equator traces out in the heavens a great or terrestrial is the angle which the vertical plane passing circle, namely, the celestial equator. through the object makes with the meridian. The south meridian is generally taken as the zero of azimuth. The plane touching the surface at Q is the visible horizon there a plane parallel to this through the centre of the earth being called the rational horizon. The altitude at Q of a heavenly body, as a star, is the angle which the line drawn from to the star makes with the plane of the horizon,—the zenith distance of the same star being the angle between its direction and the vertical at Q. By a degree of the meridian is meant this: if E, F are points on the same meridian such that the directions of their verticals make with each other an angle of one degree a ninetieth part of a right angle-then the distance between E and F measured along the meridian is a degree of the As the positions of points on the earth are defined with reference to the equator and a certain fixed meridian, so the positions of stars are defined by their angular distance from the celestial equator, called in this case declination, and by their right ascension, which corresponds to terrestrial longitude. Stars which are on the same meridian plane (extended to the heavens) have the same right ascension. Right ascension is expressed in time from 0 to 24. sidereal clock, going truly, indicates 24h for every revolution of the earth: at every observatory, the sidereal clock there shows, at each moment, the right ascension of the stars which at that moment are on the meridian. Thus the right ascension of the zenith is the sidereal time. A In the left hand circle of the diagram (fig. 3) two star. If we consider in the first circle the changes of distance between any one star and the zenith of Q as the latter traces out its path in the heavens, we see that the distance becomes alternately a maximum and a minimum every twelve hours, namely, when the meridian of Q passes through the This is called the star's culmination or meridian transit. It will be clear from an inspection of the figure that, if for instance the star culminate to the south of the zenith, the star's declination plus its zenith distance at culmination is equal to the latitude of the zenith, that is, of Q. A corresponding rule is easily made for a northern transit. Thus the simplest manner of determining the latitude is to measure the zenith distance of a known star at its meridian transit. The position of the zenith at any moment may be determined by simultaneous observation of the zenith distance of two known stars. For these distances clearly determine a point in the heavens (two points rather, which however need not be confounded) whose declination and right ascension can be computed by spherical trigonometry. Thus, at the same time, are obtained both the time and the latitude. For the success of this method, which is suitable for travellers exploring an unknown country, it is desirable that the stars should differ in azimuth by about a right angle. If the path of the zenith, that is, the latitude, be known, then clearly a single observation of the zenith distance of a known star, which should be towards the east or west, not towards the north or south, will fix the place or right ascension of the zenith, that is, the sidereal time, at the moment of observation. Here the pole, the zenith, and the star are the angular points of a spherical triangle, of which the three sides are known: the angle at the pole, being computed, is the difference of right ascension of the star and the zenith. Thus the sidereal time is found. The determination of the difference of longitude of the two stations AB on the earth's surface requires that the true time be kept at each. All that is necessary is a comparison of these times at any instant. For instance, the time at B may, by the transport of chronometers, be brought to A, and thus the difference of the local times be ascertained, or the indications of the clock at A may be conducted by electro-telegraphy to B. The difference of the local times at A and B is the time a star takes to pass from the meridian of the one to that of the other; and this is the difference of longitude which may be converted into angle at the rate of 360° to 24h. But the traveller in unknown lands, who seeks to fix astronomically his position, has no telegraph to count on and his expectations for longitude depend chieft on observations of the moon. In the Nautical Almanac are published the angular distances of the moon from certain stars in its path for every three hours of Greenwich time. Therefore, by actually observing the distance of the moon from one of these stars, one can infer the corresponding Greenwich time at the moment of observation. The comparison of this with the local time gives the longitude. Observations on the sun have shown that it traces out amongst the stars in the course of a year a great circle, inclined to the equator at an angle of 23°; at midsummer it attains a maximum northern declination of 234°, and at midwinter a maximum southern declination of the same amount. Hence it is inferred that the earth moves round the sun in a plane, completing one orbital revolution yearly, the axis of the earth's diurnal rotation being inclined to this plane at an angle of 66. Upon this angle of inclination depend the seasons, and in great measure the climates of the different portions of the earth's surface. It is usual to draw on globes and in maps a circle or parallel at the distance of 231 from the equator on either side; of these circles the northern is called the Tropic of Cancer, the southern is the Tropic of Capricorn. A circle drawn with a radius of 234° from the North Pole as centre is the Arctic Circle; a similar and equal circle round tho South Pole is the Antarctic Circle. When the sun is in the equator-which it crosses from north to south in September, and from south to north in March-it is in the horizon of either pole. When the sun has northern declination, the North Pole is in constant daylight and the South Pole in darkness. When the sun has southern declination the North Pole on the contrary is in constant darkness while the South Pole is illuminated by sunshine. At midsummer in the northern hemisphere the whole region within the Arctic Circle is in constant daylight, and that within the Antarctic Circle is in darkness; át midwinter this state of things is exactly reversed. The portion of the globe lying between the Tropic of Cancer and the Arctic Circle is called the North Temperate Zone; that between the Tropic of Capricorn and the Antarctic Circle is the South Temperate Zone. In the former the sun is always to the south of the zenith; in the latter it is always to the north. In the Torrid Zone, which lies between the Tropics, the sun, at any given place, passes the meridian to the north of the zenith for part of the year, and to the south for the remainder. When the sun is to the north of the equator the days are longer than the nights in the northern hemisphere, while in the southern hemisphere the nights are longer than the days; when the sun has southern declination this condition is reversed. As the sun increases his north declination from 0° to 231, not only do the days increase in length in the northern hemisphere, but the rays of the sun-in the Temperate and Arctic regions-impinge more perpendicularly on the surface; hence the warmth of summer. Even in summer the rays of the sun in the Arctic regions strike the surface very obliquely; this, combined with the protracted season of darkness, produces excessive cold. Summer in the northern hemisphere is thus contemporaneous with winter in the southern; while winter in the northern hemisphere is simultaneous with summer is the southern. The length of the day at any place at any season of the year is easily ascertained from the following considerations. Let ns (fig. 4) be the axis of rotation, eq the equator orthographically projected on a meridian plane, ab the parallel of the given place; draw the diameter fg making ng b = the angle nog equal to the sun's declination, which we | Since is -23° 18° 51°, we see from the diagram that suppose to be north, then the hemisphere gnacy is in the South Pole is at this time in total darkness, which ea sunshine, while the hemisphere gbqsf tends to all places within 5 of it. When the sun's is in darkness. As the earth rotates, declination is 9° south, the North Pole is in the centre of a point which is at a at midday is the twilight belt; thus all places whose latitude is greater carried from a towards b, which it than 81 then move in continual twilight, alternating reaches at midnight; h is reached at between clearness and dimness, never attaining either day6 o'clock P.M. and k at sunset. Now light or total darkness. The actual period during which ifo be the latitude of the place and 8 either pole is in total darkness is about two and a half the sun's declination hk = sin & tan 8; months. this in the parallel whose radius is Los corresponds to an angle whose aine is tan tan 8. Call this angle 7; Fig. 4. the time taken to rotate through it is; hence the Length of the daylight is 12h+, and the length of night 12h-37. Now vanishes when either & or 8 is zero; that is, at the equator the nights and days are equal in length throughout the year; and again when the sun is in the equator, that is, at the equinox, the nights and days are equal in all latitudes. When the sun's declination is equal to the co-latitude, ʼn is a right angle, and the sun does not actually set; this can only happen at places within the polar circle. The longest day at Gibraltar is 14h 27m, at Falmouth 16h 11", and in Shetland 18h 14m; while in Iceland it is 20h on the south coast and 24h on the north. At Washington the longest day is 14h 44m, and at Quebec 15h 40m. All this, however, is on the supposition that day ends with sunset; but the length of apparent day is increased by atmospheric refraction and reflection. When the disk of the setting sun first seems to touch the horizon it is in reality wholly below it and is only seen by refraction. After the sun has wholly set at any given place his light still continues to illuminate the upper portion of the atmosphere there, so that, instead of ending abruptly, daylight gradually fades away until the sun is 18° below the horizon. e m T In a diagram (fig. 5) similar to the last draw mi parallel to gf, and at a distance from it equal to the sine of 18°; then gbf being the hemisphere unenlightened by the direct rays of the sun, gmif will represent the twilight zone. A point in the latitude of a describing the parallel ab loses sight of the sun at k, and is in twilight until it reaches the small circle mi, when the sun's zenith distance is 108°. The duration of twilight corresponds then to the portion kl of ab, the angle rotated through being sin-1 (hl:hb)-sin-1 (hk: hb); Fig. 5. At the equator, the shortest twilight occurs at the equinox, when it is 1h 12m; the longest when the sun is in the tropics, being 1h 18m. At London, in latitude 51°, twilight continues all night from May 22 to July 21; it is shortest about three weeks after the autumnal and three weeks before the vernal equinox, when its duration is 1h 50m. At Washington the shortest twilight (being 1h 33m) occurs on the 6th of March and 7th October; at Quebec the shortest is 1h 46m, falling on the 3d March and 10th October. At page 205, fig. 19 is a perspective representation of the earth of more than a hemisphere, in fact-namely, the segment mgnafi in fig. 5. It exhibits all those regions of the earth which at Greenwich apparent noon at midsummer are in sunshine and twilight. It is very re markable how Asia and America, but especially the former, just escape going into darkness. Construction of Maps. In the construction of maps, one has to consider how a portion of spherical surface, or a configuration traced on a sphere, can be represented on a plane. If the area to be represented bear a very small ratio to the whole surface of the sphere, the matter is easy: thus, for instance, there is no difficulty in making a map of a parish, for in such cases the curvature of the surface does not make itself evident. If the district is larger and reaches the size of a county, as Yorkshire for instance, then the curvature begins to be sensible, and one requires to consider how it is to be dealt with. The sphere not being a developable surface cannot be opened out into a plane like the cone or cylinder, consequently in a plane representation of configurations on a sphere it is impossible to retain the desired proportions of lines or areas or equality of angles. But though one cannot fulfil all the requirements of the case, we may fulfil some by sacrificing others; that is to say, we may, for instance, have in the representation exact similarity to all very small portions of the original, but at the expense of the areas, which will be quite misrepresented. Or we may retain equality of arcas if we give up the idea of similarity. It is therefore usual, excepting in special cases, to steer a middle course, and, by making compromises; endeavour to this converted into time gives the duration of twilight. obtain a representation which shall not offend the eye. Here A globe gives a perfect representation of the surface of the earth; but practically, the necessary limits to its size make it impossible to represent in this manner the details of countries. A globe of the ordinary dimensions serves scarcely any other purpose than to convey a clear conception of the earth's surface as a whole, exhibiting the figure, extent, position, and general features of the continents and islands, with the intervening oceans and seas; and for this purpose it is indeed absolutely essential and cannot be replaced by any kind of map. The construction of a map virtually resolves itself into the drawing of two sets of lines, one set to represent meridians, the other to represent parallels. These being drawn, the filling in of the outlines of countries presents no difficulty. The first and most natural idea that occurs to one as to the manner of drawing the circles of latitude and longitude is to draw them according to the laws of Let perspective. But, as Lagrange has remarked, one may regard geographical maps from a more general point of view as representations of the surface of the globe, for which purpose we have but to draw meridians and parallels according to any given law; then any place we have. to fix must take that position with reference to these lines that it has on the sphere with reference to the circles of latitude and longitude. the law which connects latitude and longitude, and w, with the rectangular coordinates x and y in the representation be such that dx mdp + ndw, and dy = m'do + n'dw. Fig. 6. In fig. 6 let the lines intersecting in the parallelogram PQRS be the representations of the meridians rp, sq and parallels rs, pq intersecting in the indefinitely small rectangle pqrs on the surface of the sphere. The coordinates of P being and y, while those of p are and w the coordinates of the other points will stand thus = w+dw $ +dp φ ταφ ය w+dw x +ndw y+n'dw y+m'do x+mdp+ndw y+m'dp+n'dw. If the lines of latitude and of longitude intersect at right angles, then mn+m'n'=0. Since the length of pr is do, its representation PR is too great in the proportion of (m2+m22): 1; and pq being in length cosodo, its representation PQ is too great in the ratio of (n2+n'2): cosp. Hence the condition that the rectangle PQRS is similar to the rectangle pars is (m2 + m2) cos2p=n2+n2, together with mn + m'n'=0; or, which is the same, the condition of similarity is expressed by - n'=m cos ; n=m' cos p. Since the area of the rectangle pqrs is cos dodu, the exaggeration of area in the representation will be expressed by m'n-mn': cos p. Thus when the nature of the lines representing the circles of latitude and longitude is defined we can at once calculate the error or exaggeration of scale at any part of the map, whether measured in the direction of a meridian or of a parallel; and also the misrepresenta tion of angles. The lines representing in a map the meridians and parallels on the sphere are constructed either on the principles of true perspective or by artificial systems of developments. The perspective drawings are indeed included as a particular case of development in which, with reference to a certain point selected as the centre of the portion of spherical surface to be represented, all the other points are represented in their true azimuths,-the rectilinear distances from the centre of the drawing being a certain function of the corresponding true distances on the spherical surface. For simplicity we shall first apply this method to the projection or development of parallels and meridians when the pole is the centre. According to what has been said above, the meridians are now straight lines diverging from the pole, dividing the 360° into equal angles; and the parallels are represented by circles having the pole as centre, the radius of the parallel whose co-latitude is u being p, a certain function of u. The particular function selected determines the nature of the development. then p'qopq and 'r'-o'pr. That is to say, σ, may be regarded as the relative Fig. 7. scales, at co-latitude u, of the representation, applying to meridional measurements, to measurements perpendicular to the meridian. A small square situated in colatitude u, having one side in the direction of the meridian-the length of its side being i-is represented by a rectangle whose sides. are io and io'; its area consequently is too. If it were possible to make a perfect representation, then we should have σ-1, o'=1 throughout. This, however, is impossible. We may make σ=1 throughout by taking. p=u. This is known as the Equidistant Projection, a very simple and effective method of representation. Or we may make σ'= 1 throughout. This gives p= sin u, a perspective projection, namely, the Orthographic. Or we may require that areas be strictly represented in the development. This will be effected by making oo' 1, or pdp-sin udu, the integral of which is p= 2 sinu, which is the Equivalent Projection of Lambert, sometimes referred to as Lorgna's Projection. In this system there is misrepreOr sentation of form, but no misrepresentation of areas. we may require a projection in which all small parts are to be represented in their true forms. For instance, a small square on the spherical surface is to be represented as a small square in the development. This condition will be dp. du , the integral of attained by making σ=σ', or sin u which is, c being an arbitrary constant, p=ctanu. This, again, is a perspective projection, namely, the Stereographic. In this, though all small parts of the surface are represented in their correct shapes, yet, the scale varying from one part of the map to another, the whole is not a similar representation of the original. The scale = c secu, at any point, applies to all directions round that point. σ= These two last projections are, as it were, at the extremes of the scale; each, perfect in its own way, is in other respects very objectionable. We may avoid both extremes by the following considerations. Although we cannot make = 1 and σ'= 1, so as to have a perfect picture of the spherical surface, yet considering σ-1 and σ'-1 as the local errors of the representation, we may make (σ −1)2+ (o' - 1)2 a minimum over the whole surface to be represented. To effect this we must multiply this expression by the element of surface to which it applies, viz, sinududu, and then integrate from the centre to the (circular) limite of the map. Let ẞ be the spherical radius of the segment to be represented, then the total misrepresentation is to be. taken as The limiting radius of the map is R2C tan B. In this system, called by the Astronomer-Royal the "Projection by balance of errors," the total misrepresentation is an absolute minimum. Returning to the general case where p is any function of u, let us consider the local misrepresentation of direction. Take any indefinitely small line, length =i, making an angle a with the meridian in co-latitude u. tions on a meridiau and parallel are i cos a, i sin the map are represented by io cosa, io'sin a. be the angle in the map corresponding to a, a of representation = €, then (2-1) tan a tan e 1+ tan2 a case the plane of the drawing may be supposed to pass Fig. 8. c, of d, and of e from the diameter aa' are the radii of the successive circles representing the parallels. It is clear that, when the points of division are very close, the parallels will be very much crowded towards the outside of the map: so much so, that this projection is not much used. For an orthographic projection of the globe on a meridian plane, let qnrs (fig. 9) be the meriIts projec-dian, ns the axis of rotation, then gr a, which in is the projection of the equator. The If then a parallels will be represented by straight lines passing through the g points of equal division; these lines are, like the equator, perpendicular to ns. The meridians will in this case be ellipses described on ns as a common major axis, the distances Fig. 9. of c, of d, and of e from ns being the minor semiaxes. is a maximum when a={, and the (fig. 10) from the radius oa, equal to the latitude. Drop the Put = cot2, then e corresponding value of is € m perpendicular pP on oa, then P is the projection of the pole. For simplicity of explanation we have supposed this method of development so applied as to have the pole in the centre. There is, however, no necessity for this, and any point on the surface of the sphere may be taken as the centre. All that is necessary is to calculate by spherical trigonometry the azimuth and distance, with reference to the assumed centre, of all the points of intersection of meridians and parallels within the space which is to be represented in a plane. Then the azimuth is represented unaltered, is the pole of the meridian which passes through . This and any spherical distance u is represented by p. Thus we get all the points of intersection transferred to the repre-ence to i exactly as the equator was described with refermeridian is of course an ellipse, and is described with refersentation, and it remains merely to draw continuous lines through these points, which lines will be the meridians and parallels in the representation. Fig. 10. ence to P. Produce io to l, and make lo equal to half the The exaggeration in such systems, it is important to remember, whether of linear scale, area, or angle, is the same; then lo is the semifor a given distance from the centre, whatever be the azimuth; that is, the exaggeration is a function of the distance from the centre only. We shall now examine and exemplify some of the most important systems of projection and development, commencing with Perspective Projections. In perspective drawings of the sphere, the plane on which the representation is actually made may generally be any plane perpendicular to the line joining the centre of the sphere and the point of vision. If V be the point of vision, P any point on the spherical surface, then p, the point in which the straight line VP intersects the plane of the representation, is the projection of P. In the orthographic projection, the point of vision is at an infinite distance and the rays consequently parallel; in this |