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no component of rotation there, the pendulum wouta con- meridian. As the radius of curvature of an ellipse is tinue to move in one and the same plane. At intermediate variable, increasing from the extremity of the major axis státions the rate of rotation is easily calculated; and obser- to the extremity of the minor axis, so on the earth's surface vations confirm the calculations, and have made the earth’s a dogree of the meridian is found by geodetic measurement rotation actually visible.

to increasa from the equator to the poles. The poles of the earth are the points in which the axis The actual length of a degree of the meridian at the of rotation, or of figure, meet the surface ; and the equator equator is 362746-4 feet; at either pole it is 366479-8 is the circle in which the surface is intersected by a plane feet. The length of one degree of the equatorial circle is through the earth's centre, perpendicular to the axis of 365231•1 feet. rotation Every point of the equator is therefore equidis With regard to the figure of the earth as a whole, the tant from the poles.

polar radius is 3949.79 miles, and the radius of the equator To determine the position of a point in space three 3963:30 miles; the difference of these, called the ellipco-ordinates or measurements are necessary; they may be ticity, is şos of the mean radius. A spheroid with these three lines, or two lines and one angle, or two angles and semiaxes is equivalent in volume to a sphere having a radius one line. Thus, to define the precise position of a point on of 3958-79 miles. Without referring further here to the the earth's surface, we express it by latitude, longitude, and spheroidal figure, we shall now, having given the precise altitude; the first two are angular measures, the third a dimensions, regard the earth as a spbero whose radius is linear magnitude, namely the height above the surface of 3959 miles On snch a sphere one degree is 69.09 miles. the sea.

From the definitions given above it appears that the mdius The line in which the surface of the earth is intersected of the parallel which corresponds to all points whose latitude by a plane through the axis of rotation is called a meridian, is $ is 3959 cosd; and that one degree of this circle, and all meridians are evidently similar curves. A line i.e., one degree of longitude in the latitude ® is 69.09 cosø perpendicular to the surface at any point is called a vertical expressed in miles. line; it corresponds with the direction of gravity there; In the representation of the spherical earth (fig. 2) P being produced outwards, that is, away from the earth's is the pole, QQ the equator, EF. any two points on the centre it meets the heavens in the zenith; and produced surface, PEe, PF; the meridians of those downwards it intersects the axis of revolution ; it would points intersecting the equator in eandf of course pass through the earth’s centre were it a sphere; Join EF by a great circle; then in the as it is, it passes near the earth's centre.

spherical triangle PEF the angle at P The angle between the meridian planes of two stations is the difference of longitude of. E and as A and B is called the difference of longitude of A and F, PE is the co-latitude of E, and PF B, or the longitude of B with reference to A. In British the co-latitude of F, the latitudes being maps the longitudes of all places are expressed with reference E and fF respectively. The angle at to the Royal Observatory of Greenwichs

E, being that contained between the

Fig. 2. The latitude of any point is the angle made by the meridian there and a vertical plane passing through F, is vertical line there with the plane of the equator, or the the azimuth of F (measured in this case from the north), co-latitude is the angle between the vertical line and the while the angle at F is the azimuth of E. If, then, there axis of rotation. The surface of the earth being one of be given the latitudes and longitudes of two places, to find revolution, any intersecting plane parallel to the equator their distance apart

, and their relative bearings, it becomes cuts it in a circle. If we imagine the vortical lines drawn necessary to calculate & spherical triangle (PEF) in which at any two points, as P and Q, in such a circle it is evident two sides and the included angle are given, the calculation from the symmetry of the surface that these verticals make bringing out the third side, which is the required distance, the same angle with the equator; in other words, the with the adjacent azimuthal angles. 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 eartb 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 liue at Q star, 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 As the positions of points on the earth are defined with meridian is generally taken as the zero of azimuth. reference to the equator and a certain fixed meridian, so The plane touching the surface at Q is the visible horizon the positions of stars are defined by their angular distance there-a plane parallel to this through the centre of the earth from the celestiai equator, called in this case declination, being called the rational horizon. The altitude at Q of and by their right ascension, which corresponds to terrestrial a heavenly body, as a star, is the angle which the line longitude. Stars which are on the same meridian plane drawn from Q to the star makes with the plane of the extended to the heavens) have the same right ascension. horizon,—the zenith distance of the same star being the Right ascension is expressed in time from Of to 244. A angle between its direction and the vertical at Q.

sidereal clock, going truly, indicates 24" for every revoluBy a degree of the meridian is meant this : if E, F are tion of the earth : at every observatory, the sidereal clock points on the same meridian such that the directions of there shows, at each moment, the right ascension of the their verticals make with each other an angle of one degree stars which at that moment are on the meridian Thus a ninetieth part of a right angle—then the distance between the right ascension of the zenith is the sidereal time. E and F measured along the meridian is a degree of the In the left hand circle of the diagram (fig. 3) two

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concentric small circles are drawn such that the sum of But the traveller in unknown lands, who seeks to fix their radii is a right angle or 90°. Let the inner circle be astronomically his position, has no telegraph to count on that traced among the stars by the zenith of any given and his expectations for longitude depend chieft; on obserplace, say Q, then the outer circle encloses all those stars vations of the moon. In the Nautical Almanac are pub

lished 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 234° ; at midsummer it attains a maximum northern declination of 231°, and at midwinter a maximum southern declination of the same

amount. Hence it is inferred that the earth moves round Fig. 3.

the sun in a plane, completing one orbital revolution yearly, which are circumpolar at Q, that is, whose entire course the axis of the earth's diurnal rotation being inclined to is performed above that horizon; for clearly the zenith this plane at an angle of 664° Upon this angle of inclinadistance of none of these can exceed 90° at Q. Or if the tion depend the seasons, and in great measure the climates outer circle be that described by the zenith of Q, then the of the different portions of the earth's surface. inner circle encloses all those sturs which are circumpolar It is usual to draw on globes and in maps a circle or at Q. The second circle in the diagram shows the diürnal parallel at the distance of 231° from the equator on either paths of stars with reference to the horizon.

side ; of these circles the northern is called the Tropic of If we consider in the first circle the changes of distance Cancer, the southern is the Tropic of Capricorn. A circle between any one star and the zenith of @ as the latter drawn with a radius of 234° from the North Pole as centre traces out its path in the heavens, we see that the distance is the Arctic Circle ; a similar and equal circle round tho becomes alternately a maximum and a minimum everytwelve South Pole is the Antarctic Circle. hours, namely, when the meridian of Q passes through the When the sun is in the equator- which it crosses from star. This is called the star's culmination or meridian north to south in September, and from south to north in transit. It will be clear from an inspection of the figure March-it is in the horizon of either pole. When the sun that, if for instance the star culminate to the south of the has northern declination, the North Pole is in constant day. zenith, the star's declination plus its zenith distance at cul- light and the South Pole in darkness. When the sun has mination is equal to the latitude of the zenith, that is, of southern declination the North Pole on the contrary is in Q. A corresponding rule is easily made for a northern constant darkness while the South Pole is illuminated by transit. Thus the simplest manner of determining the sunshine. At midsummer in the northern hemisphere the latitude is to measure the zenith distance of a known star whole region within the Arctic Circle is in constant dayat its meridian transit.

light, and that within the Antarctic Circle is in darkness; The position of the zenith at any moment may be deter-át midwinter this state of things is exactly reversed. The mined by simultaneous observation of the zenith distance portion of the globe lying between the Tropic of Cancer of two known stars. For these distances clearly determine and the Arctic Circle is called the North Temperate Zone; a point in the heavens (two points rather, which however that between the Tropic of Capricorn and the Antarctic need not be confounded) whose declination and right ascen- Circle is the South Temperate Zone. In the former the sion can be computed by spherical trigonometry. Thus, at sun is always to the south of the zenith; in the latter it is the same time, are obtained both the time and the latitude, always to the north. For the success of this method, which is suitable for In the Torrid Zone, which lies between the Tropics, the travellers exploring an unknown country, it is desirable that sun, at any given place, passes the meridian to the north the stars should differ in azimuth by about a right angle. of the zenith for part of the year, and to the south for the

If the path of the zenith, that is, the latitude, be known, remainder. then clearly a single observation of the zenith distance of When the sun is to the north of the equator the days are a known star, which should be towards the east or west, longer than the nights in the northern hemisphere, while not towards the north or south, will fix the place or right in the southern hemisphere the nights are longer than the ascension of the zenith, that is, the sidereal time, at the days; when the sun has southern declination this condition moment of observation. Here the pole, the zenith, and is reversed. As the sun increases his north declination from the star are the angular points of a spherical triangle, of 0° to 23}", not only do the days increase in length in the which the three sides are known : the angle at the pole, northern hemisphere, but the rays of the sun—in the being computed, is the difference of right ascension of the Temperate and Arctic regions—impinge more perpendistar and the zenith. Thus the sidereal time is found. cularly on the surface; hence the warmth of summer.

The determination of the difference of longitude of the Even in summer the rays of the sun in the Arctic regions two stations AB on the earth's surface requires that the strike the surface very obliquely; this, combined with the true time be kept at each. All that is necessary is a com- protracted season of darkness, produces excessive cold. parison of these times at any instant. For instance, the Summer in the northern hemisphere is thus contemporane time at B may, by the transport of chronometers, be brought ous with winter in the southern; while winter in the to A, and thus the difference of the local times be ascer- northern heinisphere is simultaneous with summer in the tained, or the indications of the clock at A may be con- southern. ducted by electro-telegraphy to B. The difference of the The length of the day at any place at any season of the local times at A and B is the time a star takes to pass year is easily ascertained from the following considerations. from the meridian of the one to that of the other; and Let ns (fig. 4) be the axis of rotation, eq, the equator this is the difference of longitude which may be converted orthographically projected on a meridian plane, ab thu into angle at the rate of 360° to 24b.

parallel of the given place; draw the diameter fg making


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Fig. 4.

the angle nog equal to the sun's declination, which we Since is = 231° - 18° = 51°, we see from the diagram that suppose to be north, then the hemisphere graef is in the South Pole is at this time in total darkness, which exsunshine, while the hemisphere gbqsf

tends to all places within 51° of it. When the sun's is in darkness. As the earth rotates,

declination is go south, the North Pole is in the centre of

b & 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

9 light or total darkness. The actual period during which íf & be the latitude of the place and

either pole is in total darkness is about two and a half the sun's declination hk = sin o tand;

months. this in the parallel 'whose radius is

At the equator, the shortest twilight occurs at the 208$ corresponds to an angle whose

equinox, when it is 1b 12m; the longest when the sun is gine is tano tand. Call this anglen;

in the tropics, being 16:18". At London, in latitude 511', the time taken to rotate through it is ton; hence the twilight continues all night from May 22 to July 21; it is length of the daylight is 12h + ion, and the length of shortest about three weeks after the autumnal and three night 12-11

weeks before the vernal equinox, when its duration is lb Now n vanishes when either 6 or 8 is zero; that is, at 50m. At Washington the shortest twilight (being 16 332)

lb the equator the nights and days are equal in length occurs on the 6th of March and 7th October ; at Quebec throughoat the year; and again when the sun is in the the shortest is 1h 46m, falling on the 3d March and 10th equator, that is, at the equinox, the nights and days are October. equal in all latitudes. When the sun's declination is equal At page 205, fig. 19 is a perspective representation of to the co-latitude, n is a right angle, and the sun does not the earth-of more than a hemisphere, in fact-namely,

actually set; this can only happen at places within the the segment mgnafi in fig. 5. It exhibits all those polar circle. The longest day at Gibraltar is 14b 27m, at regions of the earth which at Greenwich apparent noon at Falmouth 16h 11", and in Shetland 186 14m; while in midsummer are in sunshine and twilight. It is very rar Iceland it is 20h on the south coast and 24h on the north. markable how Asia and America, but especially the former, At Washington the longest day is 14b 44m, and at Quebec just escape going into darkness. 15h 40m. All this, however, is on the supposition that day ends

Construction of Maps, with sunset; but the length of apparent day is increased by In the construction of maps, one has to consider how a atmospheric refraction and reflection. When the disk of portion of spherical surface, or a configuration traced on a the setting sun first seems to touch the horizon it 'is in sphere, can be represented on plane. If the area to be reality wholly below it and is only seen by refraction. represented bear a very small ratio to the whole surface of After the sun has wholly set at any given place his light the sphere, the matter is easy : thus, for instance, there is still continues to illuminate the upper portion of the no difficulty in making a map of a parish, for in such cases atmosphere there, so that, instead of ending abruptly, day- the curvature of the surface does not make itself evident. light gradually fades away until the sun is 18° below the If the district is larger and reaches the size of a county, as horizon.

Yorkshire for instance, then the curvature begins to be In a diagram (fig. 5) similar to the last draw mi parallel sensible, and one requires to consider how it is to be dealt to gf, and at a distance from it'equal

with. The sphere not being a developable surface cannot to the sine of 18°; then gbf being

be opened out into a plane like the cone or cylinder, con. the hemisphere unonlightened by tho

sequently in a plane representation of configurations on direct rays of the sun, gmif will re

a sphere it is impossible to retain the desired proportions present the twilight zone. A point

of lines or areas or equality of angles. But though one in the latitude of a describing the


cannot fulil all the requirements of the case, we may fulfil parallel ab loses sight of the sun at k,

some by sacrificing others ; that is to say, we may, for and is in twilight until it reaches the

instance, have in the representation exact similarity to all small circle mi, when the sun's zenith

very small portions of the original, but at the expense of distance is 108°. The duration of twi. Fig.-5.

the areas, which will be quite misrepresented. Or we may light corresponds then to the portion kl of ab, the angle retain equality of arcas if we give up the idea of similarity. rotated through being

It is therefore usual, excepting in special cases, to steer a sin - ?(hl: hb)- sin - (hk : hb);

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.

A globe gives a perfect representation of the surface of Here hk-sin otang; kl=sin 18° seo 8.

the earth; bút practically, the necessary limits to its size

make it impossible to represent in this manner the details At any given latitude the twilight is shortest when the of countries. A globe of the ordinary dimensions serves great circle passing through k and I passes also through the scarcely any other purpose than to convey a clear conception sun. Expressed algebraically, if – be the duration of the of the earth's surface as a whole, exhibiting the figure, shortest twilight in angular measure and 8the sun's extent, .position, and general. features of the continents and declination at the time, then

islands, with the intervening oceans and seas; and for this - sin 8 – sin o tan 90

purpose it is indeed absolutely essential and cannot be resin T = sec ø sin 9o.

placed by any kind of map. Suppose in the last diagram the sun to be at his greatest The construction of a map virtually resolves itself into aorthern declination, then ng

231°, gm

18°, and mq the drawing of two sets of lines, one set to represent = 48}'. Hence a place whose latitude is 481° N. has, meridians, the other to represent parallels. These being at midsummer, twilight lasting from sunset to midnight and drawn, the filling in of the outlines of countries presents continuing from midnight to sunrise, that is, for a few days no difficulty. The first and most natural idea that occurs there is no absolute darkness. A little further south this to one as to the manner of drawing the circles of latitude twilight is interrupted by a short period of dárkness and longitude is to draw them according to the laws of



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Fig. 7.


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perspective. But, as Lagrange has remarked, one may Let Ppq, Prs (fig. 7) be two contiguous meridians crossed by regard geographical maps from a more general point of view parallels

rp, sq, and Op'd', Or's the straight lines representing these

meridians. If the angle at P is dy, this also is the value of the as representations of the surface of the globe, for which angle at 0. Let the co-latitude purpose we have but to draw meridians

Pp-U, Pq=u+du; Op'- p, Od-p+dp, and parallels according to any given law; then any place we have to fix must take

the circular arcs p'r', q's representing the

P parallels pr, gs.

If the radius of the that position with reference to these lines

sphere be unity, that it has on the sphere with reference to

p'd-dp ; p'r' pdu, the circles of latitude and longitude. Let

Pg - du; or = sin udfo the law which connects latitude and longi

Put tude, $ and w, with the rectangular: co


'' ordinates & and y in the representation be such that dx = mdd + ndw, and

then p'Q - Opq and f'p - opr. That is to dy = m'do + n'dw. In fig. 6 let the lines

Fig. 6. say, 0, o' may be regarded as the relative

scales, at co-latitude X, of the represen. intersecting in the parallelogram PQRS be the repre- tation, o applying to meridional measurements o to measurements sentations of the meridians rp, sq and parallels rs, pq in- perpendicular to the meridian. A small square situated in cotersecting in the indefinitely small rectangle pars on the latitude u, having one side in the direction of the meridian-the surface of the sphere. The coordinates of P being and length of its side being :—is represented by a rectangle whose sides y, while those of p are $ and w the coordinates of the

are io and io'; its area consequently is 2*00'. other points will stand thus

If it were possible to make a perfect representation, then

we should have o=1, o'=1 throughout. This, however, I w + da

is impossible. We may make -l throughout by taking $ + dd

This is known as the Equidistant Projection, a very $ + do

Q . * +ndo

simple and effective method of representation.
2c + mdø y+m'do

Or we may make o'=1 throughout. This gives p=sin u,. * + mdp + ndo y + m'do + 1ddw.

a perspective projection, namely, the Orthographic. Or Thus we easily see that PR = (m2 + m2)}d; and PQ we may require that areas be strictly represented in the do- (n? + m2)!dw; also the area of the parallelogram PQRS velopment. This will be effected by making go'=1, or is equal to (m's – mn')dødw. If 90° + v are the angles pdp = sin udu, the integral of which is p= 2 sin fu, which of the parallelogram, then

is the Equivalent Projection of Lambert, sometimes referred mn + m'n'

to as Lorgna's Projection. In this system there is misrepretang= m'n -mn' sentation of form, but no misrepresentation of areas.


we may require a projection in which all small parts are to If the lines of latitude and of longitude_intersect at right be represented in their true forms. For instance, a small angles, then mn + m'n' = 0. Since the length of pris=do, square on the spherical surface is to be represented as a its representation PR is too great in the proportion of small square in the development. This condition will be (m8 + m2)} : 1; and pq being in length cos pdw, its representation PQ is too great in the ratio of (né + n'a)! : coss. attained by making o=o', or , the integral of Hence the condition that the rectangle PQRS is similar which is, c being an arbitrary constant, p=ctanju. This to the rectangle pors is (m® + m2) cost=n? + n'?, together again, is a perspective projection, namely, the Stereo with mn + m'n' =0; or, which is the same, the condition of graphic. In this, though all small parts of the surface are similaritv is expressed by

represented in their correct shapes, yet, the scale varying -n=m cos $; n-m' cos .

from one part of the map to another, the whole is not a Since the area of the rectangle pars is cos Adødus, the similar representation of the original. The scale or exaggeration of area in the representation will be expressed 2c sec°fu, at any point, applies to all directions round that by m'n -mn': cos 6. Thus when the nature of the lines point. representing the circles of latitude and longitude is defined

These two last projections are, as it were, at the extremes we can at once calculate the error or exaggeration of scale of the scale ; each, perfect in its own way, is in other reat any part of the map, whether measured in the direction spects very objectionable. We may avoid both extremes of a meridian or of a parallel ; and also the misrepresentar =1 and oʻ= 1, so as to have a perfect picture of the

by the following considerations. Although we cannot make tion of angles. The lines representing in a map the meridians and par- local errors of the representation, we may make (p-1)+

spherical surface, yet considering 0-1 and o'-1 as the allels on the sphere are constructed either of the principles 16– 1) a minimum over the whole surface to be repre of true perspective or by artificial systems of developments. sented. To effect this we must multiply this expression by The perspective drawings are indeed included as a particular case of development in which, with reference to a certain the element of surface to which it applies, viz, sin ududy. point selected as the centre of the portion of spherical sur- and then integrate from the centre to the (circular) limits face to be represented, all the other points are represented of the map. Let ß be the spherical radius of the segment in their true azimuths,-the rectilinear distances from the to be represented, then the total misrepresentation is to be

taken as centre of the drawing being a certain function responding true distances on the spherical surface. For simplicity we shall first apply this method to the projection

du or development of parallels and meridians when the pole is the centre. According to what has been said above, the which is to be made a minimum. Putting p=u+y, and meridians are now straight lines diverging from the pole, giviug to y only a variation subject to the condition 8y = 0 dividing the 360° into equal angles ; and the parallels are

when u=

= 0, the equations of solution-using the ordinary represented by circles having the pole as centre, the radius notation of the calculus of variations-are of the parallel whose co-latitude is u being p, a certain func

Ntion of u.

=0; Pg=0, The particular function selected determines the nature of the development.

Pis being the value of 2p sin u when u-B. This gives

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dp... du


sin u

the cor

S." (0-1)+(23-1)}sinudus





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B 2

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+ sin u cos u
-y-U - sin u,

case the plane of the drawing may be supposed to pass

through the centre of the sphere. Let the circle (fig. 8) 'dy -0.

represent the plane of the equator on which we propose to du B

make an orthographic representation of meridians and This method of development is due to Sir George Airy, parallels. The centre of this circle is whose original paper—the investigation is different in form clearly the projection of the pole, and from the above-—will be found in the Philosophiasl the parallels are projected into circles Magazine for December 1861. The solution of the diñer. having the pole for a common centre. ential equation leads to this result

The diameters aa', bb' being at right

angles, let the semicircle bab' be
p=2 cot. loge sec
loge sec + Ctan

divided into the required number of

equal .parts; the diameters drawn
C-2 cotloge sec
through these points are the projec-

Fig. 8.

tions of meridians. The distances of The limiting radius of the map is R=2C tan }ß. In this

c, of d, and of e from the diameter aa' are the radii of the system, called by the Astronomer-Royal the “Projection by successive circles representing the parallels. It is clear that, balance of errors," the total misrepresentation is an absolute when the points of division are very close, the parallels ininimum.

will be very much crowded towards the outside of the map: Returning to the general case where p is any function


so much so, that this projection is not much used. of u, let us consider the local misrepresentation of direction. Take any indefinitely small line, length = i, making plane, let qnrs (fig. 9) be the meri

For an orthographic projection of the globe on a meridian an angle a with the meridian in co-latitude u. tions on a meridiau and parallel are i cos a, i sin a, which in is the projection of the equator. The

Its projec- dian, ns the axis of rotation, then gr the map are represented by io cosa, io'sin a. If then a' parallels will be represented by be the angle in the map corresponding to a,

straight lines passing through the q tan auton a

points of equal division; these lines

are, like the equator, perpendicular Put

to ns.

Tłe meridians will in this pdu

case be ellipses described on ns as sin udo

a common major axis, the distances Fig. 9. and the error a' - a of representation = €, then

of c, of d, and of e from ns being the minor semiaxes. (3-1) tan a

Let us next construct an orthographic projection of the 1+tan'a

sphere on the horizon of any place. Set off the angle aop Put £= cot* %, thun e is a maximum when a = $, and the (fig. 10) from the radius oa, equal to the latitude. Drop the corresponding value of e is

perpendicular pP on oa, then P is the projection of the pole. On ao produced take ob=pP, then ob is the minor semi

axis of the ellipse representing the method of development so applied as to have the pole in which the meridians meet this ellipFor simplicity of explanation we have supposed this equator, its major axis being or at

right angles to ao. The points in the centre. There is, however, no necessity for this, and tic equator are determined by lines 9 any point on the surface of the sphere may be taken as the drawn parallel to aob through the centre. All that is necessary is to calculate by spherical points of equal subdivision cdefgh. trigonometry the azimuth and distance, with reference to Take two points, as d and g, which the assumed centre, of all the points of intersection of meri- are 90° apart, and let ik be their dians and parallels within the space which is to be repre projections on the equator; then i

Fig. 10. sented in a plane. Then the azimuth is represented unaltered, is the pole of the meridian which passes through k. 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 refer

meridian is of course an ellipse, and is described with refersentation, and it remains merely to draw continuous lines

ence to P. Produce io to l, and make lo equal to half the through these points, which lines will be the meridians and

shortest chord that parallels in the representation. The exaggeration in such systems, it is important to re- ;; then lo is the semi

can be drawn through member, whether of linear scale, area, or angle, is the samo

axis of the elliptic for a given distance from the centre, whatever be the azimuth; that is, the exaggeration is a function of the dis- meridian, and the tance from the centre only.

major axis is the diaWe shall now examine and exemplify some of the most

meter perpendicular

to iol. important systems of projection and development, commen

For the parallels : cing with

let it be required to Perspective Projections.

describe the parallel In perspective drawings of the sphere, the plane on which whose co-latitude is the representation is actually made may generally be any u; take pm= pn = U, plane perpendicular to the line joining the centre of the ap:let m'n' be the sphere and the point of vision. If V be the point of vision, projections of m and P any point on the spherical surface, then p, the point in non opa; then m'n' is Fia. 11.-Orthographic Projection. which the straight line VP intersects the plane of the the minor axis of the ellipse representing the parallel. Its representation, is the projection of P.

centre is of course midway between m' and n', and the In the orthographic projection, the point of vision is at an greater axis is equal to mn. Thus the construction is obinfinite distance and the rays consequently parallel ; in this I vious. When pm is less than pa, the whole of the ellipse

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