Chapter 5

If φ0be taken as the latitude of the centre parallel, and co-ordinates be measured from the intersection of this parallel with the central meridian, then, if ρ be the radius of the parallel of latitude φ, we have ρ = cot φ0+ φ0− φ. Also, if S be a point on this parallel whose co-ordinates are x, y, so that VS = ρ, and θ be the angle VS makes with the central meridian, then ρθ = ω cos φ; and x = ρ sin θ, y = cot φ0− ρ cos θ.

The projection has the property of equal areas, since each small element bounded by two infinitely close parallels is equal in length and width to the corresponding element on the sphere or spheroid. Also all the meridians cross the chosen parallel (but no other) at right angles, since in the immediate neighbourhood of that parallel the projection is identical with the simple conical projection. Where an equal-area projection is required for a country having no great extent in longitude, such as France, Scotland or Madagascar, this projection is a good one to select.

Sinusoidal Equal-area Projection.—This projection, which issometimes known as Sanson’s, and is also sometimes incorrectly called Flamsteed’s, is a particular case of Bonne’s in which the selected parallel is the equator. The equator is a straight line at right angles to the central meridian which is also a straight line. Along the central meridian the latitudes are marked off at the true rectified distances, and from points so found the parallels are drawn as straight lines parallel to the equator, and therefore at right angles to the central meridian. True rectified lengths are marked along the parallels and through corresponding points the meridians are drawn. If the earth is treated as a sphere the meridians are clearly sine curves, and for this reason d’Avezac has given the projection the name sinusoidal. But it is equally easy to plot the spheroidal lengths. It is a very suitable projection for an equal-area map of Africa.

Werner’s Projection.—This is another limiting case of Bonne’s equal-area projection in which the selected parallel is the pole. The parallels on paper then become incomplete circular arcs of which the pole is the centre. The central meridian is still a straight line which is cut by the parallels at true distances. The projection (after Johann Werner, 1468-1528), though interesting, is practically useless.

Polyconic Projections.

These pseudo-conical projections are valuable not so much for their intrinsic merits as for the fact that they lend themselves to tabulation. There are two forms, thesimpleorequidistant polyconic, and therectangular polyconic.

The Simple Polyconic.—If a cone touches the sphere or spheroid along a parallel of latitude φ and is then unrolled, the parallel will on paper have a radius of ν cot φ, where ν is the normal terminated by the minor axis. If we imagine a series of cones, each of which touches one of a selected series of parallels, the apex of each cone will lie on the prolonged axis of the spheroid; the generators of each cone lie in meridian planes, and if each cone is unrolled and the generators in any one plane are superposed to form a straight central meridian, we obtain a projection in which the central meridian is a straight line and the parallels are circular arcs each of which has a different centre which lies on the prolongation of the central meridian, the radius of any parallel being ν cot φ.

So far the construction is the same for both forms of polyconic. In thesimple polyconicthe meridians are obtained by measuring outwards from the central meridian along each parallel the true lengths of the degrees of longitude. Through corresponding points so found the meridian curves are drawn. The resulting projection is accurate near the central meridian, but as this is departed from the parallels increasingly separate from each other, and the parallels and meridians (except along the equator) intersect at angles which increasingly differ from a right angle. The real merit of the projection is that each particular parallel has for every map the same absolute radius, and it is thus easy to construct tables which shall be of universal use. This is especially valuable for the projection of single sheets on comparatively large scales. A sheet of a degree square on a scale of 1 : 250,000 projected in this manner differs inappreciably from the same sheet projected on a better system,e.g.an orthomorphic conical projection or the conical with rectified meridians and two standard parallels; there is thus the advantage that the simple polyconic when used for single sheets and large scales is a sufficiently close approximation to the better forms of conical projection. The simple polyconic is used by the topographical section of the general staff, by the United States coast and geodetic survey and by the topographical division of the U.S. geological survey. Useful tables, based on Clarke’s spheroid of 1866, have been published by the war office and by the U.S. coast and geodetic survey.

Rectangular Polyconic.—In this the central meridian and the parallels are drawn as in the simple polyconic, but the meridians are curves which cut the parallels at right angles.

In this case, let P (fig. 20) be the north pole, CPU the central meridian, U, U′ points in that meridian whose co-latitudes are z and z+dz, so that UU′ = dz. Make PU = z, UC = tan z, U′C′ = tan (z + dz); and with CC′ as centres describe the arcs UQ, U′Q′, which represent the parallels of co-latitude z and z + dz. Let PQQ′ be part of a meridian curve cutting the parallels at right angles. Join CQ, C′Q′; these being perpendicular to the circles will be tangents to the curve. Let UCQ = 2α, UC′Q′ = 2(α + dα), then the small angle CQC′, or the angle between the tangents at QQ′, will = 2dα. NowCC′ = C′U′ − CU − UU′ = tan (z + dz) − tan z − dz = tan2z dz.The tangents CQ, C′Q′ will intersect at q, and in the triangle CC′q the perpendicular from C on C′q is (omitting small quantities of the second order) equal to either side of the equationtan2z dz sin 2α = −2 tan zd α.−tan z dz = 2 dα / sin 2α,which is the differential equation of the meridian: the integral is tan α = ω cos z, where ω, a constant, determines a particular meridian curve. The distance of Q from the central meridian, tan z sin 2α, is equal to2 tan z tan α=2ω sin z.1 + tan2α1 + ω2cos2αFig. 21.At the equator this becomes simply 2ω. Let any equatorial point whose actual longitude is 2ω be represented by a point on the developed equator at the distance 2ω from the central meridian, then we have the following very simple construction (due to O’Farrell of the ordnance survey). Let P (fig. 21) be the pole, U any point in the central meridian, QUQ′ the represented parallel whose radius CU = tan z. Draw SUS′ perpendicular to the meridian through U; then to determine the point Q, whose longitude is, say, 3°, lay off US equal to half the true length of the arc of parallel on the sphere,i.e.1° 30′ to radius sin z, and with the centre S and radius SU describe a circular arc, which will intersect the parallel in the required point Q. For if we suppose 2ω to be the longitude of the required point Q, US is by construction = ω sin z, and the angle subtended by SU at C istan−1(ω sin z)= tan−1(ω cos z) = α,tan zand therefore UCQ = 2α as it should be. The advantages of this method are that with a remarkably simple and convenient mode of construction we have a map in which the parallels and meridians intersect at right angles.

CC′ = C′U′ − CU − UU′ = tan (z + dz) − tan z − dz = tan2z dz.

The tangents CQ, C′Q′ will intersect at q, and in the triangle CC′q the perpendicular from C on C′q is (omitting small quantities of the second order) equal to either side of the equation

tan2z dz sin 2α = −2 tan zd α.−tan z dz = 2 dα / sin 2α,

which is the differential equation of the meridian: the integral is tan α = ω cos z, where ω, a constant, determines a particular meridian curve. The distance of Q from the central meridian, tan z sin 2α, is equal to

At the equator this becomes simply 2ω. Let any equatorial point whose actual longitude is 2ω be represented by a point on the developed equator at the distance 2ω from the central meridian, then we have the following very simple construction (due to O’Farrell of the ordnance survey). Let P (fig. 21) be the pole, U any point in the central meridian, QUQ′ the represented parallel whose radius CU = tan z. Draw SUS′ perpendicular to the meridian through U; then to determine the point Q, whose longitude is, say, 3°, lay off US equal to half the true length of the arc of parallel on the sphere,i.e.1° 30′ to radius sin z, and with the centre S and radius SU describe a circular arc, which will intersect the parallel in the required point Q. For if we suppose 2ω to be the longitude of the required point Q, US is by construction = ω sin z, and the angle subtended by SU at C is

and therefore UCQ = 2α as it should be. The advantages of this method are that with a remarkably simple and convenient mode of construction we have a map in which the parallels and meridians intersect at right angles.

Fig. 22 is a representation of this system of the continents of Europe and Africa, for which it is well suited. For Asia this system would not do, as in the northern latitudes, say along the parallel of 70°, the representation is much cramped.

With regard to the distortion in the map of Africa as thus constructed, consider a small square in latitude 40° and in 40° longitude east or west of the central meridian, the square being so placed as to be transformed into a rectangle. The sides, originally unity, became 0.95 and 1.13, and the area 1.08, the diagonals intersecting at 90° ± 9° 56′. In Clarke’s perspective projection asquare of unit side occupying the same position, when transformed to a rectangle, has its sides 1.02 and 1.15, its area 1.17, and its diagonals intersect at 90° ± 7° 6′. The latter projection is therefore the best in point of “similarity,” but the former represents areas best. This applies, however, only to a particular part of the map; along the equator towards 30° or 40° longitude, the polyconic is certainly inferior, while along the meridian it is better than the perspective—except, of course, near the centre. Upon the whole the more even distribution of distortion gives the advantage to the perspective system. For single sheets on large scales there is nothing to choose between this projection and the simple polyconic. Both are sensibly perfect representations. The rectangular polyconic is occasionally used by the topographical section of the general staff.

Zenithal Projections.

Some point on the earth is selected as the central point of the map; great circles radiating from this point are represented by straight lines which are inclined at their true angles at the point of intersection. Distances along the radiating lines vary according to any law outwards from the centre. It follows (on the spherical assumption), that circles of which the selected point is the centre are also circles on the projection. It is obvious that all perspective projections are zenithal.

Equidistant Zenithal Projection.—In this projection, which is commonly called the “equidistant projection,” any point on the sphere being taken as the centre of the map, great circles through this point are represented by straight lines of the true rectified lengths, and intersect each other at the true angles.

In the general case—if z1is the co-latitude of the centre of the map, z the co-latitude of any other point, α the difference of longitude of the two points, A the azimuth of the line joining them, and c the spherical length of the line joining them, then the position of the intersection of any meridian with any parallel is given (on the spherical assumption) by the solution of a simple spherical triangle.Thus—let tan θ = tan z cos α, then cos c = cos z sec θ cos (z − θ), and sin A = sin z sin α cosec c.

In the general case—

if z1is the co-latitude of the centre of the map, z the co-latitude of any other point, α the difference of longitude of the two points, A the azimuth of the line joining them, and c the spherical length of the line joining them, then the position of the intersection of any meridian with any parallel is given (on the spherical assumption) by the solution of a simple spherical triangle.

Thus—

let tan θ = tan z cos α, then cos c = cos z sec θ cos (z − θ), and sin A = sin z sin α cosec c.

The most useful case is that in which the central point is the pole; the meridians are straight lines inclined to each other at the true angular differences of longitude, and the parallels are equidistant circles with the pole as centre. This is the best projection to use for maps exhibiting the progress of polar discovery, and is called thepolar equidistant projection. The errors are smaller than might be supposed. There are no scale errors along the meridians, and along the parallels the scale error is (z / sin x) − 1, where z is the co-latitude of the parallel. On a parallel 10° distant from the pole the error of scale is only 0.5%.

General Theory of Zenithal Projections.—For the sake of simplicity it will be at first assumed that the pole is the centre of the map, and that the earth is a sphere. 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 z being ρ, a certain function of z. The particular function selected determines the nature of the projection.Fig. 23.Let Ppq, Prs (fig. 23) be two contiguous meridians crossed by parallels rp, sq, and Op′q′, Or′s′ the straight lines representing these meridians. If the angle at P is dμ, this also is the value of the angle at O. Let the co-latitudePp = z, Pq = z + dz; Op′ = ρ, Oq′ = ρ + dρ,the circular arcs p′r′, q′s′ representing the parallels pr, qs. If the radius of the sphere be unity,p′q′ = dρ; p′r′ = ρ dμ,pq = dz; pr = sin z dμ.Putσ = dρ / dz; σ′ = ρ / sin z,then p′q′ = σpq and p′r′ = σ′pr. That is to say, σ, σ′ may be regarded as the relative scales, at co-latitude z, of the representation, σ applying to meridional measurements, σ′ to measurements perpendicular to the meridian. A small square situated in co-latitude z, having one side in the direction of the meridian—the length of its side being i—is represented by a rectangle whose sides are iσ and iσ′; its area consequently is i2σσ′.

Let Ppq, Prs (fig. 23) be two contiguous meridians crossed by parallels rp, sq, and Op′q′, Or′s′ the straight lines representing these meridians. If the angle at P is dμ, this also is the value of the angle at O. Let the co-latitude

Pp = z, Pq = z + dz; Op′ = ρ, Oq′ = ρ + dρ,

the circular arcs p′r′, q′s′ representing the parallels pr, qs. If the radius of the sphere be unity,

p′q′ = dρ; p′r′ = ρ dμ,pq = dz; pr = sin z dμ.

Put

σ = dρ / dz; σ′ = ρ / sin z,

then p′q′ = σpq and p′r′ = σ′pr. That is to say, σ, σ′ may be regarded as the relative scales, at co-latitude z, of the representation, σ applying to meridional measurements, σ′ to measurements perpendicular to the meridian. A small square situated in co-latitude z, having one side in the direction of the meridian—the length of its side being i—is represented by a rectangle whose sides are iσ and iσ′; its area consequently is i2σσ′.

If it were possible to make a perfect representation, then we should have σ = 1, σ′ = 1 throughout. This, however, is impossible. We may make σ = 1 throughout by taking ρ = z. This is theEquidistant Projectionjust described, a very simple and effective method of representation.

Or we may make σ′= 1 throughout. This gives ρ = sin z, a perspective projection, namely, theOrthographic.

Or we may require that areas be strictly represented in the development. This will be effected by making σσ′ = 1, or ρ dρ = sin z dz, the integral of which is ρ = 2 sin1⁄2z, which is theZenithal Equal-area Projectionof Lambert, sometimes, though wrongly referred to asLorgna’s Projectionafter Antonio Lorgna (b. 1736). In this system there is misrepresentation of form, but no misrepresentation of areas.

Or we may require a projection in which all small parts are to be represented in their true formsi.e.an orthomorphic projection. For instance, a small square on the spherical surface is to be represented as a small square in the development. This condition will be attained by making σ = σ′, or dρ/ρ = dz/sin z, the integral of which is, c being an arbitrary constant, ρ = c tan1⁄2z. This, again, is a perspective projection, namely, theStereographic. 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, thewholeis not a similar representation of the original. The scale, σ =1⁄2c sec21⁄2z, at any point, applies to all directions round that point.

These two last projections are, as it were, at the extremes of the scale; each, perfect in its own way, is in other respects objectionable. We may avoid both extremes by the following considerations. Although we cannot make σ = 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+ (σ′ − 1)2a 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. sin zd zd μ, and then integrate from the centre to the (circular) limits of the map. Let β be the spherical radius of the segment to be represented, then the total misrepresentation is to be taken as∫β0{ (dρ− 1)2+(ρ− 1)2}sin z dz,dzsin zwhich is to be made a minimum. Putting ρ = z + y, and giving to y only a variation subject to the condition δy = 0 when z = 0, the equations of solution—using the ordinary notation of the calculus of variations—areN −d(P)= 0; Pβ = 0,dzPβ being the value of 2p sin z when z = β. This givessin2zd2y+ sin z cos zdy− y = z − sin z(dy)β = 0.dz2dzdzThis method of development is due to Sir George Airy, whose original paper—the investigation is different in form from the above, which is due to Colonel Clarke—will be found in thePhilosophical Magazinefor 1861. The solution of the differential equation leads to this result—ρ = 2 cot1⁄2z logesec1⁄2z + C tan1⁄2z,C = 2 cot21⁄2β logesec1⁄2β.The limiting radius of the map is R = 2C tan1⁄2β. In this system, called by Sir George AiryProjection by balance of errors, the total misrepresentation is an absolute minimum. For short it may be calledAiry’s Projection.Returning to the general case where ρ is any function of z, let us consider the local misrepresentation of direction. Take any indefinitely small line, length = i, making an angle α with the meridian in co-latitude z. Its projections on a meridian and parallel are i cos α, i sin α, which in the map are represented by iσ cos α, iσ′ sin α. If then α′ be the angle in the map corresponding to α,tan α′ = (σ′ / σ) tan α.Putσ′ / σ = ρ dz / sin z dρ = Σ,and the error α′ − α of representation = ε, thentan ε =(Σ − 1) tan α.1 + Σ tan2αPut Σ = cot2ζ, then ε is a maximum when α = ζ, and the corresponding value of ε isε =1⁄2π − 2ζ.

which is to be made a minimum. Putting ρ = z + y, and giving to y only a variation subject to the condition δy = 0 when z = 0, the equations of solution—using the ordinary notation of the calculus of variations—are

Pβ being the value of 2p sin z when z = β. This gives

This method of development is due to Sir George Airy, whose original paper—the investigation is different in form from the above, which is due to Colonel Clarke—will be found in thePhilosophical Magazinefor 1861. The solution of the differential equation leads to this result—

The limiting radius of the map is R = 2C tan1⁄2β. In this system, called by Sir George AiryProjection by balance of errors, the total misrepresentation is an absolute minimum. For short it may be calledAiry’s Projection.

Returning to the general case where ρ is any function of z, let us consider the local misrepresentation of direction. Take any indefinitely small line, length = i, making an angle α with the meridian in co-latitude z. Its projections on a meridian and parallel are i cos α, i sin α, which in the map are represented by iσ cos α, iσ′ sin α. If then α′ be the angle in the map corresponding to α,

tan α′ = (σ′ / σ) tan α.

Put

σ′ / σ = ρ dz / sin z dρ = Σ,

and the error α′ − α of representation = ε, then

Put Σ = cot2ζ, then ε is a maximum when α = ζ, and the corresponding value of ε is

ε =1⁄2π − 2ζ.

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 thesurface 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, and any spherical distance z is represented by ρ. Thus we get all the points of intersection transferred to the representation, and it remains merely to draw continuous lines through these points, which lines will be the meridians and parallels in the representation.

Thus treating the earth as a sphere and applying theZenithal Equal-area Projectionto the case of Africa, the central point selected being on the equator, we have, if θ be the spherical distance of any point from the centre, φ, α the latitude and longitude (with reference to the centre), of this point, cos θ = cos φ cos α. If A is the azimuth of this point at the centre, tan A = sin α cot φ. On paper a line from the centre is drawn at an azimuth A, and the distance θ is represented by 2 sin1⁄2θ. This makes a very good projection for a single-sheet equal-area map of Africa. The exaggeration in such systems, it is important to remember, whether of linear scale, area, or angle, is the same for a given distance from the centre, whatever be the azimuth; that is, the exaggeration is a function of the distance from the centre only.

General Theory of Conical Projections.

Meridians are represented by straight lines drawn through a point, and a difference of longitude ω is represented by an angle hω. The parallels of latitude are circular arcs, all having as centre the point of divergence of the meridian lines. It is clear that perspective and zenithal projections are particular groups of conical projections.

Let z be the co-latitude of a parallel, and ρ, a function of z, the radius of the circle representing this parallel. Consider the infinitely small space on the sphere contained by two consecutive meridians, the difference of whose longitude is dμ, and two consecutive parallels whose co-latitudes are z and z + dz. The sides of this rectangle are pq = dz, pr = sin z dμ; in the projection p′q′r′s′ these become p′q′ = dρ, and p′r′ = ρhdμ.The scales of the projection as compared with the sphere are p′q′/pq = dρ/dz = the scale of meridian measurements = σ, say, and p′r′/pr = ρhdμ/sin zdμ = ρh/sin z = scale of measurements perpendicular to the meridian = σ′, say.Now we may make σ = 1 throughout, then ρ = z + const. This gives either the group ofconical projections with rectified meridians, or as a particular case theequidistant zenithal.We may make σ = σ′ throughout, which is the same as requiring that at any point the scale shall be the same in all directions. This gives a group oforthomorphic projections.In this case dρ/dz = ρh/sin z, or dρ/ρ = h dz/sin z.Integrating,ρ = k(tan1⁄2z)h,(i.)where k is a constant.Now h is at our disposal and we may give it such a value that two selected parallels are of the correct lengths. Let z1, z2be the co-latitudes of these parallels, then it is easy to show thath =log sin z1− log sin z2.log tan1⁄2z1− log tan1⁄2z2(ii.)This projection, given by equations (i.) and (ii.), is Lambert’s orthomorphic projection—commonly called Gauss’s projection; its descriptive name is theorthomorphic conical projection with two standard parallels.The constant k in (i.) defines the scale and may be used to render the scale errors along the selected parallels not nil but the same; and some other parallel,e.g.the central parallel may then be made errorless.The value h =1⁄3, as suggested by Sir John Herschel, is admirably suited for a map of the world. The representation is fan-shaped, with remarkably little distortion (fig. 24).If any parallel of co-latitude z is true to scale hk(tan1⁄2z1)h= sin z, if this parallel is the equator, so that z1= 90°, kh = 1, then equation (i.) becomes ρ = (tan1⁄2z)h/h, and the radius of the equator = 1/h. The distance r of any parallel from the equator is 1/h − (tan1⁄2z)h/h = (1/h){1 − (tan1⁄2z)h}.If, instead of taking the radius of the earth as unity we call it a, r = (a/h){1 − (tan1⁄2z)h}. When h is very small, the angles between the meridian lines in the representation are very small; and proceeding to the limit, when h is zero the meridians are parallel—that is, the vertex of the cone has removed to infinity. And at the limit when h is zero we have r = a logecot1⁄2z, which is the characteristic equation of Mercator’s projection.

The scales of the projection as compared with the sphere are p′q′/pq = dρ/dz = the scale of meridian measurements = σ, say, and p′r′/pr = ρhdμ/sin zdμ = ρh/sin z = scale of measurements perpendicular to the meridian = σ′, say.

Now we may make σ = 1 throughout, then ρ = z + const. This gives either the group ofconical projections with rectified meridians, or as a particular case theequidistant zenithal.

We may make σ = σ′ throughout, which is the same as requiring that at any point the scale shall be the same in all directions. This gives a group oforthomorphic projections.

In this case dρ/dz = ρh/sin z, or dρ/ρ = h dz/sin z.

Integrating,

ρ = k(tan1⁄2z)h,

(i.)

where k is a constant.

Now h is at our disposal and we may give it such a value that two selected parallels are of the correct lengths. Let z1, z2be the co-latitudes of these parallels, then it is easy to show that

(ii.)

This projection, given by equations (i.) and (ii.), is Lambert’s orthomorphic projection—commonly called Gauss’s projection; its descriptive name is theorthomorphic conical projection with two standard parallels.

The constant k in (i.) defines the scale and may be used to render the scale errors along the selected parallels not nil but the same; and some other parallel,e.g.the central parallel may then be made errorless.

The value h =1⁄3, as suggested by Sir John Herschel, is admirably suited for a map of the world. The representation is fan-shaped, with remarkably little distortion (fig. 24).

If any parallel of co-latitude z is true to scale hk(tan1⁄2z1)h= sin z, if this parallel is the equator, so that z1= 90°, kh = 1, then equation (i.) becomes ρ = (tan1⁄2z)h/h, and the radius of the equator = 1/h. The distance r of any parallel from the equator is 1/h − (tan1⁄2z)h/h = (1/h){1 − (tan1⁄2z)h}.

If, instead of taking the radius of the earth as unity we call it a, r = (a/h){1 − (tan1⁄2z)h}. When h is very small, the angles between the meridian lines in the representation are very small; and proceeding to the limit, when h is zero the meridians are parallel—that is, the vertex of the cone has removed to infinity. And at the limit when h is zero we have r = a logecot1⁄2z, which is the characteristic equation of Mercator’s projection.

Mercator’s Projection.—From the manner in which we have arrived at this projection it is clear that it retains the characteristic property of orthomorphic projections—namely, similarity of representation of small parts of the surface. In Mercator’s chart the equator is represented by a straight line, which is crossed at right angles by a system of parallel and equidistant straight lines representing the meridians. The parallels are straight lines parallel to the equator, and the distance of the parallel of latitude φ from the equator is, as we have seen above, r = a logetan (45° +1⁄2φ). In the vicinity of the equator, or indeed within 30° of latitude of the equator, the representation is very accurate, but as we proceed northwards or southwards the exaggeration of area becomes larger, and eventually excessive—the poles being at infinity. This distance of the parallels may be expressed in the form r = a (sin φ +1⁄3sin3φ +1⁄5sin5φ + ...), showing that near the equator r is nearly proportional to the latitude. As a consequence of the similar representation of small parts, a curve drawn on the sphere cutting all meridians at the same angle—the loxodromic curve—is projected into a straight line, and it is this property which renders Mercator’s chart so valuable to seamen. For instance: join by a straight line on the chart Land’s End and Bermuda, and measure the angle of intersection of this line with the meridian. We get thus the bearing which a ship has to retain during its course between these ports. This is not great-circle sailing, and the ship so navigated does not take the shortest path. The projection of a great circle (being neither a meridian nor the equator) is a curve which cannot be represented by a simple algebraic equation.

If the true spheroidal shape of the earth is considered, the semiaxes being a and b, putting e = √(a2− b2) / a, and using common logarithms, the distance of any parallel from the equator can be shown to be(a / M) {log tan (45° +1⁄2φ) − e2sin φ −1⁄3e4sin3φ ...}where M, the modulus of common logarithms, = 0.434294. Of course Mercator’s projection was not originally arrived at in the manner above described; the description has been given to show that Mercator’s projection is a particular case of the conical orthomorphic group. The introduction of the projection is due to the fact that for navigation it is very desirable to possess charts which shall give correct local outlines (i.e.in modern phraseology shall be orthomorphic) and shall at the same time show as a straight line any line which cuts the meridians at a constant angle. The latter condition clearly necessitates parallel meridians, and the former a continuous increase of scale as the equator is departed from,i.e.the scale at any point must be equal to the scale at the equator × sec. latitude. In early days the calculations were made by assuming that for a small increase of latitude, say 1′, the scale was constant, then summing up the small lengths so obtained. Nowadays (for simplicity the earth will be taken as a sphere) we should say that a small length of meridian adφ is represented in this projection by a sec φ dφ, and the length of the meridian in the projection between the equator and latitude φ,√φ0a sec φ dφ = a logetan (45° +1⁄2φ),which is the direct way of arriving at the law of the construction of this very important projection.Mercator’s projection, although indispensable at sea, is of little value for land maps. For topographical sheets it is obviously unsuitable; and in cases in which it is required to show large areas on small scales on an orthomorphic projection, that form should be chosen which gives two standard parallels (Lambert’s conical orthomorphic). Mercator’s projection is often used in atlases for maps of the world. It is not a good projection to select for this purpose on account of the great exaggeration of scale near the poles. The misconceptions arising from this exaggeration of scale may, however, be corrected by the juxtaposition of a map of the world on an equal-area projection.

(a / M) {log tan (45° +1⁄2φ) − e2sin φ −1⁄3e4sin3φ ...}

where M, the modulus of common logarithms, = 0.434294. Of course Mercator’s projection was not originally arrived at in the manner above described; the description has been given to show that Mercator’s projection is a particular case of the conical orthomorphic group. The introduction of the projection is due to the fact that for navigation it is very desirable to possess charts which shall give correct local outlines (i.e.in modern phraseology shall be orthomorphic) and shall at the same time show as a straight line any line which cuts the meridians at a constant angle. The latter condition clearly necessitates parallel meridians, and the former a continuous increase of scale as the equator is departed from,i.e.the scale at any point must be equal to the scale at the equator × sec. latitude. In early days the calculations were made by assuming that for a small increase of latitude, say 1′, the scale was constant, then summing up the small lengths so obtained. Nowadays (for simplicity the earth will be taken as a sphere) we should say that a small length of meridian adφ is represented in this projection by a sec φ dφ, and the length of the meridian in the projection between the equator and latitude φ,

√φ0a sec φ dφ = a logetan (45° +1⁄2φ),

which is the direct way of arriving at the law of the construction of this very important projection.

Mercator’s projection, although indispensable at sea, is of little value for land maps. For topographical sheets it is obviously unsuitable; and in cases in which it is required to show large areas on small scales on an orthomorphic projection, that form should be chosen which gives two standard parallels (Lambert’s conical orthomorphic). Mercator’s projection is often used in atlases for maps of the world. It is not a good projection to select for this purpose on account of the great exaggeration of scale near the poles. The misconceptions arising from this exaggeration of scale may, however, be corrected by the juxtaposition of a map of the world on an equal-area projection.

It is now necessary to revert to the general consideration of conical projections.

It has been shown that the scales of the projection (fig. 23) as compared with the sphere are p′q′ / pq = dp / dz = σ along a meridian, and p′r′ / pr′ = ρh / sin z = σ′ at right angles to a meridian.Now if σσ′ = 1 the areas are correctly represented, thenhρ dρ = sin z dz, and integrating1⁄2hρ2= C − cos z;(i.)this gives the whole group ofequal-area conical projections.As a special case let the pole be the centre of the projected parallels, then whenz = 0, ρ = 0, and const = 1, we have p = 2 sin1⁄2z / δh(ii.)Let z1be the co-latitude of some parallel which is to be correctly represented, then 2h sin1⁄2z1/ δh = sin z1, and h = cos21⁄2z1; putting this value of h in equation (ii.) the radius of any parallel= ρ = 2 sin1⁄2z sec1⁄2z1(iii.)This is Lambert’sconical equal-area projection with one standard parallel, the pole being the centre of the parallels.If we put z1=θ, then h = 1, and the meridians are inclined at their true angles, also the scale at the pole becomes correct, and equation (iii.) becomesρ = 2 sin1⁄2z;(iv.)this is thezenithal equal-area projection.Reverting to the general expression for equal-area conical projectionsρ = √{2 (C − cos z) / h},(i.)we can dispose of C and h so that any two selected parallels shall be their true lengths; let their co-latitudes be z1and z2, then2h (C − cos z1) = sin2z1(v.)2h (C − cos z2) = sin2z2(vi.)from which C and h are easily found, and the radii are obtained from (i.) above. This is H. C. Albers’conical equal-area projection with two standard parallels. The pole is not the centre of the parallels.

Now if σσ′ = 1 the areas are correctly represented, then

hρ dρ = sin z dz, and integrating1⁄2hρ2= C − cos z;

(i.)

this gives the whole group ofequal-area conical projections.

As a special case let the pole be the centre of the projected parallels, then when

z = 0, ρ = 0, and const = 1, we have p = 2 sin1⁄2z / δh

(ii.)

Let z1be the co-latitude of some parallel which is to be correctly represented, then 2h sin1⁄2z1/ δh = sin z1, and h = cos21⁄2z1; putting this value of h in equation (ii.) the radius of any parallel

= ρ = 2 sin1⁄2z sec1⁄2z1

(iii.)

This is Lambert’sconical equal-area projection with one standard parallel, the pole being the centre of the parallels.

If we put z1=θ, then h = 1, and the meridians are inclined at their true angles, also the scale at the pole becomes correct, and equation (iii.) becomes

ρ = 2 sin1⁄2z;

(iv.)

this is thezenithal equal-area projection.

Reverting to the general expression for equal-area conical projections

ρ = √{2 (C − cos z) / h},

(i.)

we can dispose of C and h so that any two selected parallels shall be their true lengths; let their co-latitudes be z1and z2, then

2h (C − cos z1) = sin2z1

(v.)

2h (C − cos z2) = sin2z2

(vi.)

from which C and h are easily found, and the radii are obtained from (i.) above. This is H. C. Albers’conical equal-area projection with two standard parallels. The pole is not the centre of the parallels.

Projection by Rectangular Spheroidal Co-ordinates.

If in the simple conical projection the selected parallel is the equator, this and the other parallels become parallel straight lines and the meridians are straight lines spaced at equatorial distances, cutting the parallels at right angles; the parallels are their true distances apart. This projection is thesimple cylindrical. If now we imagine the touching cylinder turned through a right-angle In such a way as to touch the sphere along any meridian, a projection is obtained exactly similar to the last, except that in this case we represent, not parallels and meridians, but small circles parallel to the given meridian and great circles at right angles to it. It is clear that the projection is a special case of conical projection. The position of any point on the earth’s surface is thus referred, on this projection, to a selected meridian as one axis, and any great circle at right angles to it as the other. Or, in other words, any point is fixed by the length of the perpendicular from it on to the fixed meridian and the distance of the foot of the perpendicular from some fixed point on the meridian, these spherical or spheroidal co-ordinates being plotted as plane rectangular co-ordinates.

The perpendicular is really a plane section of the surface through the given point at right angles to the chosen meridian, and may be briefly called a great circle. Such a great circle clearly diverges from the parallel; the exact difference in latitude and longitude between the point and the foot of the perpendicular can be at once obtained by ordinary geodetic formulae, putting the azimuth = 90°. Approximately the difference of latitude in seconds is x2tan φ cosec 1″ / 2ρν where x is the length of the perpendicular, ρ that of the radius of curvature to the meridian, ν that of the normal terminated by the minor axis, φ the latitude of the foot of the perpendicular. The difference of longitude in seconds is approximately x sec ρ cosec 1″ / ν. The resulting error consists principally of an exaggeration of scale north and south and is approximately equal to sec x (expressing x in arc); it is practically independent of the extent in latitude.

It is on this projection that the 1/2,500 Ordnance maps and the 6-in. Ordnance maps of the United Kingdom are plotted, a meridian being chosen for a group of counties. It is also used for the 1-in.,1⁄2in. and1⁄4in. Ordnance maps of England, the central meridian chosen being that which passes through a point in Delamere Forest in Cheshire. This projection should not as a rule be used for topographical maps, but is suitable for cadastral plans on account of the convenience of plotting the rectangular co-ordinates of the very numerous trigonometrical or traverse points required in the construction of such plans. As regards the errors involved, a range of about 150 miles each side of the central meridian will give a maximum error in scale in a north and south direction of about 0.1%.

Elliptical Equal-area Projection.

In this projection, which is also called Mollweide’s projection the parallels are parallel straight lines and the meridians are ellipses, the central meridian being a straight line at right angles to the equator, which is equally divided. If the whole world is represented on the spherical assumption, the equator is twice the length of the central meridian. Each elliptical meridian has for one axis the central meridian, and for the other the intercepted portion of the equally divided equator. It follows that the meridians 90° east and west of the central meridian form a circle. It is easy to show that to preserve the property of equal areas the distance of any parallel from the equator must be √2 sin δ where π sin φ = 2δ + sin 2δ, φ being the latitude of the parallel. The length of the central meridian from pole to pole = 2 √2, where the radius of the sphere is unity. The length of the equator = 4 √2.

The following equal-area projections may be used to exhibit the entire surface of the globe: Cylindrical equal area, Sinusoidal equal area and Elliptical equal area.

Conventional or Arbitrary Projections.

These projections are devised for simplicity of drawing and not for any special properties. The most useful projection of this class is theglobular projection. This is a conventional representation of a hemisphere in which the equator and central meridian are two equal straight lines at right angles, their intersection being the centre of the circular boundary. The meridians divide the equator into equal parts and are arcs of circles passing through points so determined and the poles. The parallels are arcs of circles which divide the central and extreme meridians into equal parts. Thus in fig. 26 NS = EW and each is divided into equal parts (in this case each division is 10°); the circumference NESW is also divided into 10° spaces and circular arcs are drawn through the corresponding points. This is a simple and effective projection and one well suited for conveying ideas of thegeneral shape and position of the chief land masses; it is better for this purpose than the stereographic, which is commonly employed in atlases.

Projections for Field Sheets.

Field sheets for topographical surveys should be on conical projections with rectified meridians; these projections for small areas and ordinary topographical scales—not less than 1/500,000—are sensibly errorless. But to save labour it is customary to employ for this purpose either form of polyconic projection, in which the errors for such scales are also negligible. In some surveys, to avoid the difficulty of plotting the flat arcs required for the parallels, the arcs are replaced by polygons, each side being the length of the portion of the arc it replaces. This method is especially suitable for scales of 1 : 125,000 and larger, but it is also sometimes used for smaller scales.

Fig. 27 shows the method of plotting the projection for a field sheet. Such a projection is usually called a graticule. In this case ABC is the central meridian; the true meridian lengths of 30′ spaces are marked on this meridian, and to each of these, such as AB, the figure (in this case representing a square half degree), such as ABED, is applied. Thus the point D is the intersection of a circle of radius AD with a circle of radius BD, these lengths being taken from geodetic tables. The method has no merit except that of convenience.

Summary.

The following projections have been briefly described:—

Perspective1. Cylindrical equal-area.2. Orthographic.3. Stereographic (which is orthomorphic).4. General external perspective.5. Minimum error perspective. (Clarke’s).6. Central.Conical7. Conical, with rectified meridians and two standard parallels (5 forms).8. Simple conical.9. Simple cylindrical (a special case of 8).10. Modified conical equal-area (Bonne’s).11. Sinusoidal equal-area (Sanson’s).12. Werner’s conical equal-area13. Simple polyconic.14. Rectangular polyconic.15. Conical orthomorphic with 2 standard parallels (Lambert’s, commonly called Gauss’s).16. Cylindrical orthomorphic (Mercator’s).17. Conical equal-area with one standard parallel.18. Conical equal-area with two standard parallels.19. Projection by rectangular spheroidal co-ordinates.Zenithal20. Equidistant zenithal.21. Zenithal equal-area.22. Zenithal projection by balance of errors (Airy’s).23. Elliptical equal-area (Mollweide’s).24. Globular (conventional).25. Field sheet graticule.Of the above 25 projections, 23 are conical or quasi-conical, if zenithal and perspective projections be included. The projections may, if it is preferred, be grouped according to their properties. Thus in the above list 8 are equal-area, 3 are orthomorphic, 1 balances errors, 1 represents all great circles by straight lines, and in 5 one system of great circles is represented correctly.Among projections which have not been described may be mentioned the circular orthomorphic (Lagrange’s) and the rectilinear equal-area (Collignon’s) and a considerable number of conventional projections, which latter are for the most part of little value.The choice of a projection depends on the function which the map is intended to fulfil. If the map is intended for statistical purposes to show areas, density of population, incidence of rainfall, of disease, distribution of wealth, &c., anequal-areaprojection should be chosen. In such a case an area scale should be given. At sea,Mercator’sis practically the only projection used except when it is desired to determine graphically great circle courses in great oceans, when thecentralprojection must be employed. For conveying good general ideas of the shape and distribution of the surface features of continents or of a hemisphereClarke’s perspectiveprojection is the best. For exhibiting the progress of polar exploration thepolar equidistantprojection should be selected. For special maps for general use on scales of 1/1,000,000 and smaller, and for a series of which the sheets are to fit together, theconical, with rectified meridians and two standard parallels, is a good projection. For topographical maps, in which each sheet is plotted independently and the scale is not smaller than 1/500,000, either form ofpolyconicis very convenient.The following are the projections adopted for some of the principal official maps of the British Empire:—Conical, with Rectified Meridians and Two Standard Parallels.—The 1 : 1,000,000 Ordnance map of the United Kingdom, special maps of the topographical section, General Staff,e.g.the 64-mile map of Afghanistan and Persia. The 1 : 1,000,000 Survey of India series of India and adjacent countries.Modified Conical, Equal-area(Bonne’s).—The 1 in.,1⁄2in.,1⁄4in. and1⁄10in. Ordnance maps of Scotland and Ireland. The 1 : 800,000 map of the Cape Colony, published by the Surveyor-General.Simple Polyconic and Rectangular Polyconicmaps on scales of 1 : 1,000,000, 1 : 500,000, 1 : 250,000 and 1 : 125,000 of the topographical section of the General Staff, including all maps on these scales of British Africa. A rectilinear approximation to the simple polyconic is also used for the topographical sheets of the Survey of India. The simple polyconic is used for the 1 in. maps of the Militia Department of Canada.Zenithal Projection by Balance of Errors(Airy’s).—The 10-mile to 1 in. Ordnance map of England.Projection by Rectangular Spheroidal Co-ordinates.—The 1 : 2500 and the 6 in. Ordnance sheets of the United Kingdom, and the 1 in.,1⁄2in. and1⁄4in. Ordnance maps of England. The cadastral plans of the Survey of India, and cadastral plans throughout the empire.Authorities.—SeeTraité des projections des cartes géographiques, by A. Germain (Paris, 1865) andA Treatise on Projections, by T. Craig, United States Coast and Geodetic Survey (Washington, 1882). Both Germain and Craig (following Germain) make use of the termprojections by development, a term which is apt to convey the impression that the spherical surface is developable. As this is not the case, and since such projections are conical, it is best to avoid the use of the term. For the history of the subject see d’Avezac, “Coup d’œil historique sur la projection des cartes géographiques,”Société de géographie de Paris(1863).J. H. Lambert (Beiträge zum Gebrauch der Mathematik, u.s.w.Berlin, 1772) devised the following projections of the above list: 1, 15, 17, and 21; his transverse cylindrical orthomorphic and the transverse cylindrical equal-area have not been described, as they are seldom used. Among other contributors we mention Mercator, Euler, Gauss, C. B. Mollweide (1774-1825), Lagrange, Cassini, R. Bonne (1727-1795), Airy and Colonel A. R. Clarke.

Of the above 25 projections, 23 are conical or quasi-conical, if zenithal and perspective projections be included. The projections may, if it is preferred, be grouped according to their properties. Thus in the above list 8 are equal-area, 3 are orthomorphic, 1 balances errors, 1 represents all great circles by straight lines, and in 5 one system of great circles is represented correctly.

Among projections which have not been described may be mentioned the circular orthomorphic (Lagrange’s) and the rectilinear equal-area (Collignon’s) and a considerable number of conventional projections, which latter are for the most part of little value.

The choice of a projection depends on the function which the map is intended to fulfil. If the map is intended for statistical purposes to show areas, density of population, incidence of rainfall, of disease, distribution of wealth, &c., anequal-areaprojection should be chosen. In such a case an area scale should be given. At sea,Mercator’sis practically the only projection used except when it is desired to determine graphically great circle courses in great oceans, when thecentralprojection must be employed. For conveying good general ideas of the shape and distribution of the surface features of continents or of a hemisphereClarke’s perspectiveprojection is the best. For exhibiting the progress of polar exploration thepolar equidistantprojection should be selected. For special maps for general use on scales of 1/1,000,000 and smaller, and for a series of which the sheets are to fit together, theconical, with rectified meridians and two standard parallels, is a good projection. For topographical maps, in which each sheet is plotted independently and the scale is not smaller than 1/500,000, either form ofpolyconicis very convenient.

The following are the projections adopted for some of the principal official maps of the British Empire:—

Conical, with Rectified Meridians and Two Standard Parallels.—The 1 : 1,000,000 Ordnance map of the United Kingdom, special maps of the topographical section, General Staff,e.g.the 64-mile map of Afghanistan and Persia. The 1 : 1,000,000 Survey of India series of India and adjacent countries.

Modified Conical, Equal-area(Bonne’s).—The 1 in.,1⁄2in.,1⁄4in. and1⁄10in. Ordnance maps of Scotland and Ireland. The 1 : 800,000 map of the Cape Colony, published by the Surveyor-General.

Simple Polyconic and Rectangular Polyconicmaps on scales of 1 : 1,000,000, 1 : 500,000, 1 : 250,000 and 1 : 125,000 of the topographical section of the General Staff, including all maps on these scales of British Africa. A rectilinear approximation to the simple polyconic is also used for the topographical sheets of the Survey of India. The simple polyconic is used for the 1 in. maps of the Militia Department of Canada.

Zenithal Projection by Balance of Errors(Airy’s).—The 10-mile to 1 in. Ordnance map of England.

Projection by Rectangular Spheroidal Co-ordinates.—The 1 : 2500 and the 6 in. Ordnance sheets of the United Kingdom, and the 1 in.,1⁄2in. and1⁄4in. Ordnance maps of England. The cadastral plans of the Survey of India, and cadastral plans throughout the empire.

Authorities.—SeeTraité des projections des cartes géographiques, by A. Germain (Paris, 1865) andA Treatise on Projections, by T. Craig, United States Coast and Geodetic Survey (Washington, 1882). Both Germain and Craig (following Germain) make use of the termprojections by development, a term which is apt to convey the impression that the spherical surface is developable. As this is not the case, and since such projections are conical, it is best to avoid the use of the term. For the history of the subject see d’Avezac, “Coup d’œil historique sur la projection des cartes géographiques,”Société de géographie de Paris(1863).

J. H. Lambert (Beiträge zum Gebrauch der Mathematik, u.s.w.Berlin, 1772) devised the following projections of the above list: 1, 15, 17, and 21; his transverse cylindrical orthomorphic and the transverse cylindrical equal-area have not been described, as they are seldom used. Among other contributors we mention Mercator, Euler, Gauss, C. B. Mollweide (1774-1825), Lagrange, Cassini, R. Bonne (1727-1795), Airy and Colonel A. R. Clarke.

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