10

k

m

is to be drawn. When pm is greater than pa, the ellipse | PoP', and vP, vP' cutting lr in pp': these are the projections touches the circle in two points; these points divide the of the poles, through which ellipse into two parts, one of which, being on the other side all the circles representing of the meridian plane aqr, is invisible. meridians have to pass. All their centres then will be in a line smn which crosses pp' at v right angles through its middle point m. Now to describe the s meridian whose west longitude is w, draw pn making the angle · opn = 90° — w, then n is the centre of the required circle, whose direction as it passes through p will make an angle opgo with pp'. The lengths of the several lines are optanju; op=cotu; om=cotu; mn=cosec u cotw.

Stereographic Projection. In this case the point of vision is on the surface, and the projection is made on the plane of the great circle whose pole is V. Let kplV (fig. 12) be a great circle through the point of vision, and ors the trace of the plane of projection. Let c be the centre of a small circle whose radius is cp=cl; the straight line pl represents this small circle in orthographic projection.

=

Fig. 12.

Fig. 14.

Again, for the parallels, take Pb = Pe equal to the co-latitude, say c, of the parallel to be projected; join vb, vc cutting Ir in e, d. Then ed is the diameter of the circle which is the required projection; its centre is of course the middle point of ed, and the lengths of the lines are

od=tan (u-c); oe=tan (u+c).

The line sn itself is the projection of a parallel, namely, that of which the co-latitude c= 180°-u, a parallel which passes through the point of vision.

a

We have first to show that the stereographic projection of the small circle pl is itself a circle; that is to say, a straight line through V, moving along the circumference of pl, traces a circle on the plane of projection ors. This line generates an oblique cone standing on a circular base, its axis being cV (since the angle pVe-angle eV); this cone is divided symmetrically by the plane of the great circle kpl, and also by the plane which passes through the axis Vc, perpendicular to the plane kpl. Now Vr.Vp, being Vo sec kVp. Vk cos kVpVo Vk, is equal to Vs.Vl; therefore the triangles Vrs, Vlp are similar, and it follows that the section of the cone by the plane rs is similar to the section by the plane pl. But the latter is a circle, hence also the projection is a circle; and since the representation of every infinitely small circle on the surface is itself a circle, it follows that in this projection the representation of small parts is (as we have before shown) strictly similar. Another inference is that the angle in which two lines on the sphere intersect is represented by the same angle in the projection. This may otherwise be proved by means of fig. 13, where Vok is the diameter of the sphere passing through the point of vision, fgh the plane of projection, kt a great circle, passing of course through V, and ouv the line of intersection of these two planes. A tangent plane to the surface at t cuts the plane of projection in the line rvs perpendicular to ov; tv is a tangent to the circle kt at t, tr and ts are any two tangents to the surface at t. Now the angle vtu (u being the projection of t) is 90° - otV = 90° - oVt ouV = tuv, therefore tv is equal to uv; and since tus and uvs are right angles, it follows that the angles vts and vus are equal. Hence the angle ris also is equal to its projection rus; that is, any angle formed by two intersecting lines on the surface is truly represented in the stereographic projection. We have seen that the projection of any circle of the sphere is itself a circle. But in the case in which the circle to be projected passes through V, the projection becomes, for a great circle, a line through the centre of the sphere; otherwise, a line anywhere. It follows that meridians and parallels are represented in a projection on the horizon of which, replacing ac by its value sin u, becomes any place by two systems of orthogonally cutting circles, one system passing through two fixed points, namely, the poles; and the projected meridians as they pass through the poles show the proper differences of longitude.

Fig. 13.

=

9

To construct a stereographic projection of the sphere on the horizon of a given place. Draw the circle vlkr (fig. 14) with the diameters kv, Ir at right angles; the latter is to represent the central meridian. Take koP equal to the co-latitude of the given place, say u; draw the diameter

-b

A very interesting connexion, noted by Professor Cayley, exists between the stereographic projection of the sphere on a meridian plane (i.e., when a point on the equator occupies the centre of the drawing) and the projection on the horizon of any place whatever. The very same circles that represent parallels and meridians in the one case represent them in the other case also. In fig. 15, abs being a projection in which an equatorial point is in the centre, draw any chord ab perpendicular to the centre meridian cos, and on ab as diameter describe a circle, when the property referred to will be observed. This smaller circle is now the stereographic projection of the sphere on the horizon of some place whose co-latitude we may call u. The radius of the first circle being unity, let ac-sinx, then by what has been proved above co=sinx cotu cosa; therefore u=x, and ac= sin 26. Although the meridian circles dividing the 360° at the pole into equal angles must be actually the same in both systems, yet a parallel circle whose co-latitude is c in the direct projection abs belongs in the oblique system to some other co-latitude as c'. To determine the connexion between c and c', consider the point t (not marked), in which one of the parallel circles crosses the line soc. In the direct system, p being the pole,

=

Fig. 15.

cos (u - c') ́1+cot u cot c' therefore tanc tante' tanju is the required relation. Notwithstanding the facility of construction, the stereographic projection is not much used in map-making. But it may be made very useful as a means of graphical interpolation for drawing other projections in which points are represented in their true azimuths, but with an arbitrary

law of distance, as p=f(u). We may thus avoid the calcu- | This vanishes when = 1, that is, if the projection be stereolation of all the distances and azimuths (with reference to graphic; or for u=0, that is, at the centre of the map. the selected centre point) of the intersections of meridians At a distance of 90° from the centre, the greatest alteration and parallels. Construct a stereographic projection of the is 90°-2 cot-1 h. (See Philosoph. Mag., April 1862.) globe on the horizon of the given place; then on this pro- The constants h and k can be determined, so that the jection draw concentric circles (according to the stereo- total misrepresentation, viz., graphic law) representing the loci of points whose distances from the centre are consecutively 5°, 10°, 15°, 20°, &c., up to the required limit, and a system of radial lines at intervals of 50. Then to construct any other projection,-commence by drawing concentric circles, of which the radii are

M

= ƒ3 { (o − 1)2 + (o' − 1)2 } sinudu,

shall be a minimum, ß being the greatest value of u, or the spherical radius of the map. On substituting the expres sions for σ and o' the integration is effected without difficulty. Put

1-cos B

λ=

h+cos B

FIG. 16. Stereographic Projection.

previously calculated by the law p=f(u), for the successive values of u, 5°, 10°, 15°, 20°, &c., up to the limits as before, and a system of radial lines at intervals of 5°. This being completed, it remains to transfer the points of intersection from the stereographic to the new projection by graphic

interpolation.

a

с

m

We now come to the general case in which the point of vision has any position outside the sphere. Let abcd (fig. 17) be the great circle section of the sphere by a plane passing through p c, the central point of the portion of surface to be represented, and V the point of vision. pj perpendicular to Vc be the plane of representation, join mV cutting pj in f, then ƒ is the projection of any point m in the circle abc, and ef is the representation of cm.

Let

Fig.

Therefore M = 4 sin2 16 - H"

and h must be determined so

as to make H2: H'a maximum. In any particular case this maximum can only be ascertained by trial, that is to say, log H2-log H' must be calculated for certain equidistant values of h, and then the particular value of h which corresponds to the required maximum can be obtained by the best possible perspective representation of a hemisphere, interpolation. Thus we find that if it be required to make the values of h and I are h=η47 and k=2·034; so that

h+ cos u
Σ=
1+h cos u

in this case is 1.61. Fig. 18 is a map of Asia having the meridians and parallels laid down on this system.

of the centre meridian and centre parallel a line perpendicular to the meridian and therefore touching the parallel. Let the coordinate x be measured from the centre along this line, and y perpendicular to it. Then the coordinates of a point whose longitude measured from the centre meridian is w are

y=2cot sin2(w sin )=x tan (w sin ), the radius of the sphere being the unit; if a degree be the unit, these must be multiplied by 57.296.

The great defect of this projection is the exaggeration of the lengths of parallels towards either the northern or southern limits of the map. Various have been the devices to remedy this defect, and amongst these the following is a system very much adopted. Having subdivided the central meridian and drawn through the points of division the parallels precisely as described above, then the true lengths of degrees are Fig. 22. set off along each parallel; the meridians, which in this case become curved lines, are drawn through the corresponding points of the parallels (fig. 22).

This system is that which was adopted in 1803 by the "Dépôt de la Guerre" for the map of France, and is there known by the title "Projection de Bonne." It is that on which the Ordnance Survey map of Scotland on the scale of one inch to a mile is constructed, and it is frequently met with in ordinary atlases. It is ill-adapted for countries having great extent in longitude, as the intersections of the meridians and parallels become very oblique-as will be seen on examining the map of Asia in most atlases. If . be taken as the latitude of the centre parallel, and co-ordinates be measured from the intersection of this parallel with the central meridian, as in the case of the conical projection, then, if p be the radius of the parallel of latitude, we have pcot 4, +☀. - 4. Also, if S be a point on this parallel whose co-ordinates are x, y, so that VS = p, and ◊ be the angle VS makes with the central meridian, then p✪ = w cos &; and x=psin 0, y=cot 。 - p cos 0.

Ρ

C

ence of longitude of which is du, and two consecutive
parallels whose co-latitudes are u and u+du. The sides
of this rectangle (fig. 24) are pq=du, pr=sinudμ,
whereas in the representation p'q'r's,
p'q=dp, p'r'=phdu, the angle at O P
being = hdp. Now, as the represen-
tation is to be similar to the original,
p'd_dp pq. du
prhpdμpr sinudμ '

- =

dp du

whence =h and integrating,

Fig. 24.

determined so as to give, upon the whole, the least amount | sphere contained by two consecutive meridians the differof exaggeration for the entire map. This idea of a cutting cone seems to have originated1 with the celebrated Gerard Mercator, who in 1554 made a map of Europe on this principle, selecting for the parallels of intersection those of 40° and 60°. The same system was adopted in 1745 by Delisle for the construction of a map of Russia. Euler in the Acta Acad. Imp. Petrop., 1778, has discussed this projection and determined the conditions under which the errors at the northern extremity, at the centre, and at the southern extremity of a map so constructed shall be severally equal. Let c, c' be the co-latitudes of the extreme northern and southern parallels, y, y' those of two intermediate parallels, which are to be truly represented in the projection. Let OC', Om' (fig. 23) be two consecutive meridians, as represented in the developed cone; the difference of longitude being w, let the G angle at O be hw. The degrees along the meri- c' dian being represented by their proper lengths, CC=c-c; and P corresponding to the pole, let Fig. 23. OP=2, then OC=z+c; and so for G, G', C'. The true lengths of G'n' and Gn, namely, w sin y' and w sin y, are equal to the represented lengths, namely, ho (z+y) and ho (z+y) respectively, whence y and y' are known when h and z are known. Comparing now the represented with the true lengths of parallel at the extremities and at the centre, if e be the common error that is to be allowed, then e=hw(z+c)-w sin c,

G

n'

m'

Thus z being known, the common centre of the circles re-
presenting the parallels is given. The value of h is given
by the equation h(c' - c)=sin c' - sin c, and y and y can
be easily computed. But there is no necessity for doing
this as we may construct the angles at O, which represent-
ing a difference of longitude are in reality equal to hw.
o
For instance, to construct a map of Asia on this system,
having divided the central meridian into equal spaces for
degrees, z must be calculated. Here we have c=20°, c' =
80°, whence z+50° = 15° tan 50° cot 15° 66° 7. Hence
in this case the centre of the circles is 16°7 beyond the
north pole; also h='6138, so that a difference of longitude
of 5° is represented at O by an angle of 3° 4' 9". The de-
grees of longitude in the parallel of 70° are in this map re-
presented too large in the ratio of 1:150:1; those in the
mid-latitude of 40° are too small in the ratio of 0.933:1;
and those in 10° latitude are too large in the ratio of 1·05
to 1.

Gauss's Projection

The

may be considered as another variation of the conical
system of development. Meridians are represented by lines
drawn through a point, and a difference of longitude w is
represented by an angle ho, as in the preceding case.
parallels of latitude are circular arcs, all having as centre
the point of divergence of the meridian lines, and the law
of their formation is such that the representations of all
small parts of the surface shall be precisely similar to the
parts so represented. Let u be the co-latitude of a parallel,
and
p, a function of u, the radius of the circle representing
this parallel. Consider the infinitely small space on the

See page 178 of Traité des Projections des Cartes Géographiques, by A. Germain, Paris, an admirable and exhaustive essay. See also the work entitled Coup d'œil historique sur la Projection des Cartes de Géographie, by M. d'Avezac, Paris, 1863,

2

where the constant h is to be determined according to the
requirements of each individual case. This investigation
was first made in 1772 by the German mathematician J. H.
Lambert, but in 1825 it was again brought forward by
Gauss in an essay written in answer to a prize question pro-
posed by the Royal Society of Sciences at Copenhagen. A
translation of this essay is to be found in the Philosophical
Magazine for 1828 (see page 112), where Lambert's projec-
blem. Again, in a general investigation of the problem of
tion comes out as a particular solution of the general pro-
"similar representation," Sir John Herschel, in the 30th
volume of the Journal of the Royal Geographical Society
(1860), deduced as a particular case this same projection.
A large map of Russia was constructed and published on
this system by the Geographical Society of St Petersburg
in 1862.
The relative scale in this development is-

where a is the radius of the sphere. It is a minimum
when u = cos-1 h. This minimum should occur in the
vicinity of the central parallel of the map; if u
be the co-
latitude of this parallel, we may put

Or if we agree that the scale of the representation shall be
the same at the extreme co-latitudes c, c', then
log sin clog sin c
h=
log tan c-log tan

To construct a map of North America extending from 10° latitude to 70°, we may take h=3, and k such as shall make the difference of radii of the extreme parallels = 60, namely k 104.315. The scales of the representation at the northern and southern limits are 1∙116 and 1·096 respectively. The radii of the parallels are these

Having drawn a line representing the central meridian, and selected a point on it as the centre of the concentric circles, let arcs be described with the above radii as parallels. For meridians, in this system a difference of longitude of 10° is represented by an angle of two-thirds that amount, or 6° 40'. The chord of this angle on the parallel of 10°, whose radius is 92-801, is easily found to be 10-792. Now stepping this quantity with a pair of compasses along the parallel, we have merely to draw lines through each of the points so found and the common centre of circles. The points of division of the parallel may be checked by taking the chord of 20°, or rather of 13° 20′,

2 Beiträge zum Gebrauche der Mathematik und deren Anwendung, vol. iii. p. 55, Berlin, 1772.