d 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 Stereographic Projection.—In this case the point of visiou their centres then will be in a is on the surface, and the projection line smn which crosses pp at v Ik is made on the plane of the great right angles through its middle circle whose pole is V. Let kplv (fig. point m. Now to describe the 12) be a great circle through the point meridian whose west longitude of vision, and ors the trace of the is w, draw pn making the angle plane of projection. Let c be the opn = 90° - w, then n is the centre centre of a small circle whose radius of the required circle, whose is cp=cl; the straight line pl repre direction as it passes through p sents this small circle in orthographic will make an angle opg=w with Fig. 14. projection. Fig. 12. pp. The lengths of the several lines are We have first to show that the stereographic projection op - tan fu; op' - cot fu; of the small circle pl is itself a circle ; that is to say, a on-cotu; min-cosec u cotw. straight line through V, moving along the circumference of Again, for the parallels, take Pó - Pc equal to the co-latitude, pl, traces a circle on the plane of projection ors. This line say c, of the parallel to be projected; join vb, vc cutting Ir generates an oblique cone standing on a circular base, its in e, d. Then ed is the diameter of the circle which is the axis being cV (since the angle pvc= angle cVl); this cone required projection; its centre is of course the middle point is divided symmetrically by the plane of the great circle of ed, and the lengths of the lines are kpl, and also by the plane which passes through the axis Vc, perpendicular to the plane kpl. Now Vr.Vp, being od-tan (u-c); oe= tan (u+c). = Vo sec kVp.Vk cos kVp = Vo.Vk, is equal to Vs.Vi; there. The line sn itself is the projection of a parallel, namely, that fore the triangles Vrs, Vlp are similar, and it follows of which the co-latitude c= 180° — U, a parallel which passes that the section of the cone by the plane rs is similar to through the point of vision. the section by the plane pl. But the latter is a circle, A very interesting connexion, noted by Professor Cayley, hence also the projection is a circle ; and since the repre-exists between the stereographic projection of the sphere sentation of every infinitely small circle on the surface is on a meridian plane (i.e, when a point on the equator itself a circle, it follows that in this projection the represen occupies the centre of the drawing) and the projection on tation of small parts is fas we have before shown) strictly the horizon of any place whatever. The very same circles similar. Another inference is that the angle in which two that represent parallels and meri. lines on the sphere intersect is represented by the same dians in the one case represent angle in the projection. This may otherwise be proved them in the other case also. In by means of fig. 13, where yok is the diameter of fig. 15, abs being a projection the sphere passing through in which an equatorial point is in the point of vision, fgh the the centre, draw any chord ab perplane of projection, kt a pendicular to the centre meridian great circle, passing of cos, and on ab as diameter describe course through V, and ouv a circle, when the property referred the line of intersection of to will be observed. This smaller these two planes. A tangent circle is now the stereographic proplane to the surface at t jection of the sphere on the horizon Fig. 15. cuts the plane of projection of some place whose co-latitude we in the line rvs perpendicular may call u. The radius of the first circle being unity, let to ov; tv is a tangent to the ac=sinx, then by what has been proved above co=sina Fig. 13. circle kat at t, tr and ts are cotu= cos x; therefore u=X, and ac=sinu. Although any two tangents to the surface at t. Now the angle otu (w the meridian circles dividing the 360° at the pole into equal u being the projection of t) is 90° - 06V = 90° - Vt=0&V= tuv, angles must be actually the same in both systems, yet a therefore tv is equal to uv; and since tos and uvs are right parallel circle whose co-latitudo-is c in the direct projection angles, it follows that the angles ots and vus are equal. abs belongs in the oblique system to some other co-latitude Hence the angle rls also is equal to its projection rus; as c. To determine the connexion between c and c', conthat is, any angle formed by two intersecting lines on the sider the point t (not marked), in which one of the parallel surface is truly represented in the stereographic projection.circles crosses the line soc. In the direct system, o being We have seen that the projection of any circle of the the pole; sphere is itself a circle. But in the case in which the circle 2 pt-1-tan 3(90°-c)to be projected passes through V, the projection becomes, 1+cot sc and in the oblique, for a great circle, a line through the centre of the sphere; otherwise, a line anywhere. It follows that meridians and a pt=ac (tan fu – tan f(x–c)), parallels are represented in a projection on the horizon of any place by two systems of orthogonally cutting circles, which, roplacing ac by its value sin 2, becomes 2 sin fu sin Hd 2 one system passing through two fixed points, namely, the poles; and the projected meridians as they pass through cos }(u - ) *1 + cot fu cot d'' the poles show the proper differences of longitude. thererore can jc = tan lc' tanļu is the required relation. To construct a stereographic projection of the sphere Notwithstanding the facility of construction, the sterooon the horizon of a given place. Draw the circle vlkr (fig. graphic projection is not much used in map-making. But 14) with the diameters kv, Ir at right angles ; the latter is it may be made very useful as a means of graphical intertoj represent the central meridian. . Take kop equal to the polation for drawing other projections in which points are co-latitude of the given place, say u; 'draw the diameter | represented in their true azimuths, but with an arbitrary λ =0. law of distance, as p=f(u). We máy thus avoid the calcu- | This vanishes when h= 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- Thi (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 tbo jection draw concentric circles (according to the stereo- total misrepresentation, viz., graphic law) representing the loci of points whose distances CB froin the centre are consecutively 5o, 10°, 15°, 20°, &c., up M = %6[10–150+60 – 1)"} sin udu, M to the required limit, and a system of radial lines at inter- shall be a minimnm, ß being the greatest value of us, or the vals of 50 Then to construct any other projection,-.com spherical radius of the map. On substituting the expresmence by drawing concentric circles, of which the radii are sions for a and othe integration, is effected without difti culty. Put ; v=(k-1), . ht1 M=4 sin 184 24H +k%H'. dM = dh dk .:. +H+H=0; 20H = 0. Н? H" maximum can only be ascertained by trial, that is to say, F10.16. -Stereographic Projection. log Hạ - log H' must be calculated for' certain equidistant previously calculated by the law p=f(u), for the successive values of h, and then the particular value of h which corvalues of 2, 5, 10o, 15*, 20°, &c., up to the limits as before, responds to the required maximum can be obtained by and & system of radial lines at intervals of 5°. This being interpolation. Thus we find that if it be required to make completed, it remains to transfer the points of intersection the best possible perspective representation of a hemisphere, from the stereographic to the new projection by graphic the values of h and £ are h = 1:47 and k = 2.034; so that in this case interpolation. 2.034 sin u We now come to the general case in which the point of 1:47 + cos U. vision has any position outside For a map of Africa or South America, the limiting radius the sphere. Let abcd (fig. 17) be 9 the great circle section of the ß we may take as 40°; then in this case 2.543 sin u sphere by a plane passing through c, the central point of the portion of surface to be represented, For Asia, B = 54 , and the distance h of the point of sight and V the point of vision. Let pj perpendicular to Vc be the plane of representation, join mV cutting pj in f, then f is the projection of any point m in the circl Fig. abc, and ef is the representation of cm. Let the angle com=U, Ve=k. Vo=h, ef=p; then, since ef:eV=mg : GV, k sin u с m RO ork which gives the law connecting a spherical distance u with its rectilinear representation p. The relative scale at any point in this system of projection is given (keeping to our previously adopted notation) by 1+h cos u k (h+cos u) u) h + cos u' the former applying to measurements made in a direction which passes through the centre of the map, the latter to the transverse direction. The product oo' gives the exaggeration of areas. With respect to the alteration of angles we have Figure 19 is a perspective representation of more than a | The co-ordinates originating at the centre, take the central hemisphere, the radius ß being 108°, and the distance h of meridian for the axis of y and a line perpendicular to it for the point of vision, 1:40. thə axis of x. Let the latitude of the point G, which is to The co-ordinates zy of any point in this perspective may occupy the centre of the map, be y; if Q, w be the latitade be expressed in terms of the latitude and longitude of the and longitude of any point P (the longitude being reckoned Corresponding point on the sphere in the following manner, from the meridian of G), u the distance PG, and a the w=20° = 0 y= 0.00 Fig. 19.-Twilight Projection. azimuth of P at G, then the spherical triangle whose sides for a map of Africa, which is included between latitudes 40° are 90° – Y, 90° - 6, and u gives these relations- nurth and 40° south, and 40° of longitudo east and weat of sin u sin pcos $ sin , a central meridian. Values of . and y. a = 10° w = 30° w = 40° Te h + sin y sin p+cos y cos o cos w cos y sin p- sin q cos o cosa = 0.00 9.69 19:43 29 25 39-17 ht sinn sin cos y cos o cosa 0.00 0.00 0'00 0.00 by which x and y can be computed for any point of the 9•60 x= 0.00 19.24 28.96 10° 38.76 •9.75 y= 9.69 9.92 10.21 10.63 sphere. If from these equations we eliminate w, we get the equation to the parallel whose latitude is $; it is an ellipse 9.32 = 0.00 18.87 20° 28.07 87.53 y=19.43 19.54 19.87 20:43 21.26 whose centre is in the central meridian, and its greater axis perpendicular to the same. The radius of curvature of this 30° 8.84 17.70 26.56 35.44 y=29.25 29.40 29.87 30-67 31.83 ellipse at its intersection with the centre meridian is k cos 8.15 x= 0.00 16.28 40° 24.39 32:44 y=39.17 h sin g+ sin $' 39.36 39.94 40.93 42.34 The elimination of between x and y gives the equation of the meridian whose longitude is w, which also is an ellipse Conical Development, whose contro and axes may be determined. The conical development is adapted to the construction The following table contains the computed co-ordinates of maps of tracts of country of no great extent in latitude cone. R but any extent in the direction of a parallel. Selecting of the centre meridian and centre parallel a line perpendithe mean parallel, or that which most nearly divides the cular to the meridian and therefore touching the parallel. àrea to be represented, we have to consider the cone Let the coordinate a be measured from the centre along which touches the sphere along that parallel. In fig. this line, and y perpendicular to it. Then the coordinates 20, which is an orthographic projection of the sphere on of a point whose longitude measured from the centre i meridian plane, let Pp be the meridian is w are parallel of contact with the x= cot'o sin ( sin o), ON being the axis of y=2cot + sin^4(e sin electan(e sine), revolution, the tangents at P the radius of the sphere being the unit; if a degree be the and p will intersect ON pro unit, these must be multiplied by 57-296. duced in V. Let Qq be a The great defect of this projection is the exaggeration of parallel to the north of Pp, Rr K the lengths of parallels towards either the northern or another parallel the same dis southern limits of the map. Various tance to the south, that is, have been the devices to remedy this PQ=PR. Take on the tan defect, and amongst these the following Fig. 20. gent PV two points H, K such is a system very much adopted. Haythat PH =PK, each being made equal to the arc PQ. It ing subdivided the central meridian and is clear, then, that the surface generated by HK is very drawn through the points of division the nearly coincident with the surface generated by RQ when parallels precisely as described above, the figure rotates round ON through any angle, great or then the true lengths of degrees are Fig. 22. small. The approximation of the surfaces will, however, set off along each parallel; the meridians, which in this be very close only if QR is very small. Suppose, now, case become curved lines, are drawn through the correthat the paths of H and K, as described in the revolution sponding points of the parallels (fig. 22). round ON, are actually marked on the surface of the cone, This system is that which was adopted in 1803 by the as well as the line of contact with the “Dépôt de la Guerre” for the map of France, and is there sphere. And further, mark the surface known by the title “ Projection de Bonne." It is that on of the cone by the intersections with it which the Ordnance Survey map of Scotland on the scale of the meridian planes through OV at of one inch to a mile is constructed, and it is frequently met the required equal intervals. Then h with in ordinary atlases. It is ill-adapted for countries let the conė be cut along a generating having great extent in longitude, as the intersections of the line and opened out into a plane, and P meridians and parallels become very oblique--a8 will be seen we shall have a representation as in on examining the map of Asia in most atlases. If $. be fig. 21 of the spherical surface con- * taken as the latitude of the centre parallel, and co-ordinates tained between the latitudes of Q and Fig. 21. be measured from the intersection of this parallel with the R. The parallels here are represented by concentric circles, central meridian, as in the case of the conical projection, the meridians by lines drawn through the common centre then, if p be the radius of the parallel of latitude $, we have of the circles at equal angular intervals. Taking the radiusp=cot $. +.-4. Also, if S be a point on this parallel of the sphere as unity, and $ being the latitude of P, we whose co-ordinates are x, y, so that VS=p, and @ be the see that VP=cot $, and if w be the difference of longitude angle VS makes with the central meridian, then po = w cos 0; between two meridians, the corresponding length of the arc and Pp is w cos 6. The angle between these meridians them. c=psin 0, y=cot $. - p cos 6. selves is w sin . Now, if we form the differential coefficients of x and y with Sappose, now, we require to construct a map on this prin- respect to $ and w, the latitude and longitude of S, we get ciple for a tract of country extending from latitude $-m m'n -mn'=cos, to $+m, and covering a breadth of longitude of 2n, m and n being expressed in degrees. In fig. 21 let HKkh - mn + m'n'= Cos (cos $-psin 0); be the quadrilateral formed by the extreme lines, so that the first of which equations proves that the areas are traly HK = hk = 2m; then the angle HVh is 2n sin • expressed represented. Moreover, if 90° + y be the angles of intersecin degrees. Now, taking the length of a degree as the unit, tions of meridians and paralléls, VP= 57.296 cot $, and VH = 57.296 cot $ -m. It may convenient in the first instance to calculate the chords Hh, tan y=8-w sin , Kk, and thus construct the rectilinear quadrilateral HKkh which indeed might have been more easily obtained. In The lengths of these chords are the case of Asia, the middle latitude 6. = 40°, and the exHh=2(57.296 cot p-m) sin (n sin o), treme northern latitude is 70°. Also the map extends 90° Kk=2(57.296 cot + m) sin (n sin o), of longitude from the central meridian; hence, at the northand the distance between them is 2m cos (n sin $). The west and north-east corners of the map the angles of interinclined sides of this trapezoid will then meet in section of meridians and parallels are 90° + 33°54'. But point at V, whose distance from P and p must corre- for comparatively small tracts of country; as France or Scotspond with the calculated length of VP. Now with land, this projection is very suitable. this centre V describe the circular arcs representing the Another modification of the conical projection consists parallels through H, K, P. Also if the parallels are to be in taking, not a tangent cone, but a cone which, having drawn at every degree of latitude, divide HK into 2m equal its vertex in the axis of revolution produced, intersects the parts, and through each point of division describe a circular sphere in two parallels,—these parallels being approximately arc from the centre V. Then divide Pp into 2n equal parts, midway between the centre parallel of the country and the and draw the meridian lipes through each of these points extreme parallels. By this means part of the error is of division and the centre V. thrown on the centre parallel which is no longer represented If the centre V be inconveniently far off, it may be by its true length, but is made too small, while the necessary to construct the centre parallel by points, that is, parallels forming the intersections of the cone are truly reby calculating the coordinates of the various points of presented in length. division. For this purpose, draw through the intersection The exact position of these particular parallels_may be be a pol de ch G de _ h4 (tan; )" a determined so as to give, upon the whole, the least amount | sphere contained by two consecutive meridians the differ of exaggeration for the entire map. This idea of a cutting ence of longitude of which is dph, and two consecutive cone seems to have originated with the celebrated Gerard parallels whose co-latitudes are u and u+du. The sides Mercator, who in 1554 made a map of Europe on this prin. of this rectangle (fig. 24) are pq=du, pr=sin udu, ciple, selecting for the parallels of intersection those of 40° | whereas in the representation pars, and 60°. The same system was adopted in 1745 by De- p'd = dp, pre ' = phdy, the angle at 0 P lisle for the construction of a mạp of Russia. Euler in being = hdj. Now, as the representhe Acta Acad. Imp. Petrop., 1778, has discussed this pro tation is to be similar to the original, jection and determined the conditions under which the dp _P9__du errors at the northern extremity, at the centre, pohod po por sin udu' and at the southern extremity of a map so con du structed shall be severally equal Let whence sinu , and integrating, PHр the co-latitudes of the extreme northern and southern parallels, y, y those of two intermediate c p=2(tan)", k Y Fig. 24. parallels, which are to be truly represented in the where the constant h is to be determined according to the projection. Let OC', Om' (fig. 23) be two consecu requirements of each individual case. This investigation tive meridians, as represented in the developed was first made in 1772 by the German mathematician J. H, cone; the difference of longitude being w, let the GHno Lambert, but in 1825 it was again brought forward by angle at O be hw. The degrees along the meri- c' m! dian being represented by their proper leugths, Gauss in an essay written in answer to a prize question pro CC' =Ć - c; and P corresponding to the pole, let Fig. 23. posed by the Royal Society of Sciences at Copenhagen. A OP=2, then OC=2+c; and so for G, G, C. The true Magazine for 1828 (see page 112), where Lambert's projec translation of this essay is to be found in the Philosophical lengths of G'n' and Gn, namely, wsin y and wsin y, are equal to the represented lengths, namely, hw (2+4) and blem. Again, in a general investigation of the problem of tion comes out as a particular solution of the general prohw (z+y) respectively, whence y and y' are known when "similar representation,” Sir John Herschel, in the 30th h and 2 are known. Comparing now the represented with volume of the Journal of the Royal Geographical Society z the true lengths of parallel at the extremities and at the (1860), deduced as a particular case this same projection. centre, if e be the common error that is to be allowed, then A large map of Russia was constructed and published on e=hw(+c) -w sin c, this system by the Geographical Society of St Petersburg •=-ha(+c+)+ w sin }(6+), in 1862. The relative scale in this development is- hk we get du 2+1(c+)=3d-c) oot (d-c) tan 30' + c). where a is the radius of the sphere. It is a minimum Thus , being known, the common centre of the circles re- 'when u = = cos-1 h, This minimum should occur in the presenting the parallels is given. The value of h is given vicinity of the central parallel of the map; if u. be the coby the equation hsc – c) = sin d' - sin c, and y and ï can latitude of this parallel, we may put (' y j p=k( tan 2 ing a difference of longitude w are in reality equal to hw. Or if we agree that the scale of the representation shall be For instance, to construct a map of Asia on this system, the same at the extreme co-latitudes c, d, then log sin d - log sin c h= 80°, whence z +50° = 15° tan 50° cot 15° = 66°:7. Hence Tog tan fc' - log tan c in this case the centre of the circles is 16°:7 beyond the To construct a map of North America extending from north pole; also h= :6138, so that a difference of longitude 10° latitude to 70°, we may take h= s, and k such as shall a of 5° is represented at O by an angle of 3° 4' 9". The de make the difference of radii of the extreme parallels = 60, grees of longitude in the parallel of 70° are in this map re: namely k = 104-315. The scales of the representation at the presented too large in the ratio of 1:150:1; those in the northern and southern limits are 1.116 and 1.096 respectmid-latitude of 40° are too small in the ratio of 0.933 : 1; ively. The radii of the parallels are theseand those in 10° latitude are too large in the ratio of 1:05 32.801 30° 72.328 43.356 82.255 to 1. 53.177 10° 92-801 Gauss's Projection 40°... 62-728 0° . . 104.315 may be considered as another variation of the conical Having drawn a line representing the central meridian, system of development. Meridians are represented by lines and selected a point on it as the centre of the concentric drawn through a point, and a difference of longitude w is circles, let arcs be described with the above radii as represented by an angle hw, as in the preceding case. The parallels . For meridians, in this system a difference of parallels of latitude are circular arcs, all having as centre longitude of 10° is represented by an angle of two-thirds the point of divergence of the meridian lines, and the law that amount, or 6° 40'. The chord of this angle on the of their formation is such that the representations of all parallel of 10°, whose radius is 92.801, is easily found to be small parts of the surface shall be precisely similar to the 10.792. Now stepping this quantity with a pair of com. parts so represented. Let u be the co-latitude of a parallel, passes along the parallel , we have merely to draw lines and p, a function of u, the radius of the circle representing through each of the points so found and the common centre this parallel. Consider the infinitely small space on the of circles. The points of division of the parallel may be checked by taking the chord of 20°, or rather of 13° 20', See page 178 of Traité des Projections des Cartes Géographiques, by A. Germain, Paris, an admirable and exhaustive essay. See also the worá entitled Coup d'oeil historique sur la Projection des Cartes de 2 Beiträge zum Gebrauche der Mathematik und deren Anruendung, Séographie, by M. d'Arezac, Parix 1863. vol. iii. p. 55, Berlin, 1772. 70°. 20° 60° 50° 2 |