Two other forms of projection which may keep the meridians or the parallels straight are that of Mollweide, in which parallels are straight lines and meridians ellipses, and areas are retained, although with great deformation of the peripheral parts of the map, and the pro jection which simply plots in cartesian co ordinates the spherical or spheroidal nates of points on the surface of the earth with reference to any given great circle or ellipse. Of the projections, one of the most useful and extensively used is that devised by Bonne. It is sometimes called Flamsteed's modified pro but the latter, which is more correctly anson's projection, is really the particular case of the former, in which the parallel of reference is the equator. Bonne's projection differs from the simple conic° in that each of the con centric parallels of latitude is divided into de grees of longitude, the lengths of which are proportional to their true lengths on the sphere, and curved meridians are drawn through each corresponding series of points. The central meridian remains a straight line and the curva ture of the arcs of the successive parallels is the same as that of the middle parallel, the radius of which is equal to the cotangent of the corresponding latitude. Thus, all the meridians intersect a parallel near the middle parallel at right angles and the outlines of areas on the projection are very nearly similar to those on the sphere. It not only preserves the propor tionate equality of the areas on the sphere and projection, but permits of the use of the same linear scale for all parts of maps only a few degrees in extent This projection was adopted as the base for the important map of France constructed by the depot de la Guerre in 1803.
The step from the simple conic to the Bonne was an important one; but, the tireat obliquity of the meridians and parallels in the higher latitudes made it unsuitable for maps of large extent and led to the invention of the °poly conic° projection, which appears to have been conceived by F. R. Hasler, superintendent of the United States Coast and Geodetic Survey, between 1816 and 1820. It was proposed by him as the most suitable base for the maps of the Atlantic Coast of the United States, the great length of which, north and south, to gether with its direction nearly diagonal to the meridians and parallels, made it subject to inadmissible deviations in magnitude and figure upon a Bonne projection.
In the simple polyconic projection, it is as sumed that each parallel of latitude is devel oped upon its own cone, the vertex of which is on the axis of the sphere at its intersection with the tangent to the meridian at the parallel. Theoretically, this involves the employment of an infinite number of tangent cones and the independent development of an infinite number of parallels of latitude. This has the effect of increasing the lengths of the successive degrees of latitudes and longitudes as their distances from the central meridian increases; but the angles at which the meridians intersect the parallels, over the entire map, very closely ap proximate to right angles, thus preserving a close similarity between the figure on the pro jection and the corresponding figures on the sphere.
The mathematical theory of this projection may be briefly stated as follows: Referring to Fig. 3, the equatorial radius being a, the eccentricity c, and the latitude L, the normal produced to the minor axis is a N = (1— e' the radius of the parallel Rp N cos L; and the sides of the tangent cone or radius of the developed parallel r = N cot L.
If is be any arc of the parallel to be developed, and e the angle which subtends it at the vertex when developed sin L; and as the developed parallels are circular arcs.
the co-ordinates of curvature are x== r sin 0, y.==r versin 0=2r sin' I tan 0. The radius of curvature in the meridian is a(1 Rm — sin' L)I whence the length of a degree of latitude is 3600 Rni sin r.
and that of a degree of longitude is 3600 Rp sin r.
For maps of large extent on a small scale, it is sufficient to compute r and 0, x and y, for every whole degree, but for those of small extent, on a large scale, a more detailed pro jection becomes necessary and these values have to be computed to every minute or frac tion of a minute according to the nature of the data to be represented, and the value of the scale adopted.
To make a projection from the data fur nished by a polyconic projection table, draw a straight vertical line for the central merid ian and lay off thereon the distances cor responding to the intervals between the suc cessive latitudes required, and through the points thus obtained draw horizontal lines at right angles to the central meridian. These horizontal lines will be tangent to the developed parallels.
Now suppose, that as in the case of the Mercator projection hereinbefore illustrated, the polyconic projection is required to embrace the coast of Iceland, which lies between lat. 63° and 67° north, and between long. 13° and west from Greenwich, and that the scale of the map is adopted at 1-inch 150 statute miles.
Referring to Fig. 4, inserted under Fig. 2, the central meridian of the map will corre spond to long. 19° west, and the middle parallel will correspond to 65° north. On the cen tral meridian lay off to the north from its intersection with the middle parallel, according to the adopted scale, 69.268 miles for lat. 66°, and 69277 miles for lat. 67°, and in a similar manner lay off to the south, 69.258 miles for lat. and 69248 miles for lat. 63°. Through these points draw hori zontal lines at right angles to the central meri dian, and upon each of them set off to the east and to the west of the central meridian the values of x given in the table for the corre sponding parallels of latitude. Through the points x, xi, etc., thus obtained, draw per pendiculars toward the pole, and on them set off the proper tabular values of y. Through the final points thus obtained, draw continuous curves for parallels of latitude and meridians.
Space limitations in this Encyclopedia make it absolutely impossible to insert a polyconic table computed to even full degrees only, and one giving the values of x and y for greater latitudinal and longitudinal intervals would be more or less useless. Tables of this kind may be readily obtained from the United States Coast and Geodetic Survey, and the United States Hydrographic Office, Navy Department.
Fig. 4 is placed under Fig. 2 in order to af ford a direct comparison of the two projections.
In the rectangular polyconic projection, while the parallels are the same as in the simple polyconic projection and the central parallel retains the same graduation, the merid ians are orthogonal trajectories of the parallels.
Bibliography.— For further information consult the various authorities given under the title MAP in this Encyclopedia. Consult also d'Avezac, 'Coup d'oeil historique sur la projec tion des cartes geographiques) (in the publica tions of the Societe de geographic de Paris for 1683) ; Craig, T., 'A Treatise on Projections' (Washington 1882) ; Germain, A., 'Traite des projections des cartes geographioues) (Paris 1865).
Revised by Nom= Wiasrea