Page:EB1911 - Volume 17.djvu/675

Rh projection (or with any conical projection) what the extent in longitude is. It is clear that this class of projection is accurate, simple and useful.

In the projections designated by (c) and (d) above, absolute errors of length are considered in the place of errors of scale, i.e. between any two meridians (c) the absolute errors of length of the extreme parallels are equated to the absolute error of length of the middle parallel. Using the same notation

h (z + c) − sin c = h (z + c′) − sin c′ = −h (z + c + c′) − sin (c + c′).

L. Euler, in the ''Acta Acad. Imp. Petrop.'' (1778), first discussed this projection.

If a map of Asia between parallels 10° N. and 70° N. is constructed on this system, we have c = 20°, c′ = 80°, whence from the above equations z = 66.7° and h = .6138. The absolute errors of length along parallels 10°, 40° and 70° between any two meridians are equal but the scale errors are respectively 5, 6.7, and 15%.

The modification (d) of this projection was selected for the 1:1,000,000 map of India and Adjacent Countries under publication by the Survey of India. An account of this is given in a pamphlet produced by that department in 1903. The limiting parallels are 8° and 40° N., and the parallel of greatest error is 23° 40′ 51″. The errors of scale are 1.8, 2.3, and 1.9%.

It is not as a rule desirable to select this form of the projection. If the surface of the map is everywhere equally valuable it is clear that an arrangement by which errors of scale are larger towards the pole than towards the equator is unsound, and it is to be noted that in the case quoted the great bulk of the land is in the north of the map. Projection (a) would for the same region have three equal maximum scale errors of 2%. It may be admitted that the practical difference between the two forms is in this case insignificant, but linear scale errors should be reduced as much as possible in maps intended for general use.

f. In the fifth form of the projection, the total area of the projection between the extreme parallels and any two meridians is equated to the area of the portion of the sphere which it represents, and the errors of scale of the extreme parallels are equated. Then it is easy to show that

It can also be shown that any other zone of the same range in latitude will have the same scale errors along its limiting parallels. For instance, a series of projections may be constructed for zones, each having a range of 10° of latitude, from the equator to the pole. Treating the earth as a sphere and using the above formulae, the series will possess the following properties: the meridians will all be true to scale, the area of each zone will be correct, the scale errors of the limiting parallels will all be the same, so that the length of the upper parallel of any zone will be equal to that of the lower parallel of the zone above it. But the curvatures of these parallels will be different, and two adjacent zones will not fit but will be capable of exact rolling contact. Thus a very instructive flat model of the globe may be constructed which will show by suitably arranging the points of contact of the zones the paths of great circles on the sphere. The flat model was devised by Professor J. D. Everett, F.R.S., who also pointed out that the projection had the property of the equality of scale errors of the limiting parallels for zones of the same width. The projection may be termed Everett’s Projection.

Simple Conical Projection.—If in the last group of projections the two selected parallels which are to be errorless approach each other indefinitely closely, we get a projection in which all the meridians are, as before, of the true rectified lengths, in which one parallel is errorless, the curvature of that parallel being clearly that which would result from the unrolling of a cone touching the sphere along the parallel represented. And it was in fact originally by a consideration of the tangent cone that the whole group of conical projections came into being. The quasi-geometrical way of regarding conical projections is legitimate in this instance.

The simple conical projection is therefore arrived at in this way: imagine a cone to touch the sphere along any selected parallel, the radius of this parallel on paper (Pp, fig. 17) will be r cot, where r is the radius of the sphere and is the latitude; or if the spheroidal shape is taken into account, the radius of the parallel on paper will be cot  where  is the normal terminated by the minor axis (the value  can be found from ordinary geodetic tables). The meridians are generators of the cone and every parallel such as HH′ is a circle, concentric with the selected parallel Pp and distant from it the true rectified length of the meridian arc between them.

This projection has no merits as compared with the group just described. The errors of scale along the parallels increase rapidly as the selected parallel is departed from, the parallels on paper being always too large. As an example we may take the case of a map of South Africa of the same range as that of the example given in (a) above, viz. from 15° S. to 35° S. Let the selected parallel be 25° S.; the radius of this parallel on paper (taking the radius of the sphere as unity) is cot 25°; the radius of parallel 35° S. = radius of 25° − meridian distance between 25° and 35° = cot 25° − 10/180 = 1.970. Also h = sin of selected latitude = sin 25°, and length on paper along parallel 35° of ° = h× 1.970 = × 1.970 × sin 25°,

but length on sphere of =  cos 35°,

an error which is more than twice as great as that obtained by method (a).

Bonne’s Projection.—This projection, which is also called the “modified conical projection,” is derived from the simple conical, just described, in the following way: a central meridian is chosen and drawn as a straight line; degrees of latitude spaced at the true rectified distances are marked along this line; the parallels are concentric circular arcs drawn through the proper points on the central meridian, the centre of the arcs being fixed by describing one chosen parallel with a radius of cot as before; the meridians on each side of the central meridian are drawn as follows: along each parallel distances are marked equal to the true lengths along the parallels on sphere or spheroid, and the curve through corresponding points so fixed are the meridians (fig. 18).

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 of Projection de Bonne. It is that on which the ordnance survey map of Scotland on the scale of 1 in. 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 0 be 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 be the radius of the parallel of latitude, we have  = cot 0 + 0 −. Also, if S be a point on this parallel whose co-ordinates are x, y, so that VS =, and be the angle VS makes with the central meridian, then =  cos ; and x =  sin, y = cot 0 − cos.

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 is