PROJECTION OF MAPS. The geographical positions of the vertices of the triangles having been determined by calculation, as above explained, blank maps are prepared with lines upon them representing meridians and parallels of latitude, upon which these points are accurately put down in their true positions, and the maps thus prepared being placed in the hands of the plane table parties, are filled up with the details of the ground which they represent, in the manner described at p. 235 et seq., the points marked upon them, and identified upon the ground by the sunk masonry or pottery of the signals employed in the triangulation becoming the base points or points of ́departure for the operations of the plane table. The mode of preparing these maps in practice, it will be now proper to explain. A spherical or spheroidal surface like that of the earth not being developable, it is impossible to represent upon a plane the positions of places without changing more or less their distances from one another.* When a small portion of the earth's surface is to be represented, the best mode is to conceive the earth to be enveloped by a tangent cone, the and c, computed below; the eighth the latitude L' and longitude M' of c, found by taking the algebraic sum of the two above. The next four horizontal lines of the form contain the computation of the difference of latitude dL between A and c, the first column being the computation of the logarithm of the first term, the second that of the second term, and so on, of the formula dL, at p. 325. The next two lines contain the four terms themselves,1and the next two their combination to form -dL. The first two columns of the remainder of the form contain the computation of - dz; the third column that of the difference of longitude of A and c, viz. dм, and the fourth the correction of this, which is sometimes employed. Applied here the log. dм becomes 2.6873039, and dм 486 748, differing only 0.012 from what it was without the correction. A correction is also sometimes applied to dz, as has been already stated at p. 332, the formula for which is rdм3, in which F = = 12 sin A cos? A sin 1". The computation of this correction in the present example would be as follows: dм38.0619 In which F may be taken from a table previously prepared. The last number is the logarithm of the correction to be applied to - dz. * For the ordinary modes of projecting the hemisphere, see "spherical projections" in Davies' Descriptive Geometry, and for the analytical investigations of the same, see Francœur (Geodésie, 309, et seq.) 1 z being between 90° and 270°, cos z is negative, and .. h is negative. circle of contact being the middle parallel of the region to be embraced, and to suppose the surface of this cone to coincide with that of the earth over the whole extent between the northern and southern parallels of the map. This cone, when developed, becomes the sector of a circle, the portion of which between the two extreme parallels which we have supposed to embrace the surface of contact, will represent the surface of the map. Supposing the earth to be spherical, which may always be done in the projection of maps, its oblateness being so small, and representing the latitude of the middle parallel of the map by λ, and the number of degrees of longitude to be contained in the map by D, it is evident that the absolute length of the middle parallel of the map will be expressed by (see 1st note, p. 153) In the above expression the radius of the earth is unity, and this being the case, the slant height or length of the element of the cone from the vertex to the circle of contact will evidently be the cotangent of the latitude. The arc of the sector, which is expressed by (1), divided by its radius cot λ, gives the length of the arc which measures the angle of the sector to radius unity. The result is and this, which is the absolute length of the arc, must be multiplied by 180° T *to have its value, or the measure of the angle of the sector in degrees, D sin λ (2) then will be the formula for this angle, and the construction of the map will be very simple. It will only be necessary to draw two lines forming the angle expressed by (2), and with a radius equal to cos λ and the vertex of the angle as a centre, the arc representing the mean parallel may πα 180° be described. If the map is to contain d degrees of latitude, then will express the distance between the extreme parallels, and by describing arcs from the vertex of the sector with radii greater and less than 1800 = 59°, 29573 =3437', 74677=206264", 80625. cot by the half of this expression, the extreme parallels of the map will be constructed. The distance between the parallels is then divided into any number of equal parts at pleasure, and arcs described with the vertex as a centre, and passing through the points of division. As to the meridians, they are drawn as straight lines through the vertex, and through points of division equally distant from one another upon the arc of the middle parallel. This construction is so simple, that it is generally preferred to any other, and the greater part of maps of kingdoms and states are drawn upon this system. For greater precision, the cone, instead of being taken tangent to the sphere, is partially inscribed in it by making it pass through the two extreme circles of latitude, so that these circles shall be sections of the cone perpendicular to its axis. Imagine a meridian section of the cone and sphere, the angle a formed by the element of this section with the axis will be measured by half the difference of the arcs included between its sides (Geom. Ex. 30, p. 48). Supposing a and a' to be the points in which the element intersects the meridian section, and X and X' their latitudes, N being the place of the north pole, and s the south, the expression for the measure of the above angle will be Now in the right angled triangle formed by the element of the cone, the axis and the radius of the parallel, which last is equal to cos λ, we have for the length of the element terminating at a and for the length of the element terminating at a', The lengths of the elements of the developed sector being thus known, the rest is as above. Still better, the cone may be made to pass through two parallels, situated at half distance between the middle parallel and the extremes; the cone would then be partly internal and partly external to the sphere.* It was in this way that Delisle constructed the great map of Russia. PROJECTION OF FLAMSTEED. This consists in drawing a straight line vertically to represent the central meridian of the map, laying off upon it equal distances say 10, and through the points of division drawing perpendiculars to this meridian line, which represent parallels of latitude; then laying off upon these parallels distances bearing the same proportion to the distances on the meridian as the cosine of each latitude does to radius unity; finally, drawing through the points of the same graduation, thus determined, curved lines which will represent the other meridians. The oblateness of the earth may be taken into the account in this method, by laying off in the central meridian not equal distances, but increasing towards the poles in the same proportion as the degrees of the meridian increase. For the demonstration of the formula see App. VI., p. 367. The formula itself is δ πα 180° (1 — w 3w cos 2λ)* In which a = 6377397.15 metres, log = 6.8046434637; log w = λ = latitude. 7.5233789824; The objection to the method of Flamsteed is that it distorts somewhat the regions distant from the central meridian. METHOD OF THE FRENCH DÉPÔT DE LA GUERRE. This is a modification of the conic projection already given, and is that now in use on the coast survey of the United States. The radii of the arcs of circles representing the parallels upon the map being too long to be conveniently described from a centre, they are determined by points. Let there be drawn in the middle of the sheet the perpendiculars CA, NN'; NAN' represents the middle parallel of the map. Then is known the radius ? = CA = R cot λ, R being the radius of the earth. Suppose that the map is to N IYP N A M For a table which gives the length of a meridional arc in any latitude in yards, and a table which gives the length of a parallel, see Lee's Tables and Formulæ, Part II., p. 84. = D sin λ embrace D degrees of longitude, the angle c is then known Representing half the chord NN' by a, cr by B, we have in the triangle CNI a = r sinc, B = r cos c, AI = ↑ (1 cosc) The extreme points N and N of the arc to be described NN', are thus determined, and the point a, in which it intersects the meridian. Now for other points, such as м, a distance IP = y is laid off from 1, and a perpendicular PM is drawn in length equal to x, the value of x being expressed by the following formula x = √(r+y) (r—y) — ß* * The demonstration of this formula, which requires a knowledge of Analytical Geometry, is as follows:-The equation of the circle, the origin of co-ordinates being at c, is x + y2= r2. Transferring the origin to I, the formula for transformation will be x = x + B, and the equation of the circle becomes (x+6)2 = r2 — y2 . . . x = √ (r + y) (r — y) — B The formula in the text. The above method does not take into account the earth's oblateness; the following is the generalization of the theory. Let c be the centre of the projection, AK the middle parallel, the latitude of which represent by ; BM another parallel, whose latitude is ; м the point in question, whose co-ordinates are AP, PM y, AX being tangent at A, and perpendicular to the principal meridian CA. We have AB8, the distance in latitude between the two parallels, this length s being known by equation (5) p. 367, App. VI. The radius car is also known, being equal to KM in the diagram on p. 365, which, in the right angled triangle KмN, where мN is the normal N, has for its value r N cot l. Representing the angle ACM by 0, and cм by p. we have QM=x=p sin 0, cq=p cos 0 y=P M = BQ + 8 = 8+ BC — cq=s+ p p cos 0=s+p (1 — cos 0) =s+2p sin2 0 [see (8) Art. 72]. We may eliminate p from this last by means of the first xp sin 0. It becomes |