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 λ 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

απ (sa Na')

But sa =

90°, and Na'= 90° —λ'

.. a = f (λ + λ')

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

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 = CAR cot λ, R being the radius of the earth. Suppose that the map is to

І УР

N

x

A

M

For a table which gives the length of a meridional are 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.

embrace D degrees of longitude, the angle c is then known = D sin λ Representing half the chord NN' by a, ci by B, we have in the triangle

CNI

a = r sinc, B = r cos c, AI = r (1
= 2r sin c [see (7) p. 100].

cos c)

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 x2 + y2 = r2. Transferring the origin to 1, the formula for transformation will be x=x+, and the equation of the circle becomes (x+6)2 = r2 — y3

.. x = √ (r + y) (r — y) — 8

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 A; M the point in question, whose co-ordinates are AP=x, PM=y, AX being tangent at A, and perpendicular to the principal meridian CA. We have AB 8, 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 KMN, where MN is the normal N, has for its value ry cot l. Representing the angle ACM by 0, and cм by p, we have

QM=x=p sin 0, cq=p cos 0

Y = PM = BQ + 8 = 8+ BC —— cq=s+ p p cos 0=s+p (1 — cos 0)

=s+2 p sin2 0 [see (8) Art. 72].

We may eliminate p from this last by means of the first xp sin . It becomes

Dividing therefore N into equal parts, and for each point of division finding the corresponding value of x from the above formula, so many points in the arc NN' will be determined.

p is known since pr— 8. It remains only to find @ in order to have for each point like м the values of the co-ordinates x and y, viz:

x=p sin 0, y =8+x tan

N. B. That s must be taken negative when λ < l.

The longitude of м estimated from the central meridian suppose to be A. This will also be the number of degrees in the arc of the parallel.

But N being the normal at the point м, terminating at the polar axis, the radius of this parallel will be (see diagram, p. 365, App. VI.),

N COS À

Moreover, the arc Bм of the projection is of the same length with the arc of the parallel, but the number of degrees in two arcs of the same length will be in the inverse ratio of their radii

which formula serves to determine in the same denominations that A is given. N is known in terms of A from formula

(see App. VI., p. 368), in which e= :0.0816967, log. e= 8.9122052271.

It is easy to perceive now how a map would be drawn according to the projection under consideration. Two lines AC and Ax are first drawn at right angles to each other, intersecting at the middle of the sheet A. Setting out from A, we lay off above and below distances such as AB, respectively equal to the values of s, that is to say to arcs of the meridian corresponding to 1o, 2o, 3o, . . . of distance from a, arcs which go on increasing towards the pole. Next we compute the values of the normals N, N', . . . from degree to degree, the radii p of the projected parallels, and finally the amplitudes of the angles 8, which correspond to values of A and λ, varying also by degrees, whence result the co-ordinates x and y of the vertices of quadrilaterals in which meridians and parallels of latitude distant from each other, respectively the space of 10 intersect. It remains only to lay off these co-ordinates by a scale of equal parts. The sides of the quadrilaterals joining these vertices thus determined may be drawn without sensible error as straight lines.

The territory to be represented by the map is ordinarily too extended to be placed

1 For their values see p. 367, App. VI., also Lee's Tables and Formulas, p. 84, Part II.

LATITUDE BY ASTRONOMIC OBSERVATIONS.

OBSERVATIONS FOR LATITUDE WITH ZENITH SECTOR.

The zenith sector employed on the coast survey of the United States for determining latitude astronomically, is the same as the mural circle already described, p. 306, except that only two small portions of the limb, the one above, the other below the centre, are retained, the rest being conceived to be cut away, to render the instrument more portable. The limb and telescope, instead of being sustained by a wall, are attached to a vertical flat beam of iron, which is capable of reversal about a vertical axis, and also end for end. Long spirit levels can be attached to the

upon a single sheet. It is customary to form the map by the union, border to border, of a series of sheets, the dimensions of which are 8 decimetres by 5 To find the positions of the vertices of the quadrilaterals upon these sheets, the origin of co-ordinates is transferred to one of the corners of the sheet, an operation which consists simply in adding or subtracting 1, 2, 3, . . . times 8 decimetres in the direction of the x, and as many times 5 decimetres in the direction of the y*, according to the place which the sheet ought to occupy in the assemblage. The order of the sheets is marked upon them. Thus the sheet 3 is the one which is second in the horizontal direction, and third in the vertical, estimating from A the intersection of the middle meridian and parallel.

2

As to the inverse problem, to find the latitude and longitude of a point given upon the map, it will be sufficient to draw through the point lines parallel to the sides of the quadrilateral within which it falls, and to determine upon the scale of equal parts, the values of the fractions which the lines represent.

The following formulas are used on the United States Coast Survey, when the extent of the map is not more than 40 of latitude and longitude.

Here the cone, instead of being assumed tangent to one of the parallels of the map, is supposed successively tangent to each, that it may be required to draw upon