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On any given azimuth (or compass-bearing) to find the horizontal distances of all the successive pairs of points which have a difference of altitude of a foot.
It is not always possible, from the interference of obstacles, either to vision or motion, to obtain such observations by means of direct levelling; but when this is not possible, the survey can be so arranged that, when laid down for an adequate number of properly selected observations, these points can be assigned by numerical or graphic interpolation, whichever we choose to employ.
11. If we suppose a sufficient number of these points to be found and plotted on the drawing, a curve line may be traced by hand and humoured to suit our memory of the ground, through all of those marked of the same altitude; this will be a representation of that horizontal section of the ground which is at the assigned altitude from the plane to which the observations had reference. This line is technically called a contour.
This method of delineation has long been in use for marking the soundings or depths of water on coasts. It led, therefore, to the adoption of the plane of mean low water as that from which depths were estimated, and of course from which the heights of the adjacent coasts were likewise estimated. Still, though so long practised in such coast-charts, its principles and convenience of application were not perceived by surveyors and topographers till a recent date. It was, indeed, proposed (the suggestion being from a different source) a century ago to the French Academie ; but it was only subsequent to the general peace that it was adopted by any public body. In 1818 the Bureau de Cadastre laid it down as the plan upon which the general topographical survey of France should be delineated ; and in 1838, it was adopted in the Ordnance Survey of Ireland. Amongst the French scientific writers on projection, I have found only two who have given even a sketch of it-Leroy and Olivier; and in English, not one who has given even an intelligible description of the process, either as a mental theory or a practical operation.
12. The plane of mean low water is not, however, found to be a convenient one for inland surveys, on account of the great difficulty of finding the real height above that plane at any one point in the survey. For military plans a plane above the highest point of ground is often used, as that to which the points of the survey shall be referred; and the system is estimated by depths below this plane, instead of heights above a horizontal plane taken on the ground. The one or the other may be used in our reasonings, inasmuch as when the distance of the two supposed planes is known, the distance of a point from one of them being given, its distance from the other is found by simply subtracting the given distance from the distance of the two planes.
13. The plane to which the system is referred is called the plane of comparison : and the reference of a system of points given in respect to one plane, to another given plane, is called changing the plane of comparison.
14. As it will preserve a closer analogy to the projections with which we are most familiar, we shall in our investigations refer the system to a horizontal plane below the system itself,
15. Let us suppose, then, that from actual surveys we have obtained the following lines of 100, 120, 140, 160, and the highest point of 175 feet above the plane of comparison. It will appear at once that the nearer these lines approach to each other, the more steep the ground is in that region, and the more distant the less steep.
16. It will also be apparent that if at any point a, the declivity be required, we have only to draw a perpendicular to the curve at a, (or to its tangent), and making a, a, perpendicular to a, a, and equal to 20 feet of the scale, and joining a, a, the angle a, a a, is the elevation of the hill-side at a; or which comes to the same thing, its declivity (or depression below the horizon) at the point of which a, is the horizontal projection.
17. When the declivity is considerable, or the lines of contour numerous, the projections become inconveniently close to each other. In this case it is usual to measure the vertical and horizontal distances by different scales, the horizontal being 10 or 100 times that of the vertical. This, though it alters the appearance to the eye of the general declivity of the ground, renders the constructions which are often necessary to be made on such a map for engineering purposes more distinct and consequently more certain. We shall only require a final transformation of the results, which is easily understood and readily made.
18. In the same manner, if the declivity be very small, and both horizontal and vertical measures be made on the same scale, the contours will expand themselves to an inconvenient extent. This is remedied by taking the horizontal scale tho or to of the vertical one: for though to the eye it conveys the notion (if unapprised) of greater than the actual declivity of the ground, its operations are more easily effected than otherwise. The same remark applies to it as to the preceding case, respecting the final result.
19. When both scales are the same, we shall call it the natural system of contours : when the horizontal scale is diminished, we shall call it the compressed system of contours : and finally, when it is enlarged, the expanded system of contours.
THE GRADUATION OF HORIZONTALLY PROJECTED LINES.
1. Let the line here graduated be that by which altitudes above XY are laid down ; and let AB be a segment of a line, the plane AB ba
its projecting plane on the horizontal, and ab its actual projection. Then Xa and Bb the altitudes above the plane XY, are assumed as being measured or deduced from actual survey, and likewise the position of a and b. Then, obviously the line AB is given in all respects; and any required particular can be obtained either by construction or calculation.
It will not be necessary to actually construct the line in its real position for any of the practical uses to be made of it, though for the purposes of reasoning we must sometimes recur to the eidograph and turn the system upon the horizontal plane.
If, in the natural system, Aa, be a unit of the altitude scale, then a, b, or aß, will be the unit of the projection scale corresponding to it, and Ab, a unit of the declivity-scale, also corresponding to Aa. Instead of using the same scale-unit for all the three lines, this method employs the same number of units in designating any three corresponding seg. ments of the lines. Our first business is, the graduation of such lines adapted to the purposes in view.
Suppose that, without numerically specifying the length of ab, we merely represent it on any given scale, as below; and write the numbers 11.2 and 5.8, which designate the altitudes at a and b of points A, B, in ga the line: then this is considered to represent the line and points A, B, in it—these numbers being either referred to the fore-mentioned scale, or to another which has a specified relation to it.
The relation of the scales being given, together with the position of
a and b, with their respective numbers 11.2 and 5:8; the line AB is said to be figured: and these numbers are called the figures of the line; and a, b, the projections of the figures.
2. In our general investigations we shall view them as referred to an arbitrary point p in ab, as the origin of abscissas, and the altitudes as their corresponding ordinates. We shall thus, denoting A by x, y, B by X,Y2, any other point by xy, and the inclination by o, obtain at once the following relations :
(y – y.) (x; – x) = (x − x) (y. - y). ... (1),
X, - Xg The two first of these are identical, and either may be used. From either of them x or y may be found when the other five are given. The third gives the inclination of the figured line to the plane of comparison.
Again, if we make y = 0, in either the first or second equation, we obtain the value of x, which corresponds to the trace of the line upon the plane of comparison. It is
me Y,ks – Yzili . . . . . . . (4) ;
Yi - Yo and giving to x this value, we obtain the zero point of the projected line.
Let u be a unit on the vertical scale : then on the projection and on the line itself the corresponding units will be respectively,
Up = u cot 0; and u, = u cosec ... (5). If, however, the scale be compressed or expanded n times, the projective unit will respectively be
_ u cot e.
2 ; or Un = n. u cot 0 . ... (6); and the corresponding units on the line itself Uy = cosec 0, and
· · · · · · · (7). U1 = nu cosec o. The units on the line itself occur less frequently than in the other case, in actual researches.
One or two examples adapted to numerical data will suffice to show the use of these formulæ.
3. We shall now proceed to show the constructions which are involved in the use of this mode of viewing a straight line by means of its projection and figuring.
Denote by a', b' the figuring of the points, a, b. Set off aA, 6B perpendicular to ab from the selected scale; and draw BA to meet ab in 0. Then 0 is the zero point of the line AB, as represented on its projection ab.
To graduate it, make BB equal to the vertical unit, and draw Bl parallel to AB; then 01 is the projecting S t oritet
unit. Step this distance from 1 to 2, from 2 to 3, etc.; then we have the scale to which we must recur in all that relates to this line, and to all others in the same drawing which have the same inclination to the plane of comparison.
If the vertical scale be increased (suppose tenfold, as is usual in delineations of ground of slight inclination), then these divisions 01, 12, etc., will be, not units of projection, but as many units as the vertical scale is increased. Thus, if BB were 10 vertical units, 01 would require sub-division into 10 parts, and so on. Also, if the drawing were expanded to represent very steep ground, the units of projection would be enlarged in the same ratio.
The line so graduated is called the scale of the line AB; and it is the first step to be performed in any problem in which the line is concerned.
It must be carefully recollected, however, that all the lines in the same drawing must be referred to the same vertical unit. It will, however, be obvious that no difference of constructive process results from using one scale rather than another; but a difference in the ultimate estimation of the values of the quæsita will be necessary, and a difference in the visual result will also ensue. In the following problems the natural system will be used in general; and where otherwise, it will be specially notified.
PROPOSITION I. Through a given point, to draw a line making a given angle with the plane of comparison, and to graduate it.
This problem is evidently indeterminate in one respect, viz., that such lines will radiate in all directions round the given projection ; but whichever be selected, the process will be the same; and if found for one, may be transferred to all the others by concentric circles, whose centre · is the projection of the given point.
Let a be the projection; draw any line aa; perpendicular to which
make a A the scale length of the altitude of the given point A ; and in it take aa, a unit of that scale ; and make ab the cotangent of the given angle to that unit; join a,b and parallel to it draw A0. Then O is the zero point of the line required, and if from 0, the distance ab be set off to 1, 2, 3, . . . successively, the projection of the line is constructed and graduated.