2. A cylinder is generated by a line, which is always parallel to a given line, and which has its horizontal or vertical trace in a given

curve.

3. A cone is generated by a line which always passes through a given point, and has its horizontal or vertical trace in a given curve. 4. A sphere is generated by the revolution of a given semicircle about a diameter which is given in position.

5. Although we have employed the generating line and the generating circle respectively for the ordinary constructions of Descriptive Geometry, in representing these surfaces, these are not the only views under which we may consider these surfaces generated. In fact, when we come to consider surfaces generally in reference to the practical applications of Descriptive Geometry to the arts of construction, we shall seldom find these definitions to be the best adapted to our purpose, and hence I have dwelt upon them under this view but very slightly. Even this slight view of them under such an aspect has been adopted more for the sake of an easy transition from one view to the other than for any other reason; though it must he remarked, that even these afford facilities in certain special cases for the investigation of particular problems that are greatly simpler than the general mode of construction that is usually employed; this is, however, only casual, and is no argument in favour of its supplanting generally the more effective method which will be hereafter developed.

6. The most simple of general methods of representation of surfaces for the purposes of plan-drawing is that of supposing the surface, however generated, to be cut by a series of planes parallel to one of the planes of projection; and the curve of section to be represented by its projections on the two planes. The sections are in practice made, most commonly, by planes parallel to the horizontal plane of projection : though, as in Descriptive Geometry, there is no preference of one over the other in theory, and the horizontal is chosen from its being more in accordance with our earlier practice in this class of drawings.

7. For the purpose of showing this more clearly, I shall resume the representation of a cone as already given, and as given by the means here spoken of.

Let v, v2 be the vertex, and a, a, a point in the surface: then, if the point b1 be in the trace of the cone on the horizontal plane of projection, any point in the surface is represented if we join the point b, in the trace, and either of the projections a, or a2 of the point through which the generatrix passes. The construction, and therefore exhibition of the defining elements, of any point in the conical surface, can be effected, provided we have the requisite data for fixing that point.

A

8. If, instead of the preceding representation, we employ the method of horizontal sections, we shall have a series of curves 0, 1, 2, etc., upon the horizontal plane of projection similar to one another; whilst

the vertical projections are horizontal lines bounded by the lines which are the projections of the extreme generatrices, and which are also marked in corresponding numbers 0, 1, 2, etc.

It will be obvious that any point b on the surface of the cone will be as distinctly defined, by its projection on one of these planes, as by the other method.

It will be seen that whether b1 or bą be given, the other can, by means of known processes, be found. A very rational inquiry in this case will be, whether, since the vertical plane contains only straight lines, the constructions effected by means of them might not, possibly, be more simply performed without their aid by the substitution of other operations whose results shall be coeffective with them. In many cases it will be found to be possible, but not in all cases. The principal practical advantage is in those cases where the projections on the vertical plane are upon a small scale in comparison with the extent of those on the horizontal plane, or vice versa. To the engineer it

occurs in military drawings, where the relief is small in comparison with the ground fortified; and to the civil engineer, in the delineation of ground which varies inconsiderably from the horizontal level. To him it is of importance for the purposes of railroad surveys, those of canals and turnpikes, the disposition of buildings, sewerage, supply of water, and parochial maps.

9. To show the general principle, let us suppose these horizontal sections to be made at distances from each other equal to any given unit of altitude. Then the horizontal projections, being drawn and numbered 0, 1, 2, etc., will suffice for the adequate representation of the sections themselves. The vertical projections may be omitted from the drawing, so long as our object is merely to enable the mind to form a general conception of the form of the surface; as it is easy to connect together the ideas of altitude and form-it being, in fact, nothing more than conceiving the several projections of the section (which are equal to them in all respects) to be raised up to the heights 1, 2, 3, etc., designated by the number of units affixed to the curve of projection on the borizontal plane.

10. In actual surveys, however, our object is an inverted one. The form of the surface is not given, but is required to be found by observation in each case. It cannot in general be a regular surface, that is, one whose geometrical genesis can be assigned. The surveys for this purpose will be explained elsewhere: and I shall assume that they can be made in what I am now discussing. The problem, as one of surveying, is:

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.

140

175

150

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 or 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.

SECTION II.

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 Aa 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 segments 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 the line: then this is considered to re

5.8

11.2

present 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