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THE subjects of the following pages have been taught orally at the Military Academy for many years; but, for the saving of time, and the convenience of the pupils, it has been thought best to clothe them in a printed dress; and as, in this form, the volume might be found useful in other schools, as an application of descriptive geometry to practical questions, it was also thought well to have it published.
ONE PLANE DESCRIPTIVE GEOMETRY
AS APPLIED TO
1. THE method now in general use, among military engineers, for delineating the plans of permanent fortifications, is similar to the one which had been previously employed for representing the natural surface of ground in topographical and hydrographical maps; and which consists in projecting, on a horizontal plane at any assumed level, the bounding lines of the surfaces and also the horizontal lines cut from them by equidistant horizontal planes, the distances of these lines from the assumed plane being expressed numerically in terms of some linear measure, as a yard, a foot, &c.
2. Plane of Reference or Comparison. The assumed horizontal plane upon which the lines are projected is termed the plane of comparison or plane of reference, as it is the one to which the distances of all the lines from it are referred, and as it serves to compare these distances with each other and also to determine the relative positions of the lines.
3. References. The numbers which express the distances of points and lines from the plane of comparison are termed references. The unit in which these distances are expressed is usually the linear foot and its decimal divisions.
As the position assumed for the plane of comparison is arbitrary, it may be taken either above or below every point of the surfaces to be projected. In the French military service it is usually taken above, in our own below the surfaces. The latter seems the more natural and is also more convenient, as vertical distances are more habitually estimated from below upwards than in the contrary direction. Each of these methods has the advantage of requiring but one kind of symbol to be used, viz: the numerals expressing the ref
erences; whereas, if the plane of comparison were so taken that some of the points or lines projected should lie on one side of it and some on the other, it would be then necessary to use, in connection with the references, the algebraic symbols plus or minus to designate the points above the plane
from those below it.
As the distances of all points are estimated from the plane of comparison, the reference of any point or line of this plane will therefore be zero, (0.0); that of any point above it is usually expressed in feet; decimal parts of a foot being used whenever the reference is not an entire number. When the reference is a whole number it is written with one decimal place, thus (25.0); and when a broken number with at least two decimal places, thus (3.70), (15.63). In writing the reference the mark used to designate the linear unit is omitted, in order that the numbers expressing references may not be mistaken for those which may be put upon the drawing to express the horizontal distances between points.
The references of horizontal lines are written along and upon the projections of these lines. All other references are written as nearly as practicable parallel to the bottom border of the drawing, for the convenience of reading them without having to shift the position of the sheet on which the drawing is made.
This method of representing the projections of objects on one plane alone has given rise to a very useful modification of the one of orthogonal projections on two planes, and has been denominated one plane descriptive geometry; the plane of comparison being the sole plane of projection; and the references taking the place of the usual projections on a vertical plane. By this modification the number of lines to be drawn is less; the graphical constructions simplified; and the relations of the parts is more readily seized upon, as the eye is confined to the examination of one set of projections alone.
But the chief advantage of it consists in its application to the delineation of objects, like works of permanent fortification, where, from the great disparity of the horizontal extent covered and the vertical dimensions of the parts, a drawing, made to a scale which would give the horizontal distances with accuracy, could not in most cases render the vertical dimensions with any approach to the same degree of accuracy; or, if made to a scale which would admit of the vertical dimensions being accurately determined, would require an area of drawing surface, to render the horizontal dimensions to the same scale, which would exceed the con
venient limits of practice. Taking for example an ordinary scale used for drawing the plans of permanent fortifications of one inch to fifty feet, or the scale, the details of all the bounding surfaces can be determined with accuracy to within the fractional part of a foot, whereas a vertical projection to the same scale would be altogether too small for the same purposes.
4. Point and Right Line. To designate the position of a point, Pl. 1, Fig. 1, the projection of the point and its reference are enclosed within a bracket, thus (28.50). This expresses that the vertical distance of the point from the plane of reference is 28 feet and fifty-hundredths of a foot. The position of a right line oblique to the plane of reference is designated by the projection of the line, and the references of any two of its points. Thus in Fig. 1 the points a and 3, upon the projection of the right line, with their respective references (25.15) and (28.50), determine the position of the line with respect to the plane of reference.
When the line is horizontal, or parallel to the plane of reference, its projection, with the reference of one of its points, will be sufficient to designate it, and fix its position with respect to the plane of reference. Thus in Fig. 1 the reference (25.15), written upon the projection of the line, expresses that the line is horizontal, and 25.15 feet from the plane of reference.
5. For the convenience of numerical calculation, the position of a line, with respect to the plane of reference, is often expressed in terms of the natural tangent of the angle it makes with this plane; but as this angle is the same as that between the line and its projection, its natural tangent can be expressed by the difference of level between any two points of the line, divided by the horizontal distance between the points. Now, as the difference of level between any two points of the line is the same as the difference of the references of the points, and the horizontal distance between them is the same as the horizontal projection of the portion of the line between the same points, it follows, that the natural tangent of the angle which the line makes with the plane of reference is found by dividing the difference of the references of the points by the distance in horizontal projec iton between them.
The vulgar fraction which expresses this tangent is termed the inclination, or declivity of the line. Thus the fraction would express that the horizontal distance between any two points is six times the vertical distance, or difference of their references; the fraction, that the vertical distance