it cuts ni in one point, and ni produced in another, the point on the other side of sb furnishes no solution. When the circle touches the line ni, the two solutions are reduced to one. Finally, when the circle and the line ni do not meet, the problem is impossible. 2. If from the points p and q perpendiculars be let fall upon gi, and from their intersections with gi, perpendiculars be drawn to sb, these lines will respectively pass through p' and q'; because p and q being in their true positions when brought down as p' and q' in the plane asb, by the planes in which they are turning about sb, these points will move in vertical planes perpendicular to sb. HORIZONTAL PROJECTION. 1. In this projection, which has already been referred to (Des. Geo. 3), the positions of different points in a body are determined by their projections on a horizontal plane, and their distances from that plane. It is specially adapted to military drawing of a country and to the general representation of military works, because in these, heights and horizontal distances are commonly given numerically, and when sought are frequently to be so determined; and this projection allows us to call in arithmetic to the aid of geometry. Descriptive geometry is better adapted to particular details of military works. 2. A horizontal plane being assumed as that to which all points are referred by perpendiculars from them to the plane, the height of a point above that plane is called the Index of Height or Index of Level of that point, or sometimes simply its Index. The position, therefore, of a point is determined by its projection on the horizontal plane, and its index of level; and in order to express this position in a plan, the number denoting this index is placed on one side of the projection of the point. Thus the point 13.0 (Plate IX. fig. 1) denotes a point whose index of height is 13′0, or which is at the height 13.0 above the zero horizon plane, its position on the plan showing its horizontal projection. When a point is to be referred to in a plan by a letter, the letter is placed on one side of the point, and the index of height may be placed either on the other side of the point or below the letter: thus the point a.13'0 or a13 would refer to the point whose horizontal projection or position on the plan is marked by the letter a, and its height above the zero horizontal plane is 13. For greater distinctness, the letter and its index of height may be inclosed in brackets: thus the point [a.13] or [a1]. In naming points, the capital letters A, B, C, &c. will designate the points themselves in space; the small letters a, b, c, &c. their horizontal projections or positions on the plan; and the Greek letters a, B, y, &c. their indices when their numerical values are not stated, so that the Greek letters will always represent numbers. 3. To avoid negative indices to points on the plan, it is necessary that the horizontal plane assumed as the zero of level should be either above, or else below every point represented on it. By assuming the zero level below every point to be represented on the plan, the vertical co-ordinates are measured from below upwards; the indices are those of height; and we adhere to the method adopted in maps and topographical plans, where the prominent points are referred to the sea level: whereas, by assuming the zero level above the highest point, the vertical co-ordinates are measured downwards the indices are those of depression; and our assumption is not only opposed to the principle adopted in topographical plans, but to that always employed in Descriptive Geometry; a serious objection as regards the connexion of these methods. For these reasons, in all that follows, the zero of level is assumed below the lowest point in the plan, and the indices are always those of height*. 4. The position of a straight line in space is determined when its horizontal projection and the vertical ordinates to two points in it are known; and this position is denoted in the plan by the projection of the line, and by annexing the indices to two points in it. A straight line in space is designated in a manner similar to that adopted for a point: thus the straight line [a.a, b.ẞ], or [αa, bg], or simply a be, denotes the straight line joining the points [a.a] and [b.3], or a and b in space (fig. 2); or the straight line joining the points A and B in space, whose horizontal projections are the points a and b in a plan, and whose indices of height are a and B respectively. 5. The position of a curve in space is determined when its horizontal and vertical projections are known, or instead of the latter when the vertical ordinates or indices to every point in it are known, but this position cannot be represented by the horizontal projection and indexed points, because the indices can only be annexed to points at certain intervals. However, the curve may be considered as known when these points are so near that it may be traced by means of them without sensible error, should this be required. 6. When the curve is to be contained in a plane given in position, its horizontal projection with the position of the plane will be sufficient to determine it completely. If all the points of the curve be on the same level, a single index and the horizontal projection are sufficient to determine its position. * Should cases occur in the construction of works where it may be more convenient to assume the zero level above the highest point, no difficulty can present itself to the engineer who well understands the principles of the method, in making the changes which this assumption will require. ; 7. Since a plane is completely determined when the position of three points in it, not in a straight line, are given, the horizontal projections of three points with their indices are sufficient to determine the position of the plane in space. In this manner the position of a plane might be represented by a triangle on the plane of projection, having annexed to each angular point its index of height. This mode of representation, however, is ill-adapted to the determination of any number of points or of lines in the plane; but if the horizontal trace of the plane with the projection and index of any point in the plane be given, this object will be more readily attained, for by drawing from the projection of the given point a perpendicular on the trace, this, with the given index of the point, will determine the plane's inclination. The same object will be attained by having given the projection and index of any horizontal line in the plane and those of a point in the plane, or the projection and index of each of two horizontal lines in the plane. A plane may, on this principle, be very conveniently and usefully represented by the projections of two horizontal lines in it with their indices annexed, and the common perpendicular to them, as in figure 3. n 8. If in the projection of a perpendicular to horizontal lines in a plane, points be taken at equal distances from each other, the indices of these points will be in arithmetic progression. For let the plane, and also a horizontal plane intersecting it, be cut by a vertical plane, perpendicular to its horizontal trace; let AoA and Aon (fig. 4) be the sections of these planes, the plane of the paper being in this case supposed vertical; then AA, being perpendicular to the plane's horizontal trace, will be a line of greatest inclination in the plane, and Aoa, will be the horizontal projection of AA, Taking then a1a2, 23, 34, &c. equal to each other, drawing a,A1, α2A2, a.Ag, a4A4, &c. perpendicular to A ̧„, and Ã ̧С1, Â1⁄2Ñ2, AC3, &c. parallel to it, AC1, Ã ̧С2, A4С ̧, &c., which are the differences of the indices to the points A1, A2, A3, A4, &c., are all equal. 19 9. When the distances a a2, a2a, ɑɑ4, &c. are so taken that the differences AC1, A3C2, A4C3, &c. of the indices to the points A,, A2, A3, A4, &c. are the unit of length, then the ratio of A2С1 to A1C1 or a1a; of AC2 to aa; &c. is a measure of the plane's inclination; and the line aan (the projection of a line of greatest inclination on the plane) having the indices of the equidistant points a1, a2, ag, a4, &c. marked on it, is called the Scale of Slope of the plane. 2 10. We may thus represent a plane by a single straight line drawn in the direction of the greatest inclination, that is, at right angles to the horizontal straight lines in it, and marking, in this line, the points the differences of whose indices is the unit of length. To distinguish, however, the representation of a plane from that of a line, it is customary to draw two parallel lines at right angles to the horizontal lines in the plane, with divisions on them corresponding to points whose indices differ by an unit. Thus the double line a10, a16 (fig. 5) represents a plane the greatest slope on which is in the direction of this line, and in which, therefore, the horizontal trace and all horizontal lines are perpendicular to the line a10, a16 It is to be borne in mind that the vertical ordinate or index of height to the point a10 is 100, and that to the point a16 is 16.0, as marked in the scale, that is, the line a10, a16 being drawn in a plan which is constructed to a certain scale of feet, yards, or other unit of length, the height of the point a10 is 10 such units above the horizontal plane taken as the zero plane, and the point a16 is 16 such units above the same plane. So that dividing the line a10, a16 into 6 equal parts in the points 110, 120, 130, 140, 15'0, each of these points is one of the units higher above the zero plane than the preceding. The distances between these points as measured on the scale of the plan give the degree of slope of the plane represented by a10 @16 Thus if each of the intervals a.10, 11·0; 11·0, 12·0; 12·0, 13'0; &c. be 2 on the scale of the plan, the slope of the plane will be 1 in 2; if each interval be 3, the slope will be 1 in 3; if each interval be 5·7, the slope will be 1 in 5-7, or 10 in 57; and so on. It is necessary to distinguish clearly between the scale of a plan and the scale of slope of a plane represented on it. The intervals between the divisions on the scale of slope are always to be measured on the scale of the plan, and the numbers attached to these divisions simply indicate the heights of the points in the plane, of which the points of division are the projections. The angle of inclination of a plane may be deduced immediately from its scale of slope, since the ratio which gives the intervals in the scale of slope is in fact the tangent of the angle of inclination. We should observe here that a plane may be represented by its scale of slope in any part of a drawing where it may be most convenient to place it, or where it may best serve to indicate the particular plane to be represented by it. 11. When horizontal lines, or, as we may frequently term them, horizontals, in the plane are required to be represented on the plan, this is of course done by drawing them through the required points perpendicular to the scale of slope, as in figure 6. We may remark with reference to the scale here and in the last figure, that if these scales occurred, as they are represented, in the same plan, they would denote planes inclined in opposite directions, the one being a descending, the other an ascending plane; and the interval between two indices in fig. 5 being two-thirds of the interval in fig. 6, the slope, as measured by the differences of level in a given horizontal interval, is in the former plane one and a half times as great as in the latter. 12. There are two cases in which the scale of slope cannot be applied when the plane is horizontal; and when it is vertical. În the first case, a single index fixes the plane's position; and it may be conveniently represented by two straight lines making any angle with each other, and having the common index at the angular point and at their further extremities. In the second case, the trace of the plane indicates its position, and we have no other means of representing it on a plan. 13. The scales of slope of two planes represented on a plan give immediately the connexion of the planes; for the perpendiculars to the scales being the projections of horizontal lines in the planes, the intersections of those which are on the same level, or which have the same index, must meet in the straight line which is the projection of the intersection of the two planes. Thus let ab, cd (fig. 7) be the scales of slope of two planes: produce ab, cd, and extend the scales so as to have the points with corresponding indices in each; the perpendiculars to the scales drawn through these points will be the horizontals in the two planes, and each pair will meet in the straight line pq, which is therefore the projection of the intersection of the planes. The divisions on the line pq, with their indices, give the scale of slope of the line of intersection of the two planes. It is to be understood that the angles 444, 555, 666, &c. are the projections on the plan, of the intersections of the two planes, with horizontal planes at the heights 4, 5, 6, &c. from the zero plane. 14. If instead of two planes there be several, their connexion will be represented in the plan by the projections of horizontal lines in the planes at equal intervals of height, or the projections of the intersections of the planes with horizontal planes at equal intervals from each other; and the intersections of the corresponding projections will be the projections of the intersections of the respective planes. The straight lines at right angles to the projections of the horizontal lines in the planes will give the scales of slope in the respective planes. Thus fig. 8 represents on a plan the connexion of several planes with their intersections and scales of slope, by means of the projections of horizontal lines in them at equal intervals of height. 15. As nothing limits the number of planes that may be thus represented, we may conceive their number to be so increased, both laterally and upwards or downwards, as approximately to coincide with any curved surface, and thus to represent that surface, whether regular, as a geometric surface, or irregular, as an undulating or hilly country. Such a representation of a surface would, however, only be complicated by the introduction of the intersections of the approximating planes. Without these the surface would be represented simply by the projections of the intersections of its approximating planes with equidistant horizontal planes. Thus supposing the planes in fig. 8 to touch a curved surface so as to form a first approximation to it, the projections of the level lines without the |