the surface of a portion of country, the lines drawn from one contour to another may be considered to represent roads which, in every part, have the same degree of inclination. Of two of these commencing at the same point, one is carried down in a succession of zigzags. PROBLEM XXIII. To determine the intersection of a given plane with a surface given by its horizontal contours. This is determined on the same principle as the intersection of two planes (Prob. IX.); the several horizontals in the plane, having the same index as the given contours of the surface, being drawn at right angles to the plane's scale of slope, their intersections with the corresponding horizontal contours of the surface will evidently give points of intersection of the plane with the contoured surface; and the line joining these points will be their line of intersection. In joining the points of intersection it is, however, to be observed, that those correspond to the same intersection of the plane with the surface, in which the horizontals of the plane, taken in the same direction, having been exterior to the surface, penetrate it, and become interior; or having been interior, pass out and become exterior; and it is the succeeding corresponding points which are to be joined, to obtain the intersections of the plane with the surface. In fig. 29 the intersections of a plane, given by its scale of slope, with a surface given by its horizontal contours, are represented by the lines joining the corresponding intersections of the dotted horizontals of the plane with contours of the surface, which have the same index. PROBLEM XXIV. To find the intersection of a given straight line with a surface given by its horizontal contours. If through the given straight line we conceive a plane to pass, of which this line is the scale of slope, and find the intersections of this plane with the given surface, the point where the projection of the given straight line meets the projection of the line of intersection of the plane and the surface will evidently be the projection of the point of intersection of the given straight line with the given surface (fig. 29). PROBLEM XXV. To find the intersection of two surfaces given by their horizontal contours. This will evidently be determined by finding the intersections of the horizontal contours of one surface with the corresponding ones of the other, and drawing a line through those points of intersection (fig. 30) in a manner precisely similar to that by which the intersections of a plane with a surface have been determined (Prob. XXIII.). 103 ISOMETRIC PERSPECTIVE. 1. The leading feature in descriptive geometry is that all lines drawn on the paper are represented in their true dimensions, and, when in the same plane, in their true relative positions. Although the method we are about to describe has not this advantage, and, besides, cannot, conveniently at least, be applied to the solution of problems, it has that of exhibiting a conventional picture, readily understood, and in which lines in the three principal directions are represented to the same scale. In machinery*, in astronomical and other instruments, and in buildings, the principal parts to be represented are very commonly in planes at right angles to each other, and it is therefore to such objects that the method is peculiarly applicable. 2. The principle of Isometric Perspective will be best understood by conceiving a cube to be so placed on the plane of the paper, that its diagonal is perpendicular to that plane. In this position, the three lower edges of the cube (and also its other edges) make equal angles with the plane of the paper, and consequently their orthographic projections on it will be equal straight lines radiating from a point; as the angles between these edges are equal, and their planes are equally inclined to the plane of projection, the angles between the projections of these edges will be equal, and therefore each of them will be a third of four right angles. The diagonal of the cube being at right angles to the plane of projection, calling an edge of the cube 1, its isometric projection is equal to the perpendicular let fall from the right angle, on the opposite side, of a right-angled triangle of which the sides are 1, 2 and 3: it is therefore equal to 3. ABDEFGC (Plate XI. fig. 1) represents such a projection of a cube as has been described. One diagonal of this cube being perpendicular to the plane of the paper, its orthographic projection on that plane will be a point. O denoting the projection of the lower extremity of this diagonal, and C that of its upper extremity, OD, OF, OA will represent the projections of the cube's lower edges; CB, CG, CE those of its upper edges; and AB, BD, DE, EF, FG, GA those of its other edges which here bound its picture. All these edges being equally inclined to the plane of projection, their projections must all be equal. * The method originated, with its inventor, the late Professor Farish, in attempts to give such pictures of machinery, as should enable a person, acquainted with the principles on which the pictures were drawn, to put the several parts in their proper relative positions, without any other assistance. Cambridge Transactions, vol. i. p. 1. 4. The angle contained by any two intersecting edges is a right angle, but the angle contained by their projections cannot be the same in all cases: the projections of all right angles about the points O and C, and of all angles opposite to these, will be a third of four right angles, or 120°; and those of all others will be a third of two right angles, or 602, since the projection of each face of the cube is a parallelogram. 5. All the faces of the cube are projected in, and represented by equal rhombuses, but the projections of the diagonals of these faces are not equal. The projections of diagonals which are parallel to the plane of projection are equal to those diagonals, that is, are each equal to 2, an edge of the cube being 1; and the projections of diagonals which are inclined to the plane of projection are equal to the projections of an edge, that is, to 6. Thus each of the dia 1 3 gonals AF, FD, DA, EB, BG, GE, which are projections of diagonals parallel to the plane of projection, is equal to 2; and each of the diagonals GO, EO, BO, DC, AC, FC, which are projections of inclined diagonals, is equal to 6. 1 to✔ 6. If OD, OF, OA, the projections of the lower edges of the cube, be produced, then OX, OY, OZ represent similar projections of three axes at right angles to each other, with reference to which the position of any point in space may either be given or be required to be determined. Lines measured along these axes, and likewise any lines parallel to them, will all be to the same scale; so that the position of a point in space, with reference to three rectangular axes, being given in terms of any unit of length, its position will be correctly represented by measuring, by means of a scale of equal parts, from O along the respective axes or lines parallel to them, lines expressing the distances from the origin or point of the intersection of the axes. Thus let the distances along the axes OX, OY, OZ, which determine the position of a point, be 7, 5 and 8 inches respectively; taking from a scale, OA=7, OB=5 and OC=8 (fig. 2), and drawing AD, DE parallel and equal to OB, OC respectively, E is the position of the given point. Completing the construction in the figure, we have the representation of a rectangular parallelopiped such that each edge may be measured to a given scale. 7. We should remark here, that in the figure, the lines are to a scale of an eighth of their true length, but that, to such a scale, the point E is not the orthographic projection of the given point, with respect to the point O. If it were required to give its projection to such a scale, we must take Oa=7·√/6=5·7157, Ob=5√/6=4·0825, Oc=8·√/6=6·5320 ; then constructing as before, e will be the true orthographic projection of the given point with respect to the point O, to a scale of an eighth. For the purposes of construction it is however better to consider the distances on the axes OA, OB, OC as taken of their true length on a scale. 8. The axes OX, OY, OZ are called the Isometric Axes, because lines in them are all measured to the same scale; and for the same reason all lines parallel to any one of these axes are called Isometric Lines. 9. One of the rectangular planes XOY being taken as the horizontal plane, the other two XOZ, YOZ are vertical planes: these planes may be designated as the horizontal plane XY, the vertical plane ZX, the vertical plane ZY. These and all planes parallel to any one of them are called Isometric Planes. In the representation of objects, such as buildings, portions of a fortification, &c., the plan is represented in the horizontal plane XY, and elevations or sections in the vertical plane ZX, ZY, or planes parallel to these. 10. The point in the object which is assumed as the origin of the axes is called the Regulating Point. 11. If any point in the same isometric plane with the point required to be found be already represented in the picture, that point may be assumed as a new regulating point, and the point required be found by taking two distances, from the scale of the picture; and if the new assumed regulating point be in the same isometric line with the required point, it is found by taking only one di stance. 12. It is evident that by means of the isometric projection or perspective, any objects, such as buildings or frame-work, the lines in which are in the isometric directions, may be easily and accurately represented to any scale required; and that by means of such a picture those objects may be readily and correctly constructed, without further explanation. 13. To represent lines which are not in the isometric directions, it is only necessary to determine the isometric projections of their extremities, and to join them, the positions of these extremities being supposed to be given with reference to the isometric axes, or some assumed regulating point. 14. Curved lines may be represented by determining the co-ordinates of a sufficient number of points, and taking these co-ordinates to a scale in the direction of the isometric axes. 15. In representing astronomical and other instruments, machinery in which there are wheels working into each other, and, frequently, buildings, there are circles to be represented which are very commonly in the isometric planes, and it is a matter of much convenience that the picture of all circles in these planes is an ellipse* of the same form, in whichever of the three isometric planés the circle may be. 16. If circles be inscribed on the faces of the cube to which we at first referred, it is evident that, in each, the projection of that diameter which is in the direction of the diagonal parallel to the plane of projection is the greatest; and the projection of the diameter at right angles to this, and which is in the direction of the inclined diagonal, is the least: the former is therefore the major axis of the ellipse in which the circle is projected, and the latter is its minor axis. The projections of these diameters being to each other as the projections of the corresponding diagonals, we have the major axis of the ellipse to its minor axis as 2 to 6, or as 1 to√3. 13 3 An edge of the cube being 1, the major axis of the ellipse, which is equal to the diameter of the inscribed circle, is also 1, and consequently the minor axis is 3, 1 3 17. The diameters of the ellipse which are the projections of diameters of the circle parallel to the edges of the cube, and are therefore parallel to the isometric axes, are called Isometric Diameters. Being equal to the projection of the edge, they are equal to✔ 6. 18. It follows from this, that Taking the diameter of the circle to be represented as 1, the minor axis, the isometric diameter and the major axis of the ellipse Taking the isometric diameter as 1, the minor axis, the isometric diameter and the major axis are 1 √2, 1 and √6 respectively. 2 √2, 1 and Taking the minor axis as 1, the isometric diameter and major axis are respectively √2 and √3†. * It may be necessary here to state, that if a cone or a cylinder be cut obliquely through both sides by a plane, the section is called an Ellipse. The two straight lines by which an ellipse is divided into parts which are perfectly symmetrical on each side of them, are called its Axes, the longer being called the Major Axis, and the shorter the Minor Axis. An ellipse is readily thus described: two pins are fixed upright on the paper; a thread having its two ends tied together is then passed over the pins; this thread is stretched tight by a pencil point, and a curve is traced by the pencil on the paper round the pins, keeping the thread constantly stretched: the curve thus traced is an ellipse. The species of ellipse so described depends upon the ratio which the length of the thread bears to the distance between the pins. The ordinate in an ellipse has a constant ratio to the corresponding ordinate in a circle; and as this is the case with the ordinate of a circle and the orthographic projection of that ordinate, it follows that the orthographic projection of a circle is an ellipse. ↑ To describe, by the method pointed out, an ellipse of the above form, and |