shall be with the branch AD or AD': and, similarly, if, in Fig. 2, AP be the trace of the plane PQ upon MN, it must be predetermined whether the inclination is to be estimated from the portion of the plane to the right of AP or from that to the left-in other words, whether AB is to be compared with AD or AD' in defining the inclination of PQ to MN. In speaking of the inclination as "given,” this is always, then, understood as an element in the conditions by which it is given, whether so expressed or not. 10. The simplest form in reference to one plane of projection in which a plane can be given is by means of its trace and inclination ; and in this way, when nothing to the contrary is expressed, it is to be understood as given in the following pages. It may, however, be otherwise given; as by means of two lines in it; of a point and line in it; of three points in it; of being drawn through a given point parallel to a given plane; or through a given line parallel to another given line. 11. A line is given in the following ways : (a) By its position in a given plane in reference to the trace of that plane; (6) By its trace, inclination, and the line in the plane of projection to which its inclination is referred. This line, it will be presently seen, is the projection of the given line. (c) By giving two points through which it passes, either in the ways designated by (a) or (6), or by any others. Many other methods of exhibiting the data of a line occur in practice, of which examples will hereafter occur; but for use, they are almost invariably reduced to one or other of these. 12. A point is most simply given by means of its projection, and the magnitude and direction of its projector. The direction of the projector is usually given by means of a pattern-projector; or, in other words, by giving one line to which the projectors are all parallel, or . a point through which they all pass. In the orthographic projection, this line being always perpendicular to the plane of projection, need not be further specified; and when the projector is said to be given, its magnitude alone is meant, as the projection itself being in this case given, the direction of the projector becomes fixed without further intimation. A point, however, is often most conveniently given in practice, by giving its position on a given plane in reference to the trace of that plane on the plane of projection. It occurs in this shape in almost every class of actual problems. Often, indeed, it is given by both methods simultaneously ; viz., by its projection and projector in reference to a plane which is itself given by means of its trace and inclination to the plane of projection. In this way almost all polyhedral solids are given, the plane in question to which the angular points are referred being one of the faces of the figure, and the trace of that plane on the plane of projection serving as the common trace for reference to the entire system. A point, however, may be (and often is) given as the intersection of two given lines; as the intersection of three given planes ; and in other ways; but for the most part these require subsidiary constructions to reduce the data to the simpler forms above enunciated. 13. All plane figures whatever (as angles, polygons, circles, or planè curves) are given by giving them as a line was given, by stating them in reference to the trace of that plane upon the plane of projection, and giving that plane by means of its trace and inclination. 14. In every case of the projection of a point upon a plane, the projection is a single point. For through that point only one line can be drawn either parallel to a given line, or to emanate from a given fixed point; and this line can cut the plane of projection in only one point: that is, the projection of a point is a single point. 15. In every case of the projection of a straight line upon a plane, the projection is a single straight line. For, since all the projectors are parallel to one another, or they all converge to a given point, they are all in one plane (the projecting plane) with the given line; and the trace of this plane on the plane of projection will (Art. 7) contain the projections of all the points of the given line. Moreover, this projecting plane can only cut the plane of projection in one line; and hence the projection of the given line is a single line. 16. In every case, if a given line be parallel to the plane of projection, its projection will be parallel to itself; and if the given line meet the plane of projection, its projection will always pass through its trace upon that plane. For, in the first place, the line is projected by a plane, upon a plane parallel to that line, and hence (Pls. and Sols., Chap. 1., Prop. 11.) the projection of the line itself is parallel to the original. And in the second, the trace of the line being in the plane of projection, the projector of that point also meets the plane of projection in the same point that the line itself does ; that is, the trace is one of the points in the projection of the line. 17. The parallel projections of parallel lines are parallel ; and their polar projections meet in a point. For, let Mo be the plane of projection, and AB, ab two parallel lines which are to be projected by lines parallel to a given line : then, if two lines AC, ac be drawn parallel to the projecting pattern (12) to cut MN in C and c, the lines BC, bc will be the projections of AB, ab. But, since AB, AC are parallel to ba, ac, the projecting planes ABC, abc are parallel ; and hence their sections BC, bc with MN are parallel. Again, let C be the projecting point, and AB, ab two parallel lines : draw CD to meet MN in D parallel to AB, ab. Then the projecting planes for AB, ab pass through CD, and hence through D, in which they intersect MN. Whence also the polar projections of AB, ab pass through D. 18. The traces of parallel planes are parallel ; which is only another enunciation of Prop. I., Chap. 1., Planes and A B N M D Solids. PROPOSITION I. ORTHOGRAPHIC PROJECTION. Given a point in a given figure plane to construct its projection, and the magnitude of its projector. Let ZaX (first Fig.) be the figure plane, and YaX the plane of projection ; and let A be the given point in it, whose projection on XaY is required to be found by a construction in a plane. Now, the point A being given in the given plane XaZ, the line Aa perpendicular to its trace aX is given in magnitude and position. From a draw ay in the plane XaY perpendicular to aX. Then aA and aY being perpendicular to ax, the plane AaY is perpendicular to aX, and therefore, to each of the planes XaY, XaZ passing through it (Pls. 11. 18); and YaZ is the inclination, which, by the problem, is given. Lastly, since A is in the plane Yaz, and this plane is perpendicular to the plane of projection, the projector Aa, of the point A is in this plane; and since the hypothenuse Aa of the right-angled triangle Aaa,, and the angle Aaa, are given, the sides aa, and a, A are also given. Let us now turn the plane XaZ about aX, and the plane YaZ about aY till they coalesce with the plane of projection: then aZ will coincide with a Y (either actually if brought forward, or in extension if revolved backwards), and A'a,, that is Aa, will still be perpendicular to ay, and the perpendicular A'a, will pass through the projection of A, and be equal to the projector Aa,. By a little consideration of the indications of the figure in its present state, we shall see that the proposed conditions will enable us to construct it by means of operations performed wholly in one plane. For, the point A being given with respect to the trace, the point A and perpendicular Aa are given, and can be laid down on the plan. Produce this perpendicular to Y (Fig. 2): then the projection of A is in ay. Also the inclination YaZ' is given, and the line a A'Z' being drawn to make that angle with aY, and A'a being set off equal to aA, and finally, the perpendicular A'a, being drawn to aY, we obtain the point a, which is the required projection of A viewed as in its original position. SCUOLIUM 1. It will be readily seen from the preceding proposition, that the only use that has been made of the first figure was to enable us to investigate the properties of the real figure, so as to deduce relations which should enable us to dispense with its actual use in the construction itself. The student is urged to keep this distinctly in his mind as the essential spirit of all projective methods, which have a practical object in view; and he will find that throughout this entire work the same fundamental characteristic is readily distinguishable. The figure which represents the lines, etc., in space is used for reasoning only-never used or implied in construction. This distinction has not been brought prominently forward in the previous branches of geometry. In plane geometry, the data, the construction of the problem, and the subsidiary lines employed in analysis and demonstration, are all in one plane; and hence there is no necessity for paying the least attention to the circumstances under which those lines were supposed to have been introduced, except as far as their related properties are concerned. In the consideration of properties of lines, planes, or other figures in space, the properties were absolutely deduced and the hypothetical constructions were performed, without the restriction to special means ; and hence the system required no such restrictions in the manner of performing the assigned operations. SCHOLIUM 2.-In the case which we have exemplified, the point A is above the plane of projection, and the figure plane is turned to the opposite side of the trace from that on which the projection is situated. Either of these circumstances might have been reversed, or even both of them together. The figures ordinarily given in works on projection suppose the figure plane to be turned upon the region that contains the projection : but the mixture of the lines which are involved in delineating a figure and its projection, render the separation generally more convenient in complex cases. We shall, however, sometimes employ one and sometimes the other mode, for the sake of rendering every combination familiar to the mind and eye of the student. SCHOLIUM 3.—When several points in the figure plane are to be orthographically projected, the same process which has been applied to one may be, obviously, applied to each of the others in succession : but in prac 13 tice, the work may be in a slight degree abridged, and the number of lines to be actually drawn, somewhat lessened. Let, for instance, A, B, C, be projected, each according to the method and independently of each other, as in the first figure: and produce B’b,, C'c, to meet aA' in B and y: then bp and cy are parallelograms, and aß = 1B' = bB, and ay = CC' = cc. Whence, if we take upon one of the lines making the given inclination with the profile trace, as aA' (Fig. 2), and set off upon it the distances aß, ay, aA' equal to с B BA A с B B A Bb, Cc, Aa, and draw parallels Bb,, yo,, IA A'a to the trace, meeting Bb, Cc, Aa (produced if necessary) in b, c, a,, these points will be the projections required. İt is very common, for preserving the uniformity of the figure to take instead of a line through one of the points a, b, c, concerned in the problem, one altogether unconnected with the inquiry, as in Fig. 3; the import of which will be at once apparent without verbal explanation.' SCHOLIUM 4.-A slight modification of the primary construction is sometimes employed; and it will appear obvious from the following transformation of the figure. Suppose the profile plane Aaa, of the system (Fig. 1 of the Prop.) to be turned about Aa till it coincides with Xaz, and then the compound plane figure A'aX to be made to coincide with the plane of projection. Then the projector will take the position Ac perpendicular to aA', as in the annexed diagram ; and aa will be the distance of the projection a, from the trace ax. Setting off this distance upon Aa, or upon a Y, will give the projection sought, on the upper or lower side of the trace ax respectively. That this gives the same point as the former construction is thus made evident. Make a A' = aA, and join Ala. Then the triangles Aaa, A'a,a have two sides equal each to each, A'a aA, aa = aa,, and the angle at a common : and hence, the bases are equal, and the remaining angles are equal, each to each. Whence A'a,a = Aaa; and this last is a right angle. The point a, is, therefore, the same as would have been found by the first process. Also Aa is the orthographic projector of the point A, as is obvious. SCHOLIUM 5.—The converse problem is sonietimes required in conducting a course of consecutive and dependent constructions ; and likewise in projecting upon "oblique sites," which is often employed in military works. It is,Given the projection of a point situated in a given figure plane, to fin the point itself. In this case a, is given (Fig. 2 of the Prop.); construct aZ' as before, and draw a, A perpendicular to aY: then aA' being set off above or below a upon ZY, gives A the original point. X PROPOSITION II. Given the trace and inclination of the figure plane, and the position of a given line in that plane, to project orthographically the line, and to construct the inclination of the line to the plane of projection. (a.) It is obvious that, if by the preceding proposition we project two points of the given line, then the line through these will |