To obviate the confusion that might arise when points below the line of level represent vertical projections, or when points above that line represent horizontal ones, the following notation is adopted.

The capitals A, B, C, &c. denote points in space, and rarely appear in the figures.

The small letters a, b, c.. without accents belong to the horizontal plane.

When the letters are accentuated as a', b', c'... they belong to the vertical plane.

The line of level is always denoted by xy; and most commonly the Greek letters a, ß, y . . . indicate points in this line.

a'.

Different abbreviations sanctioned by custom are also employed.
The point [a, a'] is that of which the projections are a,
The line [ab, a'b'] is that of which the projections are ab, a'b'.
The plane aad' is that of which the traces are aa, aa'.

When it is said that a point, a line, or a plane is given, it is to be understood that the projections of the point, those of the line, or the traces of the plane, are known. In the same manner, when it is proposed to determine a point, a line, or a plane, the projections of the point or of the line, or the traces of the plane, will be what it is required to find.

In the reasoning on which the construction of a problem is founded, it is always to be understood, that the points and lines referred to are considered to be in their true positions in space, with reference to the planes of projections, likewise in their true positions; that in the different operations of the construction, the vertical plane is supposed to have turned about the line of level until it coincides with the horizontal plane on that of the paper; and that to have a correct idea of the result of the construction, it is necessary mentally to restore the vertical plane, with the several points and lines on it, to its true position at right angles to the horizontal plane, so that the points and lines whose projections are represented on the two planes may, by means of these projections, be viewed in their true positions in space. The student, bearing this carefully in mind, is therefore recommended, after having performed all the operations of the construction, to fold the paper on the line of level, and having placed the portion representing the vertical plane at right angles to that representing the horizontal plane, to endeavour to obtain a clear notion of the positions in space, of the points or lines which were to be determined, by considering what must be the position of the intersection of their projecting lines or planes.

PROBLEMS.

INTERSECTIONS OF STRAIGHT LINES AND PLANES.

PROBLEM I.

A straight line being given, that is, its projections being known; to find its traces, that is, the points where the straight line meets the planes of projection.

The straight line in space is the intersection of its two projecting planes (20). But the point of meeting of the horizontal traces of these two planes is common to both planes, and is therefore a point in the straight line. As it is in the horizontal plane, it is therefore the horizontal trace of the straight line.

It appears in the same manner that the intersection of the vertical traces of the two projecting planes is the vertical trace of the straight line.

Let ab, a'b' (fig. I. 1) be the two given projections of the straight line, which meet the line of level xy in a and b'. The plane perpendicular to the plane yb'a' and passing through a'b' is one of the projecting planes of the given straight line (11); and the horizontal trace of this plane is the line b'h, perpendicular to xy (35. 3). The horizontal trace of the vertical plane passing through ab, the other projecting plane, is ab. As the intersection h of these horizontal traces is a point in the line, h is the horizontal trace of the line.

Similarly, the vertical trace of the vertical plane passing through ab is av perpendicular to xy, and the vertical trace of the other projecting plane is a'b'. As the intersection v of these vertical traces is a point in the line, v is the vertical trace of the line.

Hence we have the following general rule. To determine the horizontal trace of a given straight line, produce its vertical projection to intersect the line of level, and from the point of intersection draw a perpendicular to this line, meeting the horizontal projection: the point of intersection will be the horizontal trace of the given line.

Similarly, to determine the vertical trace of a given straight line, produce its horizontal projection to intersect the line of level, and from the point of intersection draw a perpendicular to this line, meeting the vertical projection: the point of intersection will be the vertical trace of the given line.

Remarks. In changing the projections ab, a'b', the traces h, v may take an infinity of different positions. We shall notice four principal

cases.

In fig. I. 1, the horizontal trace h is in front of the line of level, and the vertical trace v is above it.

In fig. I. 2, the horizontal trace h is still in front of the line of level; but the vertical trace v is in the lower part of the vertical plane.

In fig. I. 3, the reverse is the case: the horizontal trace h is behind the line of level, and the vertical trace v is above it.

In fig. I. 4, the horizontal trace h is behind the line of level, and the vertical trace v is below it.

By conceiving the portion of the figure above xy which represents the vertical projection of the given line, to revolve about xy, until it is perpendicular to the plane of the paper, and then in imagination supplying the projecting plane, a clear idea of the position in space of the given line, in the foregoing cases, will be obtained.

PROBLEM II.

The traces of a straight line being given, to determine its horizontal and vertical projections.

Let h and v (fig. II. 1) be the given traces of a line. Then the plane avy being in its original position, perpendicular to the plane xhy, the line joining h and v will be the line of which these points are the traces; and if va be drawn perpendicular to xy, and ha be joined, the plane hav will be its horizontal projecting plane, and therefore ha its horizontal projection. Similarly, if hb be drawn perpendicular to xy and vb' be joined, the plane hb'v will be the vertical projecting plane of the line, and b'v its vertical projection.

PROBLEM III.

To find the intersection of two given planes; that is, the traces of two planes being given, to find the projections of their intersection.

If the traces of the two planes intersect in two points, these points being the traces of the line of intersection of the two planes, the projections of this line are immediately found by the last problem.

Let pa, pa' (fig. III.) be the horizontal and vertical traces of one of the given planes, and qb, qb' those of the other, m being the intersection of the horizontal, and n' that of the vertical traces. Then m is the horizontal trace of the line of intersection of the two planes; and n' is the vertical trace of that line. Therefore, drawing mm' perpendicular to xy, m' is the vertical projection of m (35. 1), and drawing nn perpendicular to xy, n is the horizontal projection of n' (35. 1); joining therefore mn, m'n', mn is the horizontal projection of the intersection of the two planes, and m'n' is its vertical projection (Prob. II.).

Particular cases.-1. When one of the given planes, bqb', has one of its traces perpendicular to the line of level, the horizontal

trace bq (fig. III. 1), for example; then this plane is perpendicular to the vertical plane (Prop. 32), and therefore its vertical trace will be the vertical projection of its intersection with the other plane. In this case, the lines mm', m'n' (fig. III.) themselves become the horizontal and vertical traces mq, qn', of the planes: in other respects the construction is the same as before.

2. Suppose that on one of the planes of projection, the traces of the two given planes are parallel to each other. For example, let the horizontal traces pa, qb (fig. III. 2) be parallel. In this case the intersection of the two planes will be parallel to pa and qb (Prop. 19. Cor.); and therefore its horizontal projection will also be parallel to them (Prop. 23). n'n being therefore drawn, as before, perpendicular to xy, mn, drawn parallel to pa or qb, is the horizontal projection of the intersection of the two planes. Since the intersection is parallel to the horizontal plane (Prop. 21), its vertical projection is the line n'm' drawn parallel to xy (35. 2).

3. Let the two given planes cut the line of level at the same point. In this case, the general construction fails, but we may cut the planes by any other plane, and find, as above, the projections of the line of intersection of each of the given planes with this auxiliary plane. The points where these projections meet each other will be the projections of a point common to both the given planes; and as the point where these planes meet the line of level is also common to them, the projections of their intersection are completely determined.

Very commonly the auxiliary plane is taken perpendicular to the line of level. It is then considered as a new plane of projection on which the traces of the given planes being found, the construction is by this means reduced to that of the general case.

Let apa', bpb' (fig. III. 3) be the given planes, cutting the line of level ay in the same point p, their traces being ap, a'p and bp, b'p. Let an auxiliary plane be conceived to be drawn perpendicular to xy, its traces on the co-ordinate planes being ßa, Ba', which meet the traces of the plane apa' in a and a'; then if we conceive the vertical plane of projection to be in its true position, perpendicular to the horizontal plane, the line joining a and a' in the auxiliary plane will be the trace of the plane apa upon this plane. Similarly, the line joining b and b', in the auxiliary plane, will be the trace of the plane bpb' on this plane. Let us now conceive that this vertical plane is turned about ßa, down upon the horizontal plane, so that the line Ba' is upon By. The point a will not be changed, but a' will be brought down to the point a", so that Ba"-Ba'; and consequently the straight line joining a and a' in space will be brought down in aa".

=

In the same manner, the trace bb", of the other given plane, on the auxiliary plane, is found; the point d where it cuts aa" is com

mon to the two planes; and we have only now to determine the projections of this point upon the primitive co-ordinate planes.

Now if dm be drawn perpendicular to ßa, the horizontal projection of d will be m; and if dd' be drawn perpendicular to By, and Bď be brought back to ßm' on Ba', the vertical projection of d will be m'*. The projections of the required intersection of the planes will therefore be pm, pm'.

4. Let one of the given planes be parallel to the line of level.

In this case, the traces bq, b'q' (fig. III. 4) of one of the planes are parallel to the line of level ay (Prop. 23). m being the intersection of the horizontal traces of the two planes, and n' that of their vertical traces, the horizontal and vertical projections, mn and m'n', of the intersection of the planes will be found as before, by drawing mm' and n'n perpendicular to xy.

5. Let both the given planes be parallel to the line of level.

In this case, the traces of the given planes are parallel to the line of level (Prop. 23); their intersection is so likewise (Prop. 23, Cor. 2); and the general construction again fails. But this defect is obviated by means of an auxiliary plane, as in the 3rd case.

Let ap, a'p' (fig. III. 5) be the horizontal and vertical traces of one of the given planes, and bq, b'q' those of the other, all parallel to the line of level xy. Let ßa, ßa' be the horizontal and vertical traces of an auxiliary plane, which is here taken to be oblique to xy, intersecting ap, bq in c, e, and a'p', b'q' in d', f'; c and d' will be the horizontal and vertical projections of the intersection of the plane ap, a'p' with the auxiliary plane, and e and f' those of the intersection of the plane bq, b'q' with the same plane.

Draw cc and d'd perpendicular to xy, and join cd, d'c': cd is the horizontal projection of the intersection of the plane ap, a'p' with the auxiliary plane aßa', and d'c' is the vertical projection of that intersection (Case 4). Again, drawing ee and ff perpendicular to xy, and joining ef, f'e', ef and f'e' are the horizontal and vertical projections of the intersection of the plane bq, o'q' with the plane aßa'. Therefore cd and ef being the horizontal projections of the intersections of the two given planes with the auxiliary plane, m, the point where these lines intersect each other, will be the horizontal projection of the point of intersection of the two given planes on the auxiliary plane; and d'c' and f'e' being the vertical projections of those intersections, m', the point where they intersect, will be the

* The operation by which the vertical projection of the intersection of the traces of the given planes on the auxiliary vertical plane is represented in the drawing, is simply this: after the determination of the point d', the line ßa' is supposed to be in its correct position perpendicular to the plane xß, the line Bd' is then turned up on ßa', to determine the point m', and then the line 3a, with this point m' determined on it, is again brought down upon the paper, supposed throughout to be horizontal, as ßm'a'b'.