corresponding line on the other face having the batir will be parallel to ab, and at the assumed altitude froin it. Drawing, therefore, a line through the points a and n, this will be the intersection in plan.

To obtain the intersection in elevation, draw the foot of the wall A'a' in elevation, also the line m'n' at the assumed altitude. The point a will be projected in a', and the point n in n'; and the line a'n', drawn through them, will be the intersection in elevation.

Prob. 3. To construct the projections of the bounding lines and edges of the joints of a buttress against the inclined face of a given wall; the base of the buttress being given, and being in the same horizontal plane as the base of the wall; the faces of the buttress, its end, and the top to have given slopes.

Along the foot 44, set off the breadth of the buttress ad at its base, and construct the two sides ab, cd; and the end be of the base. Let the batir of the face of the wall be ; that of the two faces and the end of the buttress ; and that of its top .

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By Prob. 2 construct the projections, in plan and elevation, of the intersections ae, a'e', and dh, d'h', of the faces of the buttress and wall; and those bf, bf', and cg, c'g', of the end and faces.

To construct the projections of the top surface of the buttress.

Suppose that the top line of the buttress eh, e'h', where it joins the wall, is of the same altitude as the top of the wall; and that the top surface from this line outwards from the wall has the given slope. Now, if a line as y'z' be drawn parallel to e'h' the top line, and at any assumed distance below it, this line may be regarded as the projection of a horizontal line in the top surface of the buttress; and its corresponding projection in plan will be a line y parallel to eh, and at six times the distance from it that y'' is below e'h'. Having drawn these two lines of indefinite length, construct the projections of the horizontal in the face of the buttress which is at the same distance below the top. The projection in elevation will be a continuation of the same line y'z', and in plan its projection will be parallel to the foot ab, and at a distance from it equal to 1 its altitude above it. Drawing an indefinite line ay parallel to ab at this distance from it, the point y, where it intersects the line ya, will be a point of the projection in plan of the intersec tion of the top surface and face of the buttress. The point e is another point; joining e and y by a line and prolonging

it until it intersects the projection of the intersection of the face and end at f, this line ef will be the projection of one side of the top surface in plan. The other side hzg will be found by a like construction. The points ƒ and g being joined will be the projection of the end of the top surface.

To obtain the corresponding lines in elevation; the points y z and z are projected into y' and '; the points e', y', and h', z', are joined by lines which are prolonged to meet the lines which are the projections of the exterior edges of the buttress at f' and g', which correspond to ƒ and g in plan, and the points f' and g' are joined; e'f'g'h' is the elevation of the top surface.

The projections of the edges of the vertical joints in plan will be perpendicular respectively to the lines be and cd, and breaking joints as shown on the right portion of the plan. The projections of these lines in elevation will be found by projecting their extremities into the corresponding projections of the horizontal edges in elevation, as shown on the elevation of the face and a portion of the end, on the right.

Prob. 3, Case 2 (Pl. B, Figs. 1, 2, 3). Having given the cross section of a brook, or other small natural water way, over which a full centre arched stone culvert is to be thrown, to support an embankment of a roadway, of a given height above the natural surface of the ground, to_construct the bounding lines of a wing wall with plane bounding surfaces.

Figs. 1, 2 are the elevation and plan, or the vertical and horizontal projections of the parts; PQ being the ground line. Fig. 3 is a section and elevation, through the axis of the arch, on the vertical plane of which R S is the ground line. M, M' are the slopes of the embankment. N, N' the bottom of the brook.

Let C B Z', Fig. 1, be the cross section of the side bank of the water way, and of the adjacent level ground; O the centre of the semicircle of the full centre arch, taken on the level C B of the natural surface; L I the level of the top of the embankment; L' E', Fig. 2, the foot of the embankment.

The wing wall and arch (Figs. 1, 2), are supported upon a general substructure, the height of which is A A'; the plan of that portion of which, supporting the wing wall, is shown by A'B'C'. The faces of this substructure being vertical, and projecting a distance, represented by Z A, beyond the springing line of the arch, the foot of the wing wall, and the foot of the embankment; the point B' being taken on the crest B' K of the side bank.

Through the line Z D, Z' D', the foot of the wing wall,

the plane of its face is passed; the top point of which X, X' is found by drawing a line m n parallel to Z' D', and at a distance from it equal to the base of the batir corresponding to the height of the embankment L I above D Z, and taking its intersection X, X' with the top line of the head of the arch. Joining X Z, it will be the elevation of the intersection of the face of the wing wall with the end of the arch.

The top surface of the wing wall X"D" (Fig. 3), receives the same slope as the side slope of the embankment M, M', and is here taken to coincide with it. The wing wall is terminated at the end by a vertical plane D" F" parallel to the head wall of the arch. The thickness I X, I'X' of the wing wall at top is assumed.

Joining the points X' D' and X D, the the top line of the wing wall is found. I'H' respectively parallel to these, the found.

interior edge of Drawing I H and exterior lines are

The lower end of the wing wall is terminated by what is termed a newel stone, which serves, in this case, as a buttress. The height of this stone D" F" is arbitrary, as is also its slope F"G" on top. Assuming these, the intersection of the vertical plane, terminating the wing wall with its face, will be the lines F" D', F D, parallel respectively to X' Z' and X Z. The lines F' G', F G, and H' E', HE, which also are parallel, will be found by Prob. 3.

The vertical joints of the face of the wing wall are perpendicular to its face. Drawing the line X'Y' perpendicular to D' Z', and its corresponding projections X Y, X" Y" on Figs. 1, 3, the directions of the edges of the vertical joints, as x'y', x'y, x'' y'', will be parallel, on their respective Figs., to these lines.

The top of the wing wall, instead of a coping, is formed with elbow joints uniting with the horizontal joints. The portion of the joint forming the elbow is perpendicular to the top surface. Drawing then a line Z" W" (Fig. 3), perpendicular to D" X", and its corresponding projections Z W, Z'W' on Figs. 1, 2, these will be the directions z w, z' w', z" w", of the elbows. The depth of the elbow is arbitrarily assumed, by drawing a line p" q" on Fig. 3 parallel to D" X".

CYLINDRICAL

AND OTHER

ARCHES.

To facilitate the geometrical operations for determining the bounding surfaces and lines of the voussoirs of arches, a few preliminary problems and theorems, on which these operations are based, will first be explained.

Prob. 4, (Pl. A, Fig. 2.) Having given a semi cylinder, the right section of which is a semicircle, and its axis and two bounding elements being horizontal, to construct the projections of the intersection of the cylinder by a plane inclined to its axis and having a given inclination to the horizontal plane containing the axis; also, the projection of the intersection of this semi cylinder with another semi cylinder with a semicircle also for its right section, the axis and bounding elements of this last being in the same horizontal plane as those of the first; and then to develop the portion of the first semi cylinder which lies between the given plane and the other cylinder.

Let a'c'b' be the right section of the given cylinder, and o' its centre; the line a'b' being horizontal. Let a'Á and b'B be the horizontal projections of its bounding elements, and o' that of its axis. Let ab be the trace of the given inclined plane on the horizontal plane of the bounding elements; AB one of the bounding elements of the other cylinder, and LM its axis. The quadrant AL the half of the right section of this cylinder; Z the centre of this quadrant.

1st. Taking any two elements of the given cylinder, at the same height, as x'x, and y'y,, above a'b', they will be projected in plan parallel to the axis o' C, and will be drawn

2

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indefinitely through the points, and y. The given inclined plane will cut these elements at the same height x',, and if the projection xy of a horizontal line in this plane, at the height xx,, be drawn, the points x and y, where it cuts the two elements of the cylinder, will be two points of the required projection in plan. To construct this line xy, let the given inclination of the plane be ; the projection of this horizontal line, which is at the height x', above the foot ab of the plane, will be (Prob. 2) parallel to ab, and at a distance from it equal to of x'x,; drawing therefore xy parallel to ‡ ab, and at this distance, it will be the required projection in plan. The points x and y thus found will be two points of the projection in plan required. In the same way any number of points can be found, and the curve axcyb, traced through them, will be the required projection in plan.

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2

The construction just explained, although very simple, may be abridged as follows: Through a draw af perpendicular to a'b'; prolong y'a' to the left, and set off from m"", where it cuts af, the distance m"""" equal to mx, as before found. Through a'x"" draw the indefinite line a'e'. Now, to construct the projection of any other point in plan, as c on the element at the height c'; through e' draw a line parallel to a'b', take the part o"""""intercepted between a'f' and a'e' and set it off from o, where the projection of the element through c' cuts ab, to c along the projection of the element; c will be the required point. This is evident from the relations which the heights and horizontal distances considered bear to each other.

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2

2d. To find the projection in plan of the intersection of the cylinders. Draw AD perpendicular to AL. If a distance Ar" equal a', is set off on this line, and a parallel to AL be drawn through ", the point u" where it cuts the quadrant will give the point on it through which the element of the second cylinder, at the height x'x, of the two elements at x' and y', is drawn. Through u" drawing an indefinite line parallel to AB, the bounding element of the second cylinder, it will be the projection in plan of the element at the height Ar"=x'x,; and the points w and v, where it cuts the two projections of the elements of the first cylinder at the same height, will be two points of the required projection. In the same way other points would be found, by constructing the projections in plan of corresponding elements on the two cylinders.

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This operation, like the former, may be also abridged as follows: Through b' draw a perpendicular b'd' to a'b'. With a radius equal to AL describe a quadrant tangent to b'd' at