EXAMPLES.

EX. 12. A cylinder, 4" diameter, 2" long, has a spherical end which is divided into six segments. Draw a side and end elevation, and a development of one segment.

EX. 13.-Draw eight contour lines on a sphere of 4" diameter. EX. 14.-A sphere 33" diameter is penetrated by a square prism 2′′ edge of base and 6′′ long. The axis of the prism coincides with an axis of the sphere, the centre of the sphere being at the centre of the prisms. Draw a plan and elevation of the solids showing the curves of intersection when the axis of the prism is horizontal and parallel to the vertical plane, the sides of the prism being equally inclined to the paper. Develop one of the holes made in the surface of the sphere, and the surface of one part of the prism up to its contact with the sphere.

Projection of Helices and Screw Thread.-A helix may be defined as the curve traced out by a point moving round a cylindrical surface in such a way that its movement in the direction of the length of the cylinder shall be uniform with its movement around the surface of the cylinder. So that if a point starts from the base of a cylinder and moves in an upward direction and at the same time moving round the cylinder, sc that when it has moved up, say ", it shall have moved one-fourth the way round, and when it has moved " up and 1" up it shall have moved half round and wholly round, the path of the point would be a helix. The distance moved in the direction of the length of the cylinder during the complete revolution is called the pitch of the helix. Spiral staircases, spiral springs, and screw threads are generated by helices, but the latter example only will be explained in consideration of the great practical value of screw threads.

PROBLEM LVII. (Fig. 94).—To draw a helix upon a given cylinder having a given pitch.

Let a', b', c', d' be the elevation of the cylinder and the circle of diameter ab its plan.

Make the distance a' 12' equal to the given pitch. Then from the definition of a helix we know the curve must rise from a' to 12' uniformly with its travel around the circumference of the circle which represents the plan of the cylinder. Therefore divide the circle and the pitch distance into the same number of equal parts, and mark as shown 1, 2, 11, and 1', 2', . . . 6' Draw projectors through each of the division points on the circle. Then when the point has moved round to 1, it must have moved up one-twelfth of its pitch, and it will, therefore, be

on the horizontal line drawn through 1', and at the point where this line cuts the projector through the point 1 on the circle, similarly the second point is where the projector through point 2

on the circle cuts the horizontal line drawn through the point 2', and so on for the remaining points. The curve a'é 12' is then a helix of one convolution.

The figure also represents the geometrical projection of a square thread screw, for a further description of which see p. 166. The width of the thread a' 6' is half the pitch, therefore the curves Y beginning at 6', 12', and d' are each parallel to the half helix a' e', and are half the pitch apart, and can be drawn by setting off distances of half the pitch along the projectors already drawn, starting from points on the curve a' e- thus, m n = n o = op. The depth of the thread is the distance marked x, therefore when the thread is cut, it leaves a cylinder of the diameter shown by the smaller circle in the

plan. The inner edge of the thread is a helix of the same pitch drawn upon a cylinder of the diameter 6′′ 12′′, and is constructed in exactly the same manner as the helix representing the outer edge, the construction lines being shown dotted. Thus the curve starts from h' and rises to the line through 1', while it has moved round one-twelfth of the circumference, and the first point in the curve is found by drawing a projector through the point 1" on the circle to meet the horizontal line through the point l' in the elevation, and so on for the successive points. These inner curves disappear from sight at the centre of the screw, and are only

seen on the top part of the thread from the centre to the limit of the cylinder, as seen in the figure. As these curves are similar, it is only necessary to obtain one in the manner just described, the others being conveniently found by setting off distances along the projectors. The parts of the curves, such as the one marked 'l', are found by continuing the larger helix.

Helix upon a Conical Surface. - When a helix is traced upon a vertical cylinder, its plan is a circle, but when it is traced upon the surface of a cone

it is continually approaching the axis, and, therefore, its plan is a spiral which uniformly approaches the point representing the plan of the cone vertex. The curve of a helix upon a cone is shown in plan and elevation in Fig. 95, the distance a' 12' being the pitch. To obtain the curve, draw a number of stripes down the cone, and draw their plan and elevation as shown. Then after the first one-twelfth of its travel the point will be on the stripe v 1, and will have moved upwards from the starting point a to the height of the horizontal line through 1', its elevation will, therefore, be where the stripe v' 1" cuts the line through 1", and similarly for the second point, where the stripe v′ 2′′ cuts the line through 2',

and so on for each point. The plans of the points are found by drawing perpendiculars from the elevations to meet the plan of the corresponding stripe. For the stripes v 3, v 9 the method of sections must be adopted (see p. 125).

Spiral Springs.-When the material of the spring is of square section, it can be correctly drawn by adopting the construction of Fig. 94 for the square threaded screw, allowing for the absence of the solid cylindrical centre. The outer edge of the

spring is a helix upon a cylinder equal to the outside diameter, and the inner edge a helix upon a cylinder equal to the inside diameter. With springs of circular section, a helix should be drawn upon a cylinder equal to the mean diameter of the inside and outside of the spring, which helix will represent the path of the centre of the material of which the spring is made, then a number of circles of diameter equal to the section of the spring should be drawn upon this helix as a centre, to give points for the lines of the spring

EXAMPLES.

EX. 15.-Draw a helix of one convolution upon a cylinder 3" diameter, and develop the surface of the cylinder with the helix. Pitch of helix 11⁄2".

EX. 16.-Show three threads of a square thread, outside diameter 3", pitch 1", depth ".

EX. 17.-A square prism 4" edge of base, 3" high, is bored with a central hole and screwed internally with a square thread screw, 2" diameter, " pitch, " deep. Show a vertical section through the centre of the prism when it stands with one base upon the paper.

EX. 18. Draw a helix of one convolution upon a cone of 21′′ diameter of base, 4′′ high, and develop the cone surface with the helix. Pitch of helix 2′′.

EX. 19. A spiral spring is 2′′ outside diameter, and is made of " round wire. Draw a length of the spring showing six coils, the pitch being ". Show the two top coils in section, the section plane being vertical and passing through the centre of the spring.

SECTION XII.

ISOMETRIC PROJECTION.

THE principles of isometric projection enable the three dimensions of a solid to be shown by one drawing, which, in appearance, is somewhat similar to a perspective representation, with the additional advantage that the actual sizes of the solid can be measured direct from the drawing.

If a cube be made to rest by one corner upon the paper, so that a diagonal of the solid is vertical, its plan will be represented by the drawing of Fig. 96. For the three top faces which

meet in a solid right angle at A, are each equally inclined to the paper, therefore their plans are similar and equal figures, and for the same reason the length of the plans of all the edges are equal. It is also evident that the three lines A B, A C, and A D which represent the three edges of the solid right angle, make angles 360° with each other of or 120°, and that all other lines repre3

senting edges of the solid are parallel to one of these three lines. The figure is, therefore, very easily constructed, as the lines A C and AD make angles of 30° with the horizontal and 60° with the vertical, and can thus be drawn with the T square and 60° set square.

The above reasoning only strictly applies to oblong solids having solid right angles, but, as will be shown later on, the same construction can be very conveniently applied to irregular solids and solids with curved surfaces. For example, the drawings of the simple solids on pp. 104 and 105 are in isometric projection.

Referring again to the example of the cube in Fig. 96, it is evident that the length of the edges in the drawing, should not be equal to their real length, as they are all inclined to the

plane of the paper. The relation of their projected length to their real length can be seen on reference to Fig. 97; a, b, c, d is a face of the cube, and ac a diagonal of the face; if af and ce are drawn at right angles to a c, and each made equal to the length of an edge of the cube, then the oblong a, f, e, c represents a section of the cube containing two diagonals of the solid, and fc is one of these diagonals. But in Fig. 96 this diagonal is supposed vertical therefore, draw XY through c at right angles to fc, and the length cg is the projected plan of an edge of the cube. But if the cube edge ad or de 1, then a c = √2, and √3; also, the triangles ecg and ceƒ are similar, there

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