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 kl, 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 v1, 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 3" 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 fc = √3; also, the triangles e cg and cef are similar, there

fore ce cg√3: √2, hence, by constructing a right-angled triangle similar to the triangle fac, where the base is √2 and the perpendicular is 1, the hypotenuse will be √3, and real lengths along the hypotenuse, when projected upon the base, will give the isometric length.

The practical objection to this correct isometric projection is that it entails the use of an isometric scale, and that lengths of the object cannot be measured direct from the drawing. But if the cube in Fig. 96, or any other solid, is drawn its real size, the only alteration in the drawing is in its size, and not in its shape, and hence we see there can be no objection to making isometric projections the actual size of the objects they represent, thus dispensing with the use of an isometric scale, and making it possible to take measurements direct from the drawing. This arrangement is generally adopted in practice, and is adhered to in the following examples :

It has been said, in referring to the drawing of the cube Fig. 96, that with oblong solids all lines are parallel to one of the three lines A B, A C, and A D.

These lines are termed the "ISOMETRIC AXES,” and it is necessary in commencing any isometric projection to first set out these three lines.

We may now regard Fig. 96 not as the plan of a cube with a diagonal of the solid vertical, but as a drawing of a cube with one face lying upɔn the paper. On such a supposition the figure ACED shows the top horizontal face, and the figures A D F B and ACGB vertical faces, so that in projecting a horizontal surface isometrically its length and breadth must be set off along the two sloping isometric axes AC and A D, while for a vertical surface, its length and breadth must be set off along the vertical axis A B, and one of the sloping axes AC or AD. It is important to remember this distinction.

Either surfaces or solids can be projected isometrically, and, as before stated, the construction can be extended to surfaces or solids not of oblong form, the method by which this is done will be clearly seen in the following examples, but it will be better understood by remembering that since the isometric axes represent lines at right angles only, the projection of figures containing other angles requires that they shall be surrounded by oblong figures, thus a circle is first enclosed in a square and a hexagon in an oblong.

PROBLEM LVIII. (Fig. 98a, b).—To draw the isometric projections of a hollow square prism (a) with its axis vertical, (b) with its axis horizontal.

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Fig. 98a, draw the isometric axes ab, ac, ad, set off a e along a c, and aƒ along a d, equal in length to the edge of the prism base. Draw fg parallel to ac and eg parallel to a d, meeting at g, then the figure afge is the isometric projection of one outside square base of the prism. Set off a 1 = a 2 = g 3 g 4, equal to the thickness of the sides of block, and draw lines as shown dotted to obtain the inside square 5, 6, 7, 8. Set off the length of the prism down the axis ab from a to h, and draw lines through f and e parallel to a b. Through the point h draw lines parallel to the other axes, as shown, thus completing the projection of the prism. Dotted lines representing the bottom base, can be drawn if desired.

Fig. 986, to draw the prism with its axis horizontal, the square

representing its base must be drawn as a vertical face, and is thus shown at aefg. The completion of the projection needs no further description.

EXAMPLES.

EX. 1.-Draw the isometric projection of an oblong, sides 3" and 2", when its plan is horizontal.

EX. 2.—A cube, 21" edge, stands upon a square block 31" edge, 1′′ thick. Draw their isometric projection when the block stands upon the ground.

EX. 3.-Make an isometric projection of a wooden box 8′′ long, 6" deep, 4" broad outside, and having a flat lid opened through an angle of 120°, the thickness of the wood being " throughout. Scale, half full size. (Vict. Hon., 1892.)