Note

Penciling. Fig. A represents one turn each of two equal spirals. This particular kind of a spiral is called the Spiral of Archimedes. Make first the outside circle inches in diameter, and divide the circumference into any number of equal parts, as twelve in the figure; and draw the diameters. Divide any radius, as OL, into the same number of equal parts. Then, with 0 as center, and radius 0-1, strike an arc cutting the first radius at the left in point a; then, with the same center 0, and radii 0-2, 0-3, 0-4, etc., to 0-11, strike arcs cutting respectively radii ON, OP, OQ, etc., to OX, thus determining the various points of the curve, o—b—f—k-12. The other spiral, o—n—s—y, is constructed in a similar way.

In Fig. D the diameter of the circle is inches, and the short diameter of the ellipse is inches. The points on the ellipse may be found either by the method of Problem 12, above, or by the use of a trammel (Problem 13); and then these points are joined with the irregu lar curve.

Fig. B is for practice in drawing lines of different kinds with the compass.

In penciling, make the lines of uniform width. The square is inches on a side. The points 0 and 0' are the centers for the circles and arcs. To locate point 0, bisect the angle between the diagonal and the upper side of the square, by the line Mx, and point 0 is the inter section of this bisector with the other diagonal. Point 0' may then be located on the other side of the diagonal MN. The circles A and A', tan gent to the sides of the square and to the diag onal MN, are next drawn. Then, from point 1, mark off on the bisector Mx spaces, three to the left, and six to the right. With center 0, and radius 0-2, draw the circle next inside of A, and, before changing the setting of the compass, draw the corresponding circle from center 0'. In this same way, draw all the other circles and arcs, using each center 0 and 0' before changing the setting. Make the circles and arcs dotted, dot-and-dash, or full, as shown in the figure.

The so-called dotted lines through points 5 and 6 are really composed of short dashes a little less than inch in length. Care should be taken to make these dashes uniform in length. Similarly, the short and long clashes of the cir cles through 2 and 3 should be drawn with care, to insure a pleasing and uniform appearance.

Fig. C is intended for practice in the accu rate use of the large and small compasses, and also in the accurate use of the scale. The rect angle in which the design appears is laid out as accurately as possible 5 inches by inches.

It is then divided into squares; and on the exactness of the spacing depends largely the accuracy of the final result.

An experienced draftsman could set the scale along the side of the rectangle, and mark off accurately the 1%-inch divisions without mov ing the scale. The beginner, however, can obtain greater accuracy by using the hair-spring dividers.

First set the dividers or the scale to inches,. and test the exactness of the setting by spacing along the scale several times, noticing whether, at the last position, the point of the dividers exactly coincides with the proper divi sion of the scale. The dividers with this setting should then be used to space off on two adjoin ing sides of the rectangle. Through these points of division, the lines forming the small squares are drawn with the triangle and T-square. The centers of the circles and arcs are at the corners of the small squares, as shown in the drawing. The smallest radius is % inch, and the radii increase by inch, so that the largest radius is inches. For accuracy it is essential that all arcs or circles of the same radius be drawn with one setting of the compass.

For the penciling, draw first all the arcs, then the straight lines. Of course, in the pencil work, it is needless to attempt to stop the arcs exactly at the straight lines, as this can be done when the drawing is inked.

In Fig. E, the outside square is inches on a side. With the corners of the square as centers, draw the arcs A, B, C, and D, and through the points of intersection, 1, 2, 3, and 4, draw with triangle and T-square the smaller square. Draw the diagonals of the larger square; and with the intersection of the diag onals as center, draw the circle through 1, 2, 3, and 4.

Next, with the corners of the smaller square as centers, draw the arcs 1-4, 4-3, 3-2, and 2-1; and lastly, draw the smallest circle as in the figure, tangent to these four arcs.

Inking. The border line should be of the same weight as the heavy lines of Fig. B. Make all the full lines on the sheet of medium width, except as shown in the drawing for Fig. B; and make all construction lines fine, short-dash lines like those of Fig. B. In Fig. B, the diagonals need not be inked. In Fig. C, do not show the construction squares in the finished drawing. In Fig. D, show the axes of the ellipse as con struction lines. In Fig. E, the diagonals and the smaller square should not be inked.