The volute is an architectural figure of a geometrical nature based upon the spiral, and is of quite frequent occurrence in one form or another, consequently some remarks upon the different methods of drawing it will not be out of place.

81. To Draw a Simple Volute. - Let D A, in Fig. 225, be the width of a scroll or other member for which it is desired to draw a volute termination. Draw the line D 1, in length equal to three times D A, as shown by D A, A B and B 1. From the point 1 draw 1 2 at right angles to D 1, and in length equal to two-thirds the width of the scroll - that is, to two-thirds of D A. From 2 draw the line 2 3 perpendicular to 1 2, and in length equal to three-quarters of 1 2. Draw the diagonal line 1 3. From 2 draw a line perpendicular to 1 3, as shown by 2 4, indefinitely. From 3 draw a line perpendicular to 2 3, producing it until it cuts the line 2 4 in the point 4. From 4 draw a line perpendicular to 3 4, producing it until it meets the line 1 3 in the point 5. In like manner draw 5 6 and 6 7. The points 1, 2, 3, 4, etc., thus obtained are the centers by which the curve of the volute is struck. From 1 as center, and with 1 D as radius, describe the quarter circle D C. Then from 2 as center, and 2 C as radius, describe the quarter circle C F, and so continue using the centers in their numerical order until the curve intersects with the other curve beginning at A and struck from the same centers, thus completing the figure, as shown.

82. To Draw an Ionic Volute. - Draw the line A B, Fig. 226, equal to the hight of the required volute, and divide it into seven equal parts. From the third division draw the line 3 C, and from a point on this line at any convenient distance from A B describe ft circle, the diameter of which shall equal one of the seven divisions of the line A B. This circle forms the eye of the volute. In order to show its dimensions, etc., it is enlarged in Fig. 227. A square, D E F G, is constructed, and the diagonals G E and F D are drawn. F E is bisected at the point 1, and the line 1 2 is drawn parallel to G E. The line 2 3 is then drawn indefinitely from 2 parallel to F D, cutting G E in the point H. The distance from H to the center of the circle O is divided into three equal parts, as shown by H a b O. The triangle 2 O 1 is formed. On the line O H set off a point, as c, at a distance from O equal to one-half of one of the three equal parts into which O H has been divided. From c draw the line c 3 parallel to 1 O, producing it until it cuts 2 3 in the point 3. From 3 draw the line 3 4 parallel to G E indefinitely. From the point c draw a line c 4 parallel to 2 O, cutting the line 3 4 in the point 4, completing the triangle c 3 4 From 4 draw the line 4 5 parallel to F D, meeting 1 O in the point 5. From 5 draw the line 5 6 parallel to G E, meeting the line 2 O in the point 6. From 6 draw the line, G 7 parallel to F D, meeting the line c 3 in the point 7.

Fig. 225. - To Draw a Simple Volute.

Proceed in this manner, obtaining the remaining points, 8, 9, 10, 11 and 12. These points form the centers by which the outer line of the volute proper is drawn. From 1 as center, and with radius 1 F, Fig. 226, describe the quarter circle F G. Then from 2 as center, and with radius 2 G describe the quarter circle G D, and so continue striking a quarter circle from each of the centers above described until the last arc meets the circle first drawn. To obtain the centers by which the inner line of the volute is struck, and which gradually approaches the outer line throughout its course, proceed as follows: Produce the line 3 c, Fig. 227, until it intersects 1 2 in the point 11, which mark. This operation gives also the points 91 and 51 of intersection with the lines parallel to 1 2, which also mark. In like manner produce 4 c, 1 O and 2 O, as shown by the dotted lines, and mark the several points of intersection formed with the cross lines. Then the points l1, 21, 31, 41, etc., thus obtained are the centers for the inner line of the volute, which use in the same manner as described for producing the outer line.

Fig. 226. - To Draw an Ionic Volute.

83. To Draw a Spiral from Centers with Compasses. - Divide the circumference of the primary - sometimes called the eye of the spiral - into any number of equal parts; the larger the number of parts the more regular will be the spiral. Fig. 228 shows the primary divided into six equal parts. Fig. 229 is an enlarged view of this portion of the preceding figure. Construct the polygon by drawing the lines 1 2. 2 3, 3 4, etc.. producing them outside of the primary, as shown by A, B, D, K, C and E. From 2 as center, with 2 1 as radius, describe the are A B. From 3 as center, and 3 B as radius, describe the arc B D; and with 4 as center, with radios 4 D, describe the arc D F. In this manner the spiral may be continued any number of revolutions. In the resulting figure the various revolutions will be parallel.

84. To Draw a Spiral by Means of a Spool and Thread. - Set the spool as shown by A D B in Fig.

Fig. 227 - Eye, of the Volute in Fig. 226 Enlarged.

230 and wind a thread around it. Make a loop, E, in the end of the thread, in which place a pencil, as shown. Hold the spool firmly and move the pencil around it, unwinding the thread. A curve will be described, as shown in the dotted lines of the engraving. It is evident that the proportions of the figure are determined by the size of the spool. Hence a larger or smaller spool is to he used, as circumstances require.

Fig. 228. - To Draw a Spiral from Centers.

Fig. 229. - Enlarged View of the Eye of the Spiral in Fig. 228.

Fig. 230. - To Draw a Spiral by Means of a Spool and Thread.

85. To Draw a Scroll to a Specified Width, as for a Bracket or Modillion. - In Fig. 231, let it be required to construct a scroll which shall touch the line DB at the top, E A at the bottom and A B at the side, the length of A B, which determines the width of the scroll, being given. Bisect A B, obtaining the point C. Let the distance between the beginning and ending of the first revolution of the scroll, shown by a e, be established at pleasure. Having determined this distance, take one-eighth of it and set it off upward from C on the line A B, thus obtaining the point b. From b draw a horizontal line of any convenient length, as shown by b h. With one point of the compasses set at b, and with b A as radius, describe an arc cutting the line b h in the point 1. in like manner, from the same center, with radius b B, describe an are cutting the line b h in the point 2. Upon 1 2 as a base erect a square, as shown by 1 2 3 4. Then from 1 as center, with 1 a as radius, describe an arc, a b; and from 2 as center, with 2 b as radius, describe the arc b c. From 3 as center, with radius 3 c, describe the arc c d. From 4 as center, with radius 4 d, describe the arc d e. If the curve were continued from e, being struck from the same centers, it would run parallel to itself; but as the inner line of the scroll runs parallel to the outer line, its width may be set off at pleasure, as shown by a a1, and the inner line may be drawn by the same centers as already used for the outer, and continued until it is intersected by the outer curve. To find the centers from which to complete the outer curve, construct upon the line of the last radius above used (4 e) a smaller square within the larger one, as shown by 5 6 7 8. This is better illustrated by the larger diagram, Fig. 232, in which like figures represent the same points. Make the distance from 5 to 8 equal to one-half of the space from 4 to 1, making 4 to 8 equal the distance of 5 to 1. Make 5 to 6 equal the distance from 8 to 5. After obtaining the points 5, 6, 7, etc., in this manner, so many of them are to be used as are necessary to make the outer curve intersect the inner one, as shown at g. Thus 5 is used as a center for the arc ef, and 6 as a center for the are f g. If the distance a a1 were taken less than here given, it is easy to see that more of the centers upon the small square would require to be used to arrive at the intersection.

Fig. 281. - To Draw a Scroll to a Specified Width.

Fig. 282. - The Center of Fig. 231 Enlarged.