This section is from "The Horticulturist, And Journal Of Rural Art And Rural Taste", by P. Barry, A. J. Downing, J. Jay Smith, Peter B. Mead, F. W. Woodward, Henry T. Williams. Also available from Amazon: Horticulturist and Journal of Rural Art and Rural Taste.
The usual mode of forming a volute or spiral line is one of the simplest problems in geometry, and therefore requires no explanation here. The following method is, however, both original and better adapted for throwing up such a figure in groundwork. It is the invention ot Mr. Alexander Forsyth, and was by him first described in The Gardener's Magazine, from which source our four following figures and descriptions are taken. "Make a circle around the center of your intended volute, as much in circumference as you intend the breadth of your circuitous border to be; stick tbis circumferential line full of pegs, and tie one end of a garden line to one of them. Taking the other in your hand, go out to the point where you intend the volute to begin; and as you circumambulate, holding the line strained tight, you will delineate on the ground the annexed fig. 1.
A volute where the border is intended to be gradually narrowed towards the center as in fig. 2, may be thus formed: - "Make a circle as before, and instead of driving the pegs upright, let them form a cone; or, instead of pegs, use a large flowerpot whelmed, and, if necessary, a smaller one whelmed over it. Measure the radius of your volute, and wind that complement of line round the cone in such a manner as to correspond with the varying breadth of your intended border, and commence making the figure at the interior by unwinding the line".
A volute, the border of which widens as it approaches the center, is produced upon the same principle as the last; only, as the figure is as it were reversed, unwind the line from the other end, and fig. 3 will be produced.
The following ingenious method of forming circles or other curvilinear lines, is the invention of Mr. Forsyth, and must be of great practical use to those who have the laying-out of grounds, particularly intricate figures in geometrical gardens. Suppose a b c, fig. 4, to be three points in the curve, taken at equal distances (say fifty links): placing the cross-staff at b, with one of the sights pointing to a, make b o perpendicular to a b and measure its length. Then, removing the cross-staff to c, make c o perpendicular to b c, and equal to b o; and make the line b o d equal to a o c. Then d is a point in the curve; and in the same manner other points may be found successively.
Fig. 5 differs from the above only in this, that the angles are ing the requisite oblique angle. Setting the instrument in b fifty links from a, with one leg of the angle on the line b a, and by the other peg directing an assistant to place the peg c at the distance of fifty links; then remove to c, and so on.
To find the center of a circle, whose circumference will pass through three given points (not in a straight line), connect the three points a b c (fig. 6) together; from the middle of each, erect lines perpendicular to them, and where these perpendiculars cut each other is the center required.
To find the center of a circle, connect three points in the circumference, and from the middle of the two lines erect perpendiculars. Where these intersect each other, is the center required.
To construct a hexagon, divide the circle into three equal parts; from the middle of each line erect a perpendicular; and where these cut the circumference of the circle are the points where the sides of the hexagon meet.
To construct an octagon, divide the circle into four equal parts, by describing a square within it; erect perpendiculars from the middle of each side of the square; and where they intersect the circle are the points where the sides of the octagon meet.
To construct a pentagon, draw a line through the center of the circle, from the center of which erect a perpendicular, c d; divide the straight line from c to b into two equal parts; take c d as a radius, and describe a circle, making e the center, and when that circle cuts the straight line at f the distance from f to d is the length of the side of the pentagon.
To describe a circle the center of which is occupied with a square, say the base of the pedestal of a statue, fountain, etc., tie a cord round the square, not over tight; to that attach a line, in length equal to the radius, minus half the size of the square base; with that line describe the circle. This is a plain working plan, and near enough for all practical purposes in laying out grounds. The same rule may be applied when the base is circular, or of any equal sided figure, a pentagon, hexagon, etc.
To describe a circle when the base of the fountain, statue, etc., is oblong, lay the oblong correctly down on paper; find its center by drawing two lines diagonally through it; from that describe a circle of any size; draw two lines across the circle parallel to the longest sides of the oblong figure; from these erect perpendiculars, at equal distances, and note their respective lengths; on the ground draw two lines parallel to the longest sides of the oblong; erect perpendiculars as before, and measure their lengths from the drawing, putting in a peg at the end of each, which will describe the circle required. A line applied as in the last example, will describe an elliptical figure.
To describe an oval whose length is given, divide the length into three equal the ends of the oval; the intersecting points of these circles Will be centers to the two segments required to complete the figure 8.
To describe an oval, when the length and breadth are both given, lay down the length and breadth perpendicular to each other; combine a and d; measure the distance from c d, on the line a c from c, which will give en; measure the distance from n a, on the line d a, which will give f; divide f a into two equal parts, at the middle of which erect a perpendicular: where that perpendicular cuts the line a b will be the center h, for the end of the oval; and where it cuts the line d i at g, is the center for the side, (fig. 9.) The gardener's oval, when both the length and breadth are given, is thus formed:
Set off the length a b, and breadth c d, perpendicular to each other; take half the long diameter, and measure from c, to the line a b, with that length; when that line cuts the line a b, put in a peg; do the same on the other side, and the point e will be found; stick in there also a peg; then, with a cord passing round the pegs i e and c, with the addition of the space from a to , describe the figure with the peg c. (Figure 10.) To form an egg-shaped figure (fig. 11), the line a b being given, divide it into two equal parts; from the point c, where these lines intersect each other, construct a circle with the radius c a or c b; draw the line c d perpendicular to a b; taking a and b as centers, describe two arcs; draw a line from b through d, till it cuts the arc at; then, with d f as a radius, complete the figure.