This section is from the book "The Gardener's Monthly And Horticulturist V18", by Thomas Meehan. See also: Four-Season Harvest: Organic Vegetables from Your Home Garden All Year Long.
I think it was Hogarth who first designated the curving line as " the line of beauty." Whoever it was, he enunciated a far reaching truth. Ruskin gives utterance to very much the same thought when he says, "Nature is all made up of roundness, not the roundness of perfect globes, but of variously curved surfaces. Boughs are rounded, leaves are rounded, stones are rounded, clouds are rounded, cheeks are rounded, and curls are rounded; there is no more flatness in the natural world than there is in vacancy. The world itself is round, so is all that is in it, more or less, except human work, which is often very flat indeed." This, in nature's gentle developments, is obvious at a glance. In her sterner moods, angular lines and sharp surfaces occur; but they are blended with and softened by the more flowing lines and smoother surfaces, so that it is doubtless true " there are no sharp lines in nature."
Now it may have been that the landscape gardeners of the last century, in their strivings to get away from the constraint of the artificial lines so prevalent then in the arrangement of walks and terraces, and high walls which hampered and confined the ornamental grounds of those days, were led - more than they were aware, perhaps, - by this universal law which they saw and felt in their communings with nature, to adopt the more natural methods of arranging grounds, which a little later on so almost entirely suspended the former style. For a time at least, a line of drive or walk was hardly admirable unless it curved. Straight lines were out of fashion. Curves, and curves only, were in rogue; and there is enough of evidence to make us believe that this undue preference still exists; for the landscape gardener of to-day will hardly have practiced his profession a year before he will have come in collision with this notion in the minds of some of his clients. They will have a horror of straight lines, as nature was once said to have of a vacuum. They will want all their walks to curve, if only for the sake of curving.
He will, of course, know that this is wrong, for he will have learned, among the first axioms of his art, that straight lines still have their place, and that they cannot be laid aside without, at times, violent incongruity; yet he will often find it "hard work " to get this notion safely at a distance.
And yet, after all, it would not be difficult to prove that, as sometimes applied to the arrangement of ornamental grounds, all curves are not lines of beauty. A glance at public grounds within easy distance would convince any practical eye that better arrangements were possible than those which have been made. There seems to have been too great a use of the circular curve. This has its place, of course; but it is not universally applicable, as in some places seems to have been attempted. It is often stiff, and formal, and hard. It reverses badly, does not always fit well to the surface, and, although these faults may in a measure be modified by skillfully combining different radii, still, there is a degree of artificiality which cannot always be overcome. It has indeed proved so arbitrary and obstinate, that many have discarded the idea of laying out their roads and walks by any known law. They have preferred instead to trust to the eye altogether, and they have often secured better results by doing so than any which the circular curve could give them. And when this trusted member (the eye) has failed, other expedients have been called into use. A cart loaded with stones, and drawn by oxen around where the much sought curve was wanted, has answered the purpose.
Others have made a raid upon the nearest drying ground, and brought thence the clothes-line to their aid. But there are difficulties in these methods also. A clothes-line will not always lie exactly in the right place, especially on rough ground, through bushes, or over as yet unremoved stone walls; and in some cases it would take a large family to supply a clothes-line long enough. And so with the ox-cart. It cannot be regarded as a convenient instrument, nor one always available. It would certainly be troublesome to carry one around from place to place, no matter how well trained the oxen; and some places come back to my memory now over which drives were required to be laid out, where it would have puzzled the most sure-footed ox to have made his way, nor would the wheels' track have been exactly the curve wanted, even if he had got safely through.
And to all these methods there is one very strong objection, viz., the difficulty of mapping the grounds after they have been so laid out. or of transferring, in any of these ways, a plan already drawn, to the ground.
Is there not, then, some method which can be devised to overcome these difficulties? I do not know, but I suspect that "it is always dangerous to assert anything as a rule in matters of art." I have, however, for some years been a good deal helped in this part of my work by the use of a curve, which is very easily laid down either on a plan or on the ground. It is one which is far less stiff and formal than a circular curve, is better fitted to uneven ground, leaves a tangent line more slowly, and therefore reverses much more gracefully; and it can be laid off from tangents of unequal length, which is often practically of great convenience. This curve is the Parabola. Its name need not alarm any one by its apparent abstruseness; for, analytically considered, it is one of the simplest of all curves, and a very short acquaintance with its good qualities will convince us that it is a very graceful and obliging one, not willingly to be rejected after a little practice has made obvious its easy application.
This is not the place for a mathematical discussion of the parabola. Reference to any analytical geometry will furnish full information of its several properties; but a single diagram will suffice to show one way in which it may be located.
Tangent or straight lines are first located upon the ground, approximating as nearly as may be to the final location of the drive or walk. In ornamental work it is better to lay out one side rather than the centre. These straight lines and their angles of intersection are measured; or, if no instrument of measuring angles is available, auxiliary lines are also measured, by the help of which they may be mapped, if need be, and which will also furnish the necessary elements for constructing the curve. These tangent lines are then connected by means of some kind of curve; and it is here where the parabola is so available.
Tangent points at which to start and end the curve are selected when most convenient. In circular curves the tangents must be of equal length, unless we take the trouble to introduce a compound curve - that is, a curve of two or more centres; but a parabola can be introduced between either equal or unequal tangents, although in most cases the longer tangent should not exceed the shorter by more than one-half the length of the latter. The length of the tangents and their angles of intersection are all that are wanted to compute the elements of the curve; but if we do not have the means of measuring the angle, our tape-line will do very well, for all we will have to do will be to measure the straight line across from one tangent point to the other, instead of the angle. These remarks are elementary, but they may help further on.
Let us now suppose that we find it necessary to join two tangents, that is, two straight lines by means of a curve, which are at present connected by an angle, and that all we have to work with is a tape-line, some pegs, and a hatchet.
Measure the tangent A B - 80; BC - 120, and the chord A C - 184 feet. Find D, the middle point of A C; then measure BD - 44.8 feet, and find its middle point E. E is a point in the required cure, and B E of course equals 22.4 feet. Now one property of the parabola is this: If we divide its tangents A B and B C into any number of equal parts, the distance from any one of these points to the curve is as the square of the distance of that point from its tangent point A or C. Thus, the distance from the point marked (2) is four times the distance that (1) is from the curve; at (3) the distance is nine times, and at B it is sixteen times that of (1) to the curve.
Now we know the distance B E, because we have measured it. It is the half of C D, that is, 22.4 feet. Hence, if we, for the sake of convenience, call the distance from (1) to curve a, we have 16 a - 22.4, and a - 22.4-16 - 1.4 feet, The distance at (2) will be 4 a - 5.6 feet, and at (3) it will be 9 a - 12.6 feet. Then divide each of the tangents A B and B C into four equal parts, numbering them (1), (2), (3), etc, counting from each tangent point, and at (1) set a peg 1.4 from it, at (2) set another at 5.6 feet from it, and at (3) one at 12.6 feet from it, and the curve will be located at these points. If more points are wanted, we must increase the number of points into which we divide the tangents, always remembering that the distance of the point nearest the tangent point from the curve is found by dividing the distance of the intersection of the two tangents from the curve by the square of the number of parts into which we have divided the tangents, and that each succeeding distance is found by multiplying the first distance (which we called a), by the square of Us number from the tangent point, Thus, if in the example given we had divided our tangent into six equal parts instead of four, a would have been found by dividing 22.4 feet by 36, instead of by 16, and the other distances would have been 4 a, 9 a, 16 a, 25 a.
But there is one thing which we have not yet considered, and which we must not forget, and that is the direction in which the distances are to be laid off. They must all be in one direction, parallel to that middle line B E, and therefore parallel to each other. As we have only a tape-line to work with, the best way to secure this condition is to divide each half of the chord A C into just as many equal parts as we divide the tangents, numbering them (1), (2), (3), etc, from each tangent point. Then from (1) on the tangent range to (1) on the half chords, and lay off a. Set off 4 a from (2) on the tangents to (2) on the half chords, and so on to the end.
This is one of the several ways of locating a parabola.* It is the one which I have found most convenient in my practice, and it has proved so available that I have felt a desire to make others acquainted with it. Of course it will be understood that there is nothing new or original in it, and most civil engineers are familiar with it, but perhaps others who may have occasion to draw the "line of beauty" upon their grounds, and possibly some professional landscape gardeners may not be; they will find it applicable in many ways. As already stated, it reverses well; and it forms one of the most pleasing ovals either for carriage-sweep or flower-bed that I have met with.
* " Vide Henck's Field Book for Engineers."