CONSTRUCTION, the art of desoribing a diagram or scheme from given data.
In geometrical constructions, the accuracy of the diagram depends upon that of the points by which the lines constitut ing the figure are found. It is, therefore, of the utmost consequence to ascertain the situation of points correctly by lines crossing at right angles, or as nearly so as possible.
The choice of this is not at all times in the power of the geometer, lint when it is, he ought to avail himself of it. The situation of a point must be ascertained by the intersec tion of two iines, mid since a line cannot be without breadth, it will 1.0 all oblong. and the intersection of two lines will be a parallelogram; when the lines cross at right angles, the parallelogram will tt square ; and when at oblique angles it ill he a rhombus. In all those cases, the point regnired is in the intersection of the diagonals of the paral lelogram. Now the least of all the parallelograms !brined by the intersection of two lines of equal breadth, is a square; hut the greater the obliquity of the lines of intersection, the longer will be the rhombus; and as the drawing of the figure depends upon vision, the tome indistinct Will the angles, of the figure so formed become, and consequently the situa tion of the point must be almost guessed at. In some cases, the obliquity of the intersection is of little consequence; as, in finding- a curve by points. where the hues, which form the intersection, full very nearly in with the curve itself, or make very acute angles with the tangent, unless it be required to find the points in the curve in a given ratio; but if in finding a point, through which a line is to pass from another given point. to meet a line, of which some parts are either given or found, or to be found, it will be of the utmost consequence to determine with accuracy the situ ation of the intermediate point ; for the point asertained in the other line will vary from its true place, more or less, aecording, to the distance of the intermediate point found by the intersection.
Another source of error arising from the intersection of oblique liars. which will also be more or less accurate, as the obliquity is less or greater, is, when one or both the lines are not exactly drawn through their extremities; even the deviation of a line being drawn its own breadth, will make the intersection fall its own length (which is the diagonal of the rhombus) to the end of th., true intersection. Let it
also be considered, that the longer diagonal of the rhombus may be of any length whatever, depending upon the obliquity of angles formed by the two intersecting, lines. In the description of a diagram, when different points are ascertained in a line, in pointing out the line to the reader, it would bd better to name all the letters in the order as thy• stand, instead of pointing, out the line by two of the letters, par ticularly in a complicated diagram, where many other lines are cunecrne,l. This is still more necessary when several lines meet at the same point, as the use of all the letters not only gives a more immediate elne to identity the lines from others, but also shortens the description, as the same letters must be used again, in pointing out the other lines which cross the former line, and will thus supersede the necessity of the frequent repetition, after a line has been drawn in the required position to cut a former, of say ing, " cutting such a line in the point" or a, or whatever it map be; as the same letter cannot be in two lines, except at their inter section.
In tracing the boundaries of angular figures, it will only he necessary to name the letters progressively, as they stand at the extremities of the sides, that is, at the angles; but to trace out the whole enclosure or perimeter, it would be neces sary to name the first letter again, at the end of the series of letters. It is true, that a triangle. a quadrilateral, &c., will be understood, without naming the first letter again, by naming the figure at the same time, or the nituthei ot its sides, as in polygons the last side will always be wanting.
Though these enumerations and repetitions of letters may appear clumsy, they lead sooner to an understanding of the construction, shorten the language, and give accuracy to the des•riptbill.