Chap. VIII.

GEOMETRY.

763

Fig. 704.

work should always set them out proportionally, which sometimes is rather a difficulttask to perform. As the excavation proceeds it should be measured, as it is expedientthat this should be done before any of the marks are obliterated or destroyed, which they areconstantly subject to, particularly where horses and carts are employed in any great numbers.

Rules , Sight Vanes , fyc. — Rules are required of differentlengths and thicknesses, with straight and bevelled edges, a

either to draw lines on paper in pencil or with ink; but | ~ — H!

when a line is to be drawn on the ground, it may be done

more readily by stretching a line, as at C, from one stump_.

to another. L-!

Parallel Rules. — There are several kinds in use; thebest consists of a single rule with an axis, carrying twosmall rollers fixed at each extremity : these must bemade of precisely equal diameters, and should be as farapart as the length of the rule will permit: an instru-ment rolling on two such wheels will be moved parallelto any position it was first placed in, and consequently parallels to any line to whichits edge is set may be drawn. The edges of the wheels are grooved very truly, to preventthem from slipping instead of rolling, which would affect the parallelism.

The second variety consists of two plain rules connected by two equal pieces of brassturning on centres, and these must be truly parallel, so that in every position the fourcentres of motion may form a parallelogram : when used, the edge of one being setto any line, the other rule must be firmly held down by the hand, and the first movedtill the same edge is brought to where the required parallel to the given line is to bedrawn. The faces of these rules are generally provided with scales.

E, D, or any other very long line, may be accurately setout by means of piquets placed at short distances, andmaking use of an instrument with an eye-hole, as at B, H,and which is made to traverse along the line, and havingattached to this instrument a plummet so that its per-pendicular may always be maintained; parallel lines mayafterwards be drawn in any number. Take also the shortestdistance between the point A and the given line D E, byplacing one foot of the compasses on the point A, anddescribing with the other an arc which shall touch thegiven line D E in the point F, than the interval A F willbe the shortest distance from the point A; then with thesame distance, from the point G, strike a similar curve, andlines may be thus drawn or set out parallel to each other.

To draw through the point A a line parallel to K I,first draw from the point A the line A K, till it touchesthe right line K I in any point K; from this point A,and with any radius as O P, describe the arc O P, thenwith the same radius from A strike the arc Q. N, and bysetting off the same angle from N to Q, as that of O P,and then through Q drawing the line A Q, we have theparallel line required.

. .-d:

Fig. 795.

Fig. 796.

passes: after a straight edge is laid down, points may bemarked at the required distance, through which the linemay be afterwards traced.

If through the piquet Q a line parallel to the railingB S is to be drawn, and you then place two piquets at B,

S, and with the same length of cord strike two portions ofa circle, as at Q. T, a line drawn through two other pi-quets placed at the extremity of this radius will be atonce equally distant from the railing and parallel with it.

Perpendicular lines are those placed upon another insuch a manner that the adjacent angles formed by theirintersections are equal, and consequently each is a rightangle. A straight line is perpendicular to a curve at agiven point, when it is perpendicular to the tangent to thecurve at that point, in which case it is sometimes called anormal to that curve.

A straight line is perpendicular to a plane when it is at right angles with every straightline in the plane passing through the point of intersection: a plane is perpendicular to aplane, when any straight line in the first, which is perpendicular to the common in-tersection of the two planes, is also perpendicular to the second plane.

Fig. 797.