nished, in addition to the customary steel-points (e), with pencil and pen points, and an extension-bar, shown at g, h, and i, respectively, which are inserted as needed and securely held in place by set-screws.

The best instruments are of German silver, and have a steel friction-plate interposed between the branches or legs at the pivot, to prevent wear and secure smoothness of motion: the necessary amount of friction being obtained by means of the horizontal adjusting-screws shown in the figure-the vertical adjusting-screws serving to bind the handle to the heads of the former. With the exception of the steel-points, used principally for spacing and for setting off distances, the different points are hinge-jointed, to admit of their being set perpendicularly to the paper in describing arcs, which is particularly necessary in using the pen-point.

In the absence of leads made especially for the pencil-point, serviceable ones can be cut from a hard-lead pencil. The "shouldered " needle, shown inserted, is more rigid and convenient than a plain needle. A large radius is obtained with the extension-bar.

In describing arcs, the handle is held between the thumb and forefinger, the point of the needle is kept in place with very light pressure, and the curve is described with a continuous movement. To prevent wearing large holes in the paper, when several arcs are described from the same centre, a "horn-centre"-a thin transparent piece of horn, with points on its under surface to keep it in place-is interposed between the point and the paper.

14. Spring-Dividers (k, Fig. 2).—This instrument, also called a bow-pen or pencil, is used for describing arcs of very small radii. In the kind shown at the top of the figure the legs are formed of one piece of steel, bent until the points are about an inch asunder, and are adjusted by means of the screw and nut. Of the other two, in the form shown on the left, the pen or pencil point revolves about a steel rod, which serves both as a handle and needle-point; and as the rod is always kept perpendicular to the paper, and stationary, it is the handier form for describing very small arcs.

15. Irregular Curve. This instrument has a variety of forms, two of which are shown in the accompanying figure. It is very useful as a guide to the pen or pencil point in drawing bends of roads and other accidental curves, giving a smoothness to the drawing not otherwise easily attainable. The only difficulty attending its use is to avoid making angular junctions in the prolongation of a line.

This is obviated by being careful to make the edge of the curve, at the beginning of the prolongation, tangent to the latter part of the line already drawn.

16. Beam-Compass (m, Fig. 2).—This instrument is used for describing arcs of greater radii than can be described with the dividers and extension-bar. As shown in the figure, it consists of two metal clamps which can be attached to a straight-edge or bar, and at any required distance apart. The clamps have sockets for pen, pencil, or needle points. Some kinds require special bars to fit the clamps, while in others, termed "portable beam-compasses," the clamps can be attached to any straight-edge. The pattern (McCord's) shown in

the figure is substantial, and has exact means of adjustment afforded by the form of clamp, the part carrying the point being made to traverse the slide by means of the milled-head screw, while the points remain perpendicular to the paper.

It is best used with a graduated bar made to fit it. A support furnished with casters is often used with a beam-compass to support the bar at its middle point.

17. The Curve- Pen (Fig. 3).-This differs from the right-line pen in having curved nibs attached to a rod which turns freely within the handle. It is particularly useful in drawing curves free-hand; the pen for this purpose being held with the handle perpendicular to the surface of the paper, and moved in the direction of the required curve-the point following the pencilled tracing. By means of a nut at the upper extremity of the handle, the latter may be clamped to the rod, and the instrument used as a right-line pen.

18. Dotting, Border, and Road Pens.-These may be termed labor-saving instruments, since by their use regular spacing and even width of line are attained by a single stroke, while the eye and hand are only engaged in following the right direction.

The Dotting-Pen (Figs. 4 and 6) has a set of interchangeable wheels, with teeth or cogs corresponding to the arrangements of dots and dashes required.

In the form (Fig. 6), the teeth act as cogs upon a lever which carries a pen-point at its outer extremity; and motion is given to the cog-wheel by rolling the outer wheel, fastened to the same arbor with the latter, along the edge of a ruler.

In the other form, the wheel is supplied with ink from a reservoir above, the teeth being applied directly to the paper. This instrument is also called a roulette.

The Border and Road Pen (Fig. 5) is used for drawing either heavy or parallel lines. For the former, the space between the branches is filled with ink, and adjusted by means of the large milled-head screw; for roads, the branches only are filled, and can be independently adjusted to different widths of line. It is useful in drawing any features represented by parallels. Some draughtsmen prefer the branches bent in the same direction, to facilitate following the edge of a ruler closely.

To a

19. The principal instruments for measuring lines and angles, or for setting off horizontal distances, azimuths and bearings, are the scale of equal parts and the protractor. description of these is added that of the sector, mainly on account of its usefulness in finding proportional parts. These instruments are made of boxwood, ivory, hard rubber, or metal. Horn and paper are also used for protractors and scales.

20. Scale of Equal Parts.—A convenient form of this instrument is the "triangular scale of equal parts" (Fig. 8), so called from the shape of its cross-section. Its different edges are graduated to show 10, 20, 30, 40, 50, and 60 parts of an inch, and for finer work 80 and 100 parts. Those graduated the entire length of the edges are most convenient. Distances are laid off directly with this scale. The "scale-guard" is a sliding attachment used with the scale to save time in finding the particular edge in use, and in repeating a measurement. In the use of paper scales, which are accurately engraved or machine divided, and similarly graduated, the variations in length due to heat and moisture, are sensibly equal to corresponding changes in the drawing; besides, they are not so liable to soil the latter as scales made from other material.

Scales of equal parts, graduated to other convenient units, as chains, metres, etc., are likewise readily obtainable. When much time is required for the completion of a drawing,

greater accuracy is probably attained by the use of a scale constructed upon the drawing itself.

21. Spacing-Dividers.-The steel points of dividers are of frequent use in comparing distances, and in transferring them from a scale; but when great nicety is required and successive small distances are to be laid off, the spacing-dividers, which differ from the springdividers (k, Fig. 2) only in having two steel points instead of one, are very useful.

22. Protractors.-These instruments, in general use in plotting, are for the direct measurement of angles. The usual forms are the rectangular, semicircular, full circle, and vernier. The rectangular protractor is graduated on both faces, the front face (Fig. 9) having three edges graduated into degrees or half-degrees, the third edge being the diameter of the circle of graduation, with its centre marked as shown.

The divisions are commonly numbered from o° at the left, around to 180° on the outer edge, and repeated in the reverse direction on an inner line. This face also contains scales of equal parts of an inch, and a scale of chords marked "CHO" or "C."

The other face contains like scales, with the addition of a diagonal scale of equal parts. The Abbot Protractor, of rectangular shape, has the edges of each face graduated to halfdegrees, numbered from left to right; on one face from o° to 180°, and on the other from 180° to 360°. It contains a scale of tenths of inches, and one of hundredths of a foot. It is particularly handy in field-sketching.

23. The Semicircular Protractor is of two forms-either with the graduation extending a few degrees beyond the diameter, or with the diameter and outer edge coincident. In the latter case a straight-edge can be utilized in securing exact coincidence of the diameter with a given line, by first adjusting the straight-edge to the line, and then placing the diameter against it.

The circular protractor is simply an extension of the graduation to 360°, covering an entire circle.

24. To protract an Angle with either of the above forms: The instrument is laid flat upon the paper, and held securely in position; a line is drawn along the diameter, and points are marked upon the paper exactly beneath the centre, and the degree-division corresponding to the required angle; then a right line joining these points will make the required angle with the first line drawn. To draw a line through any point of a given line, which shall make a required angle with the line: the diameter and centre are first placed coincident with the line and point respectively, and the operation is then as above described; or the centre and degree-division corresponding to the angle may be placed coincident with the line, and a line drawn along the diameter.

To protract an angle with the scale CHO: From a given point of a right line as a centre, with the distance 0-60 as a radius, describe an arc intersecting the line; from this point of intersection as a centre, with a radius equal to the distance from o to that division of CHO corresponding to the number of degrees in the required angle, describe an arc intersecting the first arc; the right line joining the point of intersection of the arcs with the given point will make the required angle with the given line.

25. To Plot a Bearing.-In plotting bearings taken with the "surveyor's compass," the quadrant in which a course lies is apparent from the designation of the bearing: this is also the case with the improved prismatic compass (a, par. 154); but a little difficulty arises at

first with other methods of graduation used in the prismatic compass, which the following simple rule is intended to obviate.

With the compass-card graduated from o° at the N. around to the right 360°: I. Denote the difference between the reading and 180° by + R or — R, according as the reading is less or greater than 180°. II. For + R, use the direct graduation of the protractor, placing the protractor so that its centre and the degree division R shall be on a meridian line passing through or near the plotted station, and the diameter shall pass through it; a line drawn along the diameter from the station to the left will coincide with the required course. For R, use the reverse graduation and draw the line to the right.

It is apparent that, with a compass-card graduated from o° at the N. around to the right 180°, repeated in the same direction on the other semi-circumference, the only modification in this rule is that the W. readings themselves are the R values.

With the second protractor described in par. 22, and the above 0°-360° graduated card, the rule applies to readings less than 180°; but owing to the direct graduation of this protractor from 180° to 360°, readings greater than 180° are subtracted from 540°.

26. Vernier Protractor.-With protractors already described it is difficult to estimate as closely as one fourth of a degree; consequently for accurate work the instrument represented in Fig. 11 is used. The arm, movable about the centre, carries a vernier, reading in some instruments to minutes. The beveled edge of the arm is in the prolongation of the radius passing through the o of the vernier, and the exact centre is indicated by the intersection of two fine lines marked on a piece of horn let into the instrument. In protracting an angle, the arm is set by means of the vernier, the 180° diameter is placed coincident with the given line, with the centre at the given point; and using a sharp-edged pencil, the required line is drawn along the beveled edge. In some instruments the graduation extends over a full circle.

27. Among other forms of vernier protractors are Crozet's and Tabarant's. In the former the scale is let into a flat rectangular metal frame. By turning the scale about its centre, the diameter may be set at any required angle with an edge of the frame, which edge is then placed against a straight-edge coincident with the meridian or other line of direction, and the protractor is moved along it until the diameter passes through the desired point.

The special feature of Tabarant's protractor is a metal parallelogram, articulated at the corners, hinged at one corner to an extremity of the diameter and connected at an adjacent corner with the other extremity of the diameter by an arc, of which the length can be made equal to the magnetic declination. The parallelogram, of which the edge joining the other two corners is placed against a straight-edge, serves to enlarge the area of plotting for any position of the latter.

28. In a semicircular protractor, the diameter and outer edge when not coincident, should be parallel.

To test this: Draw a line along the diameter, also a radial line to any degree-division; then slide the protractor along the radial, the centre and this division remaining upon it: the outer edge should coincide with the first line drawn.

29. Sector (Fig. 10).—This is of about the form and size of the single-folding foot-rule. Radials, termed sectoral lines, arranged in pairs, one of a pair on each arm, and variously divided, are engraved on each face.

The sectoral lines on one face are a pair of scales of equal parts termed the "line of lines," a pair of scales of chords, of secants, and of polygons, marked respectively L, C, S and POL. A scale of tenths of inches is marked along the edge, and one of hundredths of a foot on the outer edge. On the other face are sectoral lines of sines, of tangents up to 45°, and another of tangents from 45° to 75° to a lesser radius, marked respectively S and T; together with "Gunter's lines" of logarithmic numbers, sines and tangents, marked respectively N, S, and T.

The solution of problems with the sector depends upon the principle that homologous sides of similar triangles are proportional, and a solution is termed simple or compound. according as it involves the use of one or two pairs of lines.

In the latter case, from the above principle, the two pairs used should make equal angles at the centre; but in some instruments the angles formed by the line T, from 45° to 75°, and of secants, are equal to each other, but unequal to those formed by the other sectoral lines; therefore when one of these circular functions is used in a compound solution, it must be expressed in terms of some other function.

A distance measured from the centre on a sectoral line is termed lateral (1); and from a point on one of a pair to the corresponding point on the other, transverse (t). The divisions of a sectoral line are contained between three parallels, and the points of the dividers are always applied to the inner parallel, or to the one that passes through the centre.

To use the line L: 1. To find a fourth proportional, as x, in the proportion a: bc: x; set off = a; open the sector until, at the extremity of 1, t = b; prolong until it is equal l to c; then at the extremity of this prolongation,t = x.

2. To bisect a right line, as A: Make 10 10 A, then 15 — 5 sake of verification is laid off from each extremity of A.

A == which for the 2 2

3. A line is divided into an even number of equal parts by first bisecting it as above, then bisecting each half, and so on; and into an uneven number by a simple application of the same principle; e.g., to subdivide A into five equal parts: make 15 — 5 = A, lay off 13 — 3 from each extremity, and then bisect the extreme parts.

4. To obtain a scale of, or 1 inch =10 feet, for measurements of feet and tenths of a foot: Make 10 10 I inch; then 13.4 3.4 3.4 feet; and so on.

5. For copying drawings to a scale differing from the original; e.g., to reduce a drawing, lineally, one fifth: Make 15 - 5 = a line of the original; then t4 4 is its reduced length, and points are found in a similar way by intersecting arcs described with reduced radii from the extremities of this line as centres, and so on throughout the drawing. The lines L may be used for interpolating heights. Thus, in the figure, the heights or references of the plotted points a and b are 20 and 25 feet respectively, and the line ab is of uniform declivity; to interpolate the (23') point. On the lines L make 15-5 ab; then 133 ac, which laid off from a gives the required point.

(20

=

(25)

b

The lines C are used to protract angles to any radius, 160 - 60. For any angle of 60° or less, say 20°: With 160 - 60 describe an arc; subtend any portion of it by a chord = t20 20; the radials drawn to the extremities of the chord include the required angle. If the angle is very small, say 20, with 160 - 60 describe an arc as before; then from any point of it set off