B

G

Let AB be the side of a triangle, the azimuthal angle of which has been ascertained with reference to NS, the meridian line. Take from an accurately divided diagonal scale, exactly 5 inches as a radius, and from A, as a centre, describe an arc CD; now the chord of an arc being equal to twice the sine of half the arc, the chord CD is equal to twice CE, the sine of half the angle CAD. Take a radius AF equal to twice AC, and describe the arc FG intersecting the radius AB in F, draw the sine FH, then by similar triangles,

S

that is, the chord of a given arc is equal to the sine of half the

arc with double the radius.

The radius of the tables of natural sines is equal to 1 or 10; and having taken the half of 10 or 5 inches for the radius AC, the natural sine of half the given angle taken from the tables will correspond to FH, the sine of half the given angle with double the radius; but FH was proved equal to CD; the natural sine therefore of half the given angle to a radius 10, will be equal to the chord of the whole angle to a radius 5. Having taken that distance from the same scale of inches as the radius, place one foot in the point C, and with the other mark the point D on the arc CD, then through D and A draw the line NS, which will be the direction of the meridian.

This method of laying off angles may also be conveniently employed in dividing a circle to be used as a protractor, and

which can be made either on the same sheet of paper, intended to receive the drawing, or on a separate sheet of card-board, when it may be preserved and used on after occasions. The great difficulty of dividing a circle accurately is well known, but if the arcs are laid off by means of their chords, the division may be performed with great exactness.

A protractor laid down upon the paper, enables the draftsman to plot the work with great rapidity, and with less chance of error, when the scale is small, than by the method of laying off angles by placing the centre of a metallic protractor at every angular point, and pricking off the angle from its circular edge.

During the time which must necessarily be occupied in plotting an extensive and minute Survey, the paper which receives the work is often sensibly affected by the changes which take place in the hygrometrical state of the air, causing much annoyance to the draftsman, as the parts laid down from the same scale at different times will not exactly correspond. To remedy in some measure this inconvenience, it has been recommended that the apartments appropriated to the purposes of drawing, should be constantly kept in as nearly the same temperature as possible and also that the intended scale of the plan should be first accurately laid down upon the paper itself; and from this scale all dimensions for the work should invariably be taken, as the scale would always be in the same state of expansion as the plot, though it may no longer retain its original dimensions.

Another method of protracting a Survey, and by which the inconveniences of the above methods are avoided, and by which also the accuracy or otherwise of the Field-work is decided with precision and certainty, will be presently treated of, in the meanwhile we refer the reader to Chap. 8 and 9, Part II., where, in describing the use of the several instruments used in plotting, further instructions are given, and close this chapter by extracting from Mr. Bradley's valuable

work on Practical Geometry, the following useful rules, applicable to Geometrical construction :

1. Arcs of circles, or right lines by which an important point is to be found, should never intersect each other very obliquely, or at an angle of less than 15 or 20 degrees; and, if this cannot be avoided, some other proceeding should be had recourse to, to define the point more precisely.

2.

When one arc of a circle is described, and a point in it is to be determined by the intersection of another arc, this latter need not be drawn at all, but only the point marked off on the first, as it is always desirable to avoid the drawing of unnecessary lines. The same observation applies to a point to be determined on one straight line by the intersection of another.

3. Whenever the compasses can be used in any part of a construction, or to construct the whole problem, they are to be preferred to the rule, unless the process is much more circuitous, or unless the first rule (above) forbids.

4. A right line should never be obtained by the prolongation of a very short one, unless some point in that prolongation is first found by some other means, especially in any essential part of a problem.

5. The larger the scale on which any problem, or any part of one, is constructed, the less liable is the result to error: hence all angles should be set off on the largest circles which circumstances will admit of being described, and the largest radius should be taken to describe the arcs by which a point is to be found through which a right line is to be drawn; and the greater attention is to be paid to this rule, in proportion as that step of the problem under consideration is conducive to the correctness of the final result.

6. All lines, perpendicular or parallel to another, should be drawn long enough at once, to obviate the necessity of producing them.

7. Whenever a line is required to be drawn to a point, in order to insure the coincidence of them, it is better to com

mence the line from the point; and if the line is to pass through two points, before drawing it the pencil should be moved along the rule, so as to ascertain whether the line will, when drawn, pass through them both. Thus, if several radii to a circle were required to pass through any number of points respectively, the lines should be begun from the centre of the circle; any error being more obvious when several lines meet in a point.

CHAPTER V.

ON OFFSETS, AND THE VARIOUS PRACTICAL METHODS OF FINDING AREAS, INCLUDING THE REDUCTION OF INCLINED TO HORIZONTAL PLANES.

THE area of the principal triangles in a Survey, should, in all cases, be computed from the length of their sides, as obtained from the Field-book. The operation is simple, by the following rule :

Rule. Add the three sides together, from half the sum of the sides subtract each side severally. Multiply the half sum and three remainders together, the square root of their product will be the area, or—

By Logarithms, which method is a much shorter calculation than the former: To the Logarithm of half the sum of the sides, add the Logarithms of the three remainders, the sum of these Logarithms divided by 2, will be the Logarithm of the

area.

To find the area of offsets by calculation: Multiply half the sum of each successive pair, by the distance on the Chain line between them, the sum of all these separate areas, will give the area of the whole offset on the Chain line.