Decimal parts of an acre.

Multiply the remainder by 40, and the outside figures will be Perches. The nearest round number is usually taken for the Perches; fractions less than a half perch being disregarded.*

Thus, 86.22 square chains = 8 Acres 2 Roods 20 Perches.

(59) The following Table gives by mere inspection the Roods

and Perches corresponding to the Decimal parts of an Acre. It

.162.412.662.912 +26

.169.419.669.919 +27

.175.425.675.925

+28

.181.431.681.931

+29

.187.437.687.937 +30

.194.444.694.944

+31

.200.450.700.950 +32

.206.456.706.956 +33

.212.462.712.962 +34

.219.469.719.969 +35

.225.475.725.975 +36

.231.481.731.981

+37

.237.487.737.987

+38

.244.494.744.994

+39

(60) Chain Correction. When a survey has been made, and the plat has been drawn, and the content calculated; and after

wards the chain is found to have been incorrect, too short or too long, the true content of the land, may be found by this proportion : As the square of the length of the standard given by the incorrect chain Is to the square of the true length of the standard So is the calculated content To the true content. Thus, suppose that the chain used had been so stretched that the standard distance measured by it appears to be only 99 links long; and that a square field had been measured by it, each side containing 10 of these long chains, and that it had been so platted. This plat, and therefore the content calculated from it, will be smaller than it should be, and the correct content will be found by the proportion 992 : 1002::100 sq. chains :: 102.03 square chains. If the chain had been stretched so as to be 101 true links long, as found by comparing it with a correct chain, the content would be given by this proportion: 1002 : 1012 : : 100 square chains : 102.01 square chains. In the former case, the elongation of the chain was 15 true links; and 1002: (101) :: 100 square chains : 102.03 square chains.*

(61) Boundary Lines. The lines which are to be considered as bounding the land to be surveyed, are often very uncertain, unless specified by the title deeds.

If the boundary be a brook, the middle of it is usually the boundary line. On tide-waters, the land is usually considered to extend to low water mark.

Where hedges and ditches are the boundaries of fields, as is almost universally the case in England, the dividing line is generally the top edge of the ditch farthest from the hedge, both hedge and ditch belonging to the field on the hedge side. This varies, however, with the customs of the locality. From three to six feet from the roots of the quickwood of the hedges are allowed for the ditches.

METHODS OF CALCULATION. (62) The various methods employed in calculating the content of a piece of ground, may be reduced to four, which may be called Arithmetical, Geometrical, Instrumental, and Trigonometrical.

(63) FIRST METHOD.-ARITHMETICALLY. From direct measurements of the necessary lines on the ground.

The figures to be calculated by this method may be either the shapes of the fields which are measured, or those into which the fields can be divided by measuring various lines across them.

The familiar rules of mensuration for the principal figures which occur in practice, will be now briefly enunciated.

(64) Rectangles. If the piece of ground be rectangular in shape, its content is found by multiplying its length by its breadth.

(65) Triangles. When the given quantities are one side of a triangle and the perpendicular distance to it from the opposite angle; the content of the triangle is equal to half the product of the side and the perpendicular.

When the given quantities are the three sides of the triangle; add together the three sides and divide the sum by 2; from this half sum subtract each of the three sides in turn; multiply together the half sum and the three remainders; take the square root of the product; it is the content required. If the sides of the triangle be designated by a, b, c, and their sum by s, this rule will give its √[(8-a) (8-6) (18-c)].*

area

Fig. 30.

B

* When two sides of a triangle, and the included angle are given, its content equals half the product of its sides into the sine of the included angle. Designating the angles of the triangle by the capital letters A,B,C, and the sides opposite them by the corresponding small letters a,b,c, the area = be sin. A. When one side of a triangle and the adjacent angles are given, its content equals the square of the given side multiplied by the sines of each of the given angles, and divided by twice the

D

C

sine of the sum of

sin. B. sin, C

these angles. Using the same symbols as before, the area =a2 2 sin. (B+C)

When the three angles of a triangle and its altitude are given, its area, referring to the above figure, = & BD2

•

sin. B
sin. A. sın. C

•

(66) Parallelograms; or four-sided figures whose opposite sides are parallel. The content of a Parallelogram equals the product of one of its sides by the perpendicular distance between it and the side parallel to it.

(67) Trapezoids; or four-sided figures, two opposite sides of which are parallel. The content of a Trapezoid equals half the product of the sum of the parallel sides by the perpendicular distance between them.

If the given quantities are the four sides a, b, c, d, of which b and d are parallel; then, making q = (a+b+c-d), the area

of the trapezoid will

b+d,

d

√[q(q-a)(q-c) (q-b+d).]*

(68) Quadrilaterals, or Trapeziums; four-sided figures, none of whose sides are parallel.

A very gross error, often committed as to this figure, is to take the average, or half sum of its opposite sides, and multiply them together for the area: thus, assuming the trapezium to be equiva lent to a rectangle with these averages for sides.

In practical surveying, it is usual to measure a line across it from corner to corner, thus dividing it into two triangles, whose sides are known, and which can therefore be calculated by Art. (65).†

* When two parallel sides, b and d, and a third side, a, are given, and also the angle, C, which this third side makes with one of the parallel sides, then the content of the trapezoid=" a. sin. C.

b+d
2

•

† When two opposite sides, and all the angles are given, take one side and its adjacent angles, (or their supplements, when their sum exceeds 180°), consider them as belonging to a triangle, and find its area by the second formula in the note on page 43. Do the same with the other side and its adjacent angles. The difference of the two areas will be the area of the quadrilateral.

When three sides and their two included angles are given, multiply together the sine of one given angle and its adjacent sides. Do the same with the sine of the other given angle and its adjacent sides. Multiply together the two opposite sides and the sine of the supplement of the sum of the given angles. Add together the first two products, and add also the last product, if the sum of the given angles is more than 180°, or subtract it if this sum be less, and take half the result. Calling the given sides, p, q, r; and the angle between p and q = A; and the angle between q and r

=

B; the area of the quadrilateral

= [p.q.sin. A + q.r.sin. B + p.r. sin. (180° — A - B)]. When the four sides and the sum of any two opposite angles are given, proceed thus: Take half the sum of the four given sides, and from it subtract each side in turn Multiply together the four remainders, and reserve the product. Multiply together the four sides. Take half their product, and multiply it by the cosine of the given sum of the angles increased by unity. Regard the sign of

(69) Surfaces bounded by irregularly curved lines. The rules for these will be more appropriately given in connection with the surveys which measure the necessary lines; as explained in Part II, Chap. III.

(70) SECOND METHOD.-GEOMETRICALLY. From measurements of the necessary lines upon the plat.

(71) Division into Triangles. The plat of a piece of ground having been drawn from the measurements made by any of the methods which will be hereafter explained, lines may be drawn upon the plat so as to divide it into a number of triangles. Four

Fig. 31.

Fig. 32.

Fig. 33.

Fig. 34.

ways of doing this are shown in the figures: viz. by drawing lines from one corner to the other corners; from a point in one of the sides to the corners; from a point inside of the figure to the corners; and from various corners to other corners. The last method is usually the best. The lines ought to be drawn so as to make the triangles as nearly equilateral as possible, for the reasons given in Part V.

One side of each of these triangles, and the length of the perpendicular let fall upon it, being then measured, as directed in Art. (43,) the content of these triangles can be at once obtained by multiplying their base by their altitude, and dividing by two.

The easiest method of getting the length of the perpendicular, without actually drawing it, is, to set one point of the dividers at the angle from which a perpendicular is to be let fall, and to