open and shut their legs till an arc described by the other point will just touch the opposite side.

Otherwise; a platting scale, (described in Art. (49) may be placed so that the zero point of its edge coincides with the angle, and one of its cross lines coincides with the side to which a perpendicular is to be drawn. The length of the perpendicular can then at once be read off.

The method of dividing the plat into triangles is the one most commonly employed by surveyors for obtaining the content of a survey, because of the simplicity of the calculations required. Its correctness, however, is dependant on the accuracy of the plat, and on its scale, which should be as large as possible. Three chains to an inch is the smallest scale allowed by the English Tithe Commissioners for plats from which the content is to be determined.

In calculating in this way the content of a farm, and also of its separate fields, the sum of the latter ought to equal the former. A difference of one three-hundredth (3ᄒᄒ) is considered allowable.

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Some surveyors measure the perpendiculars of the triangles by a scale half of that to which the plat is made. Thus, if the scale of the plat be 2 chains to the inch, the perpendiculars are measured with a scale of one chain to the inch. The product of the base by the perpendicular thus measured, gives the area of the triangle at once, without its requiring to be divided by two.

Another way of attaining the same end, with less danger of mistakes, is, to construct a new scale of equal parts, longer than those by which the plat was made in the ratio √2:13 or 1.414:1. When the base and perpendicular of a triangle are measured by this new scale and then multiplied together, the product will be the content of the triangle, without any division by two. In this method there is the additional advantage of the greater size and consequent greater distinctness of the scale.

When the measurement of a plat is made some time after it has been drawn, the paper will very probably have contracted or expanded so that the scale used will not exactly apply. In that case a correction is necessary. Measure very precisely the present length of some line on the plat of known length originally. Then make this proportion: As the square of the present length of this line Is to the square of its original length, So is the content obtained by the present measurement To the true content.

(72) Graphical Multiplication. Prepare a strip of drawing paper, of a width exactly equal to two chains on the scale of the plat; i. e. one inch wide, as in the figure, for a scale of two chains to 1 inch; two-thirds of an inch wide for a scale of 3 chains; half an inch for 4 chains; and so on. Draw perpendicular lines across the paper at distances representing one-tenth of a chain on the scale of the triangle to be measured, thus making a platting scale. Apply it to the triangle so that one edge of the scale shall pass through one corner, A, of the triangle, and the other edge through another

corner, B; and note very precisely what divisions of the scale are at these points. Then slide the scale in such a way that the points of the scale which had coincided with A and B, shall always remain on the line BA produced, till the edge arrives at the point C. Then will A'C, that is, the distance, or number of divisions on the scale, from the point to which the division A on the scale has arrived, to the third corner of the triangle, express the area of the triangle ABC in square chains.*

(73) Division into Trapezoids. A line may be drawn across

the field, as in Fig. 36, and perpendiculars drawn to it. The field will

Fig. 36.

thus be divided into trapezoids, (excepting a triangle at each end), and their content can be calculated by Art. (67).

Otherwise; a line may be drawn outside of the figure, and perpendiculars to it be drawn from each angle. In that case the difference between the trapezoids formed by lines drawn to the outer angles of the figure, and those drawn to the inner angles,

will be the content.

Fig. 37.

This method is very advantageously applied to surveys by the compass; as will be explained in Part III, Chap. VI.

(74) Division into Squares. Two sets of parallel lines, at

right angles to each other,

one chain apart (to the scale of the plat) may be drawn over the plat, so as to divide it into squares, as in the figure. The number of squares which fall within the plat represent so many square chains; and the triangles and trapezoids which fall outside

of these, may then be calcu

Fig. 38.

ب

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lated and added to the entire square chains which have been counted.

Instead of drawing the parallel lines on the plat, they may better be drawn on a piece of transparent "tracing paper," which is simply laid upon the plat, and the squares counted as before. The

same paper will answer for any number of plats drawn to the same scale. This method is a valuable and easy check on the results of other calculations.

To calculate the fractional parts, prepare a piece of tracing paper, or horn, by drawing on it one square of the same size as a square of the plat, and subdividing it, by two sets of ten parallels at right angles to each other, into hundredths. This will measure the fractions remaining from the former measurement, as nearly as can be desired.

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(75) Division into Parallelograms. Draw a series of parallel lines across the plat at equal distances depending on the scale. Thus, for a plat made to a scale of 2 chains to 1 inch, the distance between the parallels should be 21 inches; for a scale of 3 chains to 1 inch, 1 inch; for a scale of 4 chains to 1 inch, inch; for a scale of 5 chains to 1 inch, 1 inch; and for any scale, make the distance between the parallels that fraction of an inch which would be expressed by 10 divided by the square of the number of chains to the inch. Then apply a common inch scale, divided on the edge into tenths, to these parallels; and every inch in length of the spaces included between each pair of them will be an acre, and every tenth of an inch will be a square chain.*

Apply it to the
Fig. 39.

To measure the triangles at the ends of the strips between the parallels, prepare a piece of transparent horn, or stout tracing paper, of a width equal to the width between the parallels, and draw a line through its middle longitudinally. oblique line at the end of the space between two parallels, and it will bisect the line, and thus reduce the triangle to an equivalent rectangle, as at A in the figure. When an angle occurs between two parallels, as at B in the figure, the fractional part may be measured by any of the preceding methods.

A

B

* For, calling the number of chains to the inch, = n, and making the width be

10

10

tween the parallels 2 inch, this width will represent - Xn -chains; and

n

n

10

n2

10 X n = 10 square chains

as the inch length represents a chains, their product, - 10

n

A somewhat similar method is much used by some surveyors, particularly in Ireland: the plat being made on a scale of 5 chains to 1 inch, parallel lines being drawn on it, half an inch apart, and the distances along the parallels being measured by a scale, each large division of which is 1 inch in length. Each division of this scale indicates an acre; for it represents 4 chains, and the distance between the parallels is 21 chains. This scale is called the "Scale of Acres."

(76) Addition of Widths. When the lines of the plat are very

irregularly curved, as in the

figure, draw across it a number of equi-distant lines as near together as the case may seem to require. Take a straight

Fig. 40.

edged piece of paper, and apply one edge of it to the middle of the first space, and mark its length from one end; apply the same edge to the middle of the next space, bringing the mark just made to one end, and making another mark at the end of the additional length; so go on, adding the length of each space to the previous

ones.

When all have been thus measured, the total length, multiplied by the uniform width, will give the content.

(77) THIRD METHOD.-INSTRUMENTALLY. By performing certain instrumental operations on the plat.

(78) Reduction of a many sided figure to a single equivalent triangle. Any plane figure bounded by straight lines may be reduced to a single triangle, which shall have the same content. This can be done by any instrument for drawing parallel lines,