The procedure for distributing a linear error of closure (one within the allowable maximum, of course) over the directions and distances in a closed traverse is called balancing or closing the traverse. Before you can understand how to do this, you must have a knowledge of latitude and departure.

LATITUDE AND DEPARTURE. Latitude and departure are values that are employed in the method of locating a point horizontally by its plane coordinates. In the plane coordinate system, a point of origin is arbitrarily y selected for convenience. The location of a point is given in terms of its distance north or south and its distance east or west of the point of origin. The plane coordinate system will be explained later in this chapter.

The latitude of a traverse line means the length of the line as projected on the north-to-south meridian running through the point of origin. The departure of a traverse line means the length of the line as projected on the east-to-west parallel running through the point of origin. To understand this, you should examine figure 7-8. The point of origin is at O. The line NS is the meridian through the point of origin; the line EW is the parallel through the point of origin. The latitude of AB is the length of AB as projected on NS; the departure of AB is the length of AB as projected on EW. You can see that for a traverse line running due north and south, the latitude would equal the length of the line and the departure would be zero. For a line running due east and west the departure would equal the length of the line and the latitude would be zero.

AC = 520.16 cos 6125' = 248.86 ft.

The latitude of a traverse line, then, equals the product of the length of the line times the cosine of the angle of bearing.

To determine the departure, you solve the triangle for the length of the side CB shown in figure 7-10.

CB = 520.16 sin 6125' = 456.76 ft.

Figure 7-9.Latitude equals length of traverse line times twine of angle of bearing.

Figure 7-10.Departure equals length of traverse line times sine of angle of bearing.

The departure of a traverse line, then, equals the length of the line times the sine of the angle of bearing. The latitude of a traverse line is designated north or south and the departure is designated east or west following the compass direction of the bearing of the line. A line bearing northeast, for example, has a north latitude and east departure. In computations, north latitudes are designated plus and south latitudes minus; east departures are designated plus and west departures minus.

 Figure 7-11 is a graphic demonstration of the fact that, in a closed traverse, the algebraic sum of the plus and minus latitudes is zero; and the algebraic sum of the plus and minus departures is zero. The plus latitude of CA is equal in length to the sum of the two minus latitudes of AB and BC; the minus departure of BC is equal in length to the sum of the two plus departures of CA and AB.

LINEAR ERROR OF CLOSURE. In practice, as you will learn, the sum of the north latitudes usually differs from the sum of the south latitudes. The difference is called the error of closure in latitude. Similarly, the sum of the east departures usually differs from the sum of the west departures. The difference is called error of closure in departure.

From the error of closure in latitude and the error of closure in departure, you can determine the linear error of closure. This is the horizontal linear distance between the location of the end of the last traverse line (as computed from the measured angles and distances) and the actual point of beginning of the closed traverse. For example, you come up with an error of closure in latitude of 5.23 feet and an error of closure in departure of 3.18 feet. These two linear intervals form the sides of a right triangle. The length of the hypotenuse of this triangle constitutes the linear error of closure in the traverse. By the Pythagorean theorem, the length of the hypotenuse equals approximately 6.12 feet. Suppose the total length of the traverse was 12,000.00 feet. Then your ratio of linear error of closure would be 6.12:12,000.00, which approximately equates to 1:2,000.