6. Latitude and Longitude Ranges.-The algebraic difference obtained by subtracting the latitude of the beginning of a line from the latitude of the end of the line will here be called the latitude range of the line. Likewise, the algebraic difference between the longitude of the end and the longitude of the beginning of the line will be called the longitude range of the line. It should be kept in mind that in obtaining these differences, the coordinate (latitude or longitude) of the end of the line is the minuend, and that of the beginning of the line, the subtrahend; also, that the subtraction is algebraic, latitudes and longitudes having the signs explained in Arts. 3 and 4.

=

Referring again to Fig. 1, the latitude range of PQ is KQ HP = HD - HP PD. The longitude range is K'Q H'PK'Q - K'D = DQ. If the line had been run from Q to P, its latitude range would have been HP KQ = KE - KQ = −E Q = -PD; and its longitude range, H' PK'Q=H' PH'E = −EP = -DQ.

As will be observed, any line, as PQ, is the hypotenuse of a right triangle whose legs are the latitude range and the longitude range of the line (PE and EQ, in the case of PQ). The latitude range indicates how far the end of the line is north or south of the beginning; and the longitude range, how far the end of the line is east or west of the beginning, or of the meridian passing through the beginning. The latitude range is positive, and is called a north latitude-range, or a northing, whenever the line bears north; it is negative, and called a south latituderange, or a southing, whenever the line bears south. The longitude range is positive, and is called an east longituderange, or an easting, when the line bears east; it is negative, and called a west longitude-range, or a westing, when the line bears west. Thus, the latitude and the longitude range of PQ are, respectively, +PD and +DQ; those of QP are QE and -EP. Likewise, the latitude range of P. Q. is -P, D1, because the end of the line is south of the beginning. The longitude range is +D. Q., because the

1

end of the line is east of the beginning. These values may be verified by observing that the latitudes of P, and Q, are, respectively, H, P, and -H, D1, whose algebraic difference is -H, D, -(—H,P1) = −H1D1 +H,P1 = −P1 D1; and that the longitudes of P, and Q, are, respectively, +H{P. and +KQ., whose difference is equal to D. Q..

NOTE. In older books, and in some modern books, the term latitude is applied to what has here been called latitude range; while what is here called longitude range is in them called departure. The expressions latitude difference and longitude difference are sometimes used instead of latitude range and longitude range, respectively.

M

B

7. General Formulas.-Let AB, Fig. 2, be a course whose length is 7, and whose bearing is G. In the right triangle AMB, in which AM is the direction of the meridian through A, the latitude range AM and longitude range MB are denoted by t and g, respectively. From trigonometry, we have,

FIG. 2

These formulas serve to compute the ranges when the length and bearing of the course have been measured. Special care should be taken to give t and g their proper signs, t being positive when G is north (that is, either northeast or northwest) and g being positive when G is east (that is, either northeast or southeast). When G is south (that is, either southeast or southwest), t is negative; and when G is west (that is, either northwest or southwest), g is negative. If t and g are given, G is found by the formula

In applying formulas 3 and 5, the signs of t and g should

be disregarded; that is, both and g should be treated as positive.

EXAMPLE 1.-The length of a course is 896.7 feet, and its bearing N 39° 15′ W; what are the ranges of the course?

SOLUTION.-Here = 896.7 ft., and G = 39° 15'. Since the bearing is northwest, its latitude range is positive, and its longitude range, negative. We have, then, applying formulas 1 and 2,

t: = 896.7 cos 39° 15'

= 694.4 ft. Ans.

g= -896.7 sin 39° 15' = -567.4 ft. Ans.

In calculations of this kind, logarithmic functions should be employed in preference to natural functions. The work is conveniently arranged by writing first the logarithm of the length of the course, then writing the logarithmic sine of the bearing over it, and the logarithmic cosine of the bearing under it, and adding upwards in one case and downwards in the other, as follows:

log g

=

2.7 5 3 8 5; g

= 567.4 ft. Ans.

1.80 120

2.9 5 2 65

1.88896

log t = 2.8 4 1 6 1; t = 694.4 ft. Ans.

Example 2.-The latitude range and the longitude range of a course are, respectively, -13.71 and -9.38 chains; find the bearing and length of the course.

SOLUTION. Since both ranges are negative, the course bears southwest. Neglecting signs, we have, by formulas 3 and 4,

The logarithm of 9.38 is written first, and under it that of 13.71. The difference between these two gives the logarithmic tangent of G, from which G is determined. At the same time that G is taken out of the table, its logarithmic

sine is taken and written above that of 9.38; it is then subtracted from the latter logarithm to obtain the logarithm of I. For the purpose of finding 7, it is better to take G to the nearest minute, as this does not involve any additional work; but, as bearings are taken to the nearest quarter of a degree, the bearing of the line would be stated as S 34° 30′ W.

EXAMPLES FOR PRACTICE

NOTE.-Bearings are given to the nearest quarter of a degree.

1. The length of a course is 19.83 chains, and its bearing N 18° 15′ E; find the ranges of the course.

Ans.

t

=

18.83 ch. g = 6.21 ch.

2. A line 649 feet long bears S 5° 45′ E; find its ranges.

Ans.{

t = -645.7 ft.

ns. {g = 65.0 ft.

3. Find the ranges of a course 3.33 chains long and bearing N 73° 30′ W. Ans. {3.19 ch. .95 ch.

4. The length of a course is 197 feet and its bearing is S 53° 45′ W; find its ranges.

Ans.{

t = 116.5 ft. 158.9 ft.

=

5. The latitude range and longitude range of a course are, respectively, -3.17 and -4.25 chains; find the length and the bearing G of the course.

Ans.

G

= 5.30 ch.
= S 53° 15' W

BALANCING THE COMPASS SURVEY OF A

CLOSED FIELD

ERROR OF CLOSURE

8. Definitions.-Let ABCDE, Fig. 3, be a closed field, O T a reference meridian, and O G a reference parallel of latitude. It is obvious that, if all the courses and bearings were determined with absolute exactness, and a plat of the field made, the end (say A) of the last line would coincide with the beginning of the first. Under such conditions, the survey is said to close. As, however, no measurements are free from error, and as the compass is read only to the nearest quarter degree, a survey never closes. When

the notes are platted the end of the last line does not coincide with the beginning of the first. This condition is shown in Fig. 4, where A B is

the first line as platted from the notes, and EA, the last.

The distance A, A from the end of the

T

last line to the begin

ning of the first is

called the total error of closure. The ratio of the total error of closure to the sum of the lengths of all the courses expresses the error per unit of length, and will here.

E

E

G

FIG. 3

be called the relative error of closure, or the rate of For example, if the sum of the lengths of the sides

error.

ring again to Fig. 3, it will be observed that the sum E' C' of the northings, or north latitude-ranges, E' A', A'B',