Declination of the sun at Greenwich apparent noon =15° 34′ 37′′ N. Diff. in time to a. m. obsn.:

= 15° 36′ 31′′ N. 3' 44" N.

Diff.to p.m.obsn., already computed (2×112=224′′)=

Sun's decl. p. m. obsn.

a. m. obsn.

= 15° 40′ 15′′ N.

=

p. m. obsn.

log cos 49.891149 log sin d=9. 429856(+) log sin d=9. 431541(+)

The discrepancy between thé a. m. and p. m. results suggests a systematic instrumental error ordinarily eliminated by taking direct

and reversed observations, which in this instance is of opposite effect in a. m. and p. m. hours and apparently eliminated in the mean result.

123. One additional fact should be noted relative to the several reductions of the above equal altitude observations:

By above direct computation, A p. m. =66° 00′ 47′′
A a. m. 65° 53′ 02′′

This value for dAS (3′ 53′′) agrees with same function as first computed.

124. Upon concluding the subject of azimuth determinations it will be of interest to note that the weighted mean of a large number of observations gives a value of S. 0° 59′ 25′′ W. for the azimuth of the line from the Washington, D. C., transit point to the flag pole heretofore described. A comparison of the methods and results of the various observations as given on the preceding pages suggests that the surveyor should seldom be without means by which accurately to determine time, latitude and azimuth at any place in the field, however remote, and should doubt arise as to his results a "check" by independent method is nearly always available and a certain guide as to the accuracy of the determinations. It might be added that a careful surveyor will not fail to surround his methods with adequate verification to insure the accuracy required in the execution of the public-land surveys.

THE TRUE PARALLEL OF LATITUDE.

125. The base lines and standard parallels of the rectangular ́system are established on the true parallel of latitude; the random latitudinal township boundary lines are also projected on the same curve; this curve is defined by a plane at right angles to the earth's polar axis cutting the earth's surface on a small circle. At every point on the true parallel the curve bears due east and west, the direction of the line being at right angles to the meridian at every point along the line. Two points at a distance of 20 chains apart on the same parallel of latitude may be said to define the direction of the curve at either point, without appreciable error, but the projection of a line so defined in either direction, easterly or westerly,

55465°-199

would describe a great circle of the earth gradually departing southerly from the true parallel. The great circle tangent to the parallel at any origin or reference point along the parallel is known as the tangent to the parallel," and it is coincident with the true latitude curve only at the point of origin. The rate of the change of the azimuth of the tangent is a function of the latitude on the earth's surface. The azimuth of the tangent varies directly as the distance from the origin, and the offset distance from the tangent to the parallel varies as the square of the distance from the point of tangency. A great circle connecting two distant points on the same latitude curve has the same angle with the meridian at both points and the azimuth of such a line at the two points of intersection is a function of one-half the distance between the points.

There are three general methods of establishing a true parallel of latitude which may be employed independently to arrive at the same result: (1) The solar method; (2) the tangent method; and, (3) the secant method.

SOLAR METHOD.

126. The solar instruments are capable of following the true parallel of latitude without substantial offsets. If such an instrument, in good adjustment, is employed, the true meridian may be determined by observation with the solar at each transit point. A turn of 90° in either direction then defines the true parallel, and if sights are taken not longer than from 20 to 40 chains distant, the line so established does not appreciably differ from the theoretical parallel of latitude. The locus of the resulting line is a succession of points each one at right angles to the true meridian at the previous station. However, during a period each day the solar is not available, and during this time, also whenever the sun may be obscured by clouds, or on account of a disturbance of the adjustments of the solar attachment, and whenever an instrument without solar attachment is employed, reference must be made to a transit line from which to establish the true latitude curve by one of the following methods.

TANGENT METHOD.

127. The tangent method of determination of the true latitude curve consists in establishing the true meridian at the point of beginning, from which a horizontal deflection angle of 90° is turned to the east or west, as may be required, and the projection of the line thus determined is called the tangent. The tangent is projected 6

40

160

200

S.89°59./E.

5.89 58.2'E.

34

Parallel
EAST

35

162

36

125

LATITUDE 45°34.5 N.

13/

Offsets, in links, from the tangent to the parallel.

240

280

Chains distant on the tangent.

S. 89°57.3'E.

Azimuths of the tangent.

S. 89°56.4'E.

Tangent

to the Parallel

320

360

400

440

480

5.89 55.6'E.

5.89°54.7'2

miles in a straight line, and as the measurements are completed for each corner point, proper offsets are measured north from the tangent to the parallel, upon which line the corners are established.

In Table 12, Standard Field Tables, are given the bearing angles or azimuths of the tangent to the parallel, referred to the true S. point, tabulated for any degree of latitude from 25° to 70° N., for the end of each mile from 1 to 6 miles. At the point of beginning the tangent bears east or west, but as the projection of the tangent is continued the deviation to the south increases in accordance with rules already stated.

In Table 13, Standard Field Tables, are shown the various offsets from the tangent north to the parallel, tabulated for any degree of latitude from 25° to 70° N., for each half mile from to 6 miles.

The accompanying diagram illustrates the establishment of a standard parallel in latitude 45° 34′.5 N., by the tangent method. (See Fig. 14.) The form of record is shown in the specimen field notes.

Objection to the use of the tangent method in a timbered country is found owing to the requirement that all blazing is to be made on the true surveyed lines. Also, all measurements to items of topography entered in the field notes are to be referred to the true established lines. These objections to the tangent method, on account of the increasing distance from the tangent to the parallel, are largely removed in the secant method.

SECANT METHOD.

128. The designated secant is a great circle which cuts any true parallel of latitude at the first and fifth mile corners, and is tangent to an imaginary latitude curve at the third mile point. From the point of beginning to the third mile corner the secant has a northeasterly or northwesterly bearing; at the third mile corner the secant bears east or west; and from the third to the sixth mile corners the secant has a southeasterly or southwesterly bearing, respectively, depending upon the direction of projection, east or west. From the point of beginning to the first mile corner and from the fifth to the sixth mile corners the secant lies south of the true parallel, and from the first to the fifth mile corners the secant lies north of the true parallel. It will thus be seen that the secant method is a mere modification of the tangent method, so arranged that the minimum offsets can be made from the projected transit line to the established true parallel of latitude.