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Calculation of the Azimuth of the line from Initial Point of the Boundary to East Base.
VI.-Azimouth of straight line from Initial Point on the Pacific lo junction of Gila and Colorado.
In the computation for the direction of the line, the longitude of the initial point and the longitude of the junction of the Gila and Colorado are taken as determined in the field by the observations compared with the moon's place, as given in the Nautical Almanac (Greenwich.) On arriving at Washington, I obtained from Professor Airy his corresponding observations, made at Greenwich, and it will be seen that a change has been made in the absolute longitude of both places, but fortunately no material change is discoverable in the relative longitude of the two places. Now, an inspection of the formula used will show that the difference in longitude is the element used in determining the azimuth of the line connecting the two points. Hence the change deduced from comparison with the corresponding Greenwich observations does not affect this result.
The preceding pages show the process by which the latitude and longitude of Camp Riley was transferred to the initial point of the boundary on the Pacific.
A word on the subject of the transfer to the "junction” of the latitude and longitude of the observatory near that point; the distance between them was so small that the transfer may be considered as having been accomplished by direct measurement.
Lieutenant Whipple's report to me, November 30, 1849, gives
The longitude of observatory, 7" 38" 125.53 ; in arc..
These, corresponding with my computations founded on the same data, were adopted; but it must be observed, to prevent future misunderstanding, that it was impracticable to measure the azimuth from the junction of the two rivers, (then under water,) and a point, B, (see sketch,) was selected in the azimuth produced, ascertained by direct measurement to be 73.5 feet south and 1,070 feet west of the junction, and here the monument was placed and the azimuth measured from it. The geographical position of this monument is, consequently,
0 1 11 In north latitude .......
...................... ... 32 43 31.6 Uncorrected longitude......
114 33 04.3 The computation of the azimuth and length of the line of boundary extending from the initial point on the Pacific coast, near San Diego, to the junction of the Gila and Colorado, was based on the following assumption :
O 11 Latitude of initial point on the Pacific..
32 31 59.63 Latitude of junction of Gila and Colorado.......
32 43 32.2 Difference of longitude.............
2 32 24.9
We compute the azimuth and distance by the formulæ given by Puissant, Traité de Geodesie, Ed. III, Vol. 2, p. 316, using the following notation :
_ a? - 72 a = equatorial radius
e= ellipticity ="
Log. e= 7,8273187
or distance in arc.
tan. l = tan. L" In the spherical triangle formed by l', 1", and P, we now compute the angles 180° - Z' and Z" —- 180°, the arc o, and also l, = the latitude of the foot of the perpendicular from the pole,. and obtain for a first approximation : Z=85° 33' 25".05
0= 7739".75 Z" – 180° = 84° 11' 25'. 21
1= 32° 54' 22".69
Next we compute the correction d P, to be applied to the spheroidal difference of longitude P, by the formula
dP=-0(1e- } c?) cos. 1 = + 21".72 And we get................ Ptar
............. P + d P = 2° 32' 46".62 Substituting this into the former computation, we obtain . Z = 85° 34' 14".41
0 = 77581.10
T= 32° 54' 16'.71
d P= +21".77
0= 7758".15 We also compute o' and o", the arcs from l' and 1" to the foot of the perpendicular 1, viz: d' = 6° 52' 31".5; 01 = 9° 01' 49".7; and obtain the distance s in metres by the expression
=0(1+ e sin.? ? — ole sin.“ 1) + (sin. 2 o" — sin. 2 0') (1 e sin.? 1 — s'; e sin.“ 1) —...
1st term ............... = + 239187.0
.. = 84° 12' 02".95 Azimuth at junction of Gila and Colorado...........
... = 85° 34' 14".49 Length of line of boundary connecting above points........
= 148.689 miles
Determinations of boundary line from the Rio Colorado of the west, to the intersection of the 111th meridian of longitude, west of Greenwich, and parallel 31° 20' north latitude : By Lieutenant N. Michler, Topographical Engineers U. S. A.
I.--AZIMUTII LINE--ASTRONOMICAL POSITIONS.
The longitude of the initial point on the Rio Colorado, twenty English miles below its junction with the Gila, as agreed upon by Lieutenant N. Michler, Topographical Engineers United States Army, on the part of the United States, and Francisco Jimenez, first engineer of the Mexican commission, on the part of Mexico, was determined to be 114° 48' 44.53" west of Greenwich.
This result was obtained by transferring the longitude of the monument near the junction to the initial point, by Lieutenant Michler's triangulation of the otwenty English miles,” the longitude of said monument having been previously obtained from observations by Lieutenant A. W. Whipple, Topographical Engineers, Assistant United States Boundary Survey, deduced from corresponding observations at Greenwich.
This latitude was obtained by a mean of results from observations with zenith telescopes by Lieutenant Michler and Señor Jimenez, at their respective observatories near the initial point, (reduced to Lieutenant Michler's observatory,) as follows:
This latitude of the observatory was then transferred to the initial point, by triangulation, by Lieutenant Michler. By transferring the latitude of the monument near the junction to the same observatory, by two different triangulations, one by Lieutenant Michler and the other by Señor A. Diaz, second engineer Mexican commission, the following were the results :
The azimuth of the line from the junction of the Colorado and Gila, to Lieutenant Michler's observatory, is, at the latter point, 36° 14' 10" northeast, and its length 104024.34 feet.
The azimuth of the boundary line from the initial point on the Colorado to the intersection of the 111th meridian, west of Greenwich, and the parallel of 31° 20' north latitude, is at the initial point 71° 20' 43".8 southeast, and at the point of intersection of parallel and meridian 69° 19' 45'.94 northwest ; the length of this line = 382841.87 metres, = 418684.3 yards, = 237.63565 English miles.
Monument No. II, made of cast-iron plates, and pyramidal in form, was placed on the edge of the desert, at a distance from the initial point of 4522.9 yards; its latitude was computed to be 32° 29' 01".48 north ; its longitude 114° 46' 14".43 west of Greenwich. The azimuth of the boundary line at this point was computed to be 71° 19' 23".18 southeast.
The longitude of the observatory of the Mexican commission at Quitobaquita, near monu
tions, commencing July 23, 1856, and ending August 2, 1856, inclusive, was found to be 112° 52' 25".73 (77. 31m. 29.7758.) west of Greenwich.
The latitude of observatory at Quitobaquita, as determined by Señor Jimenez, first engineer, from forty observations upon eight pairs of stars with zenith telescope, was found to be 31° 56' 26".57 north.
II.-Results of calculations of geodetic latitudes and longitudes of points of triangulation of a line "twenty English miles in length,” extending
from the Monument, near the junction of the Gila and Colorado, to the Initial Point of the new azimuth line. Triangulation by Lieut. N. Michler, Topographical Engineers U. Š. A. Computer, John O'Donoghue. Instrument, Bruner Theodolite.
Letter Name of
Name of angle.
Angle of station. Length of
sides in feet.
of side. station.
Name of station.
Difference of latitude
Difference Difference of latitude 1 of longitude --- Soutlı. + west.
Difference of longitude