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The larger azimuth (at the place where the latitude is greatest) sum azimuths + diff. azimuths.
sum azimuths diff. azimuths.
These azimuths, found for a sphere, are thus corrected for the earth's spheroidal form.
From the above spherical azimuths find the spherical amplitudes by taking the difference between each of them and 90°; for each case find an angle, a, by the formula *
Then the tangent of each of the true spheroidal amplitudes cos a × tangent spherical amplitude; the azimuths being obtained by applying to these 90°, additive or subtractive, according to the
If, instead of determining astronomically and by the transmission of chronometers the absolute latitudes and the difference of longitudes of these distant stations, they had been connected by a series of triangles; and that from this triangulation it was required to obtain the true bearings of each point from the other for the purpose of running a straight line between them, the following is the simple process:—
Supposing A and B to be the two stations, connected, as in the figure, by a series of triangles, assume one side as a standard, say
AC; compute CE as in a plane triangle; from this compute CD, DE; from DE compute DF; from DF compute D B. With the two known sides AC and CD, and the angle A CD, compute
*The steps by which this formula is arrived at are shown at page 346 of the "Corps Papers," where also will be found examples of azimuths calculated by it on the survey of the boundary alluded to.
A D and the angle CD A; subtract this from the sum of the three angles CD E, E D F, and F D B, and you have the angle A DB; with this angle and the two sides, AD and DB, compute the angle DBA; this is the difference between the bearing of A from B, and that of D from B. The latter is known, or can be directly observed; whence the former is deduced.
In the same manner the azimuth of the line A B, or the bearing of B from A, can be ascertained.
On the North American boundary the azimuths were laid off with an altitude and azimuth instrument, and the line prolonged with a portable transit, by which the party sent on in front to take up the rough alignement for cutting a track through the dense forest were directed. A torch of birch bark was moved to the right or left, as required, by concerted signals from the transit, made by flashing small quantities of gunpowder in an open pan; both the lighted torch and the flashes of gunpowder, being visible for far greater distances* than were ever required.
By daylight heliostats were used for keeping the advanced party in the right direction.
The true bearings of the line of 64 miles in length were in this operation determined so accurately, that when the parties employed in marking it out from each extremity met about midway, the sum of their joint deviation from the true line was exactly 341 feet; equal, as Mr. Airy observes, to "only one-quarter of a second of time in the difference of the longitudes, or only onethird of the error which would have been committed if the spheroidal form of the earth had been neglected." This slight error was corrected by running offsets at certain points along each line, proportioned, of course, to the distances from the extreme end. The distances between two places of a ship at sea are generally resolved by plane trigonometry; the difference of latitude S L, and the azimuth represented by the angle S'SL and termed the course, forming a right-angled triangle, in which S S', the nautical distance, is determined; the other side S' L,
Diff of Lat.
* Major Robinson states as much as 40 miles. See the narrative of his operations, 2nd And 3rd Numbers of the "Corps Papers."
termed the departure, being the sum of all the meridional distances passed over.
Again, in the triangle ABC: let A B represent the meridian distance (or departure), and the angle BAC be equal to the latitude, then AC, the hypothenuse, will be equal to the difference of longitude.
Also, if DB represent the nautical distance, and CD the difference of latitudes, then B C D will be a right angle, and B C, the departure, nearly equal to the meridian distance in the middle latitude. If, then, in the triangle ABC the angle ABC be measured by that middle latitude, AB, the hypothenuse, will be nearly equal to the difference of longitude between D and B.
For further information on this subject, no better work can be consulted than Riddle's
By the use of Mercator's "Projection," most of these questions can be solved without calculation. In this ingenious system the globe is conceived to be so projected on a plane that the meridians are all parallel lines, and the elementary parts of the meridians and parallels bear in all latitudes the same proportion to each other that they do upon the globe. The uses to which this species of projection can be applied, and the vast benefit its invention has proved to the navigator, will be evident by reference to any work on navigation.
The latitude and longitude of any place being known, that of other station within a short distance can any also be determined by plane trigonometry. Suppose the latitude and longitude of G for instance to be known, from whence that of O, an adjacent station, is to be determined; the distance O G must be measured, or obtained by triangulation, and the azimuth NOG observed; then the difference of longitude G L between the stations is the sine of the angle
LOG to radius OG; and O L, the difference of latitude, is the cosine to the same angle and radius. The following example will show the application of this simple method :
The distance of a station O', 238 feet due south of the Rl. Engr. Observatory at Chatham from Gillingham Church, was ascertained to be 7547-4 feet, and the angle SOG, the supplement of the azimuth, = 78° 55′ 55′′; Gillingham Church being situated in 51° 23' 24" 12 north latitude, and 0° 33′ 49′′41 east longitude.
1448.9-3.161039 Diff. of latitude (north), in feet.
74073-869642 Diff. of longitude (west), in feet.
The lengths of one second of latitude and longitude in latitude
116"81'56" 8. Difference of longitude in arc.
It is always necessary to ascertain the variation of the compass before plotting any survey, for the purpose of protracting such parts of the interior details as have been filled in by magnetic bearings, and also of marking the direction of the magnetic meridian upon detached plans. The laws of this variation are at present but little known; and it is only by accumulating a vast num
ber of observations at different places, and at different periods, that the position of the magnetic poles and the annual variation and dip can be ascertained with anything like certainty.
A meridian line being once marked on the ground, the bearing of this line by the compass is of course the variation east or west. It can be traced with an altitude and azimuth instrument, or even a good theodolite, by observing equal altitudes and azimuths of the sun, or a star, on different sides of the meridian. With the latter object no correction whatever is required: the cross hairs are made to thread the star exactly (by following its motion with the tangent screws) two or three hours before its culmination; the vertical arc is then clamped to this altitude, and the azimuth circle read off. On the star descending to the same altitude, at the same interval of time after its transit, it is again bisected by the cross hairs, and the mean between the two readings of the azimuth circle gives the direction of the true meridian, which being marked out on the ground, its bearing is then read with the compass.
When the sun is the object observed, the altitude taken may be that of either the upper or lower, and the azimuth that of the leading or following limb; the mean of the readings of the azimuth circle does not necessarily therefore in this case give the true meridian; a correction must also be applied for the change in the sun's declination during the interval of time between the observations.
If the sun's meridian altitude is increasing, as is the case from midwinter to midsummer, his lower limb when descending will have the same altitude at a greater distance from the meridian than before apparent noon, and the reverse when it is decreasing. The following formula for this correction is taken from Dr. Pear
x= Dx sect. lat. x cosect. T, where D is the change of declination in the interval of time expressed by T.
Example:-In latitude 51° 23′ 40′′ N. on May 12, 1838, the
upper limb of the sun had equal altitudes.
* The sun's change of declination is given for every hour in the first page of each month in the Nautical Almanac.