Azimuth

“ If the Magnetical Azimuth be less than the other, then the Variation is towards the same side of the Meridian, where the Sun is; if greater, on the other.
To observe the Suns Azimuth by the Needle, and the Needles Variation to Degrees, any Needle, long enough to afford upon a Card under it, a Circle divided into Degrees, put in a Square Box, after the ordinary manner of Clinatories, will serve turn; by placing the Box so, as the Sun may shine upon any two opposite sides of it, at the same times that the Suns Height, &c. ”

“ The triangulation was not adjusted, in the course of the final simultaneous reduction, to the astronomically determined azimuths, because they are liable to be vitiated by local attractions; but the azimuths observed at about fifty stations around the primary azimuthal station, which was adopted as the origin of the geodetic calculations, were referred to that station, through the triangulation, for comparison with the primary azimuth. A table was prepared of the differences (observed at the origin computed from a distance) between the primary and the geodetic azimuths ”

“ The observation of the time when a star has a certain azimuth may also be used for determining the clock error, as the hour-angle can be found from the declination, the latitude and the azimuth. As the azimuth changes most rapidly at the meridian, the observation is most advantageous there, besides which it is neither necessary to know the latitude nor the declination accurately. The observed time of transit over the meridian must be corrected for the deviations of the instrument in azimuth, level and collimation. ”

“ Having measured the altitude and the azimuth we have the two co-ordinates which are needed to indicate accurately the place of a star in the sky. But, as we shall see in a moment, other co-ordinates beside altitude and azimuth are needed for a complete description of the places of the stars on the celestial sphere. Owing to the apparent revolution of the heavens round the earth, the altitudes and azimuths of the celestial bodies are continually changing. We shall now study the causes of these changes.
5. The Apparent Motion of the Heavens. We have likened the earth to a rotating school globe. ”

“ Since the normals of the ellipsoid of rotation do not in general intersect, there consequently arises an influence of the height on the azimuth. In the case of the measurement of the azimuth from A to B, the instrument is set to a point A′ over the surface of the ellipsoid (the sea-level) , and it is then adjusted to a point B′, also over the surface, say at a height h′. The vertical plane containing A′ and B′ also contains A but not B: it must therefore be rotated through a small azimuth in order to contain B. The correction amounts approximately to −e2h′ cos2φ sin 2α/2a ”

“ Record the difference between the respondent azimuth angle and the base azimuth angle and label it as the azimuth angle difference (Z Diff.) . Reenter the tables with the base declination and LHA arguments, but with the latitude argument 1° greater or less than the base latitude argument, depending upon whether the actual (usually DR) latitude is greater or less than the base argument. ”

“ All that is necessary is to calculate by spherical trigonometry the azimuth and distance, with reference to the assumed centre, of all the points of intersection of meridians and parallels within the space which is to be represented in a plane. Then the azimuth is represented unaltered, and any spherical distance z is represented by ρ. Thus we get all the points of intersection transferred to the representation, and it remains merely to draw continuous lines through these points, which lines will be the meridians and parallels in the representation. ”

“ An amplitude, unlike an azimuth, is restricted as to time of observation, for the body must be on the horizon either rising or setting; and should be observed when the sun is about its own diameter above the horizon and with a not excessive height of eye. The amplitude is measured from the east or west point through 90° to the north or south point. If the body observed has a south declination and is rising, the amplitude will be East so much South; if declination is north, East so much North, for a body rises in the East point when its declination is 0°—on the equator. ”

“ The time of transit of the sun or star across the vertical wire of the telescope having been observed by means of a chronometer of which the error is known, it is possible to calculate the azimuth of the sun or star, if the latitude and longitude of the place of observation are given. Hence if the readings of the verniers on the azimuth circle are made when the transit is observed we can deduce the reading corresponding to the geographical meridian. ”

“ Plot the AP first; then plot the azimuth line from the AP toward or away from the GP. Then, measure the altitude intercept along this line. At the point on the azimuth line equal to the intercept distance, draw a line perpendicular to the azimuth line. This perpendicular represents that section of the circle of equal altitude passing through the navigator’s actual position. This is the line of position.
A navigator often takes sights of more than one celestial body when determining a celestial fix. ”

“ These azimuth sub-scales are therefore placed on the side, and when the gun or mortar points south the subscale points at zero on the azimuth-circle.
Q. Give some rules for caring for the azimuth instrument.
A. Never allow any of the leveling-screws to become so tight that they cannot be easily turned by hand. When setting the instrument up over a concrete floor make little holes in the concrete [115] for the points of the tripod-legs to set in. Never wipe the lenses with anything having the least sign of dirt or grit upon it. A perfectly clean chamois is always best. ”

“ When latitude and declination are of contrary name, declination is treated as a negative quantity. No special sign convention is required for the local hour angle, as in the following azimuth angle calculations.
The azimuth angle (Z) can be calculated using the altitude azimuth formula if the altitude is known. ”

“ Another very important point to bear in mind when observing a body with the view of computing its hour angle, is its azimuth. When the bearing is nearly east or west, on the prime vertical, the body is rising or falling faster than at any other time, and an error in altitude or latitude will produce the least error in the resulting longitude. ”

“ If A is the azimuth of this point at the centre, tan A = sin α cot φ. On paper a line from the centre is drawn at an azimuth A, and the distance θ is represented by 2 sin 1/2θ. This makes a very good projection for a single-sheet equal-area map of Africa. The exaggeration in such systems, it is important to remember, whether of linear scale, area, or angle, is the same for a given distance from the centre, whatever be the azimuth; that is, the exaggeration is a function of the distance from the centre only. ”

“ If b, c be the level and collimation errors, the correction to the circle reading is b cot z ± c cosec z, b being positive when the west end of the axis is high. It is clear that any uncertainty as to the real state of the level will produce a corresponding uncertainty in the resulting value of the azimuth,—an uncertainty which increases with the latitude and is very large in high latitudes. ”

“ Azimuths were determined in the usual manner [69] by elongations of close circumpolar stars, Polaris or 51 Cephei being usually selected. The azimuth mark used was an ordinary Egyptian shamadan (candlestick with spring feed) with a glass globe, placed at a distance of one to two kilometres, with its foot firmly bedded in sand and stones to prevent [48] any motion. ”

“ Azimuthal Condensing Prisms.—Previous to 1850 all apparatus were designed to emit light of equal power in every azimuth either constantly or periodically. The only exception was where a light was situated on a stretch of coast where a mirror could be placed behind the flame to utilize the rays, which would otherwise pass landward, and reflect them back, passing through the flame and lens in a seaward direction. ”

“ Azimuthal Equidistant Projection An azimuthal equidistant projection is an azimuthal projection in which the distance scale along any great circle through the point of tangency is constant. If a pole is the point of tangency, the meridians appear as straight radial lines and the parallels as equally spaced concentric circles. If the plane is tangent at some point other than a pole, the concentric circles represent distances from the point of tangency. ”

“ Or else they are to work the Amplitudes as they do the Azimuth, reckoning the Suns Distance from the Zenith 90° 36': This, though it be of little consequence near the Æquinoctial, will make a great Error in high Latitudes, where the Sun rises and sets obliquely.
But to return to our Hypothesis, In order to explain the Change of the Variations, we have adventur'd to make the Earth hollow, and to {53} place another Globe within it; and I doubt not but this will find Opposers enough. I know 'twill be Objected, That there is no Instance in Nature of the like thing ”

“ Whilst stretching in for the shore, with the ship's head north-west-by-north (magnetic) , I took azimuths with two compasses on the binnacle; after which they were immediately placed upon a stand near the taffrel, and other azimuths taken. The variation resulting from the observations on the binnacle was 5° 59′ west, and from those near the taffrel 8° 24′ west; affording another instance of the effect produced by changing the place of the compass. ”

“ Density multiplied by volume in space gives us mass or, what appears to be the same thing, energy. But from our space-time point of view, a far more important thing is density multiplied by a four-dimensional volume of space and time; this is action. The multiplication by three dimensions gives mass or energy; and the fourth multiplication gives mass or energy multiplied by time. Action is thus mass multiplied by time, or energy multiplied by time, and is more fundamental than either.
Action is the curvature of the world. ”

“ When a body is in the zenith, since a ray of light from it enters the atmosphere at right angles to the refracting medium, it suffers no refraction. Consequently, the position of the heavenly bodies, when in the zenith, is not changed by refraction, while, near the horizon, where a ray of light strikes the medium very obliquely, and traverses the atmosphere through its densest part, the refraction is greatest. The whole amount of refraction, when a body is in the horizon, is thirty-four minutes; while, at only an elevation of one degree, the refraction is but twenty-four minutes ”

“ Why we do not rather take the distance of the star from the equinoctial, at once? I answer, that it is easy to point an instrument to the north pole, and to ascertain its exact position, and of course to measure any distance from it on the meridian, while, as there is nothing to mark the exact situation of the equinoctial, it is not so easy to take direct measurements from it. ”

“ It is independent of the observer's mesh-system, but it depends on his gauge. It is obvious that the number expressing the radius of curvature of the world at a point must depend on the unit of length; so we cannot say that the curvatures at two points are absolutely equal, because they depend on the gauges assigned at the two points. Conversely the radius of curvature of the world provides a natural and absolute gauge at every point; and it will presumably introduce the greatest possible symmetry into our laws if the observer chooses this, or some definite fraction of it, as his gauge. ”

“ If a rare medium consist of very small quiescent particles of equal magnitudes, and freely disposed at equal distances from one another: to find the resistance of a globe moving uniformly forward in this medium. Case 1. Let a cylinder described with the same diameter and altitude be conceived to go forward with the same velocity in the direction of its axis through the same medium; and let us suppose that the particles of the medium, on which the globe or cylinder falls, fly back with as great a force of reflexion as possible. ”

“ We ought to construct our geometry in such a way as to show that there are intermediate steps, and that the comparison of the interval with the ultimate standard is not a kind of action at a distance.
To compare intervals in different directions at a point in space and time does not require this comparison with a distant standard. The physicist's method of describing phenomena near a point is to lay down for comparison (1) a mesh-system, (2) a unit of length (some kind of material standard) , which can also be used for measuring time, the velocity of light being unity. ”

“ The laws of conservation of energy and momentum in mechanics can be deduced from this law of world-curvature.
16. Certain phenomena such as the FitzGerald contraction and the variation of mass with velocity, which were formerly thought to depend on the behaviour of electrical forces concerned, are now seen to be general consequences of the relativity of knowledge. That is to say, length and mass being the relations of some absolute thing to the observer's mesh-system, we can foretell how these relations will be altered when referred to another mesh-system. ”