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Sextant: The Basics 1
(I) Introduction: Concept

If the positions of two points on the earth's surface are known, the distance between them can be found.

First, a latitude and longitude near the boat's actual position, such as a dead reckoning  (DR) position, is required. With reference to the Nautical Almanac , the altitude of the sun from the DR position can be obtained. The zenith distance can then be deduced from this tabulated altitude, and the position circle obtained by converting the zenith distance to nautical miles (one minute of arc at the earth's centre=one nautical mile on earth's surface). For example, if the tabulated altitude is 29° 40'.0, the zenith distance would be 90° - tabulated altitude = 60° 20'.0, and the radius of the position circle would be
60 (degrees) * 60 (min of arc) + 20 (min of arc) = 3620 minutes of arc = 3620 nautical miles.
Dip is the angle between the horizontal plane through the observer's eye and the visible horizon (see Fig 2.1). It occurs because the eye is always above the sea level so that the observed altitude is always greater than the altitude as measured from a point at sea level, where theoretically the horizon would be in a true horizontal plane.
Fig 1.1 Zenith Distance
The position circle will pass through the DR position, however the boat is near but not on the DR position, so a second reading of the altitude of the sun will have to be obtained using a sextant. The sextant will measure the true altitude, from which the true distance from the sun's GP and the true position circle can be found.

By comparing the tabulated altitude (calculated from DR position) and the true altitude (measured by sextant from true position), the distance of the true position circle from the DR position can be determined. For example:

          True altitude                      29° 48'.0          
          Tabulated altitude          29° 40'.0          
          Difference                                  8'.0 = 8 miles (intercept)

Since the table used to find tabulated altitude also gives the bearing or azimuth of the sun's GP (the centre of the position circle), a line that is plotted in the direction of the sun's GP (line of azimuth) for 8 miles will intersect the true position circle. A small part of this circle can be plotted perpendicular to the line of azimuth: the boat will be on this small part of the position circle. It will not necessarily be on the end of the line of azimuth as the azimuth was for the DR position, and represents an angular measurement from the sun's GP which is not sufficiently accurate. A second (or more) position line from another observation is required to fix the position.
1.          Index Error
Ø          This is the particular to each sextant, and can be either plus or minus.

2.          Dip
Ø          This is found by reference to the Nautical Almanac. Dip is always subtracted.
DR position
Line of azimuth
Part of true position circle
Fig 1.2 Intercept
Fig 2.1 Dip is determined by height of eye
Fig 2.2 Nautical Almanac extract: Altitude correction table showing dip
The amount of correction depends on the height of eye (HE) of the observer above sea level. The correction for a tabulated HE is diagonally to the right of the HE. For example, the correction for a HE of 2.8m would be 2'.9. This would also apply to a HE of 2.7m.
3.          Semi-diameter/Refraction/Parallax
Ø          These are combined as a single correction which is found in the Nautical Almanac.

Ø          Semi-Diameter
The true altitude is the angle between the true horizon and the centre of the observed heavenly body (HB). Stars have no visible diameter but both the sun and moon have appreciable diameters. Sextant readings should be made by measuring the upper or lower edge (limb) on the horizon and making a correction for half the body's diameter, not by guessing where the centre of the HB is on the horizon.
Fig 2.3 Semi-diameter
Ø          Refraction
Light passing from outer space into the earth's atmosphere is refracted. Refraction is at a maximum when the HB viewed is low down near the horizon, diminishing the zero when the HB is directly overhead.
Ø          Parallax
Fig 2.4 Refraction  the heavenly body's
Altitude appears to be higher than it
actually is
The altitude of a HB as measured from the surface of the earth differs from that which would be found if it were measured from the centre of the earth, which is the condition required for true altitude. The difference is called parallax. Parallax is greatest when the altitude is low and diminishes to zero when the HB is directly overhead.
Fig 2.5 Parallax if greater when altitude is low
Parallax also varies as the distance between the HB and the earth changes. The moon's parallax can be up to 61' in arc as it is relatively near the earth. The sun's parallax is fraction, never exceeding 0'.15 and parallax of all other HBs is negligible.

Ø          Total correction
Semi-diameter, refraction and parallax are combined in a single total correction found in the Nautical Almanac for the observed HB's particular altitude. For example, if for a month in May the apparent altitude was 46° 10'.01 then the correction to apply would be 16'.7. This correction would apply to all apparent altitudes between 45° 31'.0 and 48° 55'.0.
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Fig 2.6 Nautical Almanac extract: Altitude
correction table  sun's total correction
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Note that the moon's parallax changes so markedly that it is given as a separate correction.

Example 2a: Corrections to sextant altitude
An observation of the sun's lower limb taken in November gave a sextant reading of 34 25'.0. Height of eye 2.5m. Index error 2'.0. Refer to figures 2.2 and 2.6

Sextant Altitude (SA)                              34 25'.0
Index Error (IE)                                             -  2'.0          
Dip (HE 2.5m)                                              -  2'.8          
Apparent Altitude (AA)                         34  20.2
Correction (Lower Limb/LL Nov)      + 14'.9           
True Altitude                                               34  35'.1

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A Sextant, the sailor's essential navigation equipment
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The sextant utilises two mirrors. With this sextant, one of the mirrors ( mirror A in the diagram) is half-silvered, which allows some light to pass through. In navigating, you look at the horizon through this mirror. Other sextants are operated by aligning marked line on the mirrored surface to the horizon, which is visible from the side of the mirror.
The other mirror (mirror B in the diagram) is attached to a movable arm. Light from an object, normally taken to be the sun, reflects off this mirror. The arm can be adjusted to a position where the sun's reflection off the mirror also reflects off mirror A and through the eyepiece.Looking through the eyepiece, the moving arm is adjusted such that the object appears to rest on the horizon. When this happens, one object (the sun) is superimposed on the other (the horizon).  The angle between the two objects is then read off the scale. An angle in degrees can be read off the sextant and used to calculate lunar distance, longitude and location on the Earth.
What makes a sextant so useful in navigation is its accuracy. It can measure an angle with precision to the nearest ten seconds. (A degree is divided into 60 minutes.)

II) Obtaining the True Altitude

The altitude of the heavenly body (e.g. sun) from the observer's true position is measured using a sextant. This is called the sextant altitude, and has to undergo several corrections before the true altitude can be obtained.
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Sextants Galore!