How would you take vertical and horizontal sextant angles?

Vertical sextant angles are used for two quite different purposes; terrestrial (typically for finding the distance off from a lighthouse or other shore structure), and astronomical (altitude of one or more astronomical bodies, for astro navigation).

Before even starting you will need to adjust the instrument to eliminate errors; essentially this is a matter of correctly aligning the mirrors. Unfortunately there is not the space or the time here to describe this, but any decent sextant textbook or manufacturer's handbook will explain the process in detail, and it is quite essential that you do correctly set up the instrument before you start.

Having adjusted the sextant, in both cases you first apparently bring the object down to the horizon by moving the radius arm of the sextant, then read the angle off the instrument (usually using vernier or micrometer scale, and a few instruments have both, and although I have no experience of them I am reasonably sure that digital electronic instruments are now starting to appear). Then you have to process the result.

For terrestrial use a crude approximation is to simply use the sine or tangent formula - remember school level trigonometry? However that takes no account of the curvature of the earth, so it is better to use the tables given in yachtsmen's nautical almanacs (such as Reeds), which should take full account of curvature and as a bonus removes the need for calculation. You simply enter the height of the object and the height of your eye and the angle found into the table, and be prepared to interpolate between tabulated values if necessary; then Robert is your proverbial uncle, and the distance off is read directly out of the table.

For astro navigation, having already adjusted the instrument and then read the apparent angle you will now have to apply various corrections, most notably for refraction (through the earth's atmosphere), before you can do any calculations.

Having done that, there are then several calculation methods, but the simplest is probably to use the Air Navigation Tables in conjunction with a plotting form. Unfortunately there really isn't sufficient space (or time) here to teach you how to do this, but I think you will find this the simplest method.

Of course if you prefer a more complex method you could use the haversine formula, which used to be given (and explained) in Reeds - although I haven't checked whether it still is, in conjunction with a plotting form; or if you want a seriously complex method you could go for the Marc St Hilaire method, and if you want the most complex method of all (but probably the only one that does not require a chronometer) you could navigate by lunar distances. None of these are recommended for the beginner!!

Horizontal sextant angles are normally used for position fixing from three terrestrial objects. By far the simplest method is to actually draw the position on a chart, using a specialised instrument called a station pointer. Set up the three arms of the station pointer to correspond exactly to the angles you have measured between the objects, then move it around the chart until each of the three arms lines up exactly with one of the objects as marked on the chart. Then your position is the centre of the station pointer, where there should be a small hole to enable you to make a pencil mark. Again this avoids any need for calculation.

Having said the latter, my own experience is that in a small boat it is quite sufficient (and vastly easier) to take the three bearings by compass and then plot them directly on the chart as bearings. So much so that I have never actually seen a station pointer in the flesh, never mind having needed to use one. (But I have from time to time used a sextant for vertical angles, both terrestrial and astro, and used the result for navigation.)

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Draw a right triangle like an "uphill" if you were walking from left to right. Label the long side 15 lbs. Label the vertical side y. Label the horizontal side x. Using SOH CAH TOA, sin 59 = opposite / hypotenuse = y / 15 y = 15 sin 59 = 12.858 lbs cos 59 = adjacent / hypotenuse = x / 15 x = 15 cos 59 = 7.726 lbs Answer: Horizontal Load: 7.73 lbs Vertical Load: 12.86 lbs

You don't take horizontal angles. Only vertical: The angle between the horizon and the chosen celestial object.

The easiest (and safest!) example is for latitude at night. Measure the angle between the Pole Star or Southern Cross and the horizon, and that's your latitude!

If it is zero (instrument not required!) then you are at the Equator. If it is ninety degrees, you are at the pole!

If it is fifty-three degrees then you are fifty-three degrees North of the Equator if observed from the Pole Star, or fifty-three degrees South of the Equator if observed from the Southern Cross.

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