How to calculate error of parallax and sextant based navigation? 
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*First of all, why wasn't the sextant ever used for land navigation?  The horizon is easier to see at sea, but land based sextants could be used in conjunction with artificial horizons (as at sea when horizon is hidden by fog).

*Parallax has been used by both the US army and navy to measure distance to targets.  The devices that used this principle were called coincidence rangefinders.  It seems this system was still used after the introduction of radar.  Why was this system eventually phased out?

*Finally my main question: How can one calculate the error for measurements made by sextants and parallax devices?
 A: Addressing the second part:
There are really two different aspects here: the navy and the army.
For the navy, it basically boils down to one thing: after WWII, radar got a lot better a lot faster, and quite quickly began to outperform the optical systems. The big benefit is not intrinsic range precision, but the ability to work in all weathers.
For the army, optical rangefinders were used for two purposes: indirect fire (what we ordinarily call "artillery"), and antitank guns. The need for range measurements for indirect fire diminished enormously with the widespread availability of accurate maps, which allow reference to terrain to determine range. 
Antitank gunnery, in turn, is divided into two categories: tanks and (non-tank) antitank guns. On tanks, laser rangefinders are far faster and more accurate than the old style rangefinders, as well as more reliable, particularly in marginal seeing conditions. The adoption of AGMs (antitank guided missiles) made the antitank gun obsolete.
For rangefinders, the big limit on accuracy is the ability to merge two images, particularly under low-contrast conditions. Plus, of course, they don't do well in the dark.
A: P1 - They WERE used on land, extensively. Note on land the horizon is indeterminate as hills and valleys are present. A still pool of water (aka artificial horizon) is known to perfectly parallel to the true horizon and accurate measurements can be made. (Compare the direct image of the Sun to its reflected image and note the angle between the two.)
P2 - optical systems worked well, but only for ranges where you can see the target. Weather and Night become a huge problem and also required humans to drive them. Radar doesn't suffer these issues and can operate faster and more accurately with modern computers driving them. It did take a little while to work the bugs out. Note also the introduction of the JET fighter which moves so fast the optical systems couldn't keep up. A life threatening situation really can really drive innovation for the Navy.
P3 - As with any measurement device compare and contrast known true values against measured values. When you see errors find the factors that effect results. With a Sextant the fineness of the angle measurement is a big issue, the accuracy of your timepiece is another (see Longitude Prize from about 1700). The last big factor is correctly processing the measured values such as corrects for dip, refraction, index errors and time offsets for successive measurements of the various stars (good bookkeeping).
Give a very accurate Sextant and a great timepiece to a fool and you will be lucky to get a worthwhile fix. Give them to a fastidious and detail oriented navigator and you will known where he stands.
