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    Re: Measuring Dip in the 18th Century
    From: Frank Reed
    Date: 2013 Dec 23, 12:54 -0800

    Brad, you wrote:
    "This would push back the dip table origin at least 100 years from your assessment. "

    Those earlier tables would have been tables of "geometric dip" (exactly the same basic mathematical calculation that Alex mentioned, which has been known for many centuries). Given uncertainties in the diameter of the Earth, the additional error from ignorance of the refraction effect would not have been important at that time. Refraction of nearly horizontal rays of light is literally identical to a slight change in the radius of the Earth by somewhere in the range of 12% to 15%. So when the Earth's diameter was uncertain by 10% or more, the refraction effect was hidden behind that uncertainty. By the mid-17th century, the basic physics of refraction had been established (known as Snell's Law, though it was re-discovered independently by several "natural philosophers", most notably Descartes who published his version of it in 1637 which was when the relation really became widely known). But the principle could only be applied to the atmosphere at low altitudes (where the rays are nearly horizontal) based on observations, and it was not until Bradley's work on refraction (1740s?) that it was possible to assign a value to that curvature constant for light rays. In the late 18th century, there was some debate over the exact number: was the radius of curvature 7 or 8 or 10 times the radius of the Earth? See if you can figure out what constant was used in the dip table from the 1799 "Moore-Bowditch" table that Gary posted (that table is actually from Moore with no contribution from Bowditch at this date, and Moore himself almost certainly copied it directly from the Tables Requisite, the official companion volume to the Nautical Almanac compiled by Maskelyne and his staff of astronomers).

    -FER

    PS: Bradley's law for astronomical refraction, which was very popular in the 18th century, was an implicit equation: r=(57")*tan(z - m*r) where z is zenith distance. The factor m multiplying the refraction inside the tangent function is closely related to the radius of curvature of horizontal light rays, and Bradley estimated that its value was about 3 (but 3.5 isn't bad either). By the way, this is actually a rather good equation for refraction. You solve it by running through it first without the 3*r part, then use the first estimate of r to run through a second time.

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