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One reason for perceived imprecision in using the Dunthorne formula for 
clearing lunar distances may be in its implementation by tables rather than 
in the formula itself.

It appears to me that the formula itself is exact, for a spherical Earth, 
and provided the correction that Herbert refers to is applied (it usually 
isn't) it remains exact for an oblate Earth. Another small adjustment could 
be made, for the effect of refraction on the correction for semidiameter(s), 
to the between-limb spacing, to get to the between-centres spacing. But 
these corrections apply to any clearing procedure, not just to Dunthorne's, 
if extreme precision is being sought..

But there's a factor that's used with the Dunthorne procedure, and with 
Borda's and other clearance procedures also, that is usually obtained from a 
table, and isn't absolutely precise. It appears in the Dunthorne formula , 
quoted by Kent, which was-

cos D’ = cos Δ’ +  cos m’ * cos s’  * ( cos D – cos Δ )

                               cos m * cos s



as the multiplying factor-

                               cos m’ * cos s’   or as its log, to add.

                               cos m * cos s


where the dashed quantities are the apparent altitudes of Moon and Sun, and 
the undashed quantties are the true altitudes, after they have been 
corrected for parallax and refraction. As a multiplier, it's always slightly 
less than 1, so as a (nautical-style) log, it's always around 9.99xxxx  , 
that is, to six places. If could be calculated, to the necessary precision, 
directly from the cosines of those angles, but instead, that is usually 
short-cut by using an appropriate nautical table.

I'll describe this as it appears in Raper's "Practice of Navigation" (my 
edition is 1864), as Table 73, "The logarithmic difference", which covers 9 
pages. You look up the Moon altitude, to the nearest 10', in the range 3º to 
90º, against the Moon's horizontal parallax, to the nearest arc-minute in 
the range 53' to 61', using an auxiliary interpolation table to get the 
resulting log to the sixth decimal place. By far the most important part of 
this factor results from Moon parallax, with a bit for refraction, and this 
table takes no account of non-standard atmospheric conditions, for 
refraction, stating that it applies to 30 inches of Mercury pressure, and 
50ºF. Of course, any allowance for atmospheric differences would be no more 
than a correction to a correction, and would have little effect except in 
extreme conditions. But if a navigator found himself working well away from 
those standard conditions, there would be no way to allow for the fact.

Then Raper adds a pair of liitle one-line tables at the foot of each page, 
giving a small correction to add; either from a table for the Sun (for Sun 
parallax and refraction), or from another for a star (for refraction alone).. 
For the Sun this varies between 17 units (in the 6-fig log) at low Sun 
altitudes, through a minimum of 7 units around 14º, and back up to 18 units 
at 90º.

In "Norie's Navigation" (my tables date from 1914) , the same log factor 
appears as "Logarithmic difference" in table 39.

Dunthorne's original explanation, from  1766, is to be found in Maskelyne's 
first "Tables Requisite", "A new method of computing the effect of 
refraction and parallax upon the Moon's distance from the Sun or a fixed 
star". In that, there is no table provided for correcting this factor for 
the effects of the Sun's refraction and parallax . Instead, Dunthorne seems 
to have allowed for a fixed amount, independent of Sun amplitude, and 
included it in the main table. Although Dunthorne doesnt explain this, the 
justification for it might be as follows-

The effects of Sun parallax (which increases with altitude) and refraction 
(which decreases with altitude), compensate to some extent, so the sum of 
the two, varies with altitude by an amount that, if neglected, doesn't cause 
any great inaccuracy. Dunthorne was simplifying the procedure for his 
readers, when making Sun observations, but at the expense of precision, 
reducing that precision to correspond, effectively, to the use of 5-figure 
tables rather than 6 figures. I just wonder if that may be the source of the 
imprecision that the German handbook associates with the "Dunthorne method"..

George.

contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. 


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