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While looking over my two pages of mind numbing manual interpolation I
found an error in the last step and decided to organize the process on
an Excel sheet. This led to Az of 285.16 for Example 1 and 284.65 for
Example 2, Close to George's results of 285.1 and 284.7 respectively.

John Cole

On Oct 11, 9:24 pm, John Cole  wrote:
> There are no warnings in H.O. 171 about errors or instructions as to
> interpolation. There is a single worked example given from which one
> deduces that interpolation has been used however the resulting azimuth
> was given to the nearest whole degree only.
>
> I worked George's examples two ways. First a quick and dirty run
> through rounding the LHA to 3 hr 38 min the Dec to 55 deg 30 min and
> the Alt to 61 deg 30 min and interpolating for LHA in the table.  The
> answer was Az  285 to the nearest whole degree.
>
> The second time through I did not round anything  but interpolated
> carefully throughout including the last step for Az.  Unfortunately
> after all that work I got the same answer for both examples, namely
> 285.0
>
> John Cole
>
> On Oct 11, 5:16 am, "George Huxtable" 
> wrote:
>
> > I have altered the threadname because we are now comparing various sources
> > for obtaining azimuths.
>
> > Thanks to John Cole for his late recantation about message-sizes, and for
> > providing-
>
> > | Another historic table for finding azimuths is found in H.O. No. 171 "Line
> > | of
> > | Position Tables (for working sight of heavenly body for line of position
> > by
> > | the cosine-haversine formula, Marcq Saint Hilaire Method)"  US Navy Bureau
> > | of
> > | Navigation 1915. Table V the Finding of the Azimuths, page attached.
> > |
> > | The table is entered with the dec (across the column headings) and hour
> > | angle (in hr and min, down the column) to find a tabulated number. Then
> > the
> > | tabulated number is located again in the dec column whose heading is
> > closest
> > | to Hc and the Az is read off in the hour angle column and its direction
> > | determined by the usual rules.
> > |
> > | The rest of the tables in HO 171 are altitude corrections and log sines,
> > log
> > | cosines, and log and nat haversines.
>
> > =========================
>
> > That table is the same, in principle, as the one provided by Bennett. They
> > are both tabulations of a number which corresponds to cos X sin Y, where X
> > corresponds to the column heading, and Y the row heading. Bennett writes the
> > result as a 3-figure digit, the nearest that corresponds to 1000 cos X sinY.
> > In HO 171, it's tabulated instead as log( cos X sin Y), to 5 decimal places,
> > which uses more space. [Note that those are "navigator's logs", to which 10
> > has been added to turn negative numbers into positive ones, not exactly the
> > logs you will find in school log-tables or on a calculator.]
>
> > Both tables use the formula sin Az = sin LHA cos Dec / cos Alt.
>
> > We can rewrite that as          cos Alt sin Az = cos Dec sin LHA.
> > Conveniently, both sides of this expression have the same form.
>
> > So first find a number (or perhaps its log) that corresponds to cos Dec Sin
> > LHA. Then look for that same number (or same log) in the Alt column, and it
> > will be in the row that corresponds to Az.
>
> > Rounding errors in HO171 will in general be somewhat less than Bennett's
> > because the columns are at half-degree rather than whole-degree intervals,
> > though the rows are the same, and because there's no rounding (to speak of)
> > of the tabulated number.
>
> > It would be of some interest to me if John Cole would kindly extract
> > azimuths from HO171 for the contrived extreme examples I used when testing
> > Bennett's tables, as follows-
>
> > Example 1. Dec 55d 31'N, LHA 54d 29', alt 61d 29'. The true result for
> > Azimuth should be 285.1 degrees.
>
> > Example 2. Dec 55d 29'N, LHA 54d 31', alt 61 d31'. The true result should be
> > 284.7 degrees.
>
> > Even if HO171 values for those examples are less erroneous, it's likely that
> > at other combinations, some significant errors may occur. I wonder if it
> > carries any warnings about where it should not be relied on. I wonder, also,
> > whether any advice is given about interpolating between whole-number values.
>
> > George.
>
> > contact George Huxtable at geo...@huxtable.u-net.com
> > or at +44 1865 820222 (from UK, 01865 820222)
> > or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.


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