# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Obtaining Azimuths. was: Re: Burdwood's Tables**

**From:**John Cole

**Date:**2007 Oct 14, 06:21 -0700

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 Colewrote: > 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...---.u-net.com > > or at +44 1865 820222 (from UK, 01865 820222) > > or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to NavList@fer3.com To unsubscribe, send email to NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---