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Andres Ruiz  wrote:
"Both with the best method for clearing a lunar distance, for me: the
Young�s formula."

Aqnd Frank Reed commented-

Ya know, unless you get into historical things, like working out the
calculation by logarithms or maybe using tables of so-called "logarithmic
differences," there's really no meaningful difference between "Young's
formula" and the common cosine formula. They differ only by a trig identity.
Once you put these calculations on a computer, they are identical (except
for a very small difference in computation time, on the order of one
microsecond for each cleared lunar).

The attention given to Young's formula in Cotter's "History of Navigation
and Nautical Astronomy," and repeated by some who have read that book, is
completely out of proportion to its significance. Young's formula is a
footnote in the history of lunars. This mistaken attention appears to be due
to the fact that Cotter focused on publications from the period when lunars
were obsolescent --post-1850.

======================

Comment from George-

I think the point has been missed, in Cotter's book and by Frank also, about 
what Young was trying to do. He was an academic, a prof. of mathematics at a 
college in a UK seaport, rather than a practising navigator, and he was 
interested in the teaching of navigation, and trying to get students to 
understand what they were doing.

All the other texts relied on logarithms, which was, in those days, just 
about the only way of doing precise arithmetic calculations, if you wanted 
to do anything more complex than simple addition and subtraction.. However, 
using logs presents two enormous problems.

1. Having gone "into logs", it's then easy to do multiplications, divisions, 
raising to powers, and all that. However, you can't add and subtract 
intermediate quantities, that come into an equation, without coming "out of 
logs", into the real world, doing the addition / subtraction, and then 
taking the log of that intermediate result (not without the clever trick 
that Bruce Stark uses in his lunar tables, a special table of addition / 
subtraction, or Gaussian, logs).

2. Logs are only meaningful for positive quantities.

So lunarians went to enormous trouble to fiddle their trig to allow logs to 
be used. Instead of sines and cosines, which are negative for certain 
angles, haversines were invented, always positive. And the trig formulae 
were distorted and twisted, so that any additions and subtractions, if they 
were unavoidable, were at the start or the end of the calculation, separate 
from the part where logs were used. And this is why there were so many 
different "solutions" to the lunar problem; theorists vied with each other 
to minimise the number of arithmetical steps that were called for to get an 
answer.

Young provided a solution to lunars using logs. But he advocated an 
alternative way of doing the job, that didn't call for logs at all, instead 
requiring the user to do one precise "long multiplication", to 6 figures, 
and then one equally-precise "long division". Without access to calculators, 
students were expected to be much handier at such figure-work than they are 
today.

When Andres and Frank state that they are using "Young's method", I suspect 
that both are referring to what Cotter described as "Young's method" in his 
book, "History of Nautical Astronomy", for obtaining the corrected lunar 
distance, D, from the observed lunar distance, d.

This calculation always required, as an auxiliary measurement, the observed 
moon altitude m, to which parallax and refraction adjustments were applied 
to give a corrected moon altitude M; and also, the observed sun or star 
altitude s, similarly adjusted to provide a corrected altitude S.

Let's start from the basics-

If you construct two slightly-different navigational triangles, joining the 
observer's zenith to the Sun amd the Moon; in one case to the observed 
positions m and s, and in the other case to their true positions M and S, 
and recognise that the azimuths of Moon and Sun are unchanged by the 
altitude corrections, then we can say, in one case -
cos (azimuth difference) = {cos D - sin M sin S } / (cos M cos S)

and in the other-
cos (azimuth difference) = {cos d - sin m sin s } / (cos m cos s)

so therefore-
{cos D - sin M sin S } / (cos M cos S) = {cos d - sin m sin s } / (cos m cos 
s)

which we can write as cos D =
{cos d - sin m sin s } {cos M cos S } / {cos m cos s }  + sin M sin S

That's about as basic as you can get, as an expression for the corrected 
lunar distance D. With a computer, or a pocket calculator, it presents no 
problems at all.

Young tinkered with it a bit, to turn it into-
cos D =
{cos d + cos (m + s)} {cos M cos S} / {cos m cos s } - cos (M + S)
which is what Cotter quotes, as "Young's method", and presumably what Andres 
and Frank are referring to also.

This has just one advantage. It calls only for a table of cosines: no sines 
or other functions are needed. That enabled Young to incorporate a nice big 
table of cosines in early editions of his book "Navigation and Nautical 
Astronomy". These were labelled "natural cosines" to differentiate them from 
the more-common log. cosines, and were presumably to 6 decimal places 
(unfortunately, that table has been excised from my much-later 1903 
edition). Of course that was a simplification only when large lookup tables 
were called for, and makes no difference when things are done by calculator 
or computer.

Cotter commented " The computation of cos D, using Young's method, is 
shortened considerably if a table of logarithmic differences is used to 
evaluate  {cos M cos S} / {cos m cos s }". That shows Cotter's 
misunderstanding of what Young was trying to achieve. A table of log 
differences would help only if the whole calculation was being done by logs. 
Young showed, for completeness, how that could be done, but his point was 
quite different. Instead he recommended that it should be done without using 
logs at all.

To that end, he rewrote the expression for cos D as-
{cos d +cos (m + s)} {cos (M + S) + cos (M - S)} / {cos (m + s) + cos 
(m-s)} - cos (M + S)

At first sight that looks more complicated, but you can see that each of the 
three terms in curly brackets is just the sum of two cosines, taken from the 
cosine table. First he needed the cosines of 5 angles, d, M + S, M - S, m + 
s, m - s. Note, by the way, that because cosine of a negative angle is 
exactly the same as for a positive angle, with the terms involving 
subtracted angles, you can make that subtraction in the easiest way, taking 
the lesser from the greater. Sometimes the sum of two angles will exceed 90 
degrees, resulting in a negative cosine, which sign must be observed. 
Anyway, those three terms in curly brackets could be readily arrived at, 
with just the cosine table, and no need for logs.

Next comes the difficult bit, when you try to do the job without logs, which 
asks you to multiply the first two terms in curly-brackets together, 
maintaining a six digit accuracy, then dividing the result by the third term 
in curly brackets, still maintaining that accuracy. That is, a rather long 
long-multiplication, followed by a rather long long-division, done without 
arithmetic errors. Beyond most of us now, I expect, but in the 1860s 
navigators were expected to be fluent and precise with such "figuring".

Finally, that last term cos (M + S) had to be added, and the result gleaned 
from the cosine table, using it in reverse, and taking the complement from 
180 degrees if the result was negative, as it would be for a large lunar 
distance.

That difficult arithmetic the price you had to pay for working without logs, 
and Young argued that it was a price worth paying, in terms of overall 
simplification. Whether he was right or wrong about that, Cotter 
misunderstood what he was getting at. It's all, of course, completely 
irrelevant today, with our electronic tools, just as Frank says.

George.

contact George Huxtable at george@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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