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George

I missed the below in my first reading of your response regarding the Moon
and just uncovered it a few days ago.  It was an  important piece of the
puzzle that was missing in my attempt to move beyond plug and chug in
spherical trig.

It does seem in that before the 18th century (or later) the answer to, "If
you have 5 oranges and take away 8, how many do you have," was, "You can't
do that, you're out of oranges."

As Alex pointed out when I followed up your response with him (as well as
other topics) in fact formulas were manipulated to avoid both subtraction
and addition for the reasons you pointed out--working from log and trig
tables and negatives. Log tables made multiplication and division relatively
painless and precise pre computer, and slides rules were not accurate
precise/accurate enough.

Alex also reinforced your point regarding my unwarranted concerns about
negative angles today.  In context, I was concerned about negative
star-to-star SHA's in calculating separation.  As he pointed out (before I
mentally slapped my forehead--duh) my calculator or spreadsheet doesn't care
if its -30d or +330d.  Same sine or cosine.  It did matter "in-the-day."

But back to the missing puzzle piece.  As I looked at the pole, AP and body
GP triangle and tried to related it to solving it in plane trig using the
law of cosines and sines, I kept wondering why it used cosine and sine of
latitude etc. instead of the actual length of the side (90-Lat, latitude
degrees easily converted to nautical miles for a perfect sphere).  Why were
the formulas for Hc and Z as they are?

Then your response and the light went on (finally).  Possible range of
either leg is 0d-90d. Sin x = cos (90 - x) and vice versa.  Hence sin
(90-Lat) =  cos Lat.  Avoids subtraction and fewer steps.

Another puzzle piece was provided by the clever use of a GPS unit to solve
the spherical triangle.  I did not know Hc could be converted to
great-circle distance before that post.

Then it came together.  I start out with an angle (LHA or "t"), and I have
two adjacent leg lengths (pole to AP Lat and Pole to GP).  Solve for Hc with
the spherical version of the law of cosines, which gives me the length of
the side opposite angle t.  Then using the spherical version of the law of
sines I can solve for another angle (pole to AP to GP of body), that being
Z.  Then the "if" statements you mentioned to derive Zn.

It would appear we could also determine the third angle (AP, GP, pole) by
subtracting angle Z + angle t from 180.  Possibly a silly question, but I've
reached the age that knowing I don't know is worse than publicly admitting I
don't know.  In plane geometry the sum of angles of triangle always equal
180d.  Is that always the case in spherical trig?

I imagine tutoring beginners on the web might often leave you feeling like
you've had a hard day teaching a special education student, but your time
and thoroughness are appreciated.

Bill

PS  Thanks for working "sexagesimal" into the conversation--helps me
remember it. 

Bill




> In response to my comment-
>
>>> It fits in with the notion
>>> that navigators do not understand how to subtract.
>
> Bill wrote
>> I am puzzled about the notion George has suggested that navigators do not
>> understand how to subtract. Where did it come from? 
>
> Well, our trade of celestial navigation was founded, mostly, in the 18th
> century. Although navigators were indeed taught how to subtract one
> positive number from a larger positive number (even in sexagesimal),
> everyone in that age seems to have avoided manipulating  actual negative
> quantities if it was at all possible. Or so it seems to me.
>
> Whereas a modern high-school kid would be expected to subtract -9 from -16
> and (sometimes) get the right answer, I get the picture, from reading
> 19th-century navigation manuals, that such concepts were alien to the
> mindset of those days.
>
> Instead, it was common for quantities to be given NAMES, rather than signs.
> So latitudes and declinations were labelled North and South, not + and -.
> And elaborate rules were devised which said things like "If the names
> differ add, but if they are the same, take the smaller from the larger, and
> label the result appropriately". But really such rules are only a
> complicated implementation of sign-depended subtraction.
>
> Similarly, the cleverness of logs was brought into trig computations, and
> this added its own problems, because the log of a negative number has no
> meaning. There were tricks to overcome this in a truly mathematical way, as
> used by Chauvenet, but instead, wordy rules to get round the problem and
> avoid such negatives were devised.
>
> Indeed, the practice, in navigational logs, of adding ten, or tens, to the
> characteristic of a log, was another trick to avoid negative values, at the
> expense of understanding and consistency.
>
> Such matters have only started to change over the last few years, it seems
> to me, with the introduction of calculator and computer formulae to
> navigation, these having no difficulties with signed quantities.
>
> George.

   

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