Welcome to the NavList Message Boards.

NavList:

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

Compose Your Message

Message:αβγ
Message:abc
Add Images & Files
    or...
       
    Reply
    Re: Using any star for a lunar
    From: Bill B
    Date: 2005 Apr 5, 16:38 -0500

    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.
    
    
    

       
    Reply
    Browse Files

    Drop Files

    NavList

    What is NavList?

    Join NavList

    Name:
    (please, no nicknames or handles)
    Email:
    Do you want to receive all group messages by email?
    Yes No

    You can also join by posting. Your first on-topic post automatically makes you a member.

    Posting Code

    Enter the email address associated with your NavList messages. Your posting code will be emailed to you immediately.
    Email:

    Email Settings

    Posting Code:

    Custom Index

    Subject:
    Author:
    Start date: (yyyymm dd)
    End date: (yyyymm dd)

    Visit this site
    Visit this site
    Visit this site
    Visit this site
    Visit this site
    Visit this site