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    Re: Approximate methods for clearing the Lunar distances - some details; corrected version
    From: Jan Kalivoda
    Date: 2003 Apr 21, 16:12 +0200

    Dear Arthur,
    Thank you very much for your assessment - you have encouraged me.
    I am glad that you feel exactly the same as I felt when overviewing the story 
    - how rare are the necessary ideas during the ages! How great was the passion 
    of Thomson, Lyons, Maskelyne, Norie and several (not many!) others for the 
    nautical astronomy, whose work  was used by subsequent generations of seamen 
    without a great thankfullness.
    Your picture is superb, without any faults in my eyes. And the whole website 
    is splendid - I hope that at the end of semester, I will be able to explore 
    all its content and links. I shall buy some book titles mentioned by you, 
    As for your questions, I am not a graduated mathematician, therefore you must 
    not swear by my words. My "polynom" was not a proper English word, sorry - 
    the usual word for it is the "polynomial", I hope. Forgive me, if I repeat 
    things you know : Many very complicated functions and problems in mathematics 
    can be tranposed in approximate polynomials by the differential calculus; 
    these polynomials have an infinite number of terms that shall diminish in 
    their maximal value in each subsequent step to be usable in calculations. If 
    these terms really diminish ("converge", i.e. to zero) in each step, it is 
    always possible to stop their evaluating when the necessary level of accuracy 
    of the result has been attained. B.Taylor had devised the general method for 
    this purpose in 1712/1715. All logarithms, all sines etc., all logs of sines 
    etc. had been calculated by such polynomials from the half of the 18th 
    And the Taylor's polynomial for resolving the approximate lunar problem, which 
    I have mentioned in my article, was known from the half of the 18th century, 
    too. Maskelyne, Lyons etc. used it regularly. The perpendiculars to the sides 
    of LD triangles are not necessary in deducing and evaluating it.
    They are essential in the pure trigonometrical deduction of the "approximate" 
    formula. That deduction, which is very clever and ingenious,  was introduced 
    into the handbooks rather for pedagogical purposes, so as to illustrate the 
    creation of this formula to the students not imbued by calculus. Sorry, this 
    deduction is rather long, it occupies some four pages of text and 
    trigonometrical tricks are used in it that shall be discussed furthermore 
    (mathematicians like to leave gaps in his argumentation to increase the 
    sacred apprehension of their skills in pupils and common people). If you read 
    German and want it, I will scan this explanation from the treatise of Bolte 
    (1894), which is very thorough, and I will send it to you and others.
    Best wishes
    Jan Kalivoda
    ----- Original Message -----
    From: "Arthur Pearson" 
    Sent: Sunday, April 20, 2003 4:53 AM
    Subject: Re: Approximate methods for clearing the Lunar distances - some details; corrected version
    > Jan,
    > I have re-read this posting and continue to find it fascinating.  It
    > paints a vivid picture the extraordinary efforts expended to find an
    > accurate method of clearing the distance that was also convenient for
    > use by the average sea captain. The derivation of these formulae is
    > brilliant in itself. The ingenuity required to transform them in ways
    > amenable to tabular solution is stunning.  The determination to grind
    > through 30,000 LD solutions and another 50,000 interpolations to
    > actually produce a table is beyond imagination.  Sending a man to the
    > moon offers nothing more impressive than the achievement of making
    > lunars a practical tool.
    > Because I need to condense things to understand them, I have amended my
    > diagram to illustrate all the variables you mention and copied over the
    > polynom, the substitutions, and the approximate formula just below it.
    > It is still at
    > http://members.verizon.net/~vze3nfrm/Nav_L_Graphs/ApproxLD.JPG. Please
    > let me know if you see any mistakes in my transcription of formulae or
    > placement of variables. Needless to say, links to both branches of this
    > thread are now on the Nav-L section of www.LD-DEADLINK-com.
    > My only remaining questions: What exactly is a polynom and where in the
    > derivation of formulae are those perpendiculars used?
    > Thanks again,
    > Arthur

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