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: Sight Reduction by the Cosine Haversine Method
    From: Herbert Prinz
    Date: 2004 Oct 7, 16:53 -0400

    George Huxtable wrote:
    
    > Ptolemy writes down what is effectively a table of sines (actually, chords
    > of a circle).
    >
    > At half-degree intervals of angle A Ptolemy computes the lengths of the
    > chord of a circle which subtends angle A, effectively 2 sin (A/2), from A =
    > 0 to 180 deg, to remarkably high accuracy, entirely from geometrical
    > arguments. So this is effectively a table of sines, at quarter-degree
    > intervals, between 0 and 90 degrees.
    >
    
    But take note of a subtlety. When the geometrical arguments lead to a dead end,
    Ptolemy is prepared to drop the whole apparatus of pure geometry and resorts to
    a numerical procedure. Starting out with 72d and 60d from the pentagon and
    hexagon, he obtains the chords of angles 12d, 6d,...,..3/4d by continued
    bisection. But how to get the chord of 1/2d? He knows that the trisection of the
    angle is impossible with rule and dividers. With the help of some proportions,
    he estimates an upper and lower limit and observes that they agree to the
    required number of decimals. That's good enough for his purposes. He takes
    astronomy out of the realm of philosophy and treats it like an engineer.
    
    Trying hard to give the appearance that what we are discussing here has anything
    to do with navigation, I shall mention in passing the name of Cotter. (I know,
    George, that you maintain this list...)
    
    In A History etc., p.16 he writes that "Ptolemy had the theorem
    
        "sin(A+B) = sin(A)cos(B) + cos(A)sin(B)
    
    "This theorem is usually known as Ptolemean Theorem."
    
    Now that you have seen the relevant chapter in the Almagest, would you agree
    with this?
    
    Herbert Prinz
    
    
    

       
    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