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    Bygrave slide rule
    From: Gary LaPook
    Date: 2008 Sep 27, 04:01 -0700

    Gary LaPook writes:
    British captain Leonard Bygrave invented his celestial navigation 
    computer in 1921. It consists of three concentric tubes forming a 
    cylindrical slide rule designed for the one purpose of calculating 
    azimuth  (Az)  and altitude (Hc) of celestial bodies. The inner tube is 
    covered with a spiral log cotangent scale (the patent documents and 
    other descriptions identify this as a  tangent scale but it is, in fact, 
    a cotangent scale), the second tube has a spiral log cosine scale and 
    the third tube carries two pointers used to line up the two trig scales. 
    In normal slide rules you can align the scales next to each other but 
    with spiral scales this is not possible so the need for the two 
    pointers. The advantage is that a spiral scale can be made much longer 
    than  a normal ten inch slide rule allowing for much greater accuracy 
    and precision. The cotangent scale on the Bygrave slide rule covers the 
    range of 0º 20' to 89º 45' by spiraling 44 times around a two and a half 
    inch diameter tube making this scale  28.8 feet long! The copy I 
    constructed only covers the range of 0º 55' through 89º 15' by spiraling 
    37 times around a one and a half inch diameter tube making this scale 
    14.5 feet long. Each spiral represents a .1 change in the value of the 
    log cotangent. For example, the log cotangent of 20' is 2.2352 and the 
    value directly above the 20' mark and up one spiral is 25' 10" which has 
    a log cotangent of 2.1352, exactly .1 less than the value one spiral 
    down. Going up 44 turns to the top of the scale and directly above the 
    20' mark has a log cotangent of -2.1648 exactly 4.4 less than the log 
    cotangent of 20' and marked as 89º 36.5'.
    Bygrave divided the navigational triangle into two right triangles by 
    dropping a perpendicular from the geographical position to the 
    observer's meridian. Bygrave labeled the two portions of the observer's 
    meridian created this way as "y" and "Y" (lower case "y" and upper case 
    "Y" (go figure?)) which allowed a simplified way to do the necessary 
    computations which are accomplished as follows.
    First, you use the almanac in the usual manner to find GHA and 
    declination and figure LHA ("H" in Bygrave's system) in the normal 
    fashion except you do not need to have a whole number of degrees of LHA 
    so you can work the sight from a DR position.The Bygrave needs the hour 
    angle (H) to not exceed 180º so, if necessary, subtract the LHA from 
    360º to bring it within this range. (My reconstruction avoids cluttering 
    the scales by omitting markings greater than 90º so I must get H within 
    the range below 90º but the computations work out the same.) Next you 
    calculate co-latitude by subtracting your latitude (either for your AP 
    or for your DR) from 90º.
    The azimuth and altitude are calculated in three steps using the same 
    manipulations of the slide rule for each step.
    Next calculate "y" which is found by the formula:
        tan y =  tan declination / cos H
    This is the formula listed in the patent documents and in the Bygrave 
    manual but, in fact, the slide rule does the calculation by modifying 
    this formula to allow the use of the cotangent scale. The actual 
    manipulation of the slide rule uses the re-arranged formula of:
       cotan y  =  cotan declination x cos H
    You accomplish this by setting one of the pointers (or the cursor on my 
    copy) to zero on the cosine scale and while holding it there rotate the 
    cosine scale and slide it up or down on the cotangent scale so that the 
    other pointer (or cursor) is aligned with the declination on the 
    cotangent scale. Now, holding the cosine scale still, rotate the pointer 
    (cursor) to point at the hour angle (H) on the cosine scale and then 
    read out "y" from the other pointer (cursor) where it points on the 
    cotangent scale.
    Next you find "Y" by adding "y" to co-latitude (if latitude and 
    declination have the same name) or by subtracting "y" from co-latitude 
    (if of opposite names.)
    Next we find azimuth with the formula :
        tan Az  =  (tan H x cos y ) /  cos Y
    which is re-arranged into the form:
     cot  Az = (cotan H / cos y  ) x cos Y
    Now, using the same manipulations as before, set one pointer to "y" on 
    the cosine scale and the other pointer on H on the cotangent scale, move 
    the cursor to "Y" on the cosine scale and read out azimuth from the 
    other pointer on the cotangent scale.
    The third step calculates altitude, Hc. using the formula:
     tan Hc = cos Az x tan Y
     with the formula re-arranged into the form:
        cot  Hc  = cot Y / cos Az
     set one pointer to Az on the cosine scale with the other pointer to "Y" 
    on the cotangent scale. move the pointer to zero on the cosine scale and 
    read our Hc from the other pointer on the cotangent scale.
    Done, and it only takes two minutes and produces an accuracy of one or 
    two minutes of arc, uses no electrons or large books. It is easier to do 
    than it is to describe. You can also use these formulas with a digital 
    I have attached the Bygrave manual, patent  and pictures of the instrument.
    This slide rule uses
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