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    Re: The Bygrave Slide Rule
    From: Gary LaPook
    Date: 2009 May 28, 04:23 -0700

     From my Spetember 2008 post:
    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 0� 20' to 89� 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 0� 55' through 89� 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 89� 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 180� so, if necessary, subtract the LHA from 
    360� to bring it within this range. (My reconstruction avoids cluttering 
    the scales by omitting markings greater than 90� so I must get H within 
    the range below 90� 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 90�.
    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.
    Also, check out these threads.
    NavList@fer3.com wrote:
    > I was asked recently to help a friend who wanted to understand the Bygrave 
    > Slide Rule (BSR). He sent me references to most of the internet postings, 
    > and Google helped me find more, including this forum. I have improvised such 
    a device using scales printed on transparent sheets using computer programs I 
    wrote for the purpose. A most interesting project.
    > I am posting here now because I believe that some of the recent writings on 
    > the internet about the BSR may be misleading. I refer to the assertion by 
    > Gary LaPook (GLP) and Ron van Riet (RVR) that the inner scale of the BSR is 
    > actually a cotangent scale and not a tangent scale. The patent application, 
    > and the original manual of the BSR talk about tangents, so RVR and GLP seem 
    > to be suggesting that Bygrave may have made a mistake somewhere, even if 
    > only in the documentation.  It seems quite reasonable to me that this scale 
    > can be described as a tangent scale. I think I can explain why this question 
    > arose in the first place.
    > A conventional sliderule has a linear scale showing 1.0 at the left end and 
    10.0 at the right, the distance to a specific value from the left end being 
    > proportional to the logarithm of that value. The convention is clearly that 
    > the numbers (and their logarithms) increase from left to right along the 
    > scale. The inner scale of a BSR, if unwrapped horizontally, shows near-zero 
    > degrees at the left end and near-90 degrees at the right end, and the 45 
    > degree mark is in the exact centre of the scale. We know that tangents of 
    > angles less than 45 degrees are less than one, and hence that logarithms of 
    > tangents of angles less than 45 degrees are negative. Similarly logs of tans 
    > of angles greater than 45 degrees are positive.
    > The inner scale of the BSR, considered as a tangent scale, is oriented so 
    > that the logarithm of the tangent increases left-to-right in the conventional 
    > way.  It IS reasonable to say this is a logarithmic tangent scale.
    > So why are we discussing this at all? The point is that if BOTH scales were 
    > laid out in this way, the degree markings on the two scales would run in 
    > opposite directions. This arises since the tangent function INCREASES with 
    > increasing angle whereas the cosine function DECREASES. Several people have 
    > suggested that this would be prone to user error and Bygrave must have 
    > realised this. The BSR does have both degree scales running left-to-right. 
    > So how did Bygrave do it?
    > Given that I have suggested that the inner scale is laid out in a reasonable 
    > way as a tangent scale, and that BOTH scales on the BSR have the degree 
    > markings left-to-right, I suggest that Bygrave achieved his objective by reversing the COSINE scale.
    > The clever part is the way that Bygrave gets this reversed cosine scale to 
    > give the right answer.  On a conventional slide rule, multiplication of X*Y 
    > is done by aligning the origen of the slider scale with X on the fixed 
    > scale, then reading off the product on the fixed scale opposite Y on the 
    > slider. The instructions with the BSR for the tan(X)/cos(Y) operation follow 
    > this precise procedure (think of inner=fixed and outer=sliding). Bygrave 
    > has cleverly transposed the multiply and divide instructions so that they 
    > give the right answer with the reversed log(cosine) scale. A reversed 
    > log(cosine) scale is the same as a log(1/cosine) scale so Bygrave is 
    > tricking us into multiplying by (1/cos(Y)) with the reversed cosine scale 
    > when he wants us to divide by cos(Y). A similar trick is used where Bygrave 
    wants us to multiply by cosine, instructing us to perform the actions which 
    we would normally use on a conventional slide rule to divide.
    > The argument as to whether the inner scale is a tangent or a cotangent, (or 
    > whether the outer scale is a cosine or a secant) really comes down to 
    > whether you think of 'up' and 'left-to-right' as positive or negative. 
    Bygrave used the terms 'tangent/cosine' in the theory, and 'inner/outer' in 
    the instructions. There is no need for the end-user to know that one scale 
    has been cleverly reversed so that he doesn't make mistakes reading off the 
    > regards
    > Peter Martinez
    > --------------------------------------------------------
    > [Sent from archive by: peter.martinez-AT-btinternet.com]
    > >
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