# NavList:

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

**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. gl http://groups.google.com/group/NavList/browse_thread/thread/6d5ca3c3bbe778f4?hl=en# http://groups.google.com/group/NavList/browse_thread/thread/b23e8ea45770fd69?hl=en# http://groups.google.com/group/NavList/browse_thread/thread/26367135a43126aa?hl=en# http://groups.google.com/group/NavList/browse_thread/thread/4c5f5e69473e1948?hl=en# http://groups.google.com/group/NavList/browse_thread/thread/20ac6296103b4c7e?hl=en# http://groups.google.com/group/NavList/browse_thread/thread/ac94da3086897276?hl=en# gl 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 degrees. > > regards > Peter Martinez > > -------------------------------------------------------- > [Sent from archive by: peter.martinez-AT-btinternet.com] > > > > > > --~--~---------~--~----~------------~-------~--~----~ Navigation List archive: www.fer3.com/arc To post, email NavList@fer3.com To unsubscribe, email NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---