### Modern Bygrave Slide Rule

 (Download a PDF copy of this page) I have developed a very compact and convenient method for doing the computations needed for traditional celestial navigation. This method relies on my adaptation of the long extinct Bygrave Slide Rule which is elegant in its simplicity and which produces altitudes and azimuths within one or two minutes of arc and takes less than two minutes to do the computation. The original Bygrave Slide Rule was invented by RAF Captain Bygrave in 1920 and consisted of three concentric tubes carrying a cotangent and a cosine scale which formed a cylindrical slide rule. Because the scales spiral around the tubes the scales were forty four feet long which provided the accuracy necessary to calculate altitude and azimuth for celestial navigation. A modified version was used by Germany in world war two for navigation of their U-boats. But these cylindrical slide rules were expensive to produce and they fell into disuse when precomputed tables became available. I have developed a simpler implementation of the Bygrave, one that is very easy to make since it doesn't require concentric tubes. I made this by (in effect) unwinding the two scales and printing out the cotangent scale twice on a piece of paper. This avoids the problem encountered with normal slide rules of running off the ends of the scale. I then printed the cosine scale in red on a transparent sheet so that the cosine scale can be placed directly on top of the cotangent scale and aligned much like a normal slide rule. These two simple sheets replace the multi-volume sets of tables normally used for celestial computations. You can combine this slide rule with eight photocopied pages from H.O. 249 (four pieces of paper printed on both sides) containing the long term almanac, list of stars, and the precession and nutation correction tables and you have a very compact, self contained celestial navigation system usable for ten years! I will illustrate how easy it is to use with an example. I have included pictures of the scales and of the computation form showing this computation. See also: Bygrave instructional form, Bygrave instructional form example , Bygrave Nautical Almanac Form. and Nautical Almanac Example. illustration 1 , cotangent scale illustration 2 , cosine scale illustration 3 , computation form The Bygrave doesn't require that the latitude or the LHA to be whole degrees so you can compute altitude (Hc) for your Dead Reckoned position but for convenience in this example we will use whole degrees, 34º North for latitude, 20º North for declination and a Local Hour Angle (LHA) of 346º. When using a Bygrave slide rule the azimuth and altitude are calculated in three steps using the same manipulations of the slide rule for each step. We first calculate an intermediate value, "W," in the first step and then use it to compute azimuth in the second step. Then using the azimuth we finally compute altitude in the third step. Before we begin we need to bring the LHA into a reduced range of 0º to 90º and we also figure our co-latitude. After we compute "W" we will be  combining it with co-latitude, either adding or subtracting depending on the LHA and whether the latitude and the declination have the same or different names, to form the intermediate value "X" which is then brought into the range of 0º to 90º to form "Y."  The computation form leads you through all of these steps. Illustration number 3  shows the form to be used with this simplified model of the Bygrave. The top of the form is used to compute hour angle, "H", which is LHA converted into the range of 0º through 90º. Enter the LHA in the proper column and follow the form to make the computation. You can see in our example we have placed the LHA, 346º, in the column for LHAs in the range of 270º to 360º. The form shows that in this case we subtract LHA from 360º to find "H", hour angle, in this example, 14º.  We have carried this 14º down to the "H" blank on the form and we have entered declination and latitude in the appropriate blanks ( illustration 4.) illustration 3 illustration 4 We also use the top of the form to determine if we will be adding or subtracting the intermediate value of "W" to co-latitude. This is determined by the column of the LHA and by the names of the latitude and declination. In our example we can see that we will be adding "W" since LHA is in the last column and the names of the latitude and the declination are the same. Just circle the "+" mark at the bottom of this column and also place a "+" mark on the "W" line under the co-latitude line. (Illustrations 5 and 6) Shown on the form, but not happening in practice, are the two cells for LHA between 90º and 180º and  270º and contrary declination because such combinations can never happen with the celestial body above the horizon. .So the rule simplifies to: IF 270º  < LHA < 90º  and Latitude and Declination have the same name then ADD W Otherwise SUBTRACT  illustration 5 illustration 6 Next we subtract the latitude from 90º to form co-latitude. If we were using our D.R. latitude we would subtract it from 89º 60' since this notation makes it easy to subtract degrees and minutes. In our example the co-latitude is 56º.  (Illustration 7) illustration 7 Now we calculate the intermediate value "W" which is found by the formula: cotan W  =  cotan declination x cos H We follow the zig-zag diagram along the right edge of the form to find the value of "W". See (Illustration 8) The left side of the zig-zag indicates the red cosine scale and the right side indicates the black cotangent scale. The same process is used three times and values are always taken off from the black cotangent scale, never from the red cosine scale. The zig-zag tells us to line up 0 (zero) on the red cosine scale with "D" (declination) on the black cotangent scale. We then look at "H" (hour angle) on the red scale and take out "W" from the adjacent black cotangent scale. (Illustration 9) illustration 8 illustration 9 On both scales the numbering is above the marks on the scales. I have aligned the red cosine scale slightly above the black cotangent scale for clarity. We can see that the red zero is lined up directly above the black tick mark for 20º (the declination). You use visual interpolation for the minutes of declination. Now, without allowing the scales to shift, we locate "H" (14º ) on the red scale and read out "W" from the adjacent black scale, in this case 20º 33' by visual interpolation, and write this value on the "W" line below the co-latitude. (Illustration 8) illustration 8 Next we determine "X" by adding "W" (20º 33') to co-latitude (56º), in this example "X" = 76º 33'. Then, following the rule on the form, since "X" < 90º , "Y" = "X" and we carry it down to the "Y" line. If "X" > 90º we would subtract "X" from 180º (or from 179º 60') to form "Y". (Illustrations 10 and 11) illustration 10 illustration 11 Next we find azimuth with the formula : cot  Az = (cotan H / cos W  ) x cos Y We now follow the second zig-zag on the form to find Az. (Note we start with the value computed at the last step, "W.") (Illustration 12) Using  the same manipulation as the first time, we line up "W" (20º 33') on the red scale with "H" (14º) on the black scale. We than look at "Y" (76º-33') on the red scale and take out Az from the adjacent black scale. Looking at illustration 13, we have lined up red 20º 33' with black 14º by aligning the 20º 30' mark slightly to the left of the "14" on the cotangent scale. (Note at this point on the red scale the tick marks show 1 degree.) Now, without allowing the scales to shift, we locate "Y" on the red scale (76º 33') and take out the Az from the black scale. (Illustration 14) The red 76º 30' mark is lightly to the left of the black 45º tick mark and the red 76º 40' is aligned with the black 45º 20' so by visual interpolation the red 76º 33' will align with the black 45º 09', the Az. (At this point on both scales each tick mark is 10'.) illustration 12 illustration 13 illustration 14 Next enter the Az on the form  and compute Zn by applying the rules on the form so we subtract Az from 180º (179º 60') to determine Zn, 134º 51' (or 134.9º).(Illustration 15) , in this case the celestial body is to the south east. The determination of Zn is usually not a problem in real life since you know the approximate direction of the body when you take the sight. I have included rules on the form to resolve any ambiguity. illustration 15 The third step calculates computed altitude, Hc, using the formula: cot  Hc  = cot Y / cos Az We follow the last zig-zag to calculate Hc. (Note we again start with the value computed at the last step, Az). (Illustration 16) Using the same manipulation as before, we line up Az (45º 09') on the red scale with "Y" (76º 33') on the black scale.  Next we locate zero on the red scale and take out Hc from the adjacent black scale. Looking at illustration 17 we see that we have aligned the red 45º tick mark slightly to the right of the black 76º 30' (each tick mark is 10' on both scales) and we visually interpolate red 45º 09' with black 76º 33'. Now, without allowing the scales to shift, we find zero on the red cosine scale and interpolate on the black scale to take out the Hc of 71º 17' (Illustration 18) and enter it on the form. (Illustration 19) illustration 16 illustration 17 illustration 18 illustration 19 Comparing our result with H.O. 249 (see HO 249.pdf) we find the Hc in H.O 249 of 71º 17' and Zn of 135º, the same as with the Bygrave example. Link to H.O. 249 Excerpt: H.O. 249 ----------------------------------------------------------------------------------------------------------------- SPECIAL RULES There are some unusual cases that require slightly different procedures and all of these special cases are described on the form. If "H" is less than 1º or greater than 89º simply assume a longitude to bring "H" within the range of the scales. The intercept will be longer but perfectly usable for practical navigation. If the computed azimuth angle is greater than 85º the computed altitude will lose accuracy even though the azimuth is accurate because with azimuth angles in this range, even rounding the azimuth angle up or down one half minute can change the Hc by ten minutes. So you use the azimuth you have calculated but you compute altitude by interchanging declination and latitude and then doing the normal computation a second time. You discard the azimuth derived during this second computation of altitude and use the original azimuth. When declination is less than one degree you can't begin the computation the normal way to find "W" because you have to start the process with declination on the cotangent scale and this scale doesn't extend below 1º. So in this case you just skip the computation of "W" and simply set "W" equal to declination. Using this method you arrive at an azimuth that is not exact but is a close approximation and in the worst case I have found the azimuth is still within 0.9º of the true azimuth but most are much closer. If the declination is less than one degree and the latitude is also less than one degree, follow this procedure and also assume a latitude equal to one degree. After you have computed the Az you then follow the same procedure discussed above for azimuths exceeding 85º by interchanging the latitude and declination and then computing Hc which will produce an exact value of Hc. Another rare possibility is that "Y" will be less than one degree or that it will exceed 89º after adding "W" to co-declination so it won't fit on the scale. The simple way to handle this situation is to assume a different latitude so the "Y" does fit on the scale even though the resulting intercept is longer but still usable. An extremely unlikely case (I only mention it to be complete) is that "W" exceeds the range of the cotangent scale, 89º15', so cannot be computed in the first step of the process.  This can only happen when shooting one star, Kochab, which has a declination of 74º 07' north and then only if "H" exceeds 87º 20', an extremely unlikely event. The attached form steps you through the computation and contains the rules for the special cases. The special cases are likely to come up only very rarely in practice. The first rule for H < 1º or H > 89º only involves LHAs covering 5 degrees out of 360º (LHA in the ranges of 0 -1, 89-91, 269-271, and 359-360) so only occurs by chance very rarely and these can be avoided if sights are preplanned as is the normal procedure for flight navigation. In the worst case, you have to change the time of the observation by four minutes. Rule 3 covers the case when Y is less than 1 or when Y exceeds 89º which covers a range of three degrees out of a possible 180º so is also very rare. Co-lat is in the range of 0º-90º and W is also in the same range so X comes in the range of 0º -180º. If X is less than 89º then Y is also less than 89º. If X is greater than 91º then Y is also less than 89º. Only in the case of X between 89º and 91º will Y exceed 89º. This situation can't be avoided in advance because you can't predict what the value of W will be but if it occurs then just assuming a latitude that differs at most by one degree solves the problem which will result in a longer intercept but one that is still usable. The fourth rule deals with cases of bodies bearing almost directly east or west and this situation can be avoided by choosing a different body to shoot or, if only the sun is available, by waiting a few minutes to allow the azimuth to change out of this range. The remaining situation covered by rule two (declinations less than one degree) concerns only bodies in the solar system since none of the navigational stars have declinations less than one degree. Obviously the most important body is the sun and its declination is between 1º north and 1º south for five days in March and again in September so this situation can't be avoided and this is the most important special case. The special rule handles it nicely and the Hc is completely accurate. The computed azimuth is an approximation but is never more than one degree different than the actual azimuth and is usually much closer. Since you can use your D.R. for the A.P. for this situation the intercepts can be kept short and the slight inaccuracy in the azimuth will not make a noticeable difference in the LOP. ---------------------------------------------------------------------------------------------------------------- EXPLANATION OF AZIMUTH RULES `In most situations there is no ambiguity as to which quarter the Zn lies since you know the approximate direction you are looking when you take the sight. The problem arises because the azimuth angle is limited to the range of zero to 90 degrees and when the Zn is near east or west the correct Zn might fall either side of the line so there is an ambiguity in converting from azimuth angle to Zn.One easy rule to apply first is that if the declination is greater than the latitude then the azimuth can never be in the opposite semicircle. To generalize this rule, if the declination has the same name as the latitude and the declination is greater than the latitude, then you start with the direction of the elevated pole (the nearer pole) when converting from azimuth angle to azimuth (Zn.)The second rule to apply is that if the declination is contrary then the Zn must be in the opposite semicircle. To generalize this rule, if the declination and the latitude have contrary names then you start with the direction of the depressed pole (the further pole) when converting from azimuth angle to Zn.These two rules take care of most of the cases, especially for navigators in low latitudes.The remaining ambiguity concerns situations in which the declination is the same name as the latitude but is less than the latitude. In this situation the azimuth of the body will be both north and south of the east - west line during part of each day. The rules compare "W" with latitude to resolve this remaining ambiguity.So, combining all three rules:If the declination or if "W" is greater than the latitude then combine the azimuth angle with the direction of the elevated pole.If the declination name is contrary to the name of the latitude or if "W" is less than the latitude then combine azimuth angle with the direction of the depressed pole.` ------------------------------------------------------------------------------------------ CHECKING YOUR COMPUTATIONS An easy way to check the computation on a Bygrave is to do the same computation on a calculator since this allows you to check the intermediate steps. Just use the standard Bygrave formulas in the three step process following along on the form I have posted. First calculate co-latitude and save it in a memory in the calculator. If you are using a value for hour angle that is not a whole number of degrees you might want to make the conversion to decimal degrees and save it in a memory since it will we used twice. If you are using whole degrees then this step is not necessary. Then you calculate "W" using the formula: tan W = tan D / cos H and sum it to the memory where you have saved co-latitude which is then X and then make any adjustment necessary to convert X to Y. (If you are just making trials you can avoid this step by your choice of the trial values.) There is no reason to store W itself since it is not used again. You can then convert W to degree and minute format to compare with the Bygrave derived value. Then you compute azimuth angle using the formula: tan Az = (cos W / cos Y ) x tan H. If you want you can also convert Az to degree and minute format to compare with the Bygrave. The last step is to calculate altitude with the formula: tan Hc = cos Az x tan Y. Then convert to degree and minute format to compare with the Bygrave result. (When entering values in the format of degrees minutes seconds, change decimal minutes to seconds, 6 seconds per tenth of a minute, in your head  before punching in the assumed latitude, declination  and hour angle if necessary.) Using whole degrees for declination, assumed latitude and hour angle, using a TI-30 with only 3 memory locations the key strokes are: ---------------------------------------------------------------------              (co-latitude = 90 - Assimed latitude) 90 - Assumed Lat = STO 1  (co-latitude stored in memory 1) ---------------------------------------                                (tan W = tan D / cos H) Declination   tan / H cos =    inv tan    (computed W) SUM 1  (X now stored in memory 1)(change X to Y if necessary) --------------------------------------        (tan Az = (cos W / cos Y ) x tan H) cos    (of W from prior step)   / RCL 1  (recalls Y from memory 1) cos x H tan = inv tan (computed Azimuth angle) ------------------------------------        (tan Hc = cos Az x tan Y) cos    (of Az from prior step)   x RCL 1  (recalls Y from memory 1) tan = inv tan  (computed altitude, Hc) 2nd D.D - DMS (changes Hc in decimal degrees to degrees, minutes and seconds) DONE ----------------------------------------------------------------------------------------- HINTS FOR MAKING YOUR BYGRAVE Print the cotan scale on photo paper. Print the cosine scale on ink jet  transparency film such as 3M made"Transparency film for Ink Jet Printers," stock number CG3480. To ensure that the two scales are printed at exactly the same scale remember that the cotan scale spans two cycles while the cosine scale covers only one cycle. To test the proper scale check that the cosine scale covers exactly one half of the cotan scale. An easy way to do this is to place the right hand end of the cosine scale on any value you choose on any line of the cotan scale then check that the left hand end of cosine scale, one line up,  lines up with the same value one line higher on the cotan scale. If they don't, then adjust your printer until they do. You can use plain paper for making this check, instead of the more expensive transparency sheet, by holding them up to a light. To learn how to use the Bygrave, work the example given on this page, using the instructional form. For actual navigation, use the abbreviated forms but keep a copy of the instructional form with you to refer to as it contains the rules for doing the computation. Download a PDF copy of this page to use as a training manual. For increased durability you can seal each scale in 3M "Self-Sealing Laminating Pouches," stock number LS854-5G or -10G. When sealing the transparency sheet you must be careful to work slowly to smooth out any bubbles since the transparency sheet can trap bubbles. Links to the two scales: COTAN scale COSINE scale Link to original Bygrave manual: Bygrave Manual Link to paper on Bygrave type slide rules: Position Line Slide Rules by van Riet
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