# NavList:

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

**Re: Scales used on Bygrave**

**From:**Wolfgang Hasper

**Date:**2010 Feb 19, 21:31 +0100

I'll try a quick translation of page 166 / �80. You may judge if it can compete with google's... I will try to keep it as literal as possible, so everyone can follow the original text if they want BEGIN [end of �79] The altitude slide rule manufactured by Dennert Pape has proven useful and was introduced to navy and air force. �80. The position-fix altitude slide rule The position-fix altitude slide rule is based on the following considerations: The basic celestial triangle of pole-zenith-star (fig. 105) is divided in two rectangular spherical triangles by drawing a perpendicular from the star S to the meridian, the base point of which shall be A. Denoting the side PA of the triangle PSA "90-x" and the side ZA of the triangle "90-y" one can derive the following group of formulae to calculate altitude and azimuth from these two triangles. cotan x = cotan delta *cos t cotan Az=cotan t*cos y/cos x cotan h=cotan y/cos Az In doing so x is to be taken greater than 90� if the hour angle t is greater than 90�. y equals 90�-phi+x if phi [latitude] and delta are of same name. If of contrary names, y=90�-phi-x The azimuth Az on the northern hemishere counts from the South Point, on the southern hemisphere from the North Point and is to be assumed in the same quadrant as y, thus greater 90� if y is greater than 90�. These formulae consist of the trigonometric functions cotangent and cosine only. This allows to build the slide rule with only two sliding scales, with which all three formulae can be solved. To achieve the required precision (1 minute of arc) the two scales, divided with log cos and log cotan respectively, are fitted in a spiralling manner on two concentric cylinders. Thus they can, naturally, be very much longer than on on a linear arrangement, without making the slide rule bulky. The log cotan scale fitted to the inner cylinder would unwound be about 7 metres long, the log cos scale unwound from the outer cylinder would be about 4 metres long. The scales cannot, however, be set next to each other to make a reading as on a conventional slide rule. A pair of cursors, fixed to a small outer ring [here the description does in my eyes fail] compensates for that. It resembles the hair line of an ordinary slide rule. Setting of the zero mark on the log cos scale is simplified by a fixed zero-stop for the cursor-pair.- the scales do not only show markings from 0-90� but also the respective complements, thus cumbersome side-calculations can be omitted, if one only follows the sign-rules as given above. The use of the slide rule in detail unfolds from the instructions for use printed on the device. END Well, google would have been a lot faster! Wolfgang Am Freitag, 19. Februar 2010 01:35 schrieb Gary LaPook: > A while ago we had an ongoing discussion of what > scales were actually used on the Bygrave and the > German version, the HR1. Wolfgang just posted an > excerpt of a book in German which supplies > additional information on this issue. I am attaching > pages 166 and 167 of this book. Even without > speaking German I think it is clearly stated on page > 167 that the inner scale is a log cotangent scale > and the other scale is a log cosine scale. Also page > 166 states the formulas that are actually solved > with these slide rules which identify the scales as > a cotangent scale and a cosine scale. > ---------------------------------------------------- >-------------- Some of you may be interested in the > linked pages of "Lehrbuch der Navigation" > Bremen 1943, volume II > They show different aircraft sextants and describe > their use and properties. (in German...) > > Most intersting is the last paragraph on mechanical > calculation devices for the celestial triangle, > including HR1. > It reads: > "The altitude slide rule manufactured by Dennert & > Pape has proven useful and was introduced to navy > and air force." > Another hint that MHR1 is not likely to mean > "marine" HR > > Wolfgang > > Pdf documents, zip compressed are linked ~ 5MB > ---------------------------------------------------- >------------ NavList message boards and member > settings: www.fer3.com/NavList Members may > optionally receive posts by email. To cancel email > delivery, send a message to NoMail[at]fer3.com > ---------------------------------------------------- >------------ > > Linked File: > http://www.fer3.com/arc/imgx/f1-Lehb_d_Navigation.zi >p > > > > ---------------------------------------------------- >---------------- > > > gl > ---------------------------------------------------- >------------ NavList message boards and member > settings: www.fer3.com/NavList Members may > optionally receive posts by email. To cancel email > delivery, send a message to NoMail[at]fer3.com > ---------------------------------------------------- >------------ > > Linked File: > http://www.fer3.com/arc/imgx/SKMBT_C25010021719592.p >df