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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

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[Sent from archive by: peter.martinez-AT-btinternet.com]


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