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

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The Bygrave Slide Rule
From: UNK
Date: 2009 May 27, 14:57 -0700

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]

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