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On 2016-09-25 11:38, Greg Rudzinski wrote:
> I have been doing sight reductions with a K&E 4070-3 and noticed that the ST 
scale ends at 35'. I think an additional ST scale continuing from 35' would 
be useful when doing the sine product (classic formula) when declinations or 
latitudes are less than 35'.

Don't overlook the ′ and ″ gauge marks on ST. For example, for the sine
of 25′, set cursor to 25 on the D scale, set the ′ mark (near 2° on ST)
to the cursor, read .00728 on D at the C index. (That was my slide rule
result. Calculator says .007272.)

Note that the ′ gauge mark coincides with ca. 3440 on C. That's the
number of minutes per radian. So what you did was convert 25′ to
radians. That's practically equal to its sine if the angle is small.

The ″ gauge mark works in similar fashion. Its projection on C is the
number of seconds per radian. If you can remember there are about 3400
minutes or 200,000 seconds per radian, that's enough to correctly place
the decimal point.

A decitrig ST scale can extend down by decades with parallel adjustments
of the decimal point in the angle and sine. For example, the 2.1°
graduation can also stand for .21°, .021°, etc.

This trick is possible but more difficult on a sexagesimal rule. Purely
from the mathematical standpoint there's no problem. But the resulting
angles are not as easy to see mentally. For instance, the 2°10′ (= 130′)
graduation can also stand for 13′, 2°20′ for 14′, etc.

Near the left end of the scale, the 40′ and 50′ graduations can stand
for 4′ and 5′. The space between them is divided into ten parts, so each
minor graduation would be .1′ or 6″.

At the right end, 4° (= 240′) can stand for 24′, 4°10′ for 25′. The five
graduations between them are .2′ each.

Like the S scale, each black number on ST has a corresponding red
number, though they're rarely marked. For example, black 4° corresponds
to red 96. The same graduations could also stand for black 24′ and red
89°36′.

When the cursor is on that graduation, on CI read 143.2, the tangent of
89°36′. It's one way to get tangents arbitrarily close to 90°. But the
trick is much easier on a decitrig rule.

   

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