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From: Paul Hirose
Date: 2016 Jan 22, 17:47 -0800

```On 2016-01-22 0:02, Francis Upchurch wrote:
> On another, but related subject, my Thornton manual (yes available still
free on line!) describes Sd and Td differential trig scales (on the original
slider) which it claims are more accurate throughout the angle range than
standard sines and cosines.
>
> I had never heard of or used these as boy. Does anyone know about these and
maybe we should be using these, Bob, especially on 12’ linear rules as you
intend?

I think differential trig scales were patented, and unique to Thornton
slide rules. They're on the slide in the lower image:

http://www.sliderulemuseum.com/British/S041_PIC_221.jpg

The basic idea is to set an angle on the D scale, then divide by a
correction factor to obtain the trig function of that angle. That
correction factor is obtained from the appropriate differential trig scale.

For instance, sin 50° = .766. The correction factor x is such that 50 /
x = .766. Hence x = 50° / .766 = 65.3. In the image, note that 50 on
scale Sd (differential sine) lines up with 65.3 on C.

Thus to get sin 50°, set the cursor to 50 on D, set 50 on Sd to cursor,
and read .766 on D at the C index.

At one extreme of the sine function, note that sin 90° = 1 exactly, so
dividing the angle by 90 yields its sine. And as expected, 90 on Sd
coincides with 90 on C.

At the other end of Sd, note how the small angles converge on 57.3,
which is the number of degrees per radian. That reflects the
mathematical fact that the sine of a small angle is simply its value in

In this way the basic C and D scales, plus one short auxiliary scale
which spans 57.3 to 90 on C, yield the sine of any angle from 0 to 90.

The same principle applies to the Td (differential tangent) scale.
However, whereas the sine correction factor increases with angle, the
tangent correction factor decreases:

sin   tan
10°  57.6  56.7
20°  58.5  54.9
40°  62.2  47.7
60°  69.3  34.6

That's convenient. Since both differential scales converge on 57.3 for
small angles, and go in opposite directions as angle increases, the Sd
and Td scales can share the same space on the slide rule.

An arc sine or arc tangent could be found by iteration with those
scales, but the ISd and ITd differential scales are easier. For example,
to find arc sin .766, set the cursor to .766 on D, slide .766 on ISd
under the cursor, and read 50° on D at the C index.

At small values, both inverse scales converge on the number of radians
in one degree: .01745. They go in opposite directions as angle
increases. By limiting the tangents to about 60°, all four differential
trig scales fit on one line of the slide rule.

That compactness comes at a price, though. You have to set each value
twice. I can't see how that makes for greater accuracy. And don't want
to think about multiplying, say, two cosines with such scales.

```
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