The Slide Rule and the neighborhood

I am going to venture to subject the reader to yet another post
about slide rules.  So let us launch into another problem involving
height and distance, but this time with an added twist.  Then I'll
discuss what is to be gained by going through this exercise rather
than punching the numbers in on the calculator.

http://melton.sdf-us.org/images/rt-triangle-trig.png

Our man is standing atop a 300 ft. ocean front cliff and observes a
ship passing by at a 12° depression.  Let us see if we can
determine the distance from the base of the cliff to the ship.
We'll refer back to our handy table for solutions to right
triangles.  There are a couple of approaches we can take, but in
this case since we are given the angle of depression, we'll subtract
that from 90° to give us our remaining angle which is 78°.
To determine the distance from the base of the cliff to the boat
(we'll call it 'a'), we multiply 300 * (TAN 78°) = a.  So first
let us find the tangent of 78° on the slide rule.  Hmm...the
tangent scale only goes from 5.7° to 45°.  Let us see what
we can do about that.  Thankfully, the instructions that came with
one of my slide rules (Sterling Acumath 400) provides a solution:

http://melton.sdf-us.org/images/acumath-400-tangent.png

So if we follow this path then 90°-78°=12°.  Back to
where we were, but let's proceed.  So now let us find the tangent of
12°:

http://melton.sdf-us.org/images/acumath-T-scale.webp

Now we need the reciprocal of .212.  Here we go to the 'CI' and 'D'
scales.

http://melton.sdf-us.org/images/reciprocal-value.webp

To determine the decimal point when finding reciprocals one of the
rules we can follow is:

To find y = 1/x:

- Convert x to scientific notation and read it's coefficient c and
it's exponent, p
- if the coefficient is 1 or -1 exactly, y exponent is -p
- otherwise y's exponent is -p-1

To get our decimal in the correct place, we convert .212 to
scientific notation which makes it 2.12E-1.  On this particular
Acumath slide rule the 'CI' scale is on the slide.  We place the
cursor over 2.12 on the CI scale (remembering the 'CI' scale
increases from right to left) and read down to the 'D' scale which
gives us a value around 4.71.  Since our coefficient (2.12) is
neither 1 or -1, the exponent of our answer is -(-1)-1 = 0 so our
result is 4.71.  Let's see what the calculator gives us:

http://melton.sdf-us.org/images/tan78.png

Pretty darned close.  Good enough for our needs.  Now let us finish
it out.  Now we multiply the height of the cliff (300 or 3E2) times
the tangent of 78° which we just figured out is 4.71E0.  On the
slide rule we place the right index of the 'C' scale over the 3 on
the 'D' scale:

http://melton.sdf-us.org/images/index-c-over-3-d.webp

Then we move the cursor over to 4.71E0 on the 'C' scale and read our
answer directly below on the 'D' scale, in this case, 1.42.  

http://melton.sdf-us.org/images/final-answer.webp

Now we add the exponents 0 + 2 = 2, but since we have to adjust for
the change of magnitude (3 * 4.71 > 10), We add one to the exponent
(0 + 2 + 1 = 3).  Our answer is 1.42E3 or 1420.  What does the
calculator tell us?

http://melton.sdf-us.org/images/final-answer1.png

Pretty darned close.  Most slide rules are good up to three
significant figures and it is only as accurate as it's manufacture
and calibration.  The additional factor is the user's ability to set
it and read the values.  For most applications, three significant
figures is good enough.

Now that we have gone through this seemingly tedious exercise, I am
sure the reader is asking what is to be gained.  In a sense it is
the mathematical equivalent to stopping and smelling the roses.  In
our rush for an instant answer or instant gratification, we miss out
on the intimate details of the journey.  In a similar way, it is
like the difference between hopping in our car and driving to the
local market for our groceries, or slowing down and taking the
bicycle or even walking and have more intimate contact with the
neighborhood and surroundings.  As I have mentioned in a previous
post

https://gopher.floodgap.com/gopher/gw?a=gopher%3A%2F%2Fsdf.org%2F0%2Fusers%2Fmelton%2Farchive%2F06102017

a simple tool like the scythe allows one not only to mow grass and
weeds, but it provides an opportunity for the user to become more
intimate with the subtleties of the land (and wildlife) that would
otherwise be lost when sitting in a tractor noisily masticating the
grass, weeds and brush.  The slide rule in a similar way, takes us
on a journey to an answer and on the trip, we become more familiar
with mathematical relationships (the neighborhood) that we would
probably miss using the computer or calculator.  Make no mistake, I
have no problem with technology if it is for the common good.  Even
the Luddites only destroyed machines that did not support or foster
the "commonality."  Sometimes I find myself pausing and reflecting
if a particular technology I am using meets that standard, or have I
become a slave to that particular device and all of its ancillary
requirements.