# Slide Rule Using Common and Natural Log Scales

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## Introduction: Slide Rule Using Common and Natural Log Scales

The slide rules that took men to the Moon were made without computers, so these instructions are based on computerless production. There are still books in existence that contain the log scales necessary without the use ot computers.

I never used anything except circular slide rules, so this Instructable will give directions on how to make those kind.

Numbers people do not always do crafts well, so for the sake of people who craft well, the numbers to make a Natural Log scale for a slide rule would be helpful and are included at Step 7.

I don't do pictures well, and I speak too much math and too much verbose. I will try to appreciate questions to help me clarify, if the questions are polite. An internet search ought to produce plenty of pictures of "circular slide rules".

The slide rule is based on logarithmic scales, and the accuracy of the slide rule is only as good as the precision of the markings.

1. Mark the uniform log scale.

2. Mark the C scale from the log scale.

3. Mark the Inverse, Square, Cube and / or identical D scales.

4. Mark the Natural Log scale, optional.

## Step 1: Quick Overview of the Math of Logarithms

According to programming conventions, * means multiplication and ^ means exponents, so

2 * 3 = 6

and

3 ^ 2 = 9

The slide rule is based entirely on the logarithmic scale. Logarithms are not really intimidating because they are easily calculated with calculators or software. For now.

log(2) = .30103

10 ^ .30103 = 2

log(3) = .4771

10 ^ .4771 = 3

log(50) = 1.699

10 ^ 1.699 = 50

A logarithm is defined as:

when

a ^ x = y

then

x*log(a) = log(y)

For example, because

2 ^ 3 = 8

then

3 * log(2) = log(8)

and

3 = log(8) / log(2).

Logarithms are usually only roughly equal, they usually require several places for accuracy.

10 ^ .48 = 3.01995

but

10 ^ .4771212 = 2.9999996

## Step 2: Calculations

Choose a precision of N from 1 to 9. 3 is convenient.
If N = 4 then

log(3) = .4771

when N = 2 then

log(3) = .48

The circumference will be determined by the precision N you choose and the length of the scale or markings you choose. If the precision N = 3 and the smallest unit is 1 mm, then the circumference will be

(10 ^ 3) * 1 mm = 1 meter

Because the circumference equates to PI * diameter, the diameter of the circle would be

(10 / PI)* 10 ^ (N - 1) = 3.183 * 10 ^ (N - 1)

in whichever units are being used.

When N = 3 and the units are millimeters, the diameter will be

(10 / PI) * 10 ^ (N-1) = 3.183 * 10 ^ 2

= 318.3 mm

which would be

318.3 / 25.4 inches = 12.53 inches

The diameter would then be 12.53 inches and the radius would be 6.265 inches.

A circle can be formed accurately by fastening the measuring stick in the center of the circle and then marking the radius (half of the diameter) in many directions. There are 360 degrees in a circle.

On the log scale, 360/10 = 36 degrees would be the size of each .1 increment on the log scale.

360 / 100 = 3.6 for each .01 and

360 / 100 = .36 for each .001.

On the blank circle, measure 10 ^ N units for the circumference, using measuring tape or a yardstick. If it is too large, then evenly shave off some of the circumference, but be sure to keep it a perfect circle. If it is too small, then a new circle must be cut.

Start marking the circumference and the face of the blank circle when the circumference is exactly 1,000 units. Paper is better for finding the perfect dimensions and for making patterns.

It is convenient to mark all the .001 increments of the Log scale as 1 millimeter each.

To make sure the slide rule is exactly the right diameter, it would be convenient to use a tape measure or a measuring stick to verify the circumference of the circle being used before beginning to make the markings.

Using a tape measure would also be a convenient way to mark off the 1 millimeter increments around the outside of the circle for the Log Scale.

## Step 3: Mark the Circle

There are 360 degrees in a circle.

On the log scale, 360/10 = 36 degrees would be the size of each .1 increment on the log scale.

360 / 100 = 3.6 for each .01 and

360 / 100 = .36 for each .001.

On the blank circle, measure 10 ^ N units for the circumference, using measuring tape or a yardstick. If it is too large, then evenly shave off some of the circumference, but be sure to keep it a perfect circle. If it is too small, then a new circle must be cut.

Start marking the circumference and the face of the blank circle when the circumference is exactly 1,000 units. Paper is better for finding the perfect dimensions and for making patterns.

It is convenient to mark all the .001 increments of the Log scale as 1 millimeter each.

To make sure the slide rule is exactly the right diameter, it would be convenient to use a tape measure or a measuring stick to verify the circumference of the circle being used before beginning to make the markings. Using a tape measure would also be a convenient way to mark off the 1 millimeter increments around the outside of the circle for the Log Scale.

## Step 4: Protractors

This is also a convenient way to make a protractor, which makes it a lot easier to make gears.

If we use 1 mm per each degree,

360 millimeters / 3.141592654 = 114.6 mm diameter, or 57.3 mm radius.

pi = 3.141592654

Millimeters are convenient units, but as long as they are all uniform, any unit would serve.

## Step 5: Two Dials or Two Cursor Hands?

I personally like the slide rules that have an inner turning dial somewhat better than the ones that have two clear cursors like clock hands. Using two cursor hands requires that they be turned on the face of the slide rule without moving at all relative to the other cursor hand.
If there is a rotating inner circle, then there would be a D scale on the outside (of the inner circle) to line up exactly with the C scale on the inside (of the outer ring).

Fractions work much better with two dials that can rotate relative to each other.

## Step 6: Scales: C (&D) B, K, CI

In my opinion, the log scale should be the scale on the outside of the largest outside ring of the slide rule, because it requires the highest precision and because it is the foundation on which all the other scales rest.
It makes it easier to read the scales when the smallest increments are the shortest length of marking and the 5s are one length and the 10s are another length.

The range of the C scale is 0 to 1 in increments of .1 and .01 (and .001 if large enough). The C scale is 10 raised to the log scale, so that 2 on the C scale would line up with .301 on the Log scale, and 7 on the C scale with .845.

The D scale is exactly the same as the C scale and is only used on slide rules with two dials the rotate independently.

The CI scale is the inverse of the C, so 5 on the C scale would line up with 2 on CI and 2 on the C scale would line up with 5 on CI. Also, 3 would line up with 333, and 4 would line up with 25.

The B scale is C ^ 2 and the K scale is C ^ 3. With a LogLog scale, these scales are less necessary.

Slide rules assume mental calculation of exponents.

## Step 7: Natural Log Scale

The Natural Log scale is especially useful to verify interest rates being charged. The Natural Log scale can also calculate exponents directly.
On the Natural Log Scale, e ^ .02, e ^ .2, e ^ 2, e ^ 20 all line up, one above the other, under 2 on the C scale. Decimals and magnitudes are calculated mentally. Attached are some tables of natural logs, but it would be a good idea to verify the numbers with an old book of mathematical tables.

ln(LL scale) = C scale

The Natural Log scale can be a spiral for the artistically adept. It is also shown as nesting circles with steps up.

The characteristics (exponents) are left up to mental calculations, but it is easy to calculate 10 ^ 3. To find the characteristic or magnitude of the exponent of e only requires following the line down in value to e ^ (10N) which is 1 on the C scale, 0 on log scale, to easily see the exponent of e that applies. For example, e ^ .1 = 1.1052 and e ^ .04 = 1.0408.

PRIME NUMBERS

From the logs of primes, other values can be calculated. There are about 200 prime numbers between 1 and 1,000.

A slide rule can be created with only a table of the common and natural logs of prime numbers. The other values can be calculated with each prime number and its common (base 10) or natural (base e) logarithm.

For example:

4.2 = (2 x 3 x 7) / 10

= 10 ^ [log(2) + log(3) + log(7) - log(10)]

= 10 ^ [.30103 + .47712 +.8451 - 1]

4.2 = 10 ^ .62345

However, people who most need a slide rule may be more prone to make mistakes in calculating from prime numbers.

## Step 8: Master Model

To increase precision, measure out at a large scale, then scale down to a smaller scale.

Make a very large scale outer "donut" template from which to form the portable, personal slide rules.

In the center, have a pin to put the blank slugs on to make into more portable slide rules.

## Recommendations

I had teachers who used a slide rule but no one ever taught us to use one. We just pushed buttons. Do you know a web site that I could use to learn? Got to be very user friendly. My mom tried to teach me once. The results were not pretty--we ended up not talking for a few days. Really not good memories.

I did have to use the instruction books that came with mine to learn some functions. But the basic rule of thumb to use when learning anything new is to start with what you know.

Start with an equation like 2 X 3 = 6. It is best to use different numbers when learning, rather than 2 X 2. Point one cursor to 1 and the other to 2, then move the cursor from 1 to 3.

If the cursors did not move relative to each other, the cursor that pointed to 2 should now be pointing to 6.

If your circular slide rule has a dial so that the C and D scales move relative to each other, then point one "1" cursor to 2, then on the same scale that has the "1" pointing to 2, the 3 should be pointing to 6.

This also shows that 6 / 3 = 2 and 6 / 2 = 3.

For the other scales (functions), I look at what lines up with 2. If 5 lines up with 2, then it is an inverse scale, and 4 will line up with 2.5. Be careful because this scale is reversed from the others.

If 2 lines up with 4 and 3 lines up with 9, it is a "B" scale, which means it shows squares, and square roots. If 2 lines up with .3, it is an "L" or log10 scale.

For the sine, cosine and tangent scales, I had to make a short list of sines and cosines for 30, 45, and 60 degrees.

Just play! That is the best way to learn.

Or you could use these sites:

Eric's Slide Rules
http://www.sliderule.ca/manual.htm

Greg's Slide Rules