# ⊙ AntiQuark

Truth, Beauty, Charm, Strange

## 2005/01/10

### Slide Rule Tricks

X-treme Slide-rulering circa 1976

If you thought Zippo lighter tricks were pointless, then you've never seen slide rule tricks.

The main purpose of slide rules is to do multiplication. Slide rules were never designed for addition... OR WERE THEY? If you ignore the fact that addition is trivial with a pencil and paper, and is possible to do mentally without being a superhuman, you might find the addition tricks useful (assuming you still use a slide rule.)

Another trick is that you can multiply with a slide rule. Yeah I know, slide rules were designed for multiplying. But the cool thing is, you can multiply WITH THE WRONG SCALES!

So, next time someone at a party starts with the lighter tricks, just whip out your slipstick and wow the crowd with the following maneuvers...

Adding with the C and D scales.

The C and D scales are logarithmic scales designed for multiplication. Since we want the result of x + y, we need to find a multiplier (let's call it m) such that x * m = x + y. Solving for m:

x * m = x + y
m = x/x + x/y
m = x/y + 1

Following the steps below will multiply x by x/y+1, resulting in x + y.

Example: calculate 2.3 + 4.5

- Move the leftmost '1' of the C scale to 2.3 on the D scale.
- Move the cursor to 4.5 on the D scale.
- Notice that the cursor is at 1.956 on the C scale.
- Mentally add 1 to 1.956 to get 2.956. Move the cursor to 2.956 on the
C scale.
- The cursor will now be at the sum of 2.3 + 4.5 on the D scale, or 6.8.

on a slide rule are unusual in that they are evenly spaced like a
regular measuring scale. As a result, simple addition of the distances
on the L scale is equivalent to numeric addition. Any ruler with an L scale won't do though; the L scale has to be on the slider for this trick to work. It seems that this layout was more common on Picketts than on any other brand.

Example: calculate 0.23 + 0.45

- "Reset" the rule so that all the scales are lined up.
- Move the cursor to 0.23 on the L scale.
- Move the leftmost 0 on the L scale to the hairline.
- Move the cursor to 0.45 on the L scale.
- Reset the rule again so that all the scales are lined up.
- The cursor should now be at 0.68 on the L scale, which is the sum of
0.23 + 0.45.

Multiplying with Log-Log scales

The LL (or Log Log) scales are for exponentiation, or calculating xp. To do multiplication on these scales, we have to find a power p such that xp is equal to x * y.
Solving for p:

xp = x * y
p log(x) = log(x * y)
p = log(x)/log(x) + log(y)/log(x)
p = log(y)/log(x) + 1

Notice the similarity to the addition trick above. The steps below will calculate xlog(y)/log(x) + 1 which is equal to x * y.

Example: calculate 4.5 * 6.7 on the LL3 scale.

- Move the cursor to 4.5 on the LL3 scale.
- Move the leftmost 1 (the index) of the C scale to the cursor.
- Move the cursor to 6.7 on the LL3 scale.
- Notice that the cursor is at 1.264 on the C scale.
- Mentally add 1 to 1.264 to get 2.264. Move the cursor to 2.264 on the
C scale.
- The cursor should now be at 30.15 on the LL3 scale, which is the
product of 4.5 x 6.7

Dividing with Log-Log scales

Similarly to the multiplication trick above, we want to find a power that produces the same result as a division.

xp = x / y
p log(x) = log(x / y)
p = log(x)/log(x) - log(y)/log(x)
p = 1 - log(y)/log(x)

The mental calculation here, 1-N, is a little more difficult than 1+N. I find it's easier if you don't try any mental calculations, and instead measure ticks symmetrically around 0.5.

Example: calculate 95 / 20.

- Move the cursor to 95 on the LL3 scale.
- Move the rightmost 1 on the C scale to the cursor.
- Move the cursor to 20.
- Notice that the cursor is at 6.57 on the C scale. (Actually it's at 0.657, but this explanation is infinitesmally easier if I say it's at 6.57.)
- Mentally calculate 5 - 6.57. It's easy if you think of reflecting the
value 6.67 to the other side of 5. Instead of a point 1.57 to the right
of 5, go 1.57 to the left of 5. Move the cursor to 3.43.
- The cursor should now be at 4.75 on LL3, which is the result of 95/20.

• At 6/01/2006 5:28 PM,  Anonymous said…

Adding with the C and D scales:
you mean y/x where you have x/y. Slide rules rule.

• At 6/01/2006 9:46 PM,  Derek said…

You are correct sir.

• At 9/29/2006 10:49 AM,  Dr. Sardonicus said…

Here's something you may not already know. You don't need log-log scales to calculate y to the x power. (I will denote y**x). The C D and L scales can do it just fine (and often with more precision than LL, especially for numerically large results).
The L scale is the functional inverse of the C,D scales. For which ever member of the C,D pair
that shares the same part of the rule (body or slide) with L, this relationship is fixed. For the Pickett N4 simulated on this site, L is fixed to C on the slide. In this case L=log(C) and C=10**L (ten to the Lth power).
if we want to find z, where:
z=y**x, then
log(z)=log(y**x)
and by a really groovy property of logarithms:
log(y**x)=x*log(y),
( * denoting simple multiplication)
So: log(z)=x*log(y)
and: 10**[log(z)]=10**[x*log(y)]
Don't worry, it gets simpler from here.
since 10**x and log x are inverse functions, they undo each other on the left side of the equation, bringing it back to z:
z=10**[x*log(y)]= y**x
This looks a bit convoluted, but it can all be done with C, D and L. You just have to start on the innermost parenthesis and work your way out.
Here's an example of THE C,D,L METHOD using the Pickett N4 simulator. Say you want to find z=27.6**3.49
so y=27.6 and x=3.49.

recall: z=10**[x*log(y)]

1. find log(y)=log (27.6).Set cursor to 2.76 on the C scale. Read 0.441 (the mantissa of the log)of the L scale. Since 27.6 is 10*2.76 = 2.76*(10**1), the characteristic(integer portion of the log)is 1, so log(27.6)=1.441.
for 276=2.76*(10**2)log(276)=2.441
for2760=2.76*(10**3),log(2760)3.441
Getting back on track:
log(y)= log(27.6)=1.441

2.Next we use the C and D scales to multiply x*log(y)in the ordinary way: 3.49*1.441=5.03

3.Now for the tricky part. we need to find 10**5.03.This means going from the L scale to the C scale. Remember, the L scale IS JUST FOR THE MANTISSA (DECIMAL PORTION) OF THE LOGARITHM. This means you set the cursor to .03 on the L scale and read 1.07 off the C scale. The integer (characteristic) part indicates the power of 10 to raise the 1.07 result to, to find our z.
This is what allows one set of scales to calculate numbers of any size.

ANd it is on the C scale where it can be used for further calculation.
Note that direct calculation of 27.6**3.49 on the LL scales of this same N4 rule only indicates some value slightly above 100,000.

The method works for negative exponents too, but the logarithm needs to be manipulated into a negative characteristic and purely positive mantissa before performing the final 10**x operation.

You can also do this method on the much smaller Pickett N904-T simulator. Here L is paired with D instead of C, but the process is the same.

On my ideal slide rule I'd ditch the LL scales and use the space for a scale of my own design I'm working on to calculate the precise number on next weeks winning lottery ticket.

• At 9/29/2006 12:47 PM,  Derek said…

Wow, that's a lot of steps, I see why the LL scales were invented.

For your lottery rule, you should take a look at the Schmendrolog:
http://www.sphere.bc.ca/test/frankenrule.html

Which had a scale for generating random numbers.

• At 7/12/2008 10:30 AM,  Anonymous said…

What Dr. Sardonicus said.

Any slide rule with CD, L, and a cursor will do it.
You're just using the slide rule to mimic it's own behavior.

a^b=c

a=10^x.
so convert that from L to D with the cursor.

x*b=10^y
multiply 'em together with C and D

10^y=c
convert the result back from the D scale to the L scale with the cursor.

easy-peasy :D

If I were to design a modern slide rule, it would have nothing but L, C&D, trig and conversions. Circular and rugged construction like an E-6B Dalton computer.

• At 7/12/2008 10:43 AM,  Anonymous said…

'Scuse me, said that backwards.

convert from D to L, multiply the result, then convert from L to D.

for example, 2.72^.69

2.72 0n the D scale is 4.36 on the L scale.

4.36*.69 using the C&D scales is 3.
3 on the L scale is 2 on the D scale.

2.72^.69=2. AKA the rule of 69.

• At 7/12/2008 10:44 AM,  Derek said…

Thanks for the info!