The Classic Slide Rule

George,Y · 2007-05-11 19:05:32

I've got one from my father. It is really cool! Not only can it serve as several mathematical reference tables (exponetial tables, sin tables), but it also provides many functions you have never imagined!

So far I can calculate:

Multiplication
Division
mass multiplication and division together ab/c a/(bc) abc
k√10
square, cube-square 3/2square as well as the inverse
Sin, Cos, Tan and the inverse
lg
ln
e^x

and:

a^b
a^1/b
log[sub]a[/sub]b !

Isn't that great?

Merely all the arithematric functions ly on the very trick of lgAB=lgA+lgB! Amazing, isn't it?!

Last edited by George,Y (2007-05-11 19:05:55)

George,Y · 2007-05-11 19:45:21

Anyone knowing the slide rule??

luca-deltodesco · 2007-05-11 20:05:41

nope, no idea what it is

George,Y · 2007-05-11 21:36:46

Virtue Slide Rule

Multiplication by 2.0:

Click and drag the middle bar to the right until the 1.0 on its lower scale matches the 2.0 of the scale on the lowest bar.

Notice that 2.0 on the lower scale is twice as much as 1.0 on the upper scale.

Moreover, any scale on the lower scale is twice as much as the corresponding one, the one just above the original one, on the upper scale, and the upper ones are one half of the lower ones.

You can check various matches yourself using the cursor and the red hairline yourself.

And, the multiplication is not limited to 2 alone, try other multipliers yourself.

MathsIsFun · 2007-05-11 22:09:25

I bought a Slide Rule once. They are great fun, and experts can do multiplication, powers and other things much faster than calculator users.

They are only accurate to a decimal place or two, but for many purposes that is enough. How thick to make a beam, how much concrete to order, what resistor to choose, things like that.

George,Y · 2007-05-11 22:11:29

Read this page carefully before you read on:
The Principle of Slide Rule Multiplication

A Complex Slide Rule with more scales provides functions beyond multiplication and division.

Look at the slide rule above. The C and D scales are the usual scale for multiplication.
the LL3 scale is the exponetial of C-that is, e^x setting the scale on C as x
the LL2 scale is then e^10x
the LL/3 scale e^-x
the LL/2 scale e^-0.1x

How to calculate 1.5^1.2:

1.5^1.2=S
ln(1.5^1.2)=lnS
1.2ln1.5=lnS

S=exp(1.2ln1.5)

So here comes the trick!
define a scale on L3 as L, and that on D as x.
move the cursor to match 1.5 on L3, L=1.5 and apparently x=ln1.5

Now we need to multiply ln1.5 on D by 1.2.
using the previous knowlege, move the middle bar so that the 1.0 on C matches the hairline now, the ln1.5 on D.
move the cursor to let the hairline match 1.2 on C, and now the hairline indicates (ln1.5)1.2 on D.

Also e^[(ln1.5)1.2] on L3! The final result!

This simple problem requires only
1 moving the cursor and match
2 moving the middle bar to match
3 moving the cursor again to match
altogether three movings and matches, skilled engineers do it very quickly, 3 seconds.
Not bad compared to electronic calculators, er?

Moreover it is independent of battery.
And it can help you better understand logs and exponetials, multiplications and divisions, as well as inversions, etc.

Other functions can be simply added on using more scales like L3, L2. And the back of the rule is also made full use of.

Click this sentence to see the table illustrating how many scales one slide rule can have
Here is the museum showing the glorious past of Slide Rules
Some funs even made his own slide rule

You can buy one or make one of your own!!

Last edited by George,Y (2007-05-12 00:11:38)

George,Y · 2007-05-11 22:17:16

Pickett's (do you know HP?) Slide Rule:

The second side of the slide rule has been explained in the previous post. And the first side of the slide rule has these functions:

S=arcsinx when D=x
Surely D=sinS
find "30" on S and discover the corresponding value on D yourself.

T=arctanx when D=x
ST=Tangents or Arc of Small angles (smaller than 6 degrees)

and L=logD notice L scale is equivalenly divided.

Last edited by George,Y (2007-05-12 19:07:03)

George,Y · 2007-05-11 22:21:28

Go to this page and click on the left pink directions to see different brands of slide ruls

George,Y · 2007-05-11 22:23:52

Yes MIF, very interesting!!!

Very challeging as well!!

George,Y · 2007-05-11 23:58:52

A special model for measure conversion:

The middle bar is apparently double informationed and the user has to take it off, flip and snug back if s/he needs a conversion on the other side.

Take the back side for example, notice inches-centimetres match. Note 1inch≈2.54cm. Imagine matching the inch arrow to 1 and you can see that 2.54 is pointed by the centimetres arrow.

Yes, the centimetres arrow is always 2.54 times ahead of the inch arrow. Theoritically it is log2.54 far right of the inch arrow. (Imagine a L scale exists)

This rule, I guess, can beat up computers and calculators in rough business. All you need to do is read, slide it, then read again, without any power lines or misc buttons to activate certain functions.

mikau · 2007-05-12 04:43:35

boy....

you know we take calculators for granted. Its not too long ago they were not available to anyone.

Even creepier, most of maths biggest discoveries were made before their arrival.

Last edited by mikau (2007-05-12 04:44:13)

Identity · 2007-05-12 15:55:33

Man... I gotta say, this slide rule thing is amazing! I wish I had one! How much did you buy yours for??? Do you think they would be allowed in 'non-calculator' tests? MWAHAHAHAHAHAHH!!!!!....

George,Y · 2007-05-12 21:21:46

Identity, well I got my slide rule from my father. For collection purpose I asked him to find the only remaining slide rules in his office and he suceeded.

On buying slide rules, I recommend you to search the internet.
Here are the sites I've found: (click)
Site 1
Site 2

I bet if you bring a slide rule to a physics exam, the old professor will not accuse you of laziness but
instead will praise your maths talent! If s/he ever used a slide rule once.

________________________________________

Fortunately I have finally found out this online simulated slide rule based on a REAL model!
Simulated Pickette's N-600ES LL Speed Rule
And the very comprehansive detailed full manual for it!!
The Manual

Here is a slide rule with measure conversions on its back
And here is the list of all the simulated slide rules provided

The advantages of using a slide rule

First, it is a very educational instrument. Pressing the buttons is foolproof but it deprives
you of the opportunity to learn maths. Calculating with a pencil and a paper may become
exhausting and many times disabled(2.3^1.5?). The slide rule is the means between the
two-you gotta think, but you don't need to think too much on details. To use a slide rule
well, you need to master the properties of logs, expontials, ratios, sin-cos relations, and
inverse of functions, etc. Even if you don't, under curiosity you will discover them one by
one. What the silde rule can train you is the basic properties and concepts of
intermediate-maths (compared to basics like arithmetic and advanced like calculus) and
the need to master these concepts Never Dies out. Moreover, the way it trains you is
enjoyable because the training process is implicit. You don't mean to practice those
properties intentionly like doing some specialized lgAB exercises. Instead, you get pleasure
from applying those properties each time you solve a pratical calculating problem. So you
learn the concepts with fun.

Secondly, it is good for your psychological health to use a brain-needing slide rule in
place of a pressing-buttons calculator. Scientists have discoverd that using the brain
keeps one from losing intelligence through aging. So wanna be smart? Discard the
calculator unless necessary and use a slide rule everyday! Moreover, the slide rule is only
a tool which needs the human's participation while the calculator is a highly-automatic
machine which needs only pressing buttons. Evolutionarily the calculator seems more
advanced, but only in machinary perspective. Take transportation means for example,
people used to walk, then they use bicycles, then motor-cars, and then automatic cars in
the future. The bicycles and the cars are the most successful tools for human for the first
save strenth and the latter merely eliminate the need for physical strength. However, the
car deprives people from physical exercise the way the calculator deprive people of mental
exercise, causing physical unfit. Moerover, the more advanced auto-cars get a psychological and
marketing obstacle according to the engineers working on it. They say that true,
an automatic car can avoid accidents because the computer drives, but humen prefer to be
in control-they just want to decide their own route! Feeling in control is a crutial part of
feeling good. So let the machine rules thing ain't always good for our happiness, as the
film Matrix describes. A slide rule definately can provide you with more sense of
control than a calculator, and a calculator provide you with more than a laser-beam detector
of a counter in super market! Although the productivity may increase due to more application of
machines, the individual's well-being can gain more from simplier tools.

The last additional advantage of using a slide rule is that it can be employed in
circumstances that don't require precision that much. Checking is a good occasion. And
sometimes maths formulas don't rule the real world completely. For example, in biology
field-as long as the calculation always falls short of the reality with a considerable error,
what's the point to get many digits? After all, pressing buttons causes finger sour, doesn't
it?

Last edited by George,Y (2007-05-13 00:53:39)

George,Y · 2007-05-14 01:48:02

Why does a slide rule work?

Normal scale align values proportional to the distance to the origin. Pick up a common axis, any axis, you will find the actual distance
between 0 and 3 is one half of that between 0 and 6 and twice as much as that between 0 and 1.5, in which actual distance means the
absolute distance between two points in the real world, usually measured in cm or inch.

Now look for the "L" scale on any slide rule. Notice the mark of 0, 0.1, 0.2, 0.3,...,1.0 and the distances between any neighboring two.
Yes, equal. The "L" scale behaves like any normal scale or any normal axis- the value increase is just proportional to the actual distance
passed. So they are equivalent in nature- just redifine the distance of 1.5 inches as 1 or something like that.

Then, please recheck the "D" scale and "L" scale together. Notice the "0" on L matches the "1" on D, and "1" on L matches the "10"
or "l" on D. The L scale has a "lgx" on the right and the D scale has a "x" on the right. They show any value on L is the logarithm of
the matched value on D at the base 10. Or put in simple symbols, L=lg(D). Inversely, D=10^L. D is actually 10 raised to power L.

D is usually called as "logarithm scale". It is true that the logarithm of D, L, is linear. But a value on D is exponetial to the actual
distance from the origin, or index, or begining. So perhaps it is wiser to call it as "exponetial". On a normal linear scale, equal
actual distance represents equal difference between the two values on the two ends. However, on a exponetially increasing
scale like D, equal distance represents equal times between the two instead. When L2-L1=L4-L3 on the L, 10[sup]L2[/sup]/10[sup]L1[/sup]=10[sup]L4[/sup]/10[sup]L3[/sup]
on the D. Look and pick up 1, 2, 4 and 8 on the D and see the distances separated by them and it is true. Moving a finger from
1 to 2, then to 4, then to 8, it feels like each time it doubles. This is indeed expontially increasing.

In addition, if one distance is twice as much as another, the multiplier of two values for the former is the square of the latter.
three times, cube ... Every distance represents a multiplier of the value pair, and the ratio between two distance represents
how much the multiplier is raised. The proof is either from the property of equal length, equal multiplier
a,b,c,d |ab|=|bc|=|cd|, b=ka, c=kb, d=kc, so d/a=k³
when |ad|=3|xy| , |xy|=|ab|, thus y/x=k while d/a=k³.

Or from the L scale
L2-L1=b(L4-L3),
10[sup]L2[/sup]/10[sup]L1[/sup]=10[sup](L4-L3)b[/sup]={10[sup]L4[/sup]/10[sup]L3[/sup]}[sup]b[/sup]

The property above is crutial when tracking the increase of sth like stock price logarithmatical scale(exponetial scale).
However, it's not a necessity to understand the slide rule.

When the middle bar slides to the right.
Simulated Pickette's N-600ES LL Speed Rule
supposedly 1 on C matches 2.5 on D.
find somewhere C1 on C and the actual distance between 1 and C1 is exactly the same as that between 2.5 and D1,
right under C1. Since C and D are in fact the same scale. C1/1= D1/2.5 thus D1=2.5(C1).

Last edited by George,Y (2007-05-14 01:51:01)

Anthony.R.Brown · 2007-05-19 01:45:54

Hope you dont Mind George,Y

I have to say that the Abacus! Is an amazing Calculating Tool!

A.R.B

George,Y · 2007-05-20 03:40:11

Yes, amazing. But it need around 100 aditional rules, which are related to the machine itself rather than being universal maths rules, to fully manupulate, whereas a slide rule needs less than 10.

I will show you many other maths magic a slide rule can perform. Wait...

Last edited by George,Y (2007-05-20 21:42:58)

Anthony.R.Brown · 2007-05-21 00:40:02

To George,Y

What puzzles me is how other Counting systems came about! Humans are Born with Ten fingers!
you would have thought this would have been the first counting System!

A.R.B

Last edited by Anthony.R.Brown (2007-05-21 00:40:34)

mathsyperson · 2007-05-21 02:19:54

Bases 2 and 16 are commonly used in computing, because computers can only recognise things as 'on' or 'off'.
Technically, computers can only count in base 2 (0,1,10,11,100,101,110,111,1000...), but Hex is used as well
so that numbers aren't quite as big as they would be otherwise. But each Hex digit is basically just a symbol that
contains information about 4 binary digits.

Sekky · 2007-05-21 03:21:43

Anthony.R.Brown wrote:

What puzzles me is how other Counting systems came about! Humans are Born with Ten fingers!
you would have thought this would have been the first counting System!
A.R.B

It was, sexagesimal wasn't in recorded use until 2000BC, and even sexagesimal is a mixed radix system of decimal and heximal.

Anthony.R.Brown · 2007-05-24 00:11:50

Quote:" It was, sexagesimal wasn't in recorded use until 2000BC, and even sexagesimal is a mixed radix system of decimal and heximal. "

A.R.B

Children counted with their Fingers! long before any known named counting system!..........

Sekky · 2007-05-24 01:32:26

Anthony.R.Brown wrote:

Children counted with their Fingers! long before any known named counting system!..........

Just the children?

(By the way, you just reiterated exactly what I just said, minus the part about the children)

Anthony.R.Brown · 2007-05-25 00:16:09

Quote:" Just the children? "

A.R.B

Just the children? as in Before the Adults! as in FIRST!

George,Y · 2007-05-29 01:45:33

Back to the topic, the slide rule.

From Post 14, we know every distance on a logarithm scale represents a ratio between the two values on each end.

Now look at the slide rule above.
Measure with your eyes the distances on D scale between 1 and 2, between 2 and 4, between 4 and 8-are they the same?
Correct, the same, the reason for this has been proved in Post 14.

Now another trick:
Measure with your eyes the distances on D scale between 1 and 1.5, between 2 and 3, between 4 and 6, between 6 and 9.
Are they the same?

The same, too. This result verify the second sentence in this post again. And, we know a new method to do multiplication.
We don't have to move the index on C above the a and read the number on D under C to get 1.5a.
Instead, we can move 2 on C above a, and read 3/2a out right under 3 on C.
That's of great importance because sometimes we need to calculate 4/3a, etc. 1.33 is not accurate, whereas 3 and 4 are.

Sim Pickette's N-600ES LL Speed Rule
Click above to do some multiplication with a fraction your self.

From the same property we get another clue: the ratio? the ratios?
Yes, between two different numbers we can get two ratios, reciprocal to each other, one larger than 1, one less than 1.
Back to the previous example. Knowing how to evaluate 2/3a?
Simple match 3 on C to a on D, and read the value on D right under 2 on C.
Ratios are less than 1 when distance is from right to left, or more accurately, displacement from right to left.

By this method, can you calculate 2 times 6/10?
Yes, this is the only way to calculate 2 times 6 when you have only C and D.
Just try 2 times 6 in conventional way. You will find 6 on C matches blank or no scale on D.
Actually conventional method inhibits 2 times any number larger than 5 less than 10.
Only by fraction multiplication can you calculte those forbidden numbers.

The last for this post
can you calculate 6/5?
Note: 6/5= 6*(1/5)
can you calculate 6/7?
Note: 6*(10/7)=10(6/7)

George,Y · 2007-05-29 01:49:11

The post above finished all multiplications and divisions by C and D scale.
Though together 5 situations (the first in post 14), you can master them easily as long as you understand thefirst property of
a logarithm scale in Post 14.

George,Y · 2007-05-29 02:23:32

Note some slide rules mark 10 on C and D as "1", at the right end.
It is for convience, telling you that it can be treated as "1" regardless of decimal point.
Indeed, when you match 10 with any number, it means the number divided by 10.

Tips for "off scale" problem of fraction multiplication.

2*7 is off scale, to solve it, we calculate 2*7/10

How about 2*22/3?
Too large.
So 1/10 is needed for calculation.
we just calculate one tenth of it.

you can do 2*2.2/3
or 2*22/10*(1/3) etc.

How about 2*2/9?
This time too small, we calculate ten times of it instead
2*2*(10/9) etc

And there is also a way to begin with a fraction. But this fraction should be larger than 1.

For example, 3/4*9/5
inverse its sequence to let a fraction larger than 1 sit the first
9/5*3/4
OK, now think 9/5 as 1*9/5, how do you do?
match 5 on C to 1(left 1) on D, and 9 on C points out the result on D. Move the cursor to 9 on C, and the
cursor now record exact value of 9/5. Now just multiply this value by 3/4.

Here is a point. The cursor can play as a physical storage for one number, but this number is usuallly
on D scale.

Just multiply 1.4*2.4*2.9
You will notice you don't have to read 1.4*2.4 out or check it, the cursor helps you.
Now swap the roles of C and D.
match 1.4 on C to 1 on D(slide leftwards), the value on C above 2.4 on D is the value of 1.4*2.4
Now what?
If you want to multiply 2.9 then you have to match 1.4*2.4 on C to 1 on D and this time the cursor should
move leftwards. The cursor has to follow the C scale, the sliding part. So it's a huge disadvantage
to get the result on C.

Therefore, let's derive a principle:
Usually get the result on D, so that the cursor can stay on static part of the rule and store the result.
By this principle you can do successive multiplications and divisions such as 1.4*2.4*2.9 etc.

There is exception however, in occasions that the next calculation is on the sliding part of the rule.
Just check the middle bar and the functions on them.
For example, the Log function is on the sliding part.
Find log(5/3)
If we get 5/3 on C it's convenient to read the corresponding log on Log scale.
So this time 1 on C-3 on D, and 5 on D-the result 5/3 on C, move the cursor to the result 5/3 and log(5/3) is
on log scale.
Or,
match 5 on C - 3 on D, 1 on D - 5/3 on C. It means 5 divided by 3 in conventional ways.
But look, 5-3, doesn't it look like
5
3?

Yes, (5/3):1=5:3
or
5/3 = 5
1 3

This means C[sub]1[/sub]/D[sub]1[/sub]=C[sub]2[/sub]/D[sub]2[/sub]
which can be easily derived by C[sub]1[/sub]/C[sub]2[/sub]= D[sub]1[/sub]/D[sub]2[/sub]

Last edited by George,Y (2007-05-30 01:13:11)

Math Is Fun Forum

#1 2007-05-11 19:05:32

The Classic Slide Rule

#2 2007-05-11 19:45:21

Re: The Classic Slide Rule

#3 2007-05-11 20:05:41

Re: The Classic Slide Rule

#4 2007-05-11 21:36:46

Re: The Classic Slide Rule

#5 2007-05-11 22:09:25

Re: The Classic Slide Rule

#6 2007-05-11 22:11:29

Re: The Classic Slide Rule

#7 2007-05-11 22:17:16

Re: The Classic Slide Rule

#8 2007-05-11 22:21:28

Re: The Classic Slide Rule

#9 2007-05-11 22:23:52

Re: The Classic Slide Rule

#10 2007-05-11 23:58:52

Re: The Classic Slide Rule

#11 2007-05-12 04:43:35

Re: The Classic Slide Rule

#12 2007-05-12 15:55:33

Re: The Classic Slide Rule

#13 2007-05-12 21:21:46

Re: The Classic Slide Rule

#14 2007-05-14 01:48:02

Re: The Classic Slide Rule

#15 2007-05-19 01:45:54

Re: The Classic Slide Rule

#16 2007-05-20 03:40:11

Re: The Classic Slide Rule

#17 2007-05-21 00:40:02

Re: The Classic Slide Rule

#18 2007-05-21 02:19:54

Re: The Classic Slide Rule

#19 2007-05-21 03:21:43

Re: The Classic Slide Rule

#20 2007-05-24 00:11:50

Re: The Classic Slide Rule

#21 2007-05-24 01:32:26

Re: The Classic Slide Rule

#22 2007-05-25 00:16:09

Re: The Classic Slide Rule

#23 2007-05-29 01:45:33

Re: The Classic Slide Rule

#24 2007-05-29 01:49:11

Re: The Classic Slide Rule

#25 2007-05-29 02:23:32

Re: The Classic Slide Rule

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