Slide rule anyone?

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MP173
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Slide rule anyone?

Post by MP173 » Fri Nov 23, 2012 9:02 am

A couple of slide rules are in my desk and are used from time to time, primarily for mental exercize.

Anyone else out there in Bogelheadland use slide rules? It is a great way to combine mental math with an early calculator. Back in the day ( early 70's) I used it for basic multiplication and division, primarily to check my work on long hand calculations. Now, it is interesting to review exponential and logarithems. I never really understood logs except it made x and / much easier on complex problems.

Anyone know of great sites for self help on slide rules?

Ed

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BL
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Re: Slide rule anyone?

Post by BL » Fri Nov 23, 2012 9:17 am

I did a Google search and came up with this site which looks helpful or at least interesting. I plan to check it out more later.
http://sliderulemuseum.com/SR_Course.htm

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Re: Slide rule anyone?

Post by tomd37 » Fri Nov 23, 2012 10:54 am

I have a K & E slide rule sitting on the shelf above my computer that was used by both my father and older brother in their engineering studies at Lehigh University. My Dad graduated in 1920 and my brother in 1952. My memory is not too good, but I seem to remember using or being introduced to slide rule use in high school in the early 1950s.
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Re: Slide rule anyone?

Post by MP173 » Fri Nov 23, 2012 10:59 am

Lehigh is a great school. My wife was born in Bethlehem and her uncle graduated from LU. We live in NW Indiana (her family were part of the Bethlehem - Burns Harbor migration in the 60's) and return each year. We drove thru the campus last summer.

I would like to use the SR for finanical calculations...shouldnt be too hard, just need to familiarize myself with the formulas. The issue now is the review of math from 40 years ago. My son is taking calculus in HS and just rolls his eyes when I pull the SR out.

Ed

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Re: Slide rule anyone?

Post by nisiprius » Fri Nov 23, 2012 11:18 am

MP173 wrote:Anyone else out there in Bogelheadland use slide rules?
John C. Bogle does.
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Re: Slide rule anyone?

Post by whr19606 » Fri Nov 23, 2012 11:33 am

I managed to graduate as an engineer from Drexel University in 1965, using a $2.00 plastic circular slide rule. It was about 4 inches in diameter, and was quite the novelty. I told anyone who asked that I used it because I never 'ran off the ends' or had to use the 'folded scales'. It cracked when I was a Senior and I discarded it. Wish I still had it. Finished senior year using a 'Post' 6 inch rule that got me thru finals. Still have that one, but haven't used it since.

I also remember having to take a class in Bufftran, (I think that followed Fortran) and using punched cards (Hollerith code?) to load my program into the university's IBM 1620(?), which took up a significant portion of the basement in one of the science buildings.

Ah, nostalgia. It isn't what it used to be.

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Re: Slide rule anyone?

Post by Valuethinker » Fri Nov 23, 2012 11:46 am

I have just inherited my father's.

I like to tease people that they designed nuclear reactors with these things-- which they did, and those reactors are still sitting on the shores of the Great Lakes, generating power. A tribute to engineering 'margin of error' ;-).

I no longer know how to use one, but each time I pick it up I think of him.

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Re: Slide rule anyone?

Post by texasdiver » Fri Nov 23, 2012 11:53 am

My dad is a retired science teacher in his mid-70s. He collects them. He has some incredibly beautiful ones made of things like ivory and brass. Apparently there is now a collector's market for these things. I'm too young to know anything about them much less how to use them. Although I do remember the days that math and science books had trig and log tables in the back so you could do things like look up the cosine of 22 degrees instead of just typing it into the calculator. Now days even calculators are going obsolete as every smart phone has a scientific calculator included. The main reason why calculators are still sold for school use is because most schools don't let kids have their phones out during class and certainly don't let them out during testing.

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Re: Slide rule anyone?

Post by Default User BR » Fri Nov 23, 2012 11:55 am

I finished high school in the 70s, so they still taught us slide rules and I had one. However, in college the first affordable calculators were available, so I never really used one.


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Re: Slide rule anyone?

Post by wilpat » Fri Nov 23, 2012 11:58 am

We called them a "Slipstick". I used them in the 50's and 60's, but then came the HP Model-35 calculator. Essentially an electronic "Slipstick". The Model-35 cost $395.00 in 1972 when it first came out. A unit comparable to the Model-35 (that does about 1000 more things) today costs about $4.00!
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Re: Slide rule anyone?

Post by Sam I Am » Fri Nov 23, 2012 12:10 pm

Message deleted.
Last edited by Sam I Am on Mon Oct 07, 2013 12:58 pm, edited 1 time in total.

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Re: Slide rule anyone?

Post by sscritic » Fri Nov 23, 2012 12:11 pm

My wife taught the use of a slide rule during a summer orientation program for incoming college freshmen in the early 70s.

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Re: Slide rule anyone?

Post by investor » Fri Nov 23, 2012 12:13 pm

Texas Instruments soon came out with the SR-50 & SR-51 that were much cheaper than the HP-35. Might I also remind folks that we put a Man on the Moon using slide rules. I have a Pickett sitting in a desk drawer and long ago forget how to use it.

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Re: Slide rule anyone?

Post by Indicing » Fri Nov 23, 2012 12:15 pm

Even SRs were luxury items for us when I grew up. We used logarithmic and trigonometric tables at the end of math books.

I never had a SR and would really like to have one now, but couldn't find where to buy it (without paying outrageous antique price).

The basic principle of SR is actually very easy: Two ordinary rulers can be used to do addition. For example, to do 3.2+5.6, we line the 0 on the 2nd ruler with 3.2 on the 1st ruler, then read on the 1st ruler the marking lined up with 5.6 on the 2nd ruler.
Taking advantage of the properties of logarithms, we can then do multiplications when we scale 1, 10, 100, 1000, ... (instead of 0, 1, 2, 3, 4, ...) with equal spacing on the two rulers. You can see visual illustrations of these at
http://www.jamestanton.com/wp-content/u ... ocfile.pdf

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Re: Slide rule anyone?

Post by NAVigator » Fri Nov 23, 2012 12:15 pm

Here is an interesting thread about slide rules; Anybody remember slide rules?

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Re: Slide rule anyone?

Post by cheese_breath » Fri Nov 23, 2012 12:23 pm

I still have my old slide rules from my high school physics (1957) and college physics (1960) classes stuffed away in the bottom drawer of my desk.
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Re: Slide rule anyone?

Post by cheese_breath » Fri Nov 23, 2012 12:36 pm

whr19606 wrote:I also remember having to take a class in Bufftran, (I think that followed Fortran) and using punched cards (Hollerith code?) to load my program into the university's IBM 1620(?), which took up a significant portion of the basement in one of the science buildings.
I remember the 1620 well. I did my college senior thesis on a 1620 in 1964. The college didn't have a computer yet, but a few of us had permission to use the 1620 at the Leonard refinery in town after the working folks went home for the night. Also, my second job after college was developing systems on a 1620. (First job used a 1401). The 1620 plus its peripherals took up a consderable amount of space but not nearly as much as some of the other computers of the day.
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Re: Slide rule anyone?

Post by sport » Fri Nov 23, 2012 12:40 pm

The biggest problem with slide rules is that the answers are, while correct, only approximate. On a standard 10-inch slide rule, you can only get 3 signficant digits. For those who are non-technical, those are the first 3 digits of a number. So you can get an answer like 4170 or 4180, but you cannot get an answer like 4177. This level of precision is suitable for some applications, but not for others.

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Re: Slide rule anyone?

Post by LadyGeek » Fri Nov 23, 2012 12:45 pm

NAVigator wrote:Here is an interesting thread about slide rules; Anybody remember slide rules?

Jerry
Sorry, that thread doesn't fit under the current forum guidelines. See: Forum Posting Guidelines - updated 2/27/12

Neither does this thread - it's a general comment with no personal consumer benefit. This topic is locked. See: Forum Policy
Note that topics must be directly connected to your (or your friend's or family's) life as a consumer. General comments or complaints about these topics will be removed.

Note that this subforum has a much lower threshold for locking or removing posts than the financial and investing subforums. ...
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Re: Slide rule anyone?

Post by LadyGeek » Fri Nov 23, 2012 2:59 pm

After receiving a PM, I unlocked the thread.
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Re: Slide rule anyone?

Post by MP173 » Fri Nov 23, 2012 3:01 pm

LadyGeek:

You are a reasonable person.

Ed

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Re: Slide rule anyone?

Post by nisiprius » Fri Nov 23, 2012 9:29 pm

So, about the appropriate investment of time...

Now, listen my friend, I consider that the hours I spent with a slide rule in my hand were golden. Help ya cultivate horse sense, and a cool head, and a keen eye. Didja ever try to keep an exponent in your head with the slide projectin' to th' right on an inverted folded scale? There's nothin' like a K&E log log decitrig decitrig decitrig. Man-TISSA!

But just as I say it takes judgement, brains and maturity t' deal with a slug and a poundal, I say that any boob can mash a button on a calculator. And I call that sloth, my friends, the first big step on th' road to the depths of degrada--th' next thing you know they've got one that graphs. Graphs, my friend, right there on the screen! Now how ya gonna teach 'em, babes I tell ya, how ya gonna teach 'em to keep their wrist on the inside of the curve?

And the next thing ya' know they're Googling for answers, comin' home from school braggin' to friends about Wikipedia. Not a wholesome hundred-and-thirty-pound set you c'n press cucumbers with, oh no, but a bunch of bits and bytes any fool can edit. Make your blood boil, don't it?

Y'got trouble, my friends, right here in River City, with a capital T and that rhymes with T and that stands for TI!
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Re: Slide rule anyone?

Post by pascalwager » Fri Nov 23, 2012 9:42 pm

Before lost in a fire, I had a six-inch and full-size K&E and a titanium circular Pickett that fell off of a camel train in the Spanish Sahara.
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Re: Slide rule anyone?

Post by sport » Fri Nov 23, 2012 9:46 pm

nisiprius wrote:So, about the appropriate investment of time...

Now, listen my friend, I consider that the hours I spent with a slide rule in my hand were golden. Help ya cultivate horse sense, and a cool head, and a keen eye. Didja ever try to keep an exponent in your head with the slide projectin' to th' right on an inverted folded scale? There's nothin' like a K&E log log decitrig decitrig decitrig. Man-TISSA!

But just as I say it takes judgement, brains and maturity t' deal with a slug and a poundal, I say that any boob can mash a button on a calculator. And I call that sloth, my friends, the first big step on th' road to the depths of degrada--th' next thing you know they've got one that graphs. Graphs, my friend, right there on the screen! Now how ya gonna teach 'em, babes I tell ya, how ya gonna teach 'em to keep their wrist on the inside of the curve?

And the next thing ya' know they're Googling for answers, comin' home from school braggin' to friends about Wikipedia. Not a wholesome hundred-and-thirty-pound set you c'n press cucumbers with, oh no, but a bunch of bits and bytes any fool can edit. Make your blood boil, don't it?

Y'got trouble, my friends, right here in River City, with a capital T and that rhymes with T and that stands for TI!
Bravo, Nisi :sharebeer

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Re: Slide rule anyone?

Post by jwvanhoven » Fri Nov 23, 2012 10:32 pm

How about 5 place log tables when 2 to 3 figure accuracy was not good enough? What a pain.

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Re: Slide rule anyone?

Post by sport » Fri Nov 23, 2012 10:54 pm

jwvanhoven wrote:How about 5 place log tables when 2 to 3 figure accuracy was not good enough? What a pain.
I have an old German book of 20 place logarithms. Unbelievable. I never tried to use it. It's an interesting curiousity.
Jeff

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Re: Slide rule anyone?

Post by CABob » Fri Nov 23, 2012 11:13 pm

Y'got trouble, my friends, right here in River City, with a capital T and that rhymes with T and that stands for TI!
Love it, Nisi!
You folks are bringing back some memories. I recall an aluminum Pickett, a cheap plastic one, a 6" K&E bamboo (that was my uncle's who graduated college in 1912) and even a circlular one that I never really mastered.
For an online slide rule see http://www.antiquark.com/sliderule/sim/ ... -rule.html
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Re: Slide rule anyone?

Post by 555 » Sat Nov 24, 2012 12:33 am

jsl11 wrote:
jwvanhoven wrote:"How about 5 place log tables when 2 to 3 figure accuracy was not good enough? What a pain."
"I have an old German book of 20 place logarithms. Unbelievable. I never tried to use it. It's an interesting curiousity."
How many digits are the numbers they're taking the logs of?

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Re: Slide rule anyone?

Post by sport » Sat Nov 24, 2012 2:07 am

555 wrote:
jsl11 wrote:
jwvanhoven wrote:"How about 5 place log tables when 2 to 3 figure accuracy was not good enough? What a pain."
"I have an old German book of 20 place logarithms. Unbelievable. I never tried to use it. It's an interesting curiousity."
How many digits are the numbers they're taking the logs of?
Four digits. The method of using it appears to be complex. However, all the instructions are in German which I do not read. The book was published in 1880, and there is a note by the apparently original owner that it cost $1.80 in 1904.
Jeff

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Re: Slide rule anyone?

Post by Taboose » Sat Nov 24, 2012 12:28 pm

Four or five years ago I was watching a news program and the interviewer was talking to an artillary soldier (Marine maybe) in Afghanistan. He asked what was in the soldier's pocket. It was a slide rule. Why did he use that? Because it works every time when needed. Backup to the computers. Smart young man.

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Re: Slide rule anyone?

Post by VictoriaF » Sat Nov 24, 2012 12:39 pm

Taboose wrote:Four or five years ago I was watching a news program and the interviewer was talking to an artillary soldier (Marine maybe) in Afghanistan. He asked what was in the soldier's pocket. It was a slide rule. Why did he use that? Because it works every time when needed. Backup to the computers. Smart young man.
I can do most additions in my head. For other arithmetic, all I need is a piece of paper, which is lighter than a slide rule.

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Re: Slide rule anyone?

Post by nisiprius » Sat Nov 24, 2012 12:56 pm

Story told my my math teacher as true. College physics lectures, which the professor was frequently doing calculations with the aid of the standard three-place log table.

The professor would announce, "One point oh five four times ten to the eighth," and a guy with a six-inch slide rule at the back of the room would then call out "One point oh five four TWO." For every calculation, the professor would announce the answer, and then the guy with the six-inch slide rule would announce it to one more decimal place.

The puzzle, for those not old enough to see it, is that a six-inch slide rule isn't even accurate to three decimal places.

It turned out that what he was doing was using the log table, along with the professor, and then using his six-inch slide rule to perform the interpolation to get one more place.
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Re: Slide rule anyone?

Post by stratton » Sat Nov 24, 2012 1:08 pm

I was a HS freshman when this came out.

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Re: Slide rule anyone?

Post by Aptenodytes » Sat Nov 24, 2012 1:39 pm

I graduated from High School in 1977, in the hometown of Texas Instruments. Everyone wanted a TI calculator, and the thing to do was get someone's dad who worked there (I only recall dads working there) to get one for you at an employee discount. Getting a TI calculator was an act of local patriotism, a step forward on the march to a better tomorrow, as well an easier way to do math problems.

In spite of all that, my high school chemistry teacher required that we do all our calculations with a slide rule. All labs, homework, tests had to be done using only the slide rule. Even the national standardized tests that affected college placement.

We joked about it and complained about it, but I am really glad she did that. It gave me a tangible connection to a long line of historical practice that was dying out. It got me to think about math in a more visceral way -- the different scales used for different calculations helped reinforce the underlying principles. And -- this is the most lasting effect -- it forced me to think about significant digits. Whereas a calculator would spit out as many digits as the display could hold, and most people who grew up in a calculator/spreadsheet world treat them as equally significant, part of the discipline of using the slide rule was figuring out in your own head how many digits were significant and reporting only those.

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Re: Slide rule anyone?

Post by walnut » Sat Nov 24, 2012 2:19 pm

tomd37 wrote:I have a K & E slide rule sitting on the shelf above my computer that was used by both my father and older brother in their engineering studies at Lehigh University. My Dad graduated in 1920 and my brother in 1952. My memory is not too good, but I seem to remember using or being introduced to slide rule use in high school in the early 1950s.
MP173 wrote:Lehigh is a great school. My wife was born in Bethlehem and her uncle graduated from LU. We live in NW Indiana (her family were part of the Bethlehem - Burns Harbor migration in the 60's) and return each year. We drove thru the campus last summer.

I would like to use the SR for finanical calculations...shouldnt be too hard, just need to familiarize myself with the formulas. The issue now is the review of math from 40 years ago. My son is taking calculus in HS and just rolls his eyes when I pull the SR out.

Ed
I too am a Lehigh Engineer (not a Mountain Hawk) and my K&E log-log duplex decitrig hangs over my desk. Cost $18.50 plus $.50 to get engraved with full name by itinerant engraver at Lambert Hall dining facility. Carrying case discarded immediately as only EE's used them to carry it on their belt. Used daily for 18 years and still ocassionaly for a trig or exponential calculation. For those interested and in London there is a large exhibit of SRs at the Science Museum.

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Re: Slide rule anyone?

Post by Epsilon Delta » Sat Nov 24, 2012 2:56 pm

My five figure log table is sealed in a case labeled "Break glass in case of Y2K".

The "glass" is actually poly(methyl methacrylate) so accessing the table would take significant effort.

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Re: Slide rule anyone?

Post by 555 » Sat Nov 24, 2012 5:07 pm

jsl11 wrote:
555 wrote:
jsl11 wrote:
jwvanhoven wrote:"How about 5 place log tables when 2 to 3 figure accuracy was not good enough? What a pain."
"I have an old German book of 20 place logarithms. Unbelievable. I never tried to use it. It's an interesting curiousity."
How many digits are the numbers they're taking the logs of?
"Four digits. The method of using it appears to be complex. However, all the instructions are in German which I do not read. The book was published in 1880, and there is a note by the apparently original owner that it cost $1.80 in 1904."
Yeah, I figured the input couldn't have too many digits. If a book has 10^n entries, n can't be too big. And if you have x to 4 digits, I find it very hard to believe that that you could ever need log x to 20 digits.

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Re: Slide rule anyone?

Post by nisiprius » Sat Nov 24, 2012 6:37 pm

555 wrote:
jsl11 wrote:
555 wrote:
jsl11 wrote:
jwvanhoven wrote:"How about 5 place log tables when 2 to 3 figure accuracy was not good enough? What a pain."
"I have an old German book of 20 place logarithms. Unbelievable. I never tried to use it. It's an interesting curiousity."
How many digits are the numbers they're taking the logs of?
"Four digits. The method of using it appears to be complex.
Quite possibly the idea is to do an interpolation. I'm way too lazy to do the math but it wouldn't surprise me if linear interpolation between twenty-place logarithms with the mantissa values at four-place intervals might be good to twenty places, and if not maybe the "complex method" involved quadratic interpolation.

Hey, what the heck, let's try the experiment.

Log10(5.000) = 0.698970004336019
Log10(5.001) = 0.699056854547668
Midpoint of the logarithms = 0.69901342944185
Logarithm of the midpoint = 0.699013431612882

So linear interpolation in a four-place table whose values are given to twenty places should be good to about 7-8 places, anyway.
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Re: Slide rule anyone?

Post by Bonnan » Sat Nov 24, 2012 11:03 pm

Ah! the old K&E slide rule; graduated from Univ Cincinnati " 1958. Standard math tool for engineers; we M.E.'s didn't carry them on our belt either. As Nisi says made you think.
Worked for a company that designed/built equipment to put a man on the moon, nuclear containment vessels, St.Louis Arch etc. all with a slide rule.
We young engineers also stood in line to use a Monroe calculator (company only had one)that could do square roots to more significant figures. Hard to believe how the world has changed!

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Re: Slide rule anyone?

Post by MP173 » Sun Nov 25, 2012 11:47 am

Enjoying the comments about SRs and their applications. I have been using an online site - Ron Manley's Slide Rule for some advanced instruction.

Here is my question to all the math folks out there....I am looking for either a good book or an online site which explains certain math properties and functions, primarily for everyday use. I am NOT an engineer, but would like to understand how to use the SR for math applications. Without wading thru years of math, I am looking for everyday applications including formulas, logs, exponential use, etc.

Recommendations?

My background...late 50's, four year college degree with last class Trig (no Calculus) with a basic understanding of math functions.

Ed

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Re: Slide rule anyone?

Post by 555 » Sun Nov 25, 2012 12:12 pm

nisiprius wrote:Quite possibly the idea is to do an interpolation. I'm way too lazy to do the math but it wouldn't surprise me if linear interpolation between twenty-place logarithms with the mantissa values at four-place intervals might be good to twenty places, and if not maybe the "complex method" involved quadratic interpolation.

Hey, what the heck, let's try the experiment.

Log10(5.000) = 0.698970004336019
Log10(5.001) = 0.699056854547668
Midpoint of the logarithms = 0.69901342944185
Logarithm of the midpoint = 0.699013431612882

So linear interpolation in a four-place table whose values are given to twenty places should be good to about 7-8 places, anyway.
Oh, interpolation. I remember log tables (but not slide rules) as a kid, but I didn't remember about interpolation. You probably have an explanation there.

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Re: Slide rule anyone?

Post by nisiprius » Sun Nov 25, 2012 12:50 pm

What I'd like to know is where the sacred Table of Three-Place Logarithms came from.

It was always the same--same layout, same format, pretty sure it was the same typeface, and never a copyright notice or source. They'd hand them out when you took exams in many math or physics courses. It wasn't from the sacred CRC Book because that contained a forty-or-fifty-page section of four- or five-place logarithms. If I recall correctly, the standard three-place logarithm table was a single 8-1/2x11" sheet with logarithms and antilogarithms on one side, and trig functions on the other.

Parenthetically, can you imagine calculating a book-length table of logarithms by hand, without so much as a MonroeMatic calculator to help you?
walnut wrote:Cost $18.50 plus $.50 to get engraved with full name by itinerant engraver at Lambert Hall dining facility.
I wonder if that was the same guy, or if there were many of them? Was the engraving device powered by #6 ignition cells? Did it involve a metal point that scanned across metal type, closing a circuit whenever the point touched the raised outline of the letter, and powering a solenoid that drove a sharp-pointed scribing device into the slide rule? A physical raster-scan? With the size of the type reduced by the ratio of the scanning movements at source and target?
Last edited by nisiprius on Sun Nov 25, 2012 4:30 pm, edited 1 time in total.
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walnut
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Re: Slide rule anyone?

Post by walnut » Sun Nov 25, 2012 2:22 pm

nisiprius wrote:I wonder if that was the same guy, or if there were many of them? Was the engraving device powered by #6 ignition cells? Did it involve a metal point that scanned across metal type, closing a circuit whenever the point touched the raised outline of the letter, and powering a solenoid that drove a sharp-pointed scribing device into the slide rule? A physical raster-scan? With the size of the type reduced by the ratio of the scanning movements at source and target?
Pretty sure it was the same guy. Have met others from different schools with same looking ID on rule. I do remember the dry cells but not the mechanism. When finished he rubbed a red chalk or crayon into the letters. Still there along with a couple of burns courtesy of Camel cigarettes (also responsible for setting carbon paper on fire in wastebasket about once a week).
[Fixed quote --admin LadyGeek]

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Re: Slide rule anyone?

Post by 555 » Sun Nov 25, 2012 4:23 pm

walnut wrote:
nisiprius wrote:I wonder if that was the same guy, or if there were many of them? Was the engraving device powered by #6 ignition cells? Did it involve a metal point that scanned across metal type, closing a circuit whenever the point touched the raised outline of the letter, and powering a solenoid that drove a sharp-pointed scribing device into the slide rule? A physical raster-scan? With the size of the type reduced by the ratio of the scanning movements at source and target?
Pretty sure it was the same guy. Have met others from different schools with same looking ID on rule. I do remember the dry cells but not the mechanism. When finished he rubbed a red chalk or crayon into the letters. Still there along with a couple of burns courtesy of Camel cigarettes (also responsible for setting carbon paper on fire in wastebasket about once a week).
I didn't say what you quoted. How did my "name" get there.
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Re: Slide rule anyone?

Post by Nestegg_User » Sun Nov 25, 2012 10:01 pm

Anyone need my old Sterling 594? (two sided dec trig log-log-- includes 121 page instructions) or my Concise No. 320 circular?

Needed it in the mid-70's physics classes : 4 bangers were just coming out...but still expensive. I waited until later --- SR52 !!

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Re: Slide rule anyone?

Post by grabiner » Sun Nov 25, 2012 11:22 pm

555 wrote:
nisiprius wrote:Quite possibly the idea is to do an interpolation. I'm way too lazy to do the math but it wouldn't surprise me if linear interpolation between twenty-place logarithms with the mantissa values at four-place intervals might be good to twenty places, and if not maybe the "complex method" involved quadratic interpolation.

Hey, what the heck, let's try the experiment.

Log10(5.000) = 0.698970004336019
Log10(5.001) = 0.699056854547668
Midpoint of the logarithms = 0.69901342944185
Logarithm of the midpoint = 0.699013431612882

So linear interpolation in a four-place table whose values are given to twenty places should be good to about 7-8 places, anyway.
Oh, interpolation. I remember log tables (but not slide rules) as a kid, but I didn't remember about interpolation. You probably have an explanation there.
Interpolation is the process of estimating a value of a function based on nearby values. Even with computers, interpolation is still a standard technique in numerical analysis. You don't need interpolation to compute logarithms, sines, square roots, or similar functions for which you have a formula. But if you have to do a complicated calculation for each value of a function and you need to compute it for many different values, you might make a table of, say, 1000 values (from 0 to 1 by .001, or from 0 to 1000 by 1), and then use interpolation from the table to get a numerical estimate.

Interpolation can be done with any number of terms. Linear interpolation is most common: you assume the graph from 5.000 to 5.001 is a straight line. But you could also do a quadratic interpolation: the graph from 4.999 to 5.000 to 5.001 must pass through all three points, and you find the quadratic (a parabola) which passes through all three, then use that to estimate values near 5.000; this is usually more accurate. Likewise, you could do a cubic interpolation with four values.

For those familiar with algebra: Given f(a), f(b), f(c), and f(d), the formula for a cubic interpolation is:

f(x)=
f(a)(x-b)(x-c)(x-d)/(a-b)(a-c)(a-d) +
f(b)(x-a)(x-c)(x-d)/(b-a)(b-c)(b-d) +
f(c)(x-a)(x-b)(x-d)/(c-a)(c-b)(c-d) +
f(d)(x-a)(x-b)(x-c)/(d-a)(d-b)(d-c)

If you plug in x=a, only the first term is nonzero, and it simplifies to f(a); similarly for the other three values. Thus you have a cubic polynomial which agrees with f at a, b, c, and d, and should thus be very close to f for values of x between b and c.
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Re: Slide rule anyone?

Post by gatorman » Mon Nov 26, 2012 6:13 am

grabiner wrote:
555 wrote:
nisiprius wrote:Quite possibly the idea is to do an interpolation. I'm way too lazy to do the math but it wouldn't surprise me if linear interpolation between twenty-place logarithms with the mantissa values at four-place intervals might be good to twenty places, and if not maybe the "complex method" involved quadratic interpolation.

Hey, what the heck, let's try the experiment.

Log10(5.000) = 0.698970004336019
Log10(5.001) = 0.699056854547668
Midpoint of the logarithms = 0.69901342944185
Logarithm of the midpoint = 0.699013431612882

So linear interpolation in a four-place table whose values are given to twenty places should be good to about 7-8 places, anyway.
Oh, interpolation. I remember log tables (but not slide rules) as a kid, but I didn't remember about interpolation. You probably have an explanation there.
Interpolation is the process of estimating a value of a function based on nearby values. Even with computers, interpolation is still a standard technique in numerical analysis. You don't need interpolation to compute logarithms, sines, square roots, or similar functions for which you have a formula. But if you have to do a complicated calculation for each value of a function and you need to compute it for many different values, you might make a table of, say, 1000 values (from 0 to 1 by .001, or from 0 to 1000 by 1), and then use interpolation from the table to get a numerical estimate.

Interpolation can be done with any number of terms. Linear interpolation is most common: you assume the graph from 5.000 to 5.001 is a straight line. But you could also do a quadratic interpolation: the graph from 4.999 to 5.000 to 5.001 must pass through all three points, and you find the quadratic (a parabola) which passes through all three, then use that to estimate values near 5.000; this is usually more accurate. Likewise, you could do a cubic interpolation with four values.

For those familiar with algebra: Given f(a), f(b), f(c), and f(d), the formula for a cubic interpolation is:

f(x)=
f(a)(x-b)(x-c)(x-d)/(a-b)(a-c)(a-d) +
f(b)(x-a)(x-c)(x-d)/(b-a)(b-c)(b-d) +
f(c)(x-a)(x-b)(x-d)/(c-a)(c-b)(c-d) +
f(d)(x-a)(x-b)(x-c)/(d-a)(d-b)(d-c)

If you plug in x=a, only the first term is nonzero, and it simplifies to f(a); similarly for the other three values. Thus you have a cubic polynomial which agrees with f at a, b, c, and d, and should thus be very close to f for values of x between b and c.
Thank you, that was interesting.
gatorman

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Re: Slide rule anyone?

Post by nisiprius » Mon Nov 26, 2012 11:29 am

A standard table of logarithms has a little table in a narrow column on the right called "Proportional Parts." These are a precalculated interpolation table so that you can get one more decimal place than is tabulated. It relies on the fact that successive differences from one entry to the next within the table stay the same for many lines of the table; for example, if I look up log sin 21°, 21° 1', 21° 2', 21° 3', etc. I see 9.55433, 9.55466, 9.55499, 9.55532, etc. The difference between one entry and the next are all 0.00033. So I look at the right where there's a column labeled "P. P." and under the heading "33" I find
1 0.6
2 1.1
3 1.6

etc. So if I need to know log sin 21°1.3' I take the value for 21° 1' and add 1.6 (right-aligned to the edge of the digits), and I suppose round--it's not anything I really did very much--and say "9.55466 + .000016 = 9.554676 rounds to 9.55468"*

Even in the 1950s and 1960s, there were a lot of humble things one did to simplify calculations. There were IBM 1620s and such around, but they weren't for everyone. In 1975 a big-university Zoology department had exactly one Monroe rotary calculator for use by grad students, and we thought that was fairly decent of them. It was a very fancy model--it actually had a "memory" feature that would save and restore a single number. And because it was so fancy, it was constantly out of order and being repaired.

One of the more interesting things I ran across, given that the purpose of logarithms is to reduce multiplication to addition, was a book with a fifty-or-so-page "Table of Half Squares" in it. Why would you want that? You want to multiply a x b? You add them, a + b, look up the half square, then you subtract them, a - b, look up the half square, then you subtract the half-squares. See, it's just one addition and two subtractions instead of a multiplication.

It is quite a different world when multiplication and division aren't effortless, instantaneous, and free everywhere for everyone all the time.

*I do believe I screwed up. Did anybody notice? I'm staring that that antique CRC book, and do believe I looked up the proportional parts for 3 seconds of arc, not 0.3 minutes, and that 9.55468 is log sin 21° 1' 3''. And that actually they seem to be adding 10 to the logarithms for some good reason I no longer know.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

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Epsilon Delta
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Re: Slide rule anyone?

Post by Epsilon Delta » Mon Nov 26, 2012 12:30 pm

nisiprius wrote:Parenthetically, can you imagine calculating a book-length table of logarithms by hand, without so much as a MonroeMatic calculator to help you?
The general approach is:
1) Some high grade mathematician works out an infinite series that approximates the function.
2) A mid-grade mathematician generates the co-efficients for a number of finite polynomials that approximate the function with sufficient accuracy over intervals that cover the range needed.
3) A calculator (it was a job title) evaluates the polynomials for all values using only addition/subtraction.
4) Typesetter takes output and sets type and prints tables.
5) Captain runs ship aground because one of the above messed up.

This is why they invented difference engines. These avoid errors at step 3 and by automatically typesetting the results avoid errors at step 4. Furthermore because they eliminate some of the premium on short polynomials they help the first two steps as well.

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Re: Slide rule anyone?

Post by Spirit Rider » Mon Nov 26, 2012 12:38 pm

Ah, the juxtaposition between slide rules and calculators. Reminds me of a time long ago and a place far far away. Mid-seventies, "A" and "C" schools. Slide rules and newly released TI SR-50 red LED calculator (bought one for 50% of my monthly military pay).

Deployment to a body of water were we did not exist and there were no sharks :wink: Computational failure of both primary and secondary navigation systems during blind hi-speed transit. In danger of "buying" a "piece of the rock", and I don't mean Prudential. My new calculator failed, senior SINS tech did continual complex computations from raw data using a military issue circular slide rule.

That was 35 years ago. Imagine how much more dependent we are on technology. Who among the general population could survive without modern technology. I still have that circular slide rule. I haven't touched it in over thrity years. Might be time to break it out and exercise the mind.

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