Palmer's Computing Scale

Aaron Palmer patented a slide rule in Boston in 1843. I believe this is the earliest example of an American slide rule.  The "Computing Scale", as he referred to it was available in a number of formats.  In 1846 Palmer apparently sold his patent rights to John E. Fuller. Fuller, in turn, added another circular scale to the original Palmer Computer Scale on the reverse side which would compute the number of days between two dates. Fuller called his scale the "Time Telegraph".

Major variations of Palmer's Computing Scale include:

1.  A 12 inch square hard cardboard device (as shown).

2. A 12 inch square version, as above, with the the Fuller Time Telegraph on the reverse side.

3. Palmer's Pocket Scale, a 10 cm. by 15.5 cm. hardbound book with a smaller version of the 12 inch device attached to the inside rear cover. 48 pp. not including the scale.

4. Key to Palmer's Pocket Scale, the same as 3. above without the slide rule attached to the inside rear cover. This was presumably used as the instruction manual for 1. above. 50pp.

5. Improvement to Palmer's Endless Self-Computing Scale and Key by John E. Fuller. This is like 4. above, but for use with 2. above. 72 pp.

6. Fuller's Computing Telegraph by Aaron Palmer and John E. Fuller, same as 2. above as part of a 22 page book.

I saw one reference to the fact that the use of the word "computing" as associated with the Palmer Scale was the first time that that word was used when referencing a device, as opposed to a person who did that operation.  I cannot confirm this.

Most references to these Palmer devices indicate that it is rare, and the prices they are sold for seem to support this theory.  However, I personally have 12 of them. I did not pay anywhere near the prices I have seen out there.  So, I don't know what to make of this.

Once I bought one of the smaller Palmer Scales on EBay from a fellow in San Francisco. I was in Connecticut at the time and happened to be going to San Francisco about two days after the close of the auction. I asked the fellow if I could come to him in San Francisco to pick it up instead of him mailing it to me in Connecticut. I think I must have spooked him. He refused to agree and insisted that he mail it to me.

Florian Cajori

Florian Cajori (1859-1930) A Swiss American scholar who was one of the most important and prolific writers on the history of Mathematics and Physics during the latter part of the 19th century and earlier part of the 20th century. Born in Switzerland, he emigrated to the United States in 1875.  He received a Ph.D. from Tulane University in Mathematics. He was a Professor of Physics, Engineering and Mathematics at Colorado College (1889-1918) and then a Professor of Mathematics at the University of California at Berkeley (1918-1930) and held a chair there in the History of Mathematics.

I spoke with Charles Cajori (b. 1921), his son, a number of years ago. He lives in Watertown near our home in Connecticut.  Charles is an artist of some note.  We spoke about his father and the fact that his papers are housed at the U.C. Berkeley Library. Now that we live much closer to Berkeley I must go over there some day and look through those archival materials. I'm sure it will be very interesting.

Cajori, among other subjects, is known for his writing on William Oughtred and the history of the slide rule.  There is certainly much more historical interest in these areas now than when he wrote his books on those subjects just after the turn of the century.

I have listed below the more substantial works written by Florian Cajori:

A History of Mathematics

A History of Mathematical Notations I

A History of Mathematical Notations II

A History of Physics in its Elementary Branches, Including Physical Laboratories

William Oughtred A Great Seventeenth Century Teacher of Mathematics

A History of the Logarithmic Slide Rule and Allied Instruments and the History of the Gunther Scale and the Slide Rule in the Seventeenth Century

Principia, Vol. I Motion of Bodies by Isaac Newton (translated by Andrew Motte) (edited by Cajori)

Principia Vol. II A System of the World by Isaac Newton (translated by Andrew Motte( edited by Cajori)

A History of the Concepts of Limits and Fluxions in Great Britain from Newton to Woodhouse

The Teaching and History of Mathematics in the United States

An Introduction to the Modern Theory of Equations

The Chequered Career of Ferdinand Rudolph Hassler

I know that at Colorado College they had the annual Cajori Lecture for two years back in the 50s, and the Cajori Award for outstanding Engineering students at Colorado College, but I don't know if there is any continuing award of any national or international acclaim in his honor, or if there have been any substantial scholarly work on his life written. I have not found it. If you know of either, I would be glad to hear about it.

I will augment this post with additional information on Florian Cajori as time permits. Of particular note are his volumes that were published posthumously.

"If a lunatic scribbles a jumble of mathematical symbols it does not follow that the writing means anything merely because to the inexpert eye it is indistinguishable from higher mathematics." --Florian Cajori

William Oughtred and the Oughtred Society

William Oughtred (1574-1660) was an English Ordained Episcopal Minister who is best known for his mathematical work. He was a Fellow of Kings College, Cambridge.

Oughtred is credited with the invention of the slide rule in  1622.  While it was John Napier who invented logarithms and Edmund Gunther who invented the logarithmic scale, it was Oughtred who recognized that by using two Gunther scales sliding by each other he could perform multiplication and division more rapidly than on a Gunther scale alone.  He conceived of this as two circular Gunther logarithmic scales.

In addition to the slide rule he is also known for the introduction of the symbols "x" for multiplication, and "sin" and "cos" for the sine and cosine functions, and "::" for proportion. The book he wrote which he is most noted for is Clavis Mathematicae (1631). 

A very good biography of William Oughtred was written by Florian Cajori entitled William Oughtred, a Great Seventeenth-Century Teacher of Mathematics, Open Court Publishing Co., Chicago, 1916. Cajori also wrote History of the Logarithmic Slide Rule, Engineering News Publishing Co., 1909 which contains many references to Oughtred. There are in print reprints of this volume.

The Oughtred Society is a non-profit educational organization dedicated to the preservation and history of the slide rule and other calculating instruments. Their goals include the dissemination and sharing of information and encouragement for collectors. They are affiliated with organizations in the United Kingdom, Germany, and The Netherlands with similar goals. The Journal of the Oughtred Society, a scholarly publication is put out semiannually by the Society. They have a number of meetings each year, usually on the East Coast and West Coast.  They have a website: Oughtred.org and encourage membership.

When I attended the last meeting in Mountain View, CA on June 27 it was held at the Computer History Museum.  It was great to see so many familiar faces again.

The Hunt for a Curta and Discovery of a Treasure

I had decided that I wanted a Curta calculator for my collection of antique mechanical calculators. It was going to represent the last of my mechanical calculators, chronologically speaking.  I should preface this with the comment that I could have easily gone out and purchased a Curta, but that takes half the fun out of it. Curtas are fairly expensive little gadgets, it would not be hard to spend $600 on a good one.  I like to find my computing collectibles at more affordable prices. I figure, if it costs less, then I can have some money left over to buy something else for my collection.

I know that most engineers who went to school when slide rules were de rigeuer ended up keeping their slide rules even though they never used them after they got weaned onto electronic calculators and computers.  They just feel that after all that time together, you don't just throw one away. You have to have more respect for it.  So, it gets tucked away in some drawer somewhere, and every once in while they run across it and it brings back all those fond memories of years past.  The damn thing has no monetary value, so they don't sell it either.  I have found, when telling people that I collect these devices that they are all to often willing to part with theirs because they know it is going to a good home and it will be treated right and won't be destroyed.

Back to the Curta. I figured that someone who used to enter rally races would be in pretty much the same situation as an engineer and his slide rule.  So, I started tracking down rally car enthusiasts.  It isn't very hard. You find one, if he doesn't have one to spare, then ask for the names of others that he know and their phone numbers. It went fairly fast. I had been looking off and on in my spare time for about two weeks, when this one fellow referred me to another.  I called the individual.

No, he had sold his last year. He had no use for it anymore. He couldn't just throw it out and someone offered to give him $100 for it, so he took it.  I, of course, had explained that I was looking for one for my collection of computing devices.  So, then, out of the blue, he says to me, "Maybe you would be interested in these slide rules we have."  My ears went up. "Oh, slide rules! What kind of slide rules?" He replied, "My wife's grandfather, or great grandfather designed them and she might want to get rid of them. I don't know much about them. You would have to talk to her about them."

His wife eventually got on the phone after I had gotten the names of a few more potential Curta owners. When his wife got on the phone she explained that her great grandfather, Edwin Thacher, had designed these slide rules and she had the original production runs for one of them and a prototype for another.  The former was the cylindrical slide rule that I have already written about, the latter was a Scofield-Thacher Engineer's slide rule. 

I asked her if she would be interested in selling the devices. She indicated that she would. We made arrangements for me to come out to visit her.  

Two weeks later I found myself on the road again.  It was about a 4 hour trip in rural New York.  

A little historical insert is due here. Edwin Thacher (1840-1920) was born in DeKalb, Lawrence County, New York and died in New York City. He grew up as a youth in Herman, NY not too far from DeKalb. He was trained as Civil Engineer at Rensselaer Polytechnic Institute in Troy, New York in 1863 and was employed at first on the Cedar Rapids & Missouri River Railroad, and subsequently with the US Military Railroad during the Civil War. Thereafter, he worked for the Louisville, Cincinnati & Lexington Railroad, Louisville Bridge & Iron Co., and as Chief Engineer of the Keystone Bridge Co. He was most noted for his concrete compression bridges and the slide rules he designed to engineer his bridges. He was always plagued by the lack of accuracy in the calculations he performed in designing bridges.  His most substantial engineering project was the design of the Walnut Street Bridge in Chattanooga, Hamilton County, TN. It was a six-span through truss bridge over the Tennessee River built from 1889 to 1891. It has a total length of approximately 2,370 feet with the longest single span being 320 feet. It is in the National Register of Historic Places.  The Walnut Street Bridge was converted to a pedestrian bridge some time in the latter part of the 20th century and may be the longest pedestrian bridge in the world.  Among the various slide rules he designed, was the cylindrical slide rule that carries his name (in reality, Thatcher[sic] due to a typo) to solve the accuracy problems he was encountering in his bridge engineering design work.

Now, back to the rest of the story. I arrived at the home of the great granddaughter of Edwin Thacher and had a cordial meeting with them.  They produced the various Thacher devices. The K&E 4012 was serial number 1050, indicating that it was the 50th production number produced.  the Scofield-Thacher Engineer's rule was clearly a prototype. They started disclosing what information they knew about him which wasn't all that much, but then they produced their Thacher family Bible.  You should know that way back when it was very common for families to record all the births, deaths, marriages, etc. in the family Bible.  Well, there it was: Edwin Thacher, born October 12, 1839 to Seymour Thacher and Elizabeth (Smith) Thacher.  From there she showed me the family tree, showing her direct descendancy from Edwin. Wow! Was this the Holy Grail squared, or what?  Most of the biographies on Edwin were wrong on the date of his birth. He was not born in 1840 as noted!

We got down to business and discussed the sale of the items.  Their concern was that these were large devices and no one in the family had expressed interest in them and they wanted to make sure that they ended up being taken care of. Yes, yes. I'm real good at taking care of my slide rules.  We talked about the price.  We did not conclude anything at that meeting.  They wanted to think about the whole matter. I did not want to pressure them. I told them I would send them some biographical information on their great grandfather, which I did as soon as I got back home.

We had a number of conversations after that and the conclusion was that they thought they would, at least for the time being, hold onto these family mementos. DARN!!!

So close, but so far from completing the purchase.

By the way, this whole story started out looking for a Curta. Well, I got my Curta a number of months later, but not from the plan I had set out to get it. An acquaintance of mine had a friend who was an engineer who had some slide rules who wanted to give them to me, and while talking to him I learned that he used to participate in rally races. Yes, he still had his Curta, and yes, if I gave him $50 he would sell it to me.

I now have that Curta.

The Quest for the Holy Grail of Slide Rules

I will now tell an interesting story about something I uncovered.

In about 1997, I was sitting and working at my computer and an email popped into my mailbox.  Someone wanted to know if I wanted to buy a slide rule. I didn't recognize the name, but I answered anyway.  Okay I'll bite, "What kind of slide rule do you have? And, oh, by the way, where did you get my name or email address?"  Most slide rules that come up are fairly boring.

A little while later a reply came with more information. "It is a Thatcher and I saw your name on some website advertising that you want to buy slide rules."  "Interesting, a Thatcher." I thought. "I don't remember ever putting an ad anywhere for buying slide rules.  I wonder where she saw that"

There were two principal models of the Thatcher that were made. Production started in 1881. There is the 4012 model and the 4013 model.  Both are almost identical, except for one thing.  They both are almost two feet wide, they are built around a 4 inch diameter cylinder with rotating vanes around the cylinder.  The theory behind a cylindrical slide rule is that by wrapping the scales in a helical fashion around the cylinder the scale can be made considerably longer.  The longer the scale, the more accurate the computations. The principal difference between the two Thatcher models is that the 4013 has a bar across the front of the device to which is attached a magnifying glass that can be adjusted to "read" the answer to a computation.  This allowed the device to be very precise -- so much so, that it was the most accurate device available for many years.  And to get the most out of the accuracy, you really do need a magnifying glass to "read" it.  It turns out that the 4013, while more expensive than its brother 4012, had many fewer produced, and consequently, they are extremely hard to find in good condition -- and they are so very expensive.

Now, back to the story, I assumed it would be a 4012 and probably, like most, fairly deteriorated.  The response I got said it was a 4012 and after a few more back and forths, I concluded it was worth seeing.  It turned out, of all places in the world that it could be, it was about an hour away, in Springfield, MA.  I hopped in the car, went to the bank to get the required cash, and off I went to Springfield from Connecticut.

I arrived at a small house in an older section of town. I knocked at the door and this woman came and let me in.  She showed me to the table where she had on display, unbeknownst to me, two Thatchers.  One was a 4012, and... the other ---the other  was the Holy Grail -- a model 4013.  I tried to restrain myself from showing how elated I was.

The story took a very strange turn.  She was apologizing to me that she had me come all that way, but she had already sold the 4012 Thatcher that I had come to buy........ But, if I was interested, she had this other one almost the same over here.  I knew it, it was a bait and switch!

I looked it over, it was beautiful. It was well preserved, and it had all its parts. "I'll let you have this one for the same price" she said.  "I'll take it"

So, that's how I got the Thatcher and now it sits right next to the Curta in the display case in my living room. But that was not always the case.  I had not purchased the Curta at that time. I was still on the trail to find one.... and that is the next interesting adventure.

The Curta Calculator

The Curta Calculator is an amazing mechanical device.  It was first produced in 1948 as the brainchild of an Austrian, Curt Herzstark (1902-1988), based upon work of the famous 17th century mathematician Godfried Leibniz (1646-1718). It performs addition, subtraction, multiplication and division, and with some ingenuity even more complex functions.  While Herzstark had conceived the device during the 1930s, it was actually during World War II while a he was imprisoned at the Buchenwald Concentration Camp (his father was Jewish), that he worked out the actual design of the first production devices.  After being liberated from Buchenwald in 1945 and before 1948 he finalized the design and brought the device to the marketplace. The Curta factory was located in Lichtenstein.

The Curta I has 8 slides for entering one of the operands and a 6 digit revolution counter for the other operand and an 11 digit result counter. The entire Curta I weighs only 200 gm.  There were 80,000 Type I calculators produced.  The larger Curta II, first manufactured in 1964 has 11 slides for entering the first operand and a 8 digit revolution counter, and 15 digits for the results. The Curta II weighs about 370 gm. The Type II model had a production run of 60,000.  The difference between the two calculators was primarily the number of digits of significance that the devices could handle.

The first thing you should know about a Curta is that it is small enough to fit in the palm of your hand.  It is so small, in fact, that you could not imagine a device this complicated fitting in such a small space. It has such a good feel when holding it and operating it. Curtas are sometimes called "pepper grinders" because of the similarity in looks and size. 

 Numbers are entered on the slides. A rotating crank on the top allows the input number to be added to the result. By lifting the crank the input is subtracted from the result. Rotating the top will cause the the input to be either multiplied by 10 or divided by 10 depending on the direction of the rotation.  So, with a sequence of rotations and lifts and turns relatively large numbers could undergo arithmetic operations with great precision.  In the case of the Curta II, up to 15 significant digits of accuracy. There is a great video showing the operation of the Curta here. Or, see Jan Meyer's amazing Curta simulator here.

I purchased my Curta from a fellow that used it during the 60s for rally car races.  The navigator for the car had the responsibility to see that the driver "paced" the car as prescribed by the rally rules.  This required quick and reliable computation under some difficult circumstances. I don't think a slide rule would have quite made it in that type of situation.  They were also used in airplane navigation and in engineering and scientific computation.  They pretty much went out of favor at the same time as the slide rule during the 1970s.

I found my Curta many years ago, when hand held mechanical calculators were out of favor, yet before the collector craze for older mechanical computing devices had started.  I keep it in a display case in my living room right next to my Thatcher 4013 Calculator.  The reason I mention this is because I have a very interesting story about Edwin Thacher and his Cylindrical Calculator that I stumbled upon while searching for a Curta for my collection.  But that is going to have to be for another day. By the way, no, I didn't make a mistake on spelling the Thatcher/Thacher names above. Edwin spelled his name without a "t" (i.e. Thacher), but the engraver making the device for production misspelled his name on the engraving plate by putting in an extra "t".  The device was so precise and it would have cost so much to redo the engraving that they went into production with the wrong spelling for his name anyway, so the device is a "Thatcher".

Georg Ernst Stahl and the Phlogiston Theory

Just to keep things balanced, I have an engraving of Georg Stahl hanging on my office wall right near my Priestley likenesses, some of these likenesses I have blogged about earlier.  I have Georg hanging there to put things in perspective.  

Georg Ernst Stahl (1659-1734) was born at Ansbach. He was an eminent chemist and physician and is best know as the foremost proponent of the now defunct and discredited phlogiston theory.  

The phlogiston theory was originally postulated by Johann Becher in 1667. It hypothesized that flammable materials contained phlogiston, a substance without any discernible properties, including, mass, or any other physical properties whatsoever.  When the flammable material was combusted the phlogiston dissipated and the material returned to its "dephlogisticated" state called the calyx or true state.

Joseph Priestley was also a lifelong proponent of the phlogiston theory, and, in fact, upon isolating a substance from mercuric oxide with heat, called it "dephlogisticated air" (oxygen).

It was Antoine Lavoisier in France who showed that combustion requires a gas that has weight. This substance he named oxygen, and in demonstrating that it did have mass he had made the phogiston theory obsolete.  The fact that Lavoisier became obsolete after the French Revolution by losing his head is the subject of a blog yet to be composed.

In any case, the conclusion is that even great minds make mistakes. In Priestley's case it was his clinging to an obsolete theory even after it had been clearly discredited.  In Scientific Correspondence of Joseph Priestley the letters Priestley wrote of his observations from experiments he had performed show his numerous mistakes and ill formed conclusions. Yet, through it all, through all the imperfect observations and incorrect thinking, he was able to accomplish so much. It is a lesson to be learned by all.  Mistakes are normal in the course of discovery. They are essential!  And every assumption must be treated with suspicion.