Ptak Science Books

A History Blank, Empty and Missing Things #77--Blank Enumerators in Sand, Pebble and Dust Computers

JF Ptak Science Books   Post 1512

 This is a short note on the blank nature of the "checkers", the place-holders, the missing numbers, of ancient computing machines,  the counters (jetton, or jeton¹) used in early/ancient arithmetical reckoners, the material pieces used to aid in addition and multiplication "devices". These bits were sometimes pebbles or shards of pottery or rocks, and placed on the ground with a grid drawn in dust, or in sand, or on any surface that could hold a line.  There are fancier and more permanent types of these instruments, as we see in the right side of this iconic woodcut image from a 1503 universal compendium of knowledge--a simple wooden table with incised counting lines, with the jettons being blank disks.

The book that this  beautifully-illustrated counting board is found is in Gregor Reisch's  Margarita Philosophica,  and depicts (amidst much else in the greatly humanist volume) representations of the mathematicians Boethius and Pythagoras working math problems on the given tools of their day. (The Reisch book is remarkable: it is basically a Renaissance encyclopedia of general knowledge, divided into twelve books:  grammar, dialectics, rhetoric, arithmetic, music, geometry, astronomy, physics, natural history, physiology, psychology,  and ethics.)  We can see in his expression that Boethius, on the left, is rather enjoying himself, knowing the superiority of his system of counting, which was the the Hindu-Arabic number notation--he definitely has a sly, self-appreciating smile on his face.  Pythagoras, working with the old counting table, definitely looks worried, or at least unhappy, unsettled.  Never mind that Pythagoras (570-495 b.c.e., none of whose works exist in the original, another sort of entry in our Blank History category) was at a definite disadvantage in the calculating department, being dead and all that for hundreds of years before the Arabic notation was more widely introduced in the West, probably being introduced by Pisano/Fibonnaci in the 12th century.  But it does fall to Boethius, the smirker, to have introduced the digits into Europe for the very first time, deep into the history of the Roman Empire, in the 6th century. 

The numerical stand-ins in the Reisch book with which Pythagoras worked were blank, coin-like slugs used as placeholders, and would be used in place of rocks or pebbles or whatever other material was at hand. It is interesting to note that the Latin expression, "calculos ponere", which basically means "to calculate"or "to compute", is more literally translated into  "to set counters" or "to place pebbles" (upon a counting board) or to set an argument²,  which is exactly what some of the Roman daily reckoners would do at their work. And also used, in this case, by the unhappy Pythagoras. 

Here's a small close-up of the blanks:

The problem that the Pythagoras-person was working on (flipped 180  degrees) shows him adding the numbers 1241 and 82.  A jetton occupying the the top of the mark on the counting board, counting as one unit of 1000; followed by two units of hundreds; four tens; 2 on the ones.  The other number interestingly keep one jetton in between two lines, signifying an easy wade of enumerating 5 of any one kind, in this case depicting 8 tens units, coupled with two ones., thus making the number 82.)

For an excellent explanation of the many and varied methods of ancient calculations see the article by Steve  Stephenson³ on how ancient computers worked, found in the IEEE Global History Network site,  here. 

It is a simple observation, but interesting to me nevertheless, that there was so much about these early counting systems that even though extremely useful they were also highly ephemeral--the counting boards being  inscribed in dust or sand or dirt, the counters/numerical placeholders being blank disks or pebbles or pottery shards--and so so much of the counting world depended on so little.

Notes

1. I should point out that these markers seem to have been decorated more often than not--its really just those items that appear in the Reisch book that I am addressing.   On jettons, in general, see here ;  also,  Jetons, Their Use in History.

2. See Jen  (1998, April 25). "Roman Counting Instruments", in The Math Forum at Drexel University. Also see Karl Mennigner's  (1969) Number Words and Number Symbols: A Cultural History of Numbers, a big, academic/coffee table book on the history of numbers.  Also in general, see: Barnard, Francis Pierrepont, (1916). The Casting-Counter and the Counting-Board, A Chapter in the History of Numismatics and Early Arithmetic. Oxford University Press, London.

3. This is a deep and nicely-explained effort looking at the development and different sorts of these instruments, as well as how they functioned.  Regarding the Reiwch woodcut, Stephenson says:

1. Pythagoras has his abacus oriented with a vertical median line;
2. The lines are equally divided so the same number of jettons can be accommodated on either side of the median line;
3. The top horizontal line is marked with an X, (perhaps) to indicate the unit line; and
4. On the right side of the abacus is a jetton in a space, all other jettons are on horizontal lines.

On pp.313-314, Prof. Barnard describes and quotes from Legendre, François, 1753. L'Arithmétique en sa perfection, Paris, pp. 497-528, Traité de l'arithmetiqué par les jetons:

It was permissible to set and to work the jettons of the sum without using the spaces [between lines], ... But it was much more convenient to anyone who was expert at the practice, and less confusing to the eye, to reduce the number of jettons by using the spaces.

Standardizing Precision and Beautiful Technical Prints: Ramsden's Dividing Engines

JF Ptak Science Books   Post 1452.223221 and a quarter

 

I like the idea of having something called a Dividing Engine--perhaps it would divide seeds, or complex problems, or simple problems, or perhaps it would divide division. 

What it really refers to here is a precision tool that whose effects were extremely wide felt but about which we don't really hear about today.  This is the dividing engine of Jesse Ramsden,and Englishman who invented a circular instrument that would incise precise values on precision instruments like surveying compasses, delineate the linear and circular scales of measuring instruments used in astronomy and navigation,   Before Ramsden, the engraved marks on instruments would have been made by each individual manufacturer, each of those depending on marks that they made in the past, generations of such things in a line, over and over, a quite individualized effort, necessarily dependent upon the precision of each manufacturer. In 1777 Ramsden produced an instrument¹ of exception precision that was correct and fine and dependable². [The image above was found in Abraham Rees' monumental Encyclopedic Dictionary..., and is entitled "Engine for cutting the screw of Ramsden's Circular Dividing Engine" and was published in 1814.  Below is a detail of the plan for the winding of the gear works. In order for his machine to work at such a high degree of accuracy the component parts must also have been of a very high calibre--to that end Ramsden developed what is essentially the modern screw cutting lathe from which he manufactured the gear works for his Straight and Circular Dividing Engines.³]

And then there was the "Engine for cutting the Screw of Ramsden's Straight Line Dividing Engine", which is an absolutely gorgeous piece of drawing.  A person could crawl all over this image in varying degrees of microscopic inspection and find all sorts of beautiful internal images, a large example appears here:

And its fantastic detail:

Basically though Ramsden's inventions were key additions to the developing scientific technologies of the Industrial Revolution, integral improvements necessary for integral improvements. 

Notes:

1.  He published a Description of an Engine for dividing Mathematical Instruments in 1777.

2. "The dividing engine was simple to operate. The instrument being divided was fixed to a large wheel on top of the engine. When the treadle was pressed, the wheel and the instrument were turned through a fixed angle. Then with the right hand, a cutting tool guided by a system of swinging links was used to mark the instrument scale. The process was repeated until the complete scale had been divided. Although this was many times faster than hand-dividing, it was backbreaking work having to lean over the engine to work on a small instrument."-"Dividing Engine." Smithsonian National Museum of American History. americanhistory.si.edu/collections/navigation/object.cfm?recordnumber=694508

3. From WIki--"The first truly modern screw-cutting lathe was likely constructed by Jesse Ramsden in 1775. He appears to have been the first person to put a leadscrew into actual use (although, as Leonardo's drawings show, he was not the first person ever to think of the idea), and he was the first to use diamond-tipped cutting tools.^[2] His device also included a slide rest and change gear mechanism. These form the elements of a modern (non-CNC) lathe and are in use to this day. Ramsden was able to use his first screw-cutting lathe to make even more accurate lathes. With these, he was able to make an exceptionally accurate dividing engine and in turn, some of the finest astronomical, surveying, and navigational instruments of the 18th century.

The First Photo Inside of the Information Revolution--the Transistor, 1949

JF Ptak Science Books  Post 1370

One of the things I love about working my way through old scientific journals is when I find the issue that I'm looking for and scroll down the list of contributors to find the significant article that I want.  Long list, usually; and then, after making my way through 30 or 40 lines of tight type of the index I find it. [This by the way is one of those experiences that is being replaced by the digital library.]   Even though the paper on pp 1208 through 1226  of the 15 April 1949 issue of The Physical Review looks like any other, it is today seen as revolutionary. The entry for "Physical Principles Involved in Transistor Action" by John Bardeen (two-time Nobel in physics) and Walter Brattain (Nobel '72) shows up about halfway down the index, sandwiched in some very good company (Enrico Fermi's "Origins of Cosmic Radiation" and a number of others), and does not show up bolded, or highlighted, or with an asterisk.  Such is the nature of publication in the academic journal world, everything delivered with equal weight. (The original publication is available for purchase here at our blog blookstore.)

It makes me wonder though how it would've felt to open this journal for the first time back there in mid-April '49, turning to page 1210 to see the microphotograph of the cutaway of a model of the transistor.  This was the defining technical publication on the transistor¹, which was the first massive step towards microminiaturization and the explosive new growth in the computer, allowing far more powerful machines to be designed in far less space, in far less amounts of time, and on and on.  It is one of the first steps in the Information Revolution, moving the computer from massive racks of electronic tubes to more simple, elegant, nimble and by-far faster circuit boards with transistors (and resistors, capacitors, inductors, diodes, etc.) to make an electronic circuit.  This would be the standard for computer construction, only supplemented by Jack Kilby (TI)  and Robert Noyce (Fairchild Camera) in 1958/9 with the integrated circuit, where transistors are made smaller still and produced in groups on circuit boards rather than individually.

The photo above shows a cutaway of the transistor, and is the first time it was published--the first photo of what was one of teh 20th century's greatest inventions. 

Notes

1.  The paper was published simultaneously in the Bell System Technical Journal; Bardeen and Brattain were with the Bell Labs.  The Bell journal also contained another revolutionary paper in the same volume, Claude Shannon's "Communication Theory of Secrecy Systems", which is one of the most important early papers on electronics and cryptology.  (We also have a copy of this classic paper for sale at our blog bookstore site.)

Selling and Costing-Out Digital Computers, 1959--When "Computers Didn't Sell Themselves"

JF Ptak Science Books   Post 1296

"Computers do not as yet sell themselves."--Lehman, 1959

Question:  what had 1000 transistors, 5000 diodes, 2000 resistors and 1000 capacitors, a team of ten and took 18 months to build?  

Answer: an inexpensive, $12,000 digital computer in 1959.  

Mind you this is legions better than what was happening in the mid-1940's, with teams of hundreds and costs in the many millions, and much better than in the early 1950's, when these numbers were of an order of magnitude greater.

It seems to me that this short 1959 paper is at the upper end of a beginning concern--costing -out the price of a digital computer for second-tier interests.  "The Specification Development of a Cost-Limited Digital Computer" (by M. Lehman of the Israeli Ministry of Defense, and available here for purchase) was written for those businesses and schools with a definite low-end budget, the upper range of which was set by Lehman at $12,000 for "peripherals, hardware and programming  (or about $125,000 in 2010 dollars, sort of¹). This figure was also exclusive of maintenance outside of the cost (at 15-20% of the construction costs overall) of designing and building the machine.

This was some new thinking on providing lower-cost digital computers to a new market:  "in just ten years there has emerged a multimillion dollar industry largely dominated....by the giants of the electronics and data-processing industries...".  Lehman was saying that there was a new opportunity for business to supply computers to "smaller research groups" who were "finding it increasingly difficult to obtain the backing which would enable them...to build an actual machine".  There were other ways of doing it, and Lehman laid out the basic understanding of that procedure.  

Lehman's leading quote for this post was accurate--computers needed to be "sold", as in salesmen and businesses actively engaged in contacting clients who would ("might") benefit from having computeriaed part of their business. The computer manufacturer's "staff will often spend many months investigating customer's problems, possibly reorganizing his techniques and generally preparing teh ground for the installation of a Digital System..."

 

(The figures in the right column are indeed dollars.)

Lehman was doing some groundbreaking work here at the end of the '50's, still far ahead of the time--and probably a full generation--where computers didn't necessarily have to be shown as being 'needed" let alone
"necessary".  The computer would be "selling itself" on the low-end of the market soon enough, but really not until the Reagan years when it was beyond question that the computer could be used by virtually any person or business--at affordable rates.  

Notes

1.  Sort of, indeed.  Straight CPI translation would put this figure at about $125,000, but the other cost of things in 1959 compared to 2010 is a different structure.  Sure, it might cost out to 125k, but that $12k in  1959 could've been traded for a decent working-class house in a big city; you couldn't trade that $125k for the same thing today, no way, no how.  

Logic Trees, Magic Squares, Magic Circles and Magic Square of Squares

JF Ptak Science Books LLC  Post 1238

[Associated posts: The Mother of all Renaissance Logical Graphs, The Knight's Tour, Porphyry and Boethius and Census Art and the Display of Quantitative Data, 1860.]

Well now:  I don’t know what the provocation or inducement is here to hurtle this axe-swinging monk to attack Porphyry’s Tree¹, though it would be interesting in a forensic sort of way to know what the tree’s section might reveal. The “tree” was a diagrammatic creation of a 3^rd century Syrian mathematician/logician/philosopher named Porphyry who-- much taken with Aristotle (and with the Categories in particular)-- developed a systematic approach to the organization of thought in diagrammatic form.

What’s inside a tree of logic and memory?  Is there a  xylem-y/phloem-y stuff besides a three-dimensional representation of the structure of organizational thinking?  Or is the 2-dimensional rip a fatal blow to other dimensions, and like Eddington’s Turtles, it’s a simple slice of Flatland all the way down?  

Perhaps Porphyry’s tree rings would look like this, a magic circle or spiral, which would make some sense, and would bring to bear an associated use of turtles—or tortoises, I should say.  It turns out that perhaps the very first use of the magic circle, rolling back its origins through the Islamic world to India and to Persia and then to Japan,  and then finally to China where, in about 2000 BCE, the magic square appears in an image with the Emperor Yu,  inscribed on the back of a tortoise.

 

[Magic circle source:  Abraham Rees Encyclopedic Dictionary , printed 1805-1815. The originals are available from our blog bookstore.]

Notes:

1. This image appears in the rare Destructio sive eradicatio totius arboris Porphirii : magni philosophi ac sacrae theologiae doctoris eximii Augustini Anchonitani ordinis fratrum Heremitarum Sancti Augustini, cũ quadã decretali eiusde, published in 1503.

A Do-It-Yourself Paper Digital Computer, 1959.

JF Ptak Science Books    Post 1210

This wonderful cut-away and paste-up template for a digital computer comes to us from the Communications of the Association for Computing Machinery, volume 2, issue 9 for September 1959.  The PAPAC-00 is a “2-register, 1-bit, fixed-instruction binary digital computer” and was submitted to the journal by Rollin P. Mayer (of the MIT Lincoln Lab).  There’s a hunk of me that wants to make this thing really big–cut out the individual pieces and then crash them out to 50" widths, pasted on found bits of cardboard packing from the neighborhood frame shop, and then piece the thing together as the world’s largest pre-1960 1-bit paper computer.  Or maybe not.  In any event I thought to share this with readers here who might actually print out these templates and try to construct the thing themselves–warning: you’ll need t be able to cut pins in order to make the model work properly. [The original paper may be purchased from our bookstore, here.] 


Mayer also wrote an interesting article on including children in the scope of the computer industry...in 1956–something I find to be a very early piece of thinking on Little Humans and Big Computing.  His abstract identifies the three main points of his paper: “(1) That the digital computing field needs, and will continue to need, not only more people who are capable of designing and programming digital computers, but more people who understand the basic limitations and potential uses of digital computers; (2) that the computer industry should take an active interest in providing a basic computer training to the largest number of people, in addition to more extensive training to those who show an interest in designing and programming computers; and (3) that the typical 12-year-old youngster has the interest, skill and basic knowledge necessary to build and understand simple working models of practically anything”. –“A proposal for training youngsters in digital computing techniques” in Proceeding ACM '56 Proceedings of the 1956 11th Annual Meeting.

Lastly I should also mention that in this same monthly issue (which ranged all of 52 pages) was an article by Julien Green, “Remarks on ALGOL and Symbol Manipulation”, a three-page paper that comes from the near-dawn of the ALGOL (short for ALGOrithmic Language). (It can be argued that the beginnings of ALGOL were about 1955, but for the sake of simplicity here I’ll note that the first committee appointed to study the matter for the ACM was in September, 1957, and the first group to formalize it met at the ETH Zurich in 1958, producing the language’s first version, known as ALGOL 58.  Julien Green was a member of the 13-person panel that created the ALGOL 60that met in Paris in January 1960.   And so the Green paper did appear within the first few years of institutional study in America, and it was written by one of the six American members of the ALGOL 60 team.)

The Kryha Cipher Machine. 1929 [Quick Post]

JF Ptak Science Books  Post 1169

[I wrote an earlier post on this blog on ciphers here: Books of Shadows—"Trithemius, Bacon, Porta, & Falconer vs. a Make-Believe Derrida on How to Make Something Unintelligible. Secret Writing and Ciphers".]

The Kryha ciphering/enciphering machine was invented by Ukranian Alexander Kryha (d. 1955) and was capable of coding about 350 characters a minute.  It was used to enforce secrecy in business transactions, and was used in conjunction with (at least) Siemens telegraphic equipment.  I found these images (except for the patent drawings) of the Kryha machines in several issues of the Illustrirte Zeitung (Leipzig) for 1929, all of which are unusual and evidently not findable as good copies online--and so I'm simply reposting them here.

 

Footnotes in the Future--Swift, Lull, Borges and the End of Writing

JF Ptak Science Books   Post 1155

“The most ignorant person at a reasonable charge, and with little bodily labor, may write books in philosophy, poetry, law, mathematics, and theology, without the least assistance from genius or study.”   Jonathan Swift, in Gulliver’s Travels

"Hell is the kingdom of the animal that swallows up the memory of all things... Between life and death there is no destiny except memory. Memory weaves the destiny of the world.."  Carlos Fuentes, Terra Nostra

"A people without history/ is not redeemed from time, for history is a pattern of timeless moments" -- T.S. Eliot, Four Quartets.

The future history of writing--it will tell us--was already written before it was written.   In the future, when everything that can be written has been written and everything spoken has already been spoken, perhaps the first footnote in the history of already-done things will be to Jonathan Swift.  The experience belongs to us through Lemuel Gulliver¹, who after surviving two trips (one to Lilliput in which Gulliver is twelve times the size of

everything else, and the second to Brobdingnag where he is one-twelfth the size of everything) comes to a third (of four²) in the land of Laputa.  It is here where he encounters a conversation machine that, when cranked, produces anything and everything, the known and unknown, the original and the copied, and have the thought and spoken word belong to the manipulator of the machine.  Everything that can be said is in that box, waiting to come.  The 20' square was occupied by cubes with words on their sides, with 40 cranks and 40 handles and 40 operators to manipulate the mechanical vocabulary and associated scribes to record anything that made sense, and would be the springboard of “knowledge”operating without a manual or an intelligent primum mobile. (The image of the machine--which bears some resemblance to an integrated chip--appears in the third edition of Swift's book, and as you can see is not a well-drawn thing, missing a row and a column and a crank here and there. A fuller description is found below.)

The idea of a universal knowledge or an expanding, forever-library is very old, particularly if you expand the concept a bit to include Aristotle and the Tree of Porphyry³.  While not exactly a generator in the sense of Jorge Borges' Library of Babel or the Swiftian machine, the ancients did produce very interesting, elegant, beautiful ways of story information and ordering information.  The Tree of Porphyry is far more concise–an abbreviation-compared to the infinitely expanding hexagonal rooms filed with books and attendant librarians in the Borges' universe.  Or the Ars Magna/Thinking Machine of the 13th century Ramon Lull (also known as Ramon, Raimundo and Raymond, Raimundus and Raymundus Lull, Lully and Lullus and Lulio), the ultimate organizer of how sentences can be made and knowledge produced/uncovered (particularly if you want to please the logic of the church and the Creator), and which was almost certainly known to Swift (and which was also written about, described and worried-over by Borges⁴).  There are many other early figures in this category to be sure (Lewis Carroll, William Jevons, and even in a way Mr. Venn, not to mention Leibniz and his calculator,  and the changes in scientific method and (English/Dutch) mathematical concentration on applied mathematics of the 17th century that made thinking about the industrial revolution possible)), but that would be left to (at least) a longer post.  (The Lull wheel, below.)
                           

How long will it take to digitize everything that has ever been written, or recorded, or composed? Several generations? Ten?  It may well come to pass in shorter time than we think; and when you compare this control of recorded human history to the centuries that will follow in developing “computers”, how can we possibly know what will become of all of it?  In the coming centuries and their attack on intelligence, when in the future the brain is a harvest of the computer, perhaps intelligence finally submits, and everything that can be done has been done, the progress of humanity snubbed to a stub. Or expanded exponentially.  It is such a tough call to make...

Notes:

1.  Jonathan Swift, Travels into Several Remote Nations of the World, in Four Parts. By Lemuel Gulliver, First a Surgeon, and then a Captain of several Ships.  1726.

2. The third voyage was Part III. A Voyage to Laputa, Balnibarbi, Luggnagg, Glubbdubdrib, and Japan. The fourth finds humans as pets/inferiors in the island of the Houyhnhnms, a land of intelligent horses.

3. The “tree” was a diagrammatic creation of a 3rd century Syrian mathematician/logician/philosopher named Porphyry who-- much taken with Aristotle (and with the Categories in particular)-- developed a systematic approach to the organization of thought in diagrammatic form.

4. Borges J. L. (1999) 'Ramon Lull’s thinking Machine' in The total library: non-fiction 1922-1986 (Edited by) Weinberger, E., trans. Allen, E. Levine, S. J., and Weinberger, E. London, Penguin, on pp 155-160.

It is interesting to note that Lull, who gave up everything to follow his calling into the church (and who was antagonized by the belief that his very act of missing his left-behind life to be a sin against god's choice of him to follow his "career path") was demonized by the church soon after his death, his works prohibited reading for hundreds of years.  He was resurrected in 1958 though on a path towards sainthood, a doctor of the church.  

A description of the Swift machine:

A New Perspective on Multitudes of Soldiers

JF Ptak Science Books LLC  Post 1019

Yesterday while trying to entertain our six-year-old , sitting in a small waiting room to an office with nothing to do, I estimated the number of holes there were in the boring acoustic ceiling tiles. A short distraction later I realized that if each one of those holes was a dollar bill, that the national deficit would require those titles to be laid out over an area roughly the size of D.C. and Alexandria (about a hundred square miles).  That sort of brought some perspective to what trillions of dollars might mean, but not really—it was still too weird and vast a number.

 

I had the same sort of feeling seeing this picture essay (yesterday as well) on soldiers and war.  The photos appeared in the July 1 1920 issue of the Illustrated London News and spoke of the new management of manufacturing British war medals.  The vastness of the casualties of that war are well known, but somehow this story seemed to put the big numbers into a more understandable perspective.

I learned that there were 300 men who worked full-time on producing these medals, making about 67,000 a week--in 1920, two years after the end of the war.  Not only that, since the Royal Mint and the Royal Ordnance Factory would be working jointly on the project, the Army Council expected production to increase to 120,000 medals a week.  That’s six million medals a year. 

It seems extraordinary that even after the fighting that production levels would need to be so high.  But there were millions who fought, and so these 300 workers would still have to work for several more years to produce enough medals to honor those who fought in WWI.

  

I read somewhere—a citation now long lost—that in America every Purple Heart that has been given out since 1945 was struck in that same year.  This  mountain was made in preparation for the hundreds of thousands of soldiers who were expected to be made casualties in the Allied assault on the Japanese homeland in 1945/6—a figure that would come close to half of the total number of 1.6 million Purple Hearts that have been awarded since 1932

 

The Data Mine: the French Census of 1911 and Storing Numbers on Paper

JF Ptak Science Books LLC  Post 835  Blog Bookstore

Today’s post on the tabulating office of the French 1911 Census seems quite natural following yesterday’s of the reading-and-writing (French) fireproof man...

 

Like many other countries, the French were certainly not playing to the technological audience in conducting their census of 1911, the physical, tabulating part of it looking much as it did in the 1880’s.  Granted, the United States was dragged somewhat into the new high-tech age of tabulating for its 1890 census, but it was absolutely and clearly shown that the Herman Hollerith machines and methods were vastly superior to those previous used, giving the government extraordinary new insights into the way in which the country functioned—a Hubble Telescope-like impact for those interested in more and more-manipulable data.   

What interests me in these photos are the endless stacks of paper, and what I imagine was the quiet of the job of the paper-stack-classifier, all of whom seem to be heavily dressed…I guess that you couldn’t keep a dry heat around exposed paper like that for fear of (a) well, possible fire and (b) drying the documents out and making them fragile and unusable.   

The image below shows two men using the Thacher Calculating Instrument, a large, cylindrical slide rule¹.     

This man is using an arithmometer, invented  by Charles Xavier Thomas de Colmar in 1820.  It was employed, as we can clearly see, well into the 20^th century even though it had been far superseded.   

The U.S. government, on the other hand, wasn’t so much taken by the astronomical range of new statistics that flowed from the Hollerith machines as it was the astronomical bill for their use.  The 1880 census had cost about $6 million and took 9 years to tabulate; the 1890 census using the Hollerith machines cost $10 million and took seven years.  The main focus of many in government was the cost differential—not the incredible amounts of new controllable information.  The rent of the Hollerith machines was only $750,000 for the conduct of the entire census, so the differential must’ve been in the extra utility costs (for electricity, for example, which was used for the first time to run the tabulators) and for the small army of statisticians and data entry people.  Be that as it may, the government was not amused, particularly when Hollerith figured that he had actually saved the government $5 million.    The two parties left each other grumbling, though the roar of the trickle down from the Hollerith success drowned it out.  The tabulating system was quickly exported, and large private concerns in the U.S. saw a savior in the system that would soon rescue them from the sea of paper in which they were beginning to drown.   The Hollerith company did very, very well for itself, and soon merged with three other companies (in 1911) to ease the burden of success.  The resulting company was called the Computing-Tabulating-Research Company (CTR), which after a short while became the International Business Machine Corporation (IBM).  

Image source:  The Illustrated London News, 11 March 1911.  

 Notes:

1.The following description is from the wonderful Slide Rule Museum site;

“Thacher's Calculating Instrument - Patented in 1881 by Edwin Thacher. Originally made by W.F. Stanley, in London, but, by 1897, Keuffel & Esser had taken over production. an 1884 instruction book notes, "The original rule in use is 12 inches long, with radii of II and 5 1/2 inches, the divisions of which are cut by hand, copying from a machine divided plate. In the present instrument the radii are 60 and 30 feet, the divisions of which are printed directly from machine divided plates. Those plates contain over 33,000 divisions, calculated to seven places of decimals from Babbage's tables by using a common multiplier, every line being subjected to correction for error of screw and temperature variations, so that possibly every line center is within .0001 inch of its true place." The instrument consists of a cylindrical slide, which admits of both rotary and longitudinal movement within an open metallic framework of 20 equidistant triangular bars. The bars are connected to rings at their ends which admit rotation within standards attached to the base. Upon the slide are wrapped two complete logarithmic scales, each of which is divided into 40 parts of length equal to half that of the slide. The parts follow each other in regular order around the cylinder, and the figures and divisions which constitute any part of the right are repeated on the left, one line in advance. By the rotary and longitudinal movement of the slide any of its divisions may be brought opposite to or in contact with any division on the fixed scales. The divisions on the upper lines are transferred to the slide by means of a pointer fitting over the bars, which is also convenient for retaining the position of any division on either line while the slide is being revolved into the required position. Near the commencement of each scale on the slide is a heavy black mark designed to catch the eye.”

Big Mathematics 1816: Manuscript Notes and Found Math Art

JF Ptak Science Books LLC  Post 757  Blog Bookstore

In my science bookstore business one of my principle interests is antique manuscript notebooks in the sciences.  It is a pleasure to see someone working through a problem, or finding what the writer thought to be the most interesting point in a lecture on a particular subject, and to read the occasional doodles and see the timely gloss every now and then.  I’ve had notebooks from an advanced and very careful student for a course given by the great Robert Bunsen that was filled with hundreds of pages of what the master was trying to communicate to his pupils.

There’s also a  series of notebooks from the spectacular period of Cornell physics history (1948-1949), including 6 notebooks of classes with Richard Feynman, including several hundred pages from his 1948/9 course in advanced quantum mechanics, which included (a very early) use of the Feynman diagrams.  The thing that makes these observant, advanced notes taken during Feynman’s classes have more than a pedagogical interest (which is substantial and enough on its own)—they have a capacity to show the logic behind the man’s thinking and how he presented his ideas over a period of some months.  Also, and curiously, notebooks like these seem to have not survived. 

Then there’s the notebooks taken by the team of Charlotte and Fritz John when they were pursing doctoral degrees at Goettingen (1929-1933), until forced to leave with the enforcement of the Nazi laws against non-Aryans.  Goettingen happened (and happens) to be the seat of a long, great mathematical heritage, home of Felix Klein, Emmy Noether, David Hilbert and Bernhard Riemann (among many others).  The notebooks—some two linear feet of them—represented four years of advanced interactions with some of the finest mathematical minds in Germany.  I was told by Mrs. John that they were required to take such full notes—but let me tell you, these notebooks were typed transcriptions of the daily handwritten notes, compiled over the course of a class, and then bound.  They are beautiful, and go far beyond my concept of “required”.  Mrs. John told me the story of the class notes that she and Dr. John took while taking a course with the great Hermann Weyl (who had taken his doctorate under the iconic David Hilbert, and who was also forced to flee Goettingen because his wife was Jewish):  everyone of course was required to take extensive notes during the day and work on them during the night, turning them in the following day for appraisal.  She told me that she still vividly remember placing their notes on a long wooden table at the front of the class, and then picking that set of note up the next day.  After a month of this, she, said, they became convinced that Weyl was not looking at the notes at all, as everything seemed always to be in the spot that they left it in.  One week they decided to see if the notes were being moved, and placed a hair carefully on the interior page, reasoning that if the notes were reviewed, the hair would be gone.  For a week, the hairs stayed in place.  “And so, what did that mean to you?” 

Mrs. John replied that it really didn’t mean anything, and that they continued to do the notes as they would, and leave them on the long wooden table.  They never slacked off (which I can fully imagine).  She said that they did it for themselves, of course, but also they did it for Dr. Weyl, fully expecting that the day that they missed handing in their notes would be the day that they were looked for by Dr. Weyl.  Plus they were studying with a famous and fabulous thinker, and did not want to disappoint themselves or their instructor. 

I do enjoy the historic or significant material like this, but I also quite the naïve, homegrown, elementary manuscript attempts by children at doing math or understanding physics.  Cipher books have always been a little hard to find in my field, wit many of them (I think) winding up in the genealogy area), and they just don’t seem to be very much on the findable side of things anymore. 

The example I’ve reproduced here are from an anonymous (but dated 1818) work on different elementary areas of mathematics.  What makes this special to me is the precision of the penmanship and the extreme effort that is being used for a simple display of division.  It would certainly convey the message if the author had used two five-figure numbers to display division rather than these enormous, never-to-be-encountered numbers. 

Surely there were pages and pages of scrap used to get to this point as the operator machined his numbers down, showing us the result rather than the real process.  But what is left to us in this effort is a thing of considerable beauty—the found art of practicing elementary math. 

Graphical Display of Quantitative Data: Anti-Submarine Warfare, 1973

JF Ptak Science Books LLC  Post 661

These original sheets were, it seems, prepared for an illustrated (1973) talk on a statistical model for the detection of anti-submarine warfare threats—unfortunately I’m at a loss to distinguish much of importance here, though I’m sure that someone who is the least bit familiar with defense modeling and simulation would instantly and easily interpret theserelatively simple images.  I’m presenting them here because they are to me an example of excellent presentation, design and logic—a fine symbol for the presentation of graphical information in a pre-PowerPoint past.

 

The work is that of Peter Luster, an analyst who seems to have worked in the 1960’s and 1970’s for the thinktank John D. Kettelle Corp of Alexandria, Virginia, a defense contracting firm.  (I’ve included titles  and abstracts of some of his published works in the "continued reading" section, below.)

 

These seem to comprise something like author/speaker’s notes for overhead illustrations for a presentation on a computerized system for the receipt and processing (and summarizing?) of tactical intelligence data on the enemy’s intent on locating and destroying submarines.

What strikes me is the number of cut-outs and paste-ups and such that are part of each slide--every data point, and virtually every notation on each page has been pasted in.  There's much more to this but these certainly give you a decent idea of the analyst/statistician/programmer/engineer bent over his desk doing surgery on his data. 

Hero's Stairs to Nowhere: an Intolerably Small Look at a Great and Important Book

JF Ptak Science Books LLC  Post 356

And so here we have a gorgeous title page for the very first treatise on automata, and the thing I'm focusing on are the stairs in the bottom corners of the design, the stairs that go nowhere.  Maybe its just because of the seemingly flat effect of the perspective that they became so obvious to me, or the  subtle sense of a twisted two-and-a-half dimensions feel to it, the very Escher-y feel of complex, impossible and improbable dimensionality.  For example, the very pre-Baroque ornamentation of this very heavy title page construction seem to bounce between convex and concave, like the background and foreground change place somehow, as it the case of Escher's trying to find a place between planes of two- and three-dimensional representation.  Although here I'm pretty sure the effect is unintentional; whereas in Escher, the effect is the aim of the art. 

The design and the printing of this large book was the work of the firm of Bernardino Baldi, of Venice, who published De gli avtomati oueri machine se moventi...in Venice in 1589, the original by Her/Heron of Alexandria, who lived ca. 10-70 ACE.   Hero was one of the most exalted of Greeks in the pantheon of ancient experimenters and scientists, an unparalleled inventor and producer of the earliest automatic and robotic devices. Have a look at  the illustrated link to Hero's fabulous work called Penumatics (below).

Palpable Arithmetic: Feeling and Seeing Representations of Numbers

JF Ptak Science Books LLC  Post 242

" Palpable Arithmetic", the sub-heading for the sheet illustrating aspects of algebra for Abraham Rees' (1743-1825) great if not problematic 45-volume Cyclopedia, is a system that  records and organizes and sometimes calculates using three dimensional objects.

For example the Egyptians (for one) calculated with pebbles; then there was the ABAX of the Greeks, and the abacus (and also called the mensa Pythagoras) of the Romans (and of the Japanese and earlier still of the Chinese), the scaccarium of the English (via the Norman conquest), and innumerable other systems that performed arithmetic and recording and
archiving functions via the employment of reeds, notches on a tree or cloth or stick (etc.), reeds, knots, fingers, beans,shells, string, sand, and on and on.  Palpable arithmetic also has a specialized meaning in places as a calculating device in which the numbers are recognized by touch and used by blind mathematicians or other parishioners. (Just for the record, there are a number of eminent blind  mathematicians including, for example,  Leonard Euler (1707–1783, who was blind in the last 17 years of his life), Nicholas Saunderson (who I wrote about in an earlier post), Louis Antoine (1888-1971), Lev Pontryagin (1908-1988.)) 

An interesting and very large philosophical issue that comes up here with the blind mathemaitican is the concept of image formation and its dependence upon sight for intuition, as with geometry or topology.    Plato for one determined for himself that image formation issues were precognate and the same in sight and non sighted people.  How would you manipulate a geometrical form if you’ve never actually seen one, or how would you extend you spatial imagination of compex forms without a reference? 

But my main issue here is the image from the hees book.  I'm by the meaning of this particular calculator or recording system--I just can't tell what it is.  Can you?  If so I'd love to hear from you.

Note on the Anthropology of Numbers:

From Levi Leonard Conant's The Number Concept Its Origin and Development we find these very descriptive definitions of words for numbers, all of which relate to the sort of implement that they were controlling
their numbers with, or calculating:

"in Javanese, Malay, and Manadu, the words for 1, which are respectively
siji, satu, and sabuah, signify 1 seed, 1 pebble, and 1 fruit
respectively. Words as natural and as much to be expected at the
beginning of a number scale as any finger name could possibly be. Among
almost all...the derivation of number words from these sources can constitute no ground
for surprise. The Marquesan word for 4 is pona, knot, from the practice
of tying breadfruit in knots of 4. The Maori 10 is tekau, bunch, or
parcel, from the counting of yams and fish by parcels of 10. The
Javanese call 25, lawe, a thread, or string; 50, ekat, a skein of
thread; 400, samas, a bit of gold; 800, domas, 2 bits of gold.The
Macassar and Butong term for 100 is bilangan, 1 tale or reckoning..."

The full sheet from the Rees book: