We started class reviewing the feedback from last class—it seems that everyone is pretty satisfied, which is great!  From now on, we will be doing about a half and half mix and reading review and class discussion, which is very similar to the previous class format.  A new element will be focused reading questions (non-graded) to aid the reading process, given out via email.

There will be a new blog post with the class’s topics for Assignment #2, as we all kicked butt on the assignment with an average of 9.3/10.

We ended with Hollerith last time, so today we continued right where we left off.  It was mentioned that IBM started on their path to creating the modern computer, at this moment in history, making calculating machines.  The company began as CTR, and became successful selling punch cards to other companies.  This was a very lucrative venture because each punch card can only be used once.  Howard Aiken designed the first real computer in the modern sense, produced by IBM, which cost $1.5 million in modern dollars, which wasn’t even a commercial venture.  The machine was given to Harvard University, and named the Harvard Mark I (although officially it was called Automatic Sequence Controlled Calculators).

This brings us to today’s real topic, Grace Hopper.  She worked as a mathematician for the navy, and when Aiken requested women from WAVES to do calculations (women were often used for calculations during this time, which I found surprising), Hopper started working for IBM.  We read her paper, “The Education of a Computer,” in which she talked about programming computers.  Programming in her time was very low level in comparison to modern programming.  She envisioned using programming languages to speed up and enhance the accuracy of computer calculations, because at the time, only raw numbers were able to be entered into computers.  To demonstrate this process, we played a game called “Robo Rally” in groups of four.

Each team of two was given a robot pieces, and told the following instructions:

000: Forward 1

101: Forward 2

010: Turn right

011: Turn left

100: Back up 1

The goal of the game is to reach the goal marker in 10 moves or less.  The conveyor belt spaces move you one or two spaces post turn, (depending on the number or arrows in the space), the gear spaces rotate you 90 degrees post turn, and the black spaces are death holes that you fall through and die.

The actual game, of course, does not work this way.  Instead of binary codes, players are given cards with symbols on them which represent possible moves.  This is better because sequences of zeros and ones have no semantic meaning to us, and it is very easy to make clerical errors.  Hopper rightly saw that a language to program computers would make programming far more human-friendly.

Grace Hopper did eventually create a programming language, which she called FLOW-MATIC.  At the time, programmers used flow charts to accurately use binary, which inspired the name FLOW-MATIC for her language.  IBM advertised the change thusly, “Mastering the knowledge of the complicated techniques and symbols of conventional computer flow charts requires a long training period.  Flow-Matic charting, however, can be easily grasped by anyone with knowledge of the application to be programmed.”

A fun anecdote about Grace Hopper was the story of the “first bug,” which happened to Harvard Mark II in 1947.  The word bug was used to describe as a flaw in physical design at the time, but after a moth was actually found in the machinery, disrupting a computer program, the term “bug” became a term for a programming flaw or mistake, and “debugging” became the process of fixing this mistake.

Finally, we watched this video:

Grace Hopper 60 Minutes Interview in 1982

For next time, the readings are available on the syllabus.  Questions will be sent out via email, including a request for a saying, less than 80 characters, to be brought in by each student.

Class today began with a short feedback session, in which we filled out short notes stating something we thought was going well in the class, and something that we thought could be improved.

From there we jumped stright into sharing interesting quotes and passages from the reading, “Johnny Builds Bombs and Johnny Builds Brains”. Topics of favorite quotes were very diverse: how von Neumann managed to win vast amounts of financial support from the government, likely due to his charisma and well-placed connections (rather unlike our old friend Babbage); the somewhat lucky rise of Mauchly and Eckert, and their fortuitous partnership with Goldstine, who had grown increasingly frustrated with army policies; von Neumann’s diverse and rather charmed existence, with his intellectually star-studded parties and major contributions to the fields of game theory, quantum physics, operational research (and later life itself and the construction of automata, to be called von Neumann machines); and finally the mess about who came up with which ideas first during this period of extremely rapid innovation.

This last topic started the discussion over rights and patents during this period. This started with the innovation of the stored program (and the infamous First Draft which lead many to give the credit solely, and perhaps unjustly, to von Neumann, who likely put only his name on the manuscript because it was only the draft version), as well as the arguments between von Neumann and co.’s ENIAC machine and Atanasoff and Berry’s ABC. The disagreement stemmed from a short visit by Mauchly to Atanasoff, where the exchange of ideas eventually leading to construction of the ENIAC may or may not have taken place While that disagreement was “solved” by Minnesota courts in 1973 (in favor of the ABC), discussion is still ongoing and unclear about who was responsible for which ideas during this time of extremely rapid innovation.

Discussion then flowed into an attempt to organize the figures and machines that took part in the computer revolution. What we came up with as a class was sort of a mish-mash of connected events and tangled ideas; however, this disarray was actually reflective of the time, in which many people were sharing ideas with others, as well as coming to similar conclusions through independent work. Dr. Wagstaff has graciously organized this information by hardware technology and chronologically:

1. Mechanical computers: Differential Analyzer (1931)

2. Electromechanical computers: Z3 (1941), Harvard Mark 1 (1944)

3. Electronic computers:

– ABC (1942, first vacuum tube logic, 300 tubes, binary representation, not programmable)

– Colossus (1944, 1500 tubes, limited programming with cables)

– ENIAC (1946, 18,000 tubes, decimal representation, programmed with cables)

– EDSAC (1949, Cambridge, 3000 tubes, binary, first stored program, using mercury delay line memory)

– Manchester Mark 1 (1949, first stored program, using cathode ray tube memory)

– ACE (1950, 1450 vacuum tubes, mercury delay line memory, 1 MHz)

– EDVAC (1951, 6,000 tubes, mercury delay lines)

– UNIVAC (1951, 5,200 tubes, mercury delay lines, first commercially available computer in US)

A bit more detail can also be found on this Wikipedia page, which includes a fully chronological table of events from the 1940’s. There were also several theoretical constructs included in this discussion, including self-replicating von Neumann machines, universal Turing machines, and how one could turn a Turing machine into a von Neumann machine by adapting the Turing machine’s output with a robotic construction device.

As for the figures, we discussed how the various groups formed and influenced one another. This ends up being somewhat of a web, so I shall arbitrarily choose Turing as our starting point. Turing made his initial contributions while at Princeton studying under Church, a mathematical logician. It was here that he initially came into contact with von Neumann, a student of Hilbert’s, though the contact didn’t foster much in the way of later partnerships; Turing returned to England to aid with code breaking during the war, and later drew up the plans for ACE. ACE was eventually built after Turing left the NPL (at the time the fastest computer in the world at 1 MHz), while Turing oversaw the construction of the Manchester Mark I. On the other side of the pond, von Neumann joined with Goldstine, Mauchly and Eckert in efforts that eventually lead to construction of the ENIAC (and this group’s interactions with those that constructed the ABC have been mentioned earlier), which later blossomed into the EDVAC. Other somewhat more independent mentions were Aiken, who was responsible for the Harvard Mark I, and the group at MIT who constructed the Differential Analyzer.

And, finally, any discussion mentioning von Neumann machines would be incomplete without the thoughts of philosopher Randall Munroe.

We picked up right from where we left off the previous Wednesday, and as part of a brief review we discussed the moral and political consequences of code breaking.

The primary example of this dilemma involved the British losing fifty to eighty of their ships a month. As this was basically as fast as they could build them it was an obvious problem, with the loss of life compounding the issue. There was an Enigma encoded message that the British intercepted from the Germans that, upon decoding, revealed the location of nine German warships. The Royal Navy didn’t want to let on that they had broken Enigma, lest the Germans change the cipher again, so the British destroyers were only told the location of seven of the ships, with command fearing that if all the ships were sunk the Germans would catch on. The destroyers, however, ran into the other two ships on their way to sink the seven, and sunk them as well. Despite this, the Germans were so sure that Enigma was unbreakable that they assumed there was a spy somewhere in their ranks. They didn’t consider the possibility that Enigma could be at fault.

Moving on, we discussed another decrypting machine that was employed in the British war effort: Colossus. Colossus was a giant computer that was designed not to crack Enigma, but to crack the Lorenz cipher, a code that was used between Hitler and his field generals. In some ways it was more advanced than the Enigma decryption techniques that Turing and the British cryptographers employed – it used 1500 vacuum tubes and was the first machine to employ them for computation – but it also looked for matching keys using brute force. Once a matching key was found, the user had to manually do the decryption.

Turning to the reading, we first reviewed Turing’s paper “On Computable Numbers.” It was quickly apparent that few people in the class understood much of Turing’s paper beyond a basic outline and what he was trying to prove, so we started breaking it down.

We first covered some background, why the paper was written in the first place. The mathematician David Hilbert wanted mathematics to be complete and encompass all knowledge, and to be a system where every presented problem could be solved. Kurt Gödel came and showed that, actually, this wasn’t possible in his paper “On Formally Undecidable Propositions of Principia Mathematica and Related Systems.” This was a difficult to understand paper, but Turing’s paper made it more comprehensible by using something he called a Turing Machine to help explain his ideas.

We then took some time to define a Turing Machine, and Dr. Wagstaff took the time to draw one on the board. A Turing Machine is made of some sort of input/output system, in this case a tape, and something that can read and write to the tape. The tape contains various symbols, all of which are contained in a possible set of symbols. In the drawing to the right, the current symbol is denoted S_i. The machine also has a configuration q_i, which we’d refer to as a ‘state.’ In this state is implicit memory, for example, if you’re in state one and see an A, do this, if you’re in state 2 and see an A, do that. This description helps to clarify the tables in the book. These tables are examples of Turing programs, and the example Dr. Wagstaff drew below. What the machine would do in reading these programs is read the state q_i, read the symbol S_i, print a symbol S_k based on certain parameters (or not), and then move to the state q’. Additionally, an entire Turing Machine can be recorded as a number. This is crucial because another Turing Machine can then read in that number and simulate that machine.

So how is this relevant to the mathematical completeness that Hilbert so wanted and Gödel and Turing showed wasn’t possible? It comes down to the Entsheidungs Problem, also called the Decision Problem. Basically, the problem asks if it’s possible to design a system that can take any logical or mathematical statement and decide if it’s true or not. Turing modified this and called it the Halting Problem. If a program halts, it ends. If it doesn’t it continues on an infinite loop. The primary question, then, of the Halting Problem is whether it’s possible to design a program that can read in another program and determine if the other program halts or not. Turing addresses this with what is essentially a more complicated version of the proof (by contradiction) Dr. Wagstaff outlined to the right. What this says is that if you assume that, like a Turing Machine, any program can be written as a number, and that you can write program H(p) that can read in any program and will return 1 or 0 if program p halts or does not halt, respectively. So then you define another program G(p) such that if program H(p) returns 0, then G(p) returns 0, and if H(p) returns something not zero, then G(p) goes into an infinite loop. Now, every machine can be written as a number, so G(G) is possible. And then the complicated part: G(G) takes action based on the output of H(G). If G does not halt, then H(G) returns 0, which would mean that G(G) would return 0. This is a contradiction because G does not halt, but returns 0 when presented with itself. Likewise, if G does halt, then H(G) returns 1, which would mean that G(G) would go into an infinite loop. This too is a contradiction, as G does not halt, but goes into an infinite loop. This means that the assumption that H(p) can be written is false. No such program is possible to write.

Moving on from complicated proofs, we looked at the reading from Diamond Age. We first went over some background on the passage to give it some context, and then went into the details of the passage. The main character, the girl, is playing a game with a “primer,” which the class likened to an iPad. Princess Nell is her character, and her character has been imprisoned by automatons. She then has to administer a Turing Test to determine if her captors are human or machine, as this will determine her method of escape. After learning that her captors are machines, she escapes to the top of the tower where she find the skeleton of the Duke of Turing. She reads his books and journals, complete with references to ‘bugs in the machine,’ and masters them. After learning how to write her own programs in the chains used to hold programs, she becomes the Duchess of Turing.  The passage mostly focused on the Turing Test, figuring out if her captors were human or not, but it does involve a Turing Machine that functions as the lock on her door. The numbers on the lock describe what state the machine is in, and with their help she was able to reverse engineer the lock by running different chains through the machine and seeing how the states changed.

There was also brief mention of the book Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter, which discusses knowledge, meaning, and thinking.

We then transitioned and watched a clip from the movie Breaking the Code, which is based on a play about Alan Turing. (Interestingly, this clip isn’t in the American cut of the film.)  The clip involves Turing explaining his work to someone reviewing it, and Turing basically explains what we were talking about for the first half of class. This person says that Turing’s paper is “baffling,” specifically pointing out the title. After a request to explain, Turing gives a very detailed and helpful explaination as to what a Turing Machine is for and how it relates to the Entsheidungs Problem. He explains that it is about trying to prove right from wrong, and in a review of the history of attempts at this he mentions someone to trying to break down everything in to pieces of pure logic. He notes that this, of course, failed, and attempts to analyze mathematical axioms led to new types of mathematics. He describes that Hilbert thought it was possible to have a fundamental system for mathematics, with consistency, completeness, and decidability. Turing then notes that Godel showed this was impossible, and that math is either inconsistent or incomplete. Turing realized that he would have to have a system of proving all mathematical statements past, present, and future for Hilbert to be correct, which is what his Turing Machine idea would be designed to do. He notes that, of course, a Turing Machine cannot do this.

Class then closed with the assignment for the next class.

Class started with a reminder that homework assignment #2 is due Monday, and due Wednesday a selection from the IEEE archives for the final project.

The anonymous self-written tests from our previous class were discussed, and praised for their depth and insights. We expanded on an answer to “why did Hollerith use punched cards instead of tape”, noting that cards could be resorted and accessed randomly, while tape constrains one to always reading data in a particular order. This is analogous to the difference today between “random access” and “serial access”.

Next, we went back in time to World War I. In May 1915, the RMS Lusitania, a British merchant ship with many American citizens aboard, was sunk by German U-boats. This was before America had entered World War I. Germany was starting to flex its naval muscles with its U-boat fleet. In 1917, Germany sent the famous Zimmerman Telegram to its ambassador in Mexico, seeking to form an alliance with Mexico. Germany would assist Mexico in reclaiming former Mexican territory including Texas, New Mexico, and Arizona; in return, Mexico would form an alliance with Germany, attacking the United States to prevent them from engaging Germany in Europe. British intelligence intercepted the note during its initial transmission, and it was decoded by their cryptographers. The note was published in American newspapers, purporting that it had been intercepted by spies when the note was in (unencrypted) transmission via telegram across Mexico. The revealing of this information prompted America’s entrance to the war. It wasn’t until 1923 however that Churchill admitted that British cryptographers had in fact intercepted and decoded the message – throughout the war, the Germans had no idea that their private communications were being intercepted.

An enigma machine being used in Russia. - German Federal Archives, via Wikimedia Commons

This dramatic breach of security for the Germans led to the development of the enigma machine. Over 30,000 of the machines were built during the course of World War II. The mechanisms of this machine were the subject of our reading for today, an excerpt from The Code Book.

To get a better sense for how the enigma machine operates, we used paper enigma simulators to decode a secret message on the board: MPXNCZJA. The scramblers were arranged in order of 1-2-3, and the day’s key was MCX. Recall that the first three letters are the message key, leaving NCZJA as the message. Decoding the message and taking it to the honors college office was rewarded with a prize.

We wrapped up our discussion by reflecting on what ultimately led to the failure of the system: not the encryption itself, but its human users. The use of repeated, predictable phrases, easy-to-guess message keys, and other patterns to exploit led to the messages being decrypted far faster than they would have been if better practices had been followed. As with many security systems, the weakest link in the chain of security is not the system itself, but those who must operate it.

In class today we talked about the results from the survey passed out during last class. It was found that people liked the concept map as well as learning about Ada and Babbage’s relationship. It was also found that people were unclear on how the analytical machine and other machines worked. Dr. Wagstaff encouraged us, if we were interested, to consider doing our final project on it, so that we may explain how it works to the class. Also, people were unclear on how much work Ada actually did on the notes and how much Babbage helped her. The fact is that we’re really not sure how much she did and probably never will. Also, the vast majority thought that the class work was about right at the moment. However, Dr. Wagstaff has decided to drop one of the assignments, so that we have two weeks to do each assignment.

In class, we had an activity where we wrote two questions that we would put on a “quiz” about the reading from last time. The first one being a “how” question and the second being a “why” questions. The questions were passed around and each person had to write down the answered and then the answers were passed around and “graded”. We then discussed things that we found interesting about the questions and answers.

1. How did Hollerith’s machine get and store data?

1890 punch-card template for Pantograph

Punched cards were used to input data. These were put in the machine which interpreted them. We thought that it was interesting that the machine had constraints on what would be considered valid data. These constraints included that cards could only be put in one way. This was done by making one of the corners of the card cut at an angle. This way the card could only be putting the reader one way. Another restraint was that the card was considered invalid if two mutually exclusive things were entered. So for example if male and female were selected the card was considered invalid. Also, if the user was only looking at one type of cards (males for instance) if another type of cards (females for example) was inputted it would be tossed out.

Woman operating a Pantograph

Cards were punched through a pantograph, which was basically a template through which holes were punched. The machine was composed of a tabulator and a sorter. Click here for some good images of the machine. The tabulator counted up everything by lowering down a tray with pins attached to it; if there were was a hole where the pin was it would go through the hole into a cup of liquid mercury. The mercury acted as a very good conductor and would make an electrical current. Part of the reason mercury was used is that the whole device was hooked up to a battery (they didn’t have a power grid during this time), so the voltage across the device was very poor. The sorter flipped open the bin to put in the card, but someone still had to manually put each card in the sorter.

2. Why was it necessary for Hollerith to make this machine

With the 1890 census it would not be able to finish the census before the next census (in 10 years). The 1880 one took 7 years to complete and there were more people in 1890 than 1880. For the 1890 census there was a race to see what method/machine could do the best, which would be used for the next census. One was Hollerith’s machine, one was a chip method which used different colored paper, and one was the slip method which used different colored ink. They had a race in which they had to process 10,000 cards. The chip method took 110.933 hours for transcription and 44.683 hours for tabulation, the slip method took 144.4167 hours for transcription and 55.367 hours for tabulation, and Hollerith’s method took 72.45 hours for transcription and 5.467 hours for tabulation. So, that makes Hollerith’s method a little over twice as fast as the chip method and a little over two and a half times as fast as the slip method.

We talked also about how single-minded Hollerith was. Hollerith only thought of one use for his machine: the census, and didn’t consider his machine’s usefulness for other scenarios, such as other data entry.

Next we went through an activity to see how long it would take for our hometown’s census to be processed back in the day. With Hollerith’s machine it took 25 seconds/person for transcription and 2 seconds/person for tabulation.

Example for my hometown: Sherwood, OR

Population 18,194

Transcription=25*18194=4.55E5 seconds=7.58E3 minutes=1.26E2 hours=5.26 days

Tabulation=2*18194=3.64E4 seconds=6.065E2 minutes=10.1 hours

Total=1.36E2 hours=5.67 days

We then talked about why the US was able to set up new things like the machine for the census, when Europe had just kept with people doing the census. One of the reasons is that the US had just gotten out of the civil war and was really far behind Europe, so the US was open to innovation. Another reason was that it would theoretically save time and money. However, for the years it saved the census, they spent it doing even further analysis on it! In the end, they ended up spending twice as much money as the 1880 census!

Images from: http://www.columbia.edu/cu/computinghistory/census-tabulator.html

Class began with Dr. Wagstaff telling us a bit more about herself. She grew up in a small town near Moab, Utah. Her undergraduate degree was in computer science at University of Utah; she followed that up with a graduate degree in computer science with a minor in Mars from Cornell University, and then worked in algorithm development for space programs at Johns Hopkins University Applied Physics Lab and finally JPL.

Dr. Wagstaff then directed us to the two following puzzles:
2, 6, 12, 20, 30, ?? = 42 = x^2+x
-3, 0, 15, 48, 105, ?? = 192 = x^3-4x

The easiest method to solve these puzzles was the one employed by Babbage’s Difference Engine No. 2: the first puzzle had a constant difference of differences (2) between each number, and the second puzzle had a constant difference of difference of differences (6). Each of these corresponded to a polynomial equation of the kind that the Difference Engine could solve. A crucial point was that in Babbage’s day, it was essential to have accurate tables of polynomials, logarithms and trigonometric functions for all kinds of calculations theoretical and practical, navigation, et cetera. This was the reason for Babbage including printing on paper and printing of plaster molds in his machine – so that the tables would never need to be copied by a human, and thus would remain reliable.

Another effect of Babbage’s desire for reliability was his demand for high quality machined parts. We discussed his disagreements with his engineer, Joseph Clement, and how most of the British government’s grant money likely ended up in the engineer’s pockets. Also mentioned was Babbage’s inability to be diplomatic, as related even in his autobiography – his funding dried up as he could not convince government officials such as the Chancellor of the Exchequer of the important of his machines. A side note to this was that a Swedish engineer, Per Georg Schultz, created a derivative Difference Engine for several governments – Sweden, England and eventually the United States. His engine was delivered over-budget but on time.

Babbage’s Difference Engine No. 2 was not fully completed until recently. We saw a video that detailed the course of his endeavors, from the first difference engine, and the analytical engine, to the complete and improved plans for the second difference engine. Most interestingly, this video featured many shots of the replica Difference Engine No. 2 in action.

Babbage’s Analytical Engine, however, has never been fully realized. The same video showed the ‘mill’ of the Analytical Engine, which Babbage’s son eventually completed. The mill was to be the area in which computations were done, as opposed to the ‘store’ of memory.

photo credit: http://www.zdnet.co.uk/i/z5/illo/nw/story_graphics/11mar/science-museum/science-museum-babbage.jpg

The Analytical Engine also incorporated three kinds of punched cards – operations, numbers, and variables (which were essentially addresses). This engine has never been completed, but a group at http://plan28.org/ is taking the first steps towards building it.

Class ended with a question to ponder: was Charles Babbage a success or a failure?

Today, Wednesday, Professor Wagstaff showed us how to use WordPress. The instructor showed us what the dashboard is and how to submit a new post. It is recommended that we save the draft regularly and preview it. Also, if we want to post a link, we can go to the preview and make sure that it works.

Next, we shared the pictures for our first assignment. It is optional to post the first assignment to the blog. Most peoples’ first memory with a computer was playing games rather than using it as a tool for calculation as it was meant for.

After that, we were shown some of the machines leading up to the computer including the Napier’s Rods and the Slide Rule. Napier’s Rods were first invented in 1617. It was made up of a bunch of rods that were lined up to make the number to multiply. Then the numbers in the parallelograms were added up to find the answer. The Slide Rule, created in 1654, was a combination of logarithms and the Napier’s Rods concepts. Next we learned about the Calculating Clock, invented by Wilhelm Schickard in 1623. It included a bunch of Napier’s Rods which were vertically placed and some other bars horizontally placed. In order to multiply, the top bar had to be twisted and the horizontal bars slid or if you wanted to add, then dials on the bottom of the machine were twisted. The clock was destroyed in a fire and “was lost to history until someone discovered Schickard’s notes.” The Pascaline was next, invented in 1642 and it was thought of as the first computer until Schickard’s notes were discovered later on. Lastly, we learned about the Stepped Reckoner, which was invented around 1673. It was a little more mechanized and worked better than the Pascaline.

When this was done, we started discussing the assigned reading about the Jacquard Loom. The weaving was operated by punch cards which set the pattern. The most important aspect of the Jacquard Loom is that it sped up weaving and it can replicate works. It eliminated the need of a “draw boy.” Modern versions weave airbags and replacement valves for blood vessels in a sterile environment. It is now fully computerized. This is a video of the Jacquard Loom: https://www.youtube.com/watch?v=NSjmFD6Q7hw.

On Monday, we went over the course syllabus and began to talk about computing. In the syllabus, we highlighted the objectives, schedule, assignments, and grading. Specifics readings are listed on the course website and should be completed before the due date, as these are essential to the class discussion. In addition, to readings there are specific assignments. These should be submitted electronically before the class in which they are due as a pdf via email to the instructor, Kiri Wagstaff. If anyone has trouble creating or cannot create a pdf, please let the instructor know. It is also recommended to post them to the class website so that everyone can see your ideas and viewpoint, but this is not required. The first assignment is due October 3^rd. If you happen to lose your schedule, it is downloadable on the Syllabus and Schedule page.

After discussing the syllabus, we discussed computers that we encounter on a regular basis. Examples included GPS, video gaming consoles, cell phones, calculators, laptops, desktops, lights, slide rule, abacus, etc. From this stemmed the question of what differentiates a computer and a calculator. An explanation given in class consisted of the following:

• A computer can store programs. These programs can be brought up later and accessed.
• A calculator has to be told each time to do a computation. It doesn’t store programs and instead has to be accessed each time.

However, during class it was noted that this is a very simple explanation and it will be covered more in depth later.

A little video was shown in class, which briefly covered the History of Computers: Computer History in 90 Seconds.

photo credit: http://711tech.com/?p=23

Also the abacus and napier’s rod were discussed. The abacus is a calculating tool that consists of two sections of rows of beads. In the top row, the beads represent the number 5, whereas the bottom represents the number 1.  Those that touch the middle bar are counted. If mastered, the abacus can greatly help in calculations. As seen in a video of Chinese school children using the abacus: Amazing Abacus Math Video.

Some background for napier’s rod was talked about during class, such as the logarithmic tables. For which numbers were converted to exponentials. This would make multiplication and division easier, since the exponents would just need to be added or subtracted.