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Monday, September 26

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Wednesday, July 7

  1. page Ancient Computers edited ... If you stare at an old mechanical calculator it just sits there. It does no computing and is, …
    ...
    If you stare at an old mechanical calculator it just sits there. It does no computing and is, therefore, not a computer. When a person starts punching the keys and turning the crank the person-device computes and is a computer. So too, an abacus is just an assembly of beads and rods or lines and pebbles, and is not a computer. But when a person uses the abacus to perform calculations, then the person-abacus is a computer. A remarkably fast and accurate computer, as demonstrated by a Japanese abacus (Soroban) operator who beat a skilled electric calculator operator in a contest in Tokyo on Nov 12, 1946 (Kojima, p.12).
    In classic Greek architecture, an abacus is a flat slab of marble on top of a column’s capital, supporting the architrave, or beam. Such an abacus (perhaps chipped beyond use in construction) makes a fine flat surface on which to inscribe lines; from which we get the name, counting board abacus. Developed later, constrained bead devices with less arithmetic functionality are also called abaci, e.g., Roman Hand Abacus, Chinese Suan Pan, and Japanese Soroban.
    “TheThe Ancient Romans were excellent practical engineers and architects. Even today we marvel over their accomplishments and wonder how they did them. For example, in a BBC2 sponsored series, Building the Impossible, http://www.materials.ac.uk/ awareness/building/index.asp, Episode 2: The Roman Catapult, structural engineer Chris Wise wonders how the Romans could do the calculations necessary to design and build the Roman Catapult used to destroy the walls of Jerusalem, when the math necessary wouldn't be developed until 1500 years later! (1st minute of http://www.youtube.com/watch? v=q_dHpLAPM5I).
    “The
    Roman expression
    ...
    pebbles.'” (Jen) Certainly the Romans would also use their abaci for engineering calculations.
    Historians have published conjectures for what ancient counting board abaci looked like and how they were used (Menninger, pp.295-306; Ifrah, pp.200-211). On p.205 of Ifrah, he concludes, “Calculating on the abacus with counters was ... a protracted and difficult procedure, and its practitioners required long and laborious training.”
    That is not true.
    (view changes)
    4:08 am

Tuesday, July 6

  1. page Ancient Computers edited ... 1st abacus design (your hands have 10 digits) (promotion factors along the left) 9834 wit…
    ...
    1st abacus design
    (your hands have 10 digits)
    (promotion factors along the left)
    9834 with 24 pebbles:
    ––––––––––– 10000s
    –000000000–10–000000000– 1000s
    ––00000000–

    10––00000000–
    100s
    –––––––000–

    10–––––––000–
    10s
    X–––––0000–

    10X–––––0000–
    1s
    X marks unit line
    2nd abacus design
    ...
    9834 with 16 pebbles:
    ––––––––––– 10000s
    02 0 5000s
    ––––––0000–

    5––––––0000–
    1000s
    0

    2 0
    500s
    –––––––000–

    5–––––––000–
    100s
    50s
    –––––––000–

    2 50s
    5–––––––000–
    10s
    5s
    X–––––0000–

    2 5s
    5X–––––0000–
    1s
    3rd abacus design
    (opposites, yin-yang, male-female, left-right)
    (promotion factors along the left)
    Noting that 9=10-1=IX, 8=10-2=IIX, 4=5-1=IV, and 3=5-2,
    9834 uses only 10 pebbles:
    ...
    What if a number is larger than the abacus grid? E.g., 9,834,000,000,000,000 = 0.9834 x 1016.
    To the ancients the exponent would be a scaling number.
    ...
    many lines isshould the unit line be shifted down?).
    Only 9 pebbles are needed to represent 9,834,000,000,000,000:
    – +
    ...
    5––––0|––––– 1/10000s
    5th abacus design
    ...
    extended to sexagesimal:sexagesimal, and the Babylonians' cuneiform numbers fit nicely into the structure:
    – +
    –––––|––––– 60s
    ...
    The Sumerians had a better reason for their septimalism. They worshiped seven gods whom they could see in the sky. Reverently, they named the days of their week for these seven heavenly bodies. (Wilson)
    The divisibility of 60 was a convenient coincidental consequence, but not the primary reason the Sumerians developed a sexagesimal number system. They did so from the periods of the two slowest moving of their seven sky gods. Jupiter and Saturn take 12 and 30 years, respectively, to track through the Zodiac. The observant Sumerians knew this. The least common multiple of 12 and 30 is 60.
    ...
    2 hands. Playing with the numbers, inIn both cases
    ...
    this as a manifestationmanifestations of the
    ...
    12 degrees. Thereby coupling the
    ...
    year is actually 365 days,days simply by watching bright stars like Sirius, so maybe they just
    Origin of The Salamis Tablet Structure
    ...
    Figure 7 (Opening slide(First scene of video
    {STstructure.jpg} Figure 7: Structural Design of The Salamis Tablet
    Computing in Sexagesimal
    (view changes)
    9:43 am
  2. page Ancient Computers edited Lütjens Possible Future Research Could the methods outlined here for using The Salamis Tablet b…
    Lütjens
    Possible Future Research
    Could the methods outlined here for using The Salamis Tablet be implemented electronically?
    (view changes)
    9:03 am
  3. page Ancient Computers edited ... 2 |0 5s 5––––0X––––– 1s But since –100=(–50)+(–50) -100=(-50)+(-50) and –10=(–5)…
    ...
    2 |0 5s
    5––––0X––––– 1s
    But since –100=(–50)+(–50)-100=(-50)+(-50) and –10=(–5)+(–5),-10=(-5)+(-5), [demotions]:
    – +
    –––––|0–––– 10000s
    ...
    2 00|0 5s
    5––––0X––––– 1s
    ...
    since -k+k=0 (cancellation):[cancellation of zero pairs]:
    – +
    –––––|0–––– 10000s
    ...
    What if a number is larger than the abacus grid? E.g., 9,834,000,000,000,000 = 0.9834 x 1016.
    To the ancients the exponent would be a scaling number.
    ...
    the lower grid and thengrid, count lines down from
    ...
    the upper grid.grid (how many lines is the unit line shifted down?).
    Only 9 pebbles are needed to represent 9,834,000,000,000,000:
    – +
    ...
    5–––––|––––– 1/3600s
    But Why Base-60, 360 Degrees in a Circle, 7 Days in a Week, etc.?
    ...
    around 2500 BC,BC (Wilson), and inherited
    The Sumerians were great innovators in matters of time. It is to them, ultimately, that we owe not only the week but also the 60-minute hour. Such things came easily to people who based their maths not on a decimal system but on a sexagesimal one.
    Why were these clever chaps, who went for 60 because it is divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30, fascinated by stubbornly indivisible seven? ...
    ...
    In 60 years Jupiter would go through 5 cycles and Saturn 2. We have 5 fingers on each of 2 hands. Playing with the numbers, in both cases 5+2=7, the number of sky gods. The mystical Sumerians would think of this as a manifestation of the sky gods reflecting themselves in our anatomy.
    The product of 12 and 30 is 360, the number of degrees in a circle; did the Sumerians define the 360 degree circle? Dividing the Zodiac into 360 parts means Jupiter would traverse 30 degrees in a year and Saturn 12 degrees. Thereby coupling the periods of the gods Jupiter and Saturn.
    ...
    into 12ths, e.g.,i.e., 12 months;
    ...
    of Saturn? Then the Sun tracks about 1 degree every day. Of course they knew that a year is 365 days, so they just added a 5 day Holiday.
    Origin of The Salamis Tablet Structure
    Both the astronomical and the anthropomorphic features of The Salamis Tablet in sexagesimal mode lead to the conclusion that the Babylonians, or their ancestors the Sumerians, were its designers; see Figure 7 (Opening slide of video 9.1, Stephenson). The Egyptians, Greeks, and Romans borrowed it for decimal and duodecimal calculations.
    ...
    On average, the calculation of each partial product would require: two additions, two table lookups, half a doubling, two subtractions, and two halvings. Combining the 19 partial products will require another 18 additions, being careful to add into the proper place value. In all, there are 56 additions, 38 table lookups, 8 doublings, 38 subtractions, and 38 halvings; a total of 178 operations!
    How would you keep track of all this if you're limited to using reeds to write cuneiform on clay tablets? How many errors would you make? How would you find them?
    ...
    and used abaciabaci, with built-in error checking, to do
    YBC 7289 sqrt(2) Calculation
    "The Babylonian clay tablet YBC 7289 (c. 1800-1600 BCE) gives an approximation of sqrt(2) in four sexagesimal figures" (retrieved 7/4/2010 from http://en.wikipedia.org/wiki/Square_root_of_2#History), a remarkably accurate achievement for the time.
    The calculation of sqrt(2) on a
    {HeronsMethod.jpg} Figure 8: Heron's Method.
    ...
    a lot longer than Stephenson's videos ... a LOT longer. TheyThe Historians should record atheir own video and post it to YouTube so we can compare its
    ...
    early arithmetic practice exercises, modern
    ...
    all their practice exercises document
    ...
    Old Babylonian period.period, where we get most of our tablet artifacts and knowledge of Babylonian mathematics (Melville).
    Conclusions
    Features of the Roman Hand Abacus indicate that the Romans used a counting board abacus exactly like The Salamis Tablet for their heavy-duty calculations; it also gives us the promotion factors between lines and spaces. The Subtractive Notation of Roman Numerals indicates that one side of The Salamis Tablet grids are used for the additive part of a number and the other side for the subtractive part of the number.
    (view changes)
    8:01 am
  4. page Ancient Computers edited ... Convert to another unit of measure; e.g., 104 yards = 312 feet; or State the answer in two pa…
    ...
    Convert to another unit of measure; e.g., 104 yards = 312 feet; or
    State the answer in two parts; e.g., 104 yards = one hundred and four yards.
    Possible HistoricalPlausible Historic Sequence in
    1st abacus design
    (your hands have 10 digits)
    ...
    2 |0 5s
    5––––0X––––– 1s
    But since 100=50+50–100=(–50)+(–50) and 10=5+5 (demotion):–10=(–5)+(–5), [demotions]:
    – +
    –––––|0–––– 10000s
    (view changes)
    6:58 am
  5. page Ancient Computers edited ... {salamisTablet.jpg} {HAmap2ST.jpg} ... Salamis Tablet, before before 300BC (Ifrah, …
    ...
    {salamisTablet.jpg}
    {HAmap2ST.jpg}
    ...
    Salamis Tablet, before
    before
    300BC (Ifrah,
    ...
    Hand Abacus mapped
    mapped
    onto The
    The mapping is perfect. It uses the bottom grid’s eleven lines exactly, no more, no less. However, to do so the Romans had to use a less preferred structure for one of the base-12 digits. Why it’s less preferred is addressed below. But the fact that they had to make an engineering compromise is indicative that they used The Salamis Tablet as a design template for their Hand Abacus.
    In Figure 4, the numbers on the left are the promotion factors that are dictated by the Roman Numerals mapped to the lines and spaces. For example, a promotion factor of 5 means that 5 pebbles on that line can be replaced by one pebble in the space above. All the spaces between lines have a promotion factor of 2. (The unused dashed line is explained below.)
    ...
    significant digits. (Stephenson, video 9.1)
    As an example of using the subtractive side, if in the year 2009 = MMIX you wanted to calculate the age of a person born in 1946 = MCMXLVI, you would first make 1946 negative by moving each pebble to the opposite side of the vertical median line. Then to make room for the next addend, you would slide each of the pebbles as far away from the vertical median line as possible, left or right. Now you would add 2009 = MMIX by placing pebbles in the empty spaces next to the median line: two on the right side of the (|) or M line, one to the left side of the I line, and one to the right side of the X line.
    Merging the pebbles and replacing the C pebble with two L pebbles and one of the X pebbles with two V pebbles (both operations being demotions), then removing zero-sum pairs on every line and space (a pebble on each side of the median is a zero-sum pair) you read the answer LXIIV = 63. (Try it with pennies as pebbles and an abacus drawn on paper.)
    (view changes)
    6:54 am
  6. page Ancient Computers edited Lütjens Possible Future Research Could the methods outlined here for using The Salamis Tablet b…
    Lütjens
    Possible Future Research
    Could the methods outlined here for using The Salamis Tablet be implemented electronically?
    ...
    In classic Greek architecture, an abacus is a flat slab of marble on top of a column’s capital, supporting the architrave, or beam. Such an abacus (perhaps chipped beyond use in construction) makes a fine flat surface on which to inscribe lines; from which we get the name, counting board abacus. Developed later, constrained bead devices with less arithmetic functionality are also called abaci, e.g., Roman Hand Abacus, Chinese Suan Pan, and Japanese Soroban.
    “The Roman expression for 'to calculate' is 'calculus ponere' - literally, 'to place pebbles'. When a Roman wished to settle accounts with someone, he would use the expression 'vocare aliquem ad calculos' - 'to call them to the pebbles.'” (Jen)
    ...
    Ifrah, pp.200-211). On p.205 of Ifrah, p.205,he concludes, “Calculating
    That is not true.
    Clues
    ...
    The two rightmost columns handle the Roman’s base-12 fractions and both count to twelve, but differently. The left column counts to five in the lower slot and carries into the upper slot on a six count, repeats to a count of eleven, then carries into the decimal units column on a twelve count. But the rightmost column breaks each six count into two three counts. Why the difference?
    Mapping the Roman Hand Abacus slot symbols onto The Salamis Tablet of Figure 3 results in Figure 4.
    {salamisTablet.jpg} Figure
    {HAmap2ST.jpg}
    Figure
    3: The
    ...
    (Ifrah, p.201)
    {HAmap2ST.jpg} Figure

    Figure
    4: Roman
    The mapping is perfect. It uses the bottom grid’s eleven lines exactly, no more, no less. However, to do so the Romans had to use a less preferred structure for one of the base-12 digits. Why it’s less preferred is addressed below. But the fact that they had to make an engineering compromise is indicative that they used The Salamis Tablet as a design template for their Hand Abacus.
    In Figure 4, the numbers on the left are the promotion factors that are dictated by the Roman Numerals mapped to the lines and spaces. For example, a promotion factor of 5 means that 5 pebbles on that line can be replaced by one pebble in the space above. All the spaces between lines have a promotion factor of 2. (The unused dashed line is explained below.)
    ...
    KojimaKojima, T. (1954). The Japanese Abacus: Its Use and Theory. Tokyo, Japan: Charles E. Tuttle Company.
    LütjensLütjens, J. (2010, April 5). Abacus-Online-Museum. (April 5, 2010) Retrieved June 21, 2010, from Jörn's Online Museum: http://www.joernluetjens.de/sammlungen/abakus/abakus-en.htm
    ...
    Mathematics: http://it.stlawu.edu/~dmelvill/mesomath/obsummary.html
    MenningerMenninger, K. (1969). Number Words and Number Symbols: A Cultural History of Numbers. (P. Broneer, Trans.) New York, New York, U.S.A.: Dover Publications.
    O'ConnorO'Connor, J. J. (2008, April 21). An overview of Babylonian mathematics. Retrieved June 21, 2010, from MacTutor History of Mathematics: http://www-history.mcs.st-and.ac.uk/history/HistTopics/Babylonian_mathematics.html
    (view changes)
    6:50 am
  7. page Ancient Computers edited ... KojimaKojima, T. (1954). The Japanese Abacus: Its Use and Theory. Tokyo, Japan: Charles E. Tut…
    ...
    KojimaKojima, T. (1954). The Japanese Abacus: Its Use and Theory. Tokyo, Japan: Charles E. Tuttle Company.
    LütjensLütjens, J. (2010, April 5). Abacus-Online-Museum. (April 5, 2010) Retrieved June 21, 2010, from Jörn's Online Museum: http://www.joernluetjens.de/sammlungen/abakus/abakus-en.htm
    MelvilleMelville, D. (1999, September 3). Old Babylonian mathematics. Retrieved July 6, 2010, from Mesopotamian Mathematics: http://it.stlawu.edu/~dmelvill/mesomath/obsummary.html
    MenningerMenninger, K. (1969). Number Words and Number Symbols: A Cultural History of Numbers. (P. Broneer, Trans.) New York, New York, U.S.A.: Dover Publications.
    O'ConnorO'Connor, J. J. (2008, April 21). An overview of Babylonian mathematics. Retrieved June 21, 2010, from MacTutor History of Mathematics: http://www-history.mcs.st-and.ac.uk/history/HistTopics/Babylonian_mathematics.html
    (view changes)
    5:37 am

Monday, July 5

  1. page Ancient Computers edited ... Clues for the true structure and methods of the ancient counting board abaci are contained in …
    ...
    Clues for the true structure and methods of the ancient counting board abaci are contained in three extant artifacts: The Japanese Soroban (Kojima; Menninger, pp.307-310), The Roman Hand Abacus (Ifrah, p.210; Menninger, p.305), and The Salamis Tablet (Ifrah, p.201; Menninger, p.299).
    Every rod on a Soroban represents one decimal digit (Figure 1). The bead above the bar represents five of the beads below the bar. Each bar can count from zero (no beads next to the bar) to nine (all beads moved next to the bar).
    {soroban.jpg} Figure
    Figure
    1: A
    ...
    Soroban
    {romanHA.jpg} Figure
    {HAsource.jpg}
    Figure
    2A: Roman Hand Abacus replica
    replica
    (Lütjens)
    {HAsource.jpg} Figure

    Figure
    2B: Source
    ...
    Figure 2A (Welser,
    (Welser,
    p.819)
    On the Roman Hand Abacus (Figure 2), each of the seven decimal digits has four beads in the lower slot and one bead in the upper slot; functioning exactly like the Soroban. It would be hard to understand why the Romans would not have developed similarly efficient methods to use the Hand Abacus as the Japanese did to use the Soroban (Kojima).
    The two rightmost columns handle the Roman’s base-12 fractions and both count to twelve, but differently. The left column counts to five in the lower slot and carries into the upper slot on a six count, repeats to a count of eleven, then carries into the decimal units column on a twelve count. But the rightmost column breaks each six count into two three counts. Why the difference?
    (view changes)
    7:40 pm

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