5 ms·

> I can multiply numbers, of arbitrary size. And I can do it in a few seconds using just a stylus and tablet. So could a Roman or Greek. He'd just need a cou

> I can multiply numbers, of arbitrary size. And I can do it in a few seconds using just a stylus and tablet. So could a Roman or Greek. He'd just need a counting board and some stones. He could probably do it faster than you, as well. Anyone using an abacus today could perform almost exactly the same algorithms, with the added convenience of having beads on rods in a frame instead of an unwieldy counting board and stones. The algorithms are the same, and you can also do division and square roots with a counting board or abacus. When done deftly moving stones around on the counting board (and Aristotle makes clear that this goes so quickly that it's possible for the person calculating to cheat an onlooker, like in a con-man's shell game), one might choose to write the result down on a tablet, and he might just finish writing before you do.

jacobolus 5y ago
> goes so quickly that it's possible for the person calculating to cheat an onlooker Indeed, the biggest advantage of paper arithmetic is that it leaves a written record: each step in the algorithm can be checked for mistakes afterward. It is otherwise significantly slower and not inherently more accurate than a counting board. The other advantage that paper arithmetic has is that it can be easily reproduced in printed books, making learning more advanced techniques more accessible / easier to spread to anyone literate, without requiring an expert teacher. (And finally, paper arithmetic [eventually] has the advantage that it can be more conveniently extended and generalized to include more kinds of operations and structures, in a way that is easier to explain and teach than adding new kinds of counting board rules. Paper arithmetic is a more natural precursor for symbolic algebra than counting-board calculation.) The big disadvantage of paper arithmetic is that it depends on widespread literacy and cheap access to paper (or similar material). In a context where paper is expensive or unavailable, written arithmetic is not very compelling.
- dredmorbius 5y ago
  Chalk (or charcoal) and slate (or other rewritable surfaces) were common before paper was cheap. Greek and Roman students typically used wax-coated boards and scratched them with styli. The surface could be heated or rubbed to restore its unmarked form.
  - jacobolus 5y ago
    Yes that is true, but if you don’t have a permanent written record of the computation, that takes away a lot of the advantage of written arithmetic. You may as well just use your counting board.
    - dredmorbius 5y ago
      Not permanent, but a record that could be shown as the computation was performed, and until the slate itself was erased. Obviously, where a permanent record is desired, permenent media would be used. My point was that there were cheap alternatives to parchment and paper for quick work.
      - jacobolus 5y ago
        A wax or clay tablet or flat patch of dirt and a stick, piece of bark or wood and charcoal, etc. don’t have enough advantage for anyone to bother organizing their accounting or calculation system around them. They are on the whole inferior to the existing system. Trying to introduce them to people 2000+ years ago might have a chance, but it would be a tough sell. The thing that really spread Hindu numbers around the world was access to paper.
        dredmorbius 5y ago
        Is there any evidence / record that Hindu / Arabic numerals and paper co-dispersed?
        jacobolus 5y ago
        I am not an expert. My impression is that in India “Hindu” numerals and arithmetic were developed using some kind of erasable board, but among Arabic speakers (and later Europeans) the spread of Hindu numerals went along with paper as a prerequisite technology. This looks relevant https://cpb-us-e1.wpmucdn.com/sites.ucsc.edu/dist/f/704/files/2019/05/Islam.atCenter.JWH_.09.pdf https://cpb-us-e1.wpmucdn.com/sites.ucsc.edu/dist/f/704/file... Here is a relevant looking book: https://yalebooks.yale.edu/book/9780300089554/paper-print https://yalebooks.yale.edu/book/9780300089554/paper-print In e.g. Britain, the transition to Hindu numerals did not occur until 1600–1800, depending on the place.
bryanrasmussen 5y ago
Ok but we know about zero!
- jacobolus 5y ago
  Anyone with a counting board also has zeros, as empty places with no counters, and ancient people had no problem understanding and using the concept of zero in calculations (though most probably would not have considered zero a “number” per se). They just didn’t write explicit zeros in their permanent serialization format. But there are certainly plenty of mathematical ideas and tools ancient people didn’t know about: they didn’t have a convenient method of manipulating algebraic expressions and equations; they had only the most rudimentary version of differential/integral calculus; they didn’t have group theory, linear algebra, complex analysis, etc.
  - ncmncm 5y ago
    Or, indeed, negative numbers.
ted_dunning 5y ago
> this goes so quickly Many people don't realize how fast it is to work on abacus (or soroban in Japanese). Here is an example of what a 7 year old can do https://www.youtube.com/watch?v=GQtqlB-jXO0 https://www.youtube.com/watch?v=GQtqlB-jXO0 For many years I always did my taxes and other accounting using an abacus because it was sooo much easier than longhand (this was before calculators). There is the famous Feynman story as well: https://www.ee.ryerson.ca/~elf/abacus/feynman.html https://www.ee.ryerson.ca/~elf/abacus/feynman.html But you wouldn't impress an ancient Greek very much by taking cube roots quickly because there wouldn't be much call for that.

jacobolus 5y ago

> goes so quickly that it's possible for the person calculating to cheat an onlooker Indeed, the biggest advantage of paper arithmetic is that it leaves a written record: each step in the algorithm can be checked for mistakes afterward. It is otherwise significantly slower and not inherently more accurate than a counting board. The other advantage that paper arithmetic has is that it can be easily reproduced in printed books, making learning more advanced techniques more accessible / easier to spread to anyone literate, without requiring an expert teacher. (And finally, paper arithmetic [eventually] has the advantage that it can be more conveniently extended and generalized to include more kinds of operations and structures, in a way that is easier to explain and teach than adding new kinds of counting board rules. Paper arithmetic is a more natural precursor for symbolic algebra than counting-board calculation.) The big disadvantage of paper arithmetic is that it depends on widespread literacy and cheap access to paper (or similar material). In a context where paper is expensive or unavailable, written arithmetic is not very compelling.

dredmorbius 5y ago
Chalk (or charcoal) and slate (or other rewritable surfaces) were common before paper was cheap. Greek and Roman students typically used wax-coated boards and scratched them with styli. The surface could be heated or rubbed to restore its unmarked form.
- jacobolus 5y ago
  Yes that is true, but if you don’t have a permanent written record of the computation, that takes away a lot of the advantage of written arithmetic. You may as well just use your counting board.
  - dredmorbius 5y ago
    Not permanent, but a record that could be shown as the computation was performed, and until the slate itself was erased. Obviously, where a permanent record is desired, permenent media would be used. My point was that there were cheap alternatives to parchment and paper for quick work.
    - jacobolus 5y ago
      A wax or clay tablet or flat patch of dirt and a stick, piece of bark or wood and charcoal, etc. don’t have enough advantage for anyone to bother organizing their accounting or calculation system around them. They are on the whole inferior to the existing system. Trying to introduce them to people 2000+ years ago might have a chance, but it would be a tough sell. The thing that really spread Hindu numbers around the world was access to paper.
      - dredmorbius 5y ago
        Is there any evidence / record that Hindu / Arabic numerals and paper co-dispersed?
        jacobolus 5y ago
        I am not an expert. My impression is that in India “Hindu” numerals and arithmetic were developed using some kind of erasable board, but among Arabic speakers (and later Europeans) the spread of Hindu numerals went along with paper as a prerequisite technology. This looks relevant https://cpb-us-e1.wpmucdn.com/sites.ucsc.edu/dist/f/704/files/2019/05/Islam.atCenter.JWH_.09.pdf https://cpb-us-e1.wpmucdn.com/sites.ucsc.edu/dist/f/704/file... Here is a relevant looking book: https://yalebooks.yale.edu/book/9780300089554/paper-print https://yalebooks.yale.edu/book/9780300089554/paper-print In e.g. Britain, the transition to Hindu numerals did not occur until 1600–1800, depending on the place.

bryanrasmussen 5y ago

Ok but we know about zero!

jacobolus 5y ago
Anyone with a counting board also has zeros, as empty places with no counters, and ancient people had no problem understanding and using the concept of zero in calculations (though most probably would not have considered zero a “number” per se). They just didn’t write explicit zeros in their permanent serialization format. But there are certainly plenty of mathematical ideas and tools ancient people didn’t know about: they didn’t have a convenient method of manipulating algebraic expressions and equations; they had only the most rudimentary version of differential/integral calculus; they didn’t have group theory, linear algebra, complex analysis, etc.
- ncmncm 5y ago
  Or, indeed, negative numbers.

ted_dunning 5y ago

> this goes so quickly Many people don't realize how fast it is to work on abacus (or soroban in Japanese). Here is an example of what a 7 year old can do https://www.youtube.com/watch?v=GQtqlB-jXO0 https://www.youtube.com/watch?v=GQtqlB-jXO0 For many years I always did my taxes and other accounting using an abacus because it was sooo much easier than longhand (this was before calculators). There is the famous Feynman story as well: https://www.ee.ryerson.ca/~elf/abacus/feynman.html https://www.ee.ryerson.ca/~elf/abacus/feynman.html But you wouldn't impress an ancient Greek very much by taking cube roots quickly because there wouldn't be much call for that.