Elheru Aran wrote: ↑2017-08-28 04:51pm
(Obligatory prefatory disclaimer: Mods, if this is in the wrong place, go ahead and punt. Thought I'd float it here as it gets more traffic and I'm more likely to get an answer)
Greetings all.
So I am
mildly interested in a variety of historic things. Recently something came to mind: zero is a relatively (in the West) recent development. So are higher maths-- algebra, calculus, and all the fancy stuff associated therewith like engineering, physics, and so forth.
Two main questions came to mind: How did the Romans do math? And how is math done without zero as a fundamental concept?
You add, subtract, divide, and multiply, and you just use common sense regarding the meaning of numbers. Like
obviously if I owe you fifty silver coins and I pay you thirty silver coins, I still owe you twenty coins. It doesn't, strictly speaking, require the concept of the number line or positive and negative numbers or whatever to do any of this.
Speculatively, I imagine it also helped that the society could function with much of the population being functionally innumerate beyond the ability to count up to two-digit numbers or something. There were a LOT of literate and numerate Romans, but in any pre-industrial society you can generally get by with much of the population having, like, a third-grade level of math education.
As to the Romans: obviously they were capable builders and engineers. But how did they, for example, work out a stress load for a span? How did they know that an arch of Y width would work with a building of X size? Did they simply empirically test everything? How did they calculate the volume of concrete they might need to pour to build something the size of the Colosseum or the Pantheon? Wikipedia suggests that they had something along the lines of an abacus, which makes sense to me... though I've never been able to wrap my mind around how those work...
Nonzero math: II + II = IV, I get that. Where it starts getting hairy is... say... how the heck do you find a circumference with Roman numbers?
Multiply by 22, divide by 7. It won't be perfectly accurate, but it'll be correct to within 0.1%. That is to say, so close that other random sources of error in your construction work are likely to present more of a problem than "we did the math wrong."
If for whatever reason that's
NOT good enough, there are other ratios you could in principle use, they'd just be more work.
Elheru Aran wrote: ↑2017-08-28 06:20pm
Okay, so I get that math without zero is mostly just a matter of 'language'. I still don't really get it, but that's a good enough explanation for now.
How does that translate into equations, though? Did the Romans have... for example, the basic length times width times breadth/depth gets you volume? Did they have the same formulas we did, in other words? The Pythagorean Theorem is a pretty obvious one, I suppose.
Yes, the Romans had the Pythagorean Theorem, as did the Greeks, as the name
Pythagoras might hint at.
The Romans and Greeks, and for that matter other civilizations from before their time like the Egyptians, actually had a pretty good idea how to compute the volumes and shapes of basic solids, things like that.
(For clarity: I've been mucking about with a time-travel idea where someone from the present goes to the past, notably Rome, and one way the traveler attempts to establish communication is via showing that he/she can do numbers... translating it into Roman math is another matter entirely though, but would be a cool gimmick)
Most math that the Romans truly did not know how to do, that has actual practical consequences, would require some fairly specific tools to be done in a useful amount of time. Like books of log tables and trig tables- or the pocket calculators that replaced them.
Elheru Aran wrote: ↑2017-08-28 07:56pm
I suppose what I'm having difficulty wrapping my head around is that we all know Rome was a heavily bureaucratized society with professional military and civilian engineers. You do not get to be a bureaucratic society without paperwork, and paperwork means accounting-- the last denarius or ounce of salt has to be accounted for-- which means a lot of numbers. Bureaucracy also means you can't just come up with a public works project and tell them you need yea much concrete, yea much stone, X quantity lead and so forth; they want exact numbers... or did they? Vitrivitus IIRC (I haven't finished his book, I should) mentions computing quantities of such and such necessities for building projects, but he never mentions HOW it was done.
The Romans had absolutely no trouble doing this. I mean, they knew how to multiply and add and compute volumes. They knew how to manipulate things algebraically on a common-sense basis, just like modern elementary school students (the smart ones) can do that well before you formally teach them algebra.
If you ask a smart child a question like "The viaduct has to span a minimum of 2200 feet, using arches that are 48 feet long, what is the smallest number of arches we can use?" They're not going to have that much trouble answering the question if you give them a little time to work on it. Even if they do something dumb like say "okay, multiply 48 by 40, is that big enough, no, try 45..." it's still
doable. And a clever child will correctly say "okay, 2200 divided by 48 is..." and get a usable answer.
That's an algebra question, but basic reasoning skills and a well-trained proficiency in arithmetic are sufficient to answer the question.
I'm basically not reconciling the difference between theory and practice here. Having a strong grasp of theory is good, essential in fact, but how does 'Pythagorean Theorem will give you a right angle if A-squared plus B-squared equals C-squared' translate to building the Aqua Marcia aqueduct? Was it a matter of actual math, or was it more... 'this much quantity of construction should use so and so amount of building materials'? And any extra material left over after construction finished was... I don't know, returned for money back?
Well, I'm sure they'd wind up with surplus construction material. We
still tend to wind up with surplus material on big modern projects, after all. It's a very bad idea to only order
just exactly enough wood to build the house, or
just exactly enough concrete to pour the dam, because you really don't want a situation where a minor imperfection or worker error means you can't finish the project.
The obvious thing to do with leftover material would be to build extra stuff. A temple, or a bridge, or whatever needs doing. Something a lot smaller than an aqueduct.
Or, to use a different craft: ship-building. Did they simply use the collective experience of generations of ship-builders and marine engineers in designing and building new types of ships? Or did they sit down and calculate stresses on the hull, the tensile strengths of different woods, and so forth? (I suspect it's more the former than the latter)
Ship-building was
overwhelmingly done by rule of thumb and collective experience, right on up through the Roman period, the medieval period, the Renaissance, and most of the Age of Sail.
Knowledge of basic trigonometry and so on would often help with shaping the beams and things, but ultimately a lot of the work was done by relatively crude methods like "scale up from this scale model I built myself" or "just build an exact copy of this other ship that survived a huge-ass storm, and the new ship will hopefully survive the next huge-ass storm."
A lot of ships just plain sank or broke up in those days, and there was a pretty dramatic difference in quality of output between a city with an experienced community of shipbuilders (say, the Phoenicians or some of the Greek cities) and a city without such shipbuilders (say, Rome at the start of the Punic Wars).
The thing is, as I've said, this didn't really change until the 1800s, because the physics required to do maritime engineering on a mathematical level is really quite hard.
