Roman Math and Math without Zero?

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Elheru Aran
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Roman Math and Math without Zero?

Post by Elheru Aran » 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?

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? Image
Are you going to like, just memorize really massive multiplication tables or something?
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Re: Roman Math and Math without Zero?

Post by Khaat » 2017-08-28 05:52pm

Um, sort of what you're looking for?
http://mentalfloss.com/article/93055/ho ... ot-numbers
I picked this one, since it includes the big thinkers in mathematics that predate Romans (with another non-Arabic numeral system).

I think first and foremost, it isn't that they "didn't have a zero", it's that they had an extra number for everything we use a zero (universal placeholder) to make: 10, 100, 1000, etc., we build with single digits (0-9) in sequence, where location encodes part of the value. They used a unique character (or mini-sums equation) for each "place". In effect, they (Greeks) encoded more data in each character (independent of overall position) than Arabic numerals do, at the cost of (to our modern experience) flexibility. Roman numerals have small-set relative position value encoding ("I before V is four, I after V is six") and other hurdles. Arabic numerals are merely a different framework with fewer, "reusable", characters. Bit depth vs sample rate.

I would liken it to the way different languages have different vocabulary - it's that they have a different way of saying it. "How did they function without 'zero'?" is like asking "How did they function without 'the' (English indefinite article)?" to which the answer would be "they used Greek, duh". :D
Smithsonian Magazine wrote:Of all the numerals, “0”—alone in green on the roulette wheel—is most significant. Unique in representing absolute nothingness, its role as a placeholder gives our number system its power. It enables the numerals to cycle, acquiring different meanings in different locations (compare 3,000,000 and 30).
From http://www.smithsonianmag.com/history/o ... 180953392/

Ok, so I led with Greek characters in math, same holds for Latin; it's just a different vocabulary. Messy? Yes, it might very well be, but that doesn't mean it didn't also work.
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Re: Roman Math and Math without Zero?

Post by Elheru Aran » 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.

(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)
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Re: Roman Math and Math without Zero?

Post by Khaat » 2017-08-28 07:22pm

Since the Romans borrowed heavily from the Greeks, and what we (modern Western European culture) have came to us from the Greeks as well, I'd have to say yes, they did have LxBxD=V for volume, but I don't know if they grouped the processing the same way I learned it (they may have had a different "grammar" to go with their vocabulary), though I'm not sure how you could get more direct.

From the first link:
The ancient Greeks were incredibly talented mathematicians—but they rarely used numbers in their math. Their particular specialty, geometry, dances around actual quantities, focusing on higher-level logic and constant relationships. Even Pythagoras, whose triangles we navigate with easy examples like “3, 4, 5,” or “5, 12, 13,” was way more interested in diagrams than in specific situations.
So strong theory, and a matter of ratios applied to the project scale? I dunno.

For A New Jersey Architect in Octavian's Court, passing oneself off as a mathematician/engineer with basic trigonometry would be a matter of ratios, and relationships, I think, rather than processing specific values. A quick challenge (where Our Hero uses Arabic numerals to come to a product faster than the counterpart can with Roman numerals) might also "wow", provided he can translate it back into something the audience can recognize as well, before his counterpart completes the task. There really is no reason not to think there was rote memorization of mathematical data in Roman times, despite the form of the numerals.
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Re: Roman Math and Math without Zero?

Post by Elheru Aran » 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.

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?

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)
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Re: Roman Math and Math without Zero?

Post by Broomstick » 2017-08-28 09:44pm

A lot of old building was empirical - that's why a lot of what survive is so overbuilt by modern standards. As for calculating things like a circumference, it was known that you had to multiply by a little bit more than three even if folks didn't know an exact value for pi.
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Re: Roman Math and Math without Zero?

Post by Simon_Jester » 2017-08-28 11:45pm

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.
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Re: Roman Math and Math without Zero?

Post by Elheru Aran » 2017-08-29 01:27pm

So the overall impression I'm getting is:

--Math, as a Thing, was not a big deal for the great majority of people because day-to-day stuff doesn't particularly require it. Not too different from today, of course (some things in life never change).

--Those that needed to use math, probably knew to a fair degree of approximation the same math that we would use for the purpose. Perhaps not at as high a level as we've taken it nowadays, but certainly enough to get the job done. It might not have LOOKED like the same math we use, but the basic principles would have been the same; it would simply have been written down differently.

--A lot of stuff was done on a rule-of-thumb and experienced-guesstimate basis built upon empirical experience, apprenticeship, family knowledge, and whatnot.

Is all that good enough to work with?
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Re: Roman Math and Math without Zero?

Post by Elheru Aran » 2017-08-29 01:35pm

I'm also noting something on Wikipedia (which I did look at before, but which didn't really make sense until now):
They did not do arithmetic with their written numerals. They used their numerals only for recording the results of calculations on an abacus.
From https://en.wikipedia.org/wiki/Subtractive_notation

So that explains why there's not much (if any) in the archaeological/historic record of, say, a work-book of Roman math equations being found-- they did the math on the abacus instead and then just wrote down the result.

Therefore it can be conjectured that less value (perhaps?) was placed upon 'doing the work' by writing it out on paper (or papyrus if you like... or clay if you're Babylonian and doing trig, apparently) and people instead learned how to do it on the abacus first?
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Re: Roman Math and Math without Zero?

Post by Broomstick » 2017-08-29 01:55pm

An abacus is a type of calculator. I built one many years ago and played around with it. The basics are pretty easy and with experience a person can become both fast an accurate. It's something better learned by actually manipulating one rather than reading about how to use one.
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Re: Roman Math and Math without Zero?

Post by Thanas » 2017-08-29 02:40pm

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.
There were civilian engineers but most of the huge proucts like aquaeducts were heavily influenced by - if not outright designed - by military engineers. We have for example an inscription from North Africa where they tried to build a tunnel, started digging from both sides and failed to meet in the middle. Oops. So they did what almost everybody did back then - whined to the emperor and asked for help. The Imperial bureaucracy responded by sending a military engineer to fix the problem, he calculated the angle they had to take to fix the problem succesfully and then they managed to complete the tunnel.

The Roman Army did a lot more building than they did fighting, so much that today those "soldiers" would probably be classified as "constructors" first and soldiers secon. In this professional environment you could pay people just to be engineers and thus they could afford to hone their skills. Same with doctors and any other form of ancient high tech.

Also we know there were instruction manuals on how to build buildings by successful architects.
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Re: Roman Math and Math without Zero?

Post by Simon_Jester » 2017-08-29 03:25pm

Elheru Aran wrote:
2017-08-29 01:27pm
So the overall impression I'm getting is:

--Math, as a Thing, was not a big deal for the great majority of people because day-to-day stuff doesn't particularly require it. Not too different from today, of course (some things in life never change).
It was more true than today, in all probability, as demonstrated by the fact that standards of living rise a lot when you enforce universal education in literacy and basic math and that this is very probably not a coincidence.

But yes, this is basically true, and if anything more true then than now.
--Those that needed to use math, probably knew to a fair degree of approximation the same math that we would use for the purpose. Perhaps not at as high a level as we've taken it nowadays, but certainly enough to get the job done. It might not have LOOKED like the same math we use, but the basic principles would have been the same; it would simply have been written down differently.
This is true up to a certain point. Speaking roughly, the Romans had about the same level of mathematical knowledge that you'd need to do pretty well on the SAT. Give a good Roman mathematician a quick course in Arabic numerals and what the notation of algebra means, hand him a pocket calculator, and translate the math section of the SAT into Latin, and he'd probably do surprisingly well for himself.

Except maybe for the probability and statistics calculations; I have no idea what the state of probability theory was in ancient Rome.

Areas where the Romans were well in advance of the SAT would include geometric proofs, mental arithmetic, and memorization of complex formulas for things we now use the general-purpose set of tools we call "algebra" for. That would actually probably be most likely to trip up your mathematically literate Roman- because they didn't have algebra as we know it, they just had common sense and constructed formulas.

For example, while the mathematician who described the formula for the greater root of a quadratic equation as "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value," that's representative of how the ancient Romans would write down a formula.

Basically, anything that a moderately smart American high school graduate couldn't do with their existing math background, at least in principle, the Romans didn't do using precise calculations. This includes, for example, designing ships. Just about anything they could do under those restrictions, they did. Except, possibly, for probability and statistics.
--A lot of stuff was done on a rule-of-thumb and experienced-guesstimate basis built upon empirical experience, apprenticeship, family knowledge, and whatnot.
Yes.
Elheru Aran wrote:
2017-08-29 01:35pm
I'm also noting something on Wikipedia (which I did look at before, but which didn't really make sense until now):
They did not do arithmetic with their written numerals. They used their numerals only for recording the results of calculations on an abacus.
From https://en.wikipedia.org/wiki/Subtractive_notation

So that explains why there's not much (if any) in the archaeological/historic record of, say, a work-book of Roman math equations being found-- they did the math on the abacus instead and then just wrote down the result.

Therefore it can be conjectured that less value (perhaps?) was placed upon 'doing the work' by writing it out on paper (or papyrus if you like... or clay if you're Babylonian and doing trig, apparently) and people instead learned how to do it on the abacus first?
In terms of what you could do with it, an abacus was roughly equivalent to a four-function calculator. If you had to multiply 82 by 17, you take your abacus, you click the beads into the position equivalent to the number "82," then you perform the operation equivalent to "multiply by 17," and you get the number "1394"

The only thing there is to write down is "82 x 17 = 1394." The rest of the computations were performed (implicitly) by the abacus and are invisible to the user. The user never needs to write down, as we would write,

1
8 2
x 1 7
-------------
5 7 4
8 2 0
-------------
1 3 9 4

Because the steps that we might write out longhand (subtotalling 524 and 820, carrying the 1, and so on) are all implicitly encoded in the operations the user performs on the abacus and are invisible in terms of the final output of the computation, much as they are today for someone who does the work on a four-function calculator.

Also, paper and paperlike products were more expensive in Roman times than today, so you were likely to see significantly less "scratch paper" and expendable documents. If you wanted to do some scratch calculations you'd probably do them by drawing in fine sand or wax with a stick, or with chalk on a piece of slate, or something like that. Reusable surfaces, rather than disposable ones.
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Re: Roman Math and Math without Zero?

Post by SpottedKitty » 2017-08-29 03:48pm

Simon_Jester wrote:
2017-08-29 03:25pm
Also, paper and paperlike products were more expensive in Roman times than today, so you were likely to see significantly less "scratch paper" and expendable documents. If you wanted to do some scratch calculations you'd probably do them by drawing in fine sand or wax with a stick, or with chalk on a piece of slate, or something like that. Reusable surfaces, rather than disposable ones.
ISTR one item used by schoolboys was two sheets of wood fixed together like a two-page loose leaf folder; the inside faces had square insets carved into them, and wax poured into the insets. Easy to write on with a pencil-like pointed stick, easy to erase and re-use. I think a few of these things have even been discovered with someone's last writing lesson still clearly visible.
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Re: Roman Math and Math without Zero?

Post by Simon_Jester » 2017-08-29 04:04pm

Precisely. Such tools remained in use up to the 19th century if not the 20th, for that matter.
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Re: Roman Math and Math without Zero?

Post by Thanas » 2017-08-29 05:36pm

Simon_Jester wrote:
2017-08-29 03:25pm
Also, paper and paperlike products were more expensive in Roman times than today, so you were likely to see significantly less "scratch paper" and expendable documents. If you wanted to do some scratch calculations you'd probably do them by drawing in fine sand or wax with a stick, or with chalk on a piece of slate, or something like that. Reusable surfaces, rather than disposable ones.
You would do those on ceramics like broken amphorae of which there were plenty to go around. And wax tablets.
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Re: Roman Math and Math without Zero?

Post by Thanas » 2017-08-29 05:37pm

Also moved the topic.
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Re: Roman Math and Math without Zero?

Post by Elheru Aran » 2017-09-01 01:23pm

Thanks all for your help.

Would anybody be able to recommend a (SIMPLE! if possible) book or two on ancient math and/or science and engineering in general? A science/engineering book doesn't need to be simple, I just suck at math :P
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Re: Roman Math and Math without Zero?

Post by madd0ct0r » 2017-09-02 04:11pm

Vitruvius would be a place to start.
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Re: Roman Math and Math without Zero?

Post by Thanas » 2017-09-02 08:38pm

madd0ct0r wrote:
2017-09-02 04:11pm
Vitruvius would be a place to start.
Or Plinius.
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Elheru Aran
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Re: Roman Math and Math without Zero?

Post by Elheru Aran » 2017-09-04 01:06am

Thanas wrote:
2017-09-02 08:38pm
madd0ct0r wrote:
2017-09-02 04:11pm
Vitruvius would be a place to start.
Or Plinius.
I resumed reading my copy of Vitruvius. By Plinius I assume you refer to the Elder and his Natural History?
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Thanas
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Re: Roman Math and Math without Zero?

Post by Thanas » 2017-09-04 06:19am

Yeah.
Whoever says "education does not matter" can try ignorance
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Re: Roman Math and Math without Zero?

Post by Surlethe » 2017-10-22 06:18pm

Elheru Aran wrote:
2017-08-29 01:27pm
So the overall impression I'm getting is:

--Math, as a Thing, was not a big deal for the great majority of people because day-to-day stuff doesn't particularly require it. Not too different from today, of course (some things in life never change).

--Those that needed to use math, probably knew to a fair degree of approximation the same math that we would use for the purpose. Perhaps not at as high a level as we've taken it nowadays, but certainly enough to get the job done. It might not have LOOKED like the same math we use, but the basic principles would have been the same; it would simply have been written down differently.

--A lot of stuff was done on a rule-of-thumb and experienced-guesstimate basis built upon empirical experience, apprenticeship, family knowledge, and whatnot.

Is all that good enough to work with?
Let's also distinguish between the math used by engineers and the math done by mathematicians. The former is concerned with the real world, the latter is concerned with patterns that arise in quantitative analysis. Let me give a modern example: In an analysis of a proposed building design, architects and engineers might conduct a finite element analysis that approximates the solution of a system of the PDEs that govern stress. An engineer might ask of this analysis: In a bad windstorm, are there any elements where the stress is within 70% of the breaking stress of the building material? A mathematician might ask: If the dihedral and verticial angles of the tetrahedral elements are constrained away from zero, how fast does the finite element approximation of the PDE's solution converge to the actual solution in the appropriate function space, and what is that convergence as a function of the smallest dihedral or verticial angle present in the mesh?

An analogous example in the ancient world might concern the number pi. A mathematician such as Archimedes might ask, "Can I approximate the area of a disk with successive regular polygons?" An engineer might ask, "How closely do I need to approximate the area of a disk to ensure that I don't have more than 5% waste?"

There is a virtuous relationship between mathematics and engineering. In addition to the usual cross-pollination of ideas in similar scenarios, engineering's problems present interesting patterns for mathematicians to explore, and mathematicians verify and expand engineers' repertoire of techniques. But the emphasis of inquiry is very different, and in a world with unsophisticated mathematics and no computers, rule-of-thumb and experienced-guesstimate analysis goes very far.

Ancient mathematics was also very geometric, and expressed verbally. While it hindered mathematical progress, I would guess it also made mathematics accessible to quantitatively sophisticated people such as military engineers. There were also a variety of tools developed to aid mathematical computation, based on experience, that have since the mid 20th century been subsumed by the computer: the abacus, for example.

I also want to add that mathematics was very different in the ancient world than it is today. When we say, for example, that Archimedes very nearly invented calculus, we do not mean that Archimedes very nearly had, say, the fundamental theorem of calculus as expressed today: "For any continuous function f defined in an interval [a, b] on the real line, the function F defined at x as the integral from a to x of f is differentiable and its derivative is equal to f." This is the fundamental theorem expressed in modern language developed in the latter half of the nineteenth century. Rather, what we mean is that Archimedes had begun to explore infinitesimal approximations, so that he was able to compute the area under a parabola by dividing it into infinitesimal rectangles. My understanding is that the notions of "differentiable," "continuous," "interval," and indeed I believe "function" did not exist in the ancient world.

So for example, when the ancient Greek mathematicians discovered the existence of irrational numbers, they did not conceptualize them as 'numbers x such that there do not exist integers m,n with x = m/n." They conceptualized irrational numbers as "there exist two line segments such that there is no smaller line segment that is commensurate with them."

(Note that I am a mathematician, not a historian of mathematics, so this is not my area of expertise.)
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