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Comment by wyldfire | Hacker News Reader

> paraphrased from an essay of reminiscence known as ‘Schooldays’, dating from 1800 BC Mesopotamia: "After he gets his packed lunch from his mother and sets off for school, everything seems to go wrong: the hall monitor beats him for being in the wrong place, another teacher beats him for tying his shirt up wrong, the Sumerian teacher beats him for speaking Akkadian and his cuneiform teacher beats him for just being terrible at cuneiform."

Whoa - I missed that "paraphrased" bit on first read and mistook this for a translation, which seemed really odd. But even paraphrased it refers to some kind of formal education for young children. Was that present in 1800 BCE Mesopotamia?

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It seems so. According to https://worldhistorycommons.org/sumerian-school-days:

>This tablet, from ancient Sumeria (as early as 2000 B.C.E.), details a day in the life of a school boy. Students learned by copying lessons on clay tablets, memorizing the lessons, and then reciting them for the school's headmaster (the "school father") or other teachers, monitors, and proctors of the school.

This web page includes a full translation if you want to read it, and cites the paper "Schooldays: A Sumerian Composition Relating to the Education of a Scribe" (https://www.jstor.org/stable/596246)

It makes sense that scribes would have formal education, like other highly-skilled professions imo, but I know nothing about history so who knows!

They also had formal education in ancient Egypt at the time. The math problems given to students at the time aren't trivial either. They need to do stuff like calculate the volume of various 3d shapes. Seems they used 256/81 (3.16) as an approximation of pi at the time.

https://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrus

That's an interesting approximation. The error is about 15 times larger than for the well-known 22/7 approximation.

They supposedly got it by using an octagon to approximate a circle. Maybe that was a simpler approximation used in education, who knows.

According to Bruno Jarrosson, the Egyptians had a marked preference for some classes of fractions, e.g. those with unitary numerator (so that, for example, representing a quantity as a composition (e.g. sum) of more fractions of the form 1/n was preferred to using single fractions of the form m/n). So, that fraction may have been privileged over others in this framework.

(One immediately notes that 256 and 81 are simple powers. As just a possibility, it could have been found intriguing that (2^8)/(3^4) could "reveal some underlying structure".)

Yes, I was thinking about the simple powers, too.

And I don't know whether they had enough math (or cared enough to do all the hard work with the math they had available to them), to figure out what's the simplest good approximation for the real value of Pi.

Well spotted that they're simple powers; riffing on that idea, if that is the aesthetic behind it, I'd guess that they're supposed to be the same base or the same power, so either (4^4)/(3^4) or (16^2)/(9^2)?

Edit:

Or perhaps (((4^2)/(3^3))^2)^1 and then it feels like an ancient aesthetic-precursor to Euler's identity?

And now I spotted the mistake:

> (((4^2)/(3^3))^2)^1

should be (((4^2)/(3^2))^2)^1

> the aesthetic behind it

It could also have been practical: "Radius to circumference? Easy: double many times, then take thirds a few times".

Certainly feels plausible, that possibly makes a lot of sense.

Some more speculation. Have a look at https://en.wikipedia.org/wiki/Ancient_Egyptian_multiplicatio...

Egyption multiplication was (implicitly) based on powers of two. In some sense weirdly similar to modern bit-twiddling.

Note that 4/3 in binary is 1.0101 0101 0101 0101 0101 0101 0101 0101...

So that suggests the following algorithm, expressed in modern day Python:

    def mul4_3(x):
      table = []
      while x > 0:
        table.append(x)
        x //= 4
      return sum(table)

I deliberately used a 'table', because that's what a scribe would do.

Repeat this function four times, and you will have multiplied by 256/81.

I have no clue whether they would have done anything resembling this procedure; this is just to show that it's plausible given how their multiplication worked.

I don't know how this relates to 'Egyption fractions'.

P.S. Just for fun the same thing for 22/7:

    def f22_7(x):
      y = (x << 1) + x
      while x > 0:
        x >>= 3
        y += x
      return y

Not actually harder to execute by hand, I'd say, but perhaps harder to come up with?

> Note that 4/3 in binary is

This is consistent with the notion from Jarrosson, because that means that 4/3 can be represented as

  1 + 1/4 + 1/16 + 1/64 + 1/256 + 1/1024 ...

which is a sum of fractions with unitary numerator - the representation ancient Egyptians are said to absolutely prefer.

The approximated ratio of the circumference to its diameter can be represented by four iterations ( (4/3)^4 ) of infinite series of sums of fractions with unitary numerator.

> [...] the representation ancient Egyptians are said to absolutely prefer.

Well, they can also represent it as 1 + 1/3.. It's just that I assumed they have an easier time dividing by two than dividing by three.

If you leave it as a fraction, instead of dividing your integers, 1/3 is fine.

Note that fractions of the form 1/n + 1/m + ... make equity obvious for non-arithmetically sophisticated populations.

Given 3 loaves of bread, and 4 workers eating lunch, we'd probably be fine if 3 of them got 3/4 loaf each and the last got the 3 1/4 slices. To make the fairness of the distribution obvious, the Egyptians might have given all 4 workers the same pair of slices: 1/2 + 1/4.

(Note that for less fungible items, there still might be some practicality in the ancient Egyptian system: if we have 3 5 meter ladders to divide between 4 people, giving everyone one 2,5 meter ladder and one 1,25m would be much fairer than giving 3 people a 3,75m ladder but the 4th three 1,25m ladders.)

> 256/81 (3.16)

Whoa, trippy coincidence. (16/3)^4 ~= 3.16

I must be misunderstanding something, because based upon my understanding (16/3)^4 ~= (5.333)^4 (or 65536/81) ~= 809.

16^4 / 3^4 = 256 / 81

That is interesting. Not as close to pi as the 22/7 that we were taught in middle school but maybe those exponentials were more meaningful to the Egyptians.

16^4 is not 256. 16^2 is 256. 16^4 is 65,536.

Oops, sorry.

Despite the math typo, (4/3)**4 is a pretty interesting composition of pi, if only because you get such a fundamental constant out of a relatively simple arrangement of low natural numbers.

Yes and no. Yes in the sense that there was a formal pattern for learning that developed from 3k bc - end (right around 1800bc) which typically included a combination of math (fractions, pemdas, etc, administrative math, enough so you could make a bill of sale) and writing (which meant learning Sumerian as well as cuneiform), among a few other "administrative" disciplines (like calculating the calendar- the Sumerian calendars often arbitrarily added a month or two whenever they felt like the calendar was too far off, so knowing when and how to add those months was critical to maintaining the cultic schedule. fun for archaeologists later to figure out what month it is..). The essential purpose was to prepare you for a life of administration of some kind, whether in a temple or household.

No in the sense that this kind of education was not available to everyone, and I would imagine the vast majority never learned to write.

Here's more Sumerian tablet jokes:

"If a scribe knows only one line, but his handwriting is good, he is indeed a scribe!"

"A scribe whose hand can follow dictation is indeed a scribe!"

"What kind of a scribe is a scribe who does not know Sumerian?"

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