Wednesday with Words: The Quadrivium is all about Math

[T]he quadrivium is essential to a liberal education in the traditional sense. And since we can normally only advance from sense-perception to intellectual intuition by way of intellectual argumentation, the quadrivium necessarily involved the study of number and its relationship to physical space or time, preparatory to the study of philosophy (in the higher sense of that word) and theology: arithmetic being pure number, geometry number in space, music number in time, and astronomy number in both space and time.
(Stratford Caldecott, Beauty for Truth’s Sake, pp. 23-24)

Elaienar made a graphic for me.
Isn't it pretty?

Squaring the Circle, part 2 -- the Mesopotamians

Plimpton 322
Image via Wikipedia

Reading about mathematics in Mesopotamia has been pretty interesting.  They seem to have done a lot of math just for the fun of it.  They were fond of tables comparing numbers and their relationships to one another – multiplication tables, tables of reciprocals, squares, cubes, and square and cube roots.  Algebra was their specialty. 

They had a base 60 system, apparently because sixty can be divided so nicely in so many different ways – into halves, thirds, fourths, fifths, sixths, tenths, twelfths, fifteenths, twentieths, and thirtieths (did I miss any?).  We still use base-60 for telling time and for measuring angles, which is interesting because it doesn’t seem that the Babylonians had the concept of measuring an angle.

For numbers above 59 they used a place value system which worked the way ours does.  The first place to the right is “ones,” the numbers from 1 to 59 in their case.  The place to the left of that is for the base units – 10 in our case, 60 in theirs.  To the left of that is the base squared – 100 for us, 3600 for them.  (This looks like x(60)² + x(60) + x in Math-ese, but I prefer English.)  They eventually developed a zero which they used as a place holder in the middle of long numbers, but they didn’t use it in the far right position if it was needed there, so 1, 60, and 3600 look like the same thing and have to be guessed from context.

This place value system was also used for fractions and is the reason for the unfamiliar notation I mentioned last week.  In the example quoted, the first number is the whole number.  The semi-colon serves the way a decimal point does for us – to separate the whole number from the “less than one” part to the right.  The commas are needed to distinguish the places since each place might have anything from a 1 to a 59 in it.

Those are some of the basics of math in Mesopotamia, but this is supposed to be about finding the area of a circle, right?

Remember all those tables and lists I mentioned in the first paragraph?  The clay tablet pictured at the top of this post is a table that describes the relationship of the squares drawn on the sides of a triangle, sort of like this:

This is such a handy way of understanding shapes that the Mesopotamian mathematicians noted this kind of relationship with all the regular (equilateral) polygons with three to seven sides, comparing the area of each shape to the square that could be made along its side.  One tablet lists all these relationships and gives very accurate ratios describing the relationships.

The ratio of the area of the pentagon, for example, to the square on the side of the pentagon, is given as 1;40, a value that is correct to two significant figures. For the hexagon and the heptagon, the ratios are expressed as 2;37,30 and 3;41, respectively.  In the same table, the scribe gives 0;57,36 as the ratio of the perimeter of the regular hexagon to the circumference of the circumscribed circle, and from this, we can readily conclude that the Babylonian scribe had adopted 3;7,30, or 3 1/8, as an approximation for π.
[Merzbach and Boyer, p. 35]

Whew!

Me, trying to learn Math.
And yes, that does mean a hexagon with a circle drawn around it.
I checked.  :-)

So it turns out that there’s a good reason for comparing various shapes to squares, and circles to various regular polygons – it helps us understand the relationships between different shapes, which helps us understand the shapes themselves.  And noting these relationships gave the Babylonians, like the Egyptians before them, a fairly accurate way of figuring out the area of any circle.

But one thing that confused me in A History of Mathematics’ chapters on the Ancient Greeks and Mesopotamians was the phrase “an approximation for π,” which the authors use several times.  They make it sound like π is a thing that everyone knows about and only needs to be clearly defined or standardized, the way a “foot” was eventually standardized as a unit of measure.  It was pretty exciting when I finally realized why they keep saying this, but that will be in the next post.

~*~ ~*~ ~*~

Click here for Part 1, which is about the Egyptians

Honeybees are awfully hard to photograph

Incidental learning is so fun

While reading the section on the area of a circle in a chapter on mathematics in Mesopotamia, I kept coming across things like this:

The ratio of the [blah blah blah] is given as 1;40 . . .. For the hexagon and the heptagon, the ratios are expressed 2;37,39 and 3;41 respectively. In the same tablet, the scribe gives 0;57, 36 as the ratio of the [blah blah blah] . . ..
[A History of Mathematics, Merzbach and Boyer, p. 35]

I've never seen notation like that before and don't know how to read it, so I decided to read the beginning of the chapter, which I had skipped over, it not being concerned with circles, and found this interesting sentence:

In modern characters, this number can be written as 1;24,51,10, where a semicolon is used to separate the integral and fractional parts, and a comma is used as a separatrix for the sexagesimal positions.
[op. cit. p. 25]

Did you catch that word separatrix?  Do you know what that means?  It means that commas are feminine.

The rest of the afternoon was spent browsing the dictionary and an online Greek grammar.

Squaring the Circle, part 1

Fragment of the Ahmes Papyrus
Image via Wikipedia

Have you heard that phrase before?  I have and I didn’t know it referred to anything other than trying to do something that’s impossible, and also rather silly, right?  I mean, why would anyone try to turn a circle into in a square?

Turns out there was actually a good reason for it.  Back in the days when the best tools mathematicians had were the compass and straightedge, they needed a fairly simple way of determining the area of a circle, which is devilishly hard to do.  It’s very easy to find the area of a square, so one solution they thought of was to figure out a simple way to draw a square that had the same area as a given circle. 

This endeavor went on for millennia and turned out to be futile.

In the mean time they did a pretty good job of estimating the area of a circle.  Here’s what they did, and how we know about it.

Back around 450 B.C. the Greek historian Herodotus traveled around Egypt interviewing priests and observing all the great work that went on there, from building monuments to farming along the Nile.  He said that geometry was invented in Egypt in part because the land had to be resurveyed every year after the Nile’s flood waters receded.  (Aristotle disagreed with him – he said that it was the priestly leisure class who did all the cool math in Egypt.  I don’t see why they can’t both be right – I guess it’s the age-old rivalry between engineers, who have practical problems to solve, and mathematicians, who are frequently philosophers and deal in the purely abstract realm.)

A piece of evidence on Herodotus’s side of the story, is the Ahmes Papyrus, which resides in the British Museum.  This papyrus was named for the scribe who around 1650 B.C. wrote it, using material from about 2000 to 1800 B.C.  In the papyrus, Ahmes says that the area of a circular field with a diameter of 9 units is the same as the area of square that’s 8 units on each side.

A little earlier on he had worked a problem that shows how this relationship was discovered.

First, you draw a circle with a diameter of 9 units.  Then you draw a square around it.  This square is 9 units on each side.

 

Then you cut off an isosceles triangle from each corner . . .  

. . . which gives you an octagon with an area of 63 units that has roughly the same area as the circle, though it’s not clear whether Ahmes thought it was exactly the same, or “close enough for practical purposes.”

From this they came up with the rule that “the ratio of the area of a circle to the circumference is the same as the ratio of the area of the circumscribed square to its perimeter [A History of Mathematics, Mertzbach and Boyer, page 15].”  Apparently this works out to a mathematical constant, 4(8/9)², which they used the way we use π, but I don’t really understand that part of the story.  Mertzbach and Boyer say that it’s “a commendably close approximation” of π and I’m taking their word for it.

The Ancient Babylonians also had a method, but I’ll have to tell y’all about that another time.

Math update... sort of.

Since I've decided to devote myself to studying math this year I've had the overwhelming desire to pick up my French studies where I left off a few years ago.  Ain't that always the way?