Sunday, March 29, 2015

Mathematics Crossing the Lines of Civilzation

There's a number of reasons for which I find in Eurocentric teachings of the history of mathematics harmful, but today I'll focus on one facet in particular: incompleteness. When I say "incomplete" I'm not saying "this isn't a compendium of the histories of all human mathematical activity". I specifically mean leaving out pieces of the story of mathematics as it is practiced today, which stretches in an unbroken (though perhaps split) line from present times all the way back to the the first civilizations of Egypt, Mesopotamia, and the Indus Valley.

While a teaching that focuses on Europe isn't necessarily bad (certainly great things in the history of mathematics have happened there...), I do see intrinsic harm in failing to contextualize these contributions. As I've stated previously on this blog, the progress of algebra over the ages can be found in:
  • Procedural Babylonian and Egyptian texts around the 19th-century BCE
  • Procedural Chaldean texts from the seventh century BCE
  • Procedural Greek texts from the fourth-century BCE
  • Procedural Indian texts from the fifth-century CE
  • Rhetorical and Syncopated Arabic texts from the ninth-century CE, and
  • Syncopated and Symbolic European texts from the 11th-century CE
But Robert! What's making you say this is all one coherent narrative? Isn't it completely possible that some of these civilizations developed mathematics independent of one-another?

That is an excellent question. It is possible... likely even, but only for one of the civilizations listed here. It is demonstrably NOT the case for the rest. This blog entry will focus on the evidence that shows how mathematics was passed from civilization to civilization (and consequently, how when the mathematics of say, Europeans, is studied in great detail, it needs to be understood as a piece of a much bigger narrative).

As an aside, up until the beginning of the Scientific Revolution in Europe, significant advances in mathematics almost invariably had close ties to astronomy.

Babylonia → Chaldea
Okay, this isn't so hard to imagine. Same geographic region. They both wrote in cuneiform and used sexagesimal. It's pretty clear Chaldean tables have a lot in common with Babylonian star catalogs. Moving on.

Chaldea → Greece
The conquests of Alexander the Great were from 335 BCE to 324 BCE and established the reaches of the Hellenistic kingdoms as far as India.

The first known trigonometric table was compiled by Hipparchus of Nicaea (180 – 125 BCE) who divided the circle into 360 degrees and fractions of chord lengths in sexagesimal (base 60). Paraphrased from page 107 of "A History of Mathematics, 3rd Ed." by Uta C. Merzbach and Carl B. Boyer: Hipparchus likely adopted this convention from Hypsicles (190 – 120 BCE), who wrote an adaptation of a work explaining a Babylonian technique for computing the rise times (at the latitude of Alexandria) of the signs of the zodiac. Both Hypsicles's adaptation and the Babylonian text divided the ecliptic into 360 degrees.

Also, on page 3 of “An Introduction to the History of Algebra” by Jacques Sesiano, it reads: "Ptolemy (A.D. 150) mentions... that since the beginning of the reign of Nabonassar (747 B.C.), "the ancient (Mesopotamian) observations are, on the whole, preserved down to our own time."

Chaldea & Greece → India
This is the hardest one, since it's not clear how much Indian mathematics originated on the subcontinent from the time of the Indus Valley Civilization (flourished between 2600 – 1900 BC) onward, and how much of it came from the Hellenistic kingdoms or prior contact with Mesopotamia.

The Āryabhaṭīya (499 CE) describes degrees and sexagesimal in verses 3.1 and 3.2:
  • 1 revolution = 12 signs
  • 1 sign = 30 degrees
  • 1 degree = 60 arc-minutes (kalās)
  • 1 arc-minute (kalās) = 60 arc-seconds (vikalās)
  • 1 arc-second (vikalās) = 60 arc-thirds (tatparās)
Based on what I've said so far, this might appear Babylonian in origin, but this is tricky because the degree (though apparently not sexagesimal) appear in the Rigveda (most likely between 1500 – 1200 BCE, though possibly between 1700 – 1100 BCE):

Twelve spokes, one wheel, navels three.
Who can comprehend this?
On it are placed together
three hundred and sixty like pegs.
They shake not in the least.
     — Dirghatamas , Rigveda 1.164.48

For what it's worth, the Egyptians and Mesopotamians (thus completing ancient trifecta of Egypt, Mesoptoamia, and the Indus Valley) were also both fond of cutting the sky into 12 sections (one for each lunar cycle or 'month') and within each section, taking particular note of 3 stars or groups of stars.

Anyway, if there was any one civilization to develop mathematics independently of the others on this list, it would have been India.

Greece & India → Arabia
The following is based almost entirely on page 205 of "A History of Mathematics, 3rd Ed." by Uta C. Merzbach and Carl B. Boyer:

The first century of Muslim Empire had been devoid of scientific achievement. Arab conquerors fought amongst themselves and with their enemies until about 750 CE when their warlike spirit subsided. Under the rule al-Mamum, the 7th caliph of the Abbasid Caliphate, a House of Wisdom comparable to the ancient Museum in Alexandria was built in Baghdad. This institution served as a focus of medieval scholarship until it was destroyed by the Mongols in the 13th century. 

In this time, Al-Mamun is said to have had a dream in which Aristotle appeared and became determined to have made Arabic versions of all of the Greek works he could lay his hands on. From the Byzantine Empire, with which the Arabs now maintained an uneasy peace, Greek manuscripts were obtained through treaties.

Delegations of scholars from India were also invited to share their knowledge of mathematics and astronomy. Probably one of the more significant examples of mathematics crossing lines of civilization is the use of Brahmangupta’s Khandakhadyaka by House of Wisdom scholar Al-Biruni (who amazingly, read it in the original Sanskrit).

Arabia → Europe
Since the fall of Western Rome in the 5th century CE, Medieval Christian Europe had been practically devoid of scientific achievement. This changed significantly in the 11th-13th centuries with the transmission of knowledge to Europe from the Muslim Empires, primarily through the Iberian Peninsula, which was known to the Arabs as Al-Andalus.

Particular points of transmission to Europe were the 1085 conquest of Toledo by Spanish Christians, the 1091 re-claiming of Sicily by the Normans (after the Islamic conquest in 965) and the Crusader battles in the Levant from 1096 to 1303. Additionally, a number of Christian scholars such as Constantine the African (1017-1087), Adelard of Bath (1080-1152) and Leonardo Fibonacci (1170-1250) traveled to Muslim lands to learn sciences.


Europe Onward
The reason the rest of the world continues this now 4000 year-long story is because Europe, from the 16th century onward, became obsessed with a little something called "colonialism". The Pan Afro-Eurasian heritage of mathematics was of course key (it still is) to the Scientific Revolution and the Industrial Revolution, both which have continued to this day.

So yeah... pretty good reasons to think it's all one narrative.

Friday, March 27, 2015

Crafting a prolate spheroid

It's time for a Mathematical Adventure!
http://i.cdn.turner.com/v5cache/CARTOON/site/Images/i70/adventure-time.png
Last year, this little guy here sent me on a journey:
https://www.youtube.com/watch?v=Lt9mjoaO2oc
As a birthday gift, my wife offered to make me a stuffed version:
I am so happy!
There was one condition: I needed to determine what fabric pattern cut so that the bee would come out as nice as possible.

I'd actually thought about a similar problem a few years prior. My college friend Warren had called me over winter break 2007 expressing frustration that a particular family tradition involved far too much guesswork and there had to be "some kind of math" that would fix it.

The tradition in question was these Christmas ornaments made from foam balls upholstered with petals of fabric; in my family we called them "tannenbaums" (though after a brief bit of Googling this might be something my brother made up).
https://craftsncoffee.files.wordpress.com/2014/11/diy-fabric-ornaments.jpg?w=640
It took some thought, but I figured the best shape of petal would be the spaced enclosed by two out-of-phase sine waves:
Made using Microsoft Math 4.0: http://www.microsoft.com/en-us/download/details.aspx?id=15702
Understanding this, I figured it couldn't be that much harder to figure out what "petal shape" would make a prolate spheroid. Wow, was I in for a surprise...

Looking at the drawing, I estimated the aspect ratio of the bee to be 1.25, meaning the bee was 1.25 times longer on one axis than it was on the other two. My first thought was just to stretch the petal shape along the horizontal direction, but immediately it was clear this wouldn't work. Since the overall shape needs to be perfectly round at the tips, the angle of each petal still needs to be the same: 360° divided by however many petals I decide to use. If I stretch the petal in the horizontal direction, this would change the angle at the ends of the petals. This would turn the ball not into a prolate spheroid, but a football.

I was not about to let my bee have the shape of a football! I'd have to find the shape of some weird curve I'd not heard of before. Hmm...

Imagine being small and standing on the surface of a proloate spheroid, marching from the equator to the north pole. The horizontal coordinate of our curve (u) should be how far we've traveled along the surface of the prolate spheroid. The vertical coordinate of our curve (v) should be a twelfth (since we're using six petals, and a petal is twice as tall as the curve we're looking for) of the circumference of the circle of latitude where you're at. Think of the cross section produced when slicing a lemon; you get circles. That's the circumference we're talking about.

I was pretty sure I'd end up using the formula for arc length.
http://tutorial.math.lamar.edu/Classes/CalcII/ArcLength.aspx
Now I just needed to plug in the equation for an ellipse:
http://www.mathwarehouse.com/ellipse/equation-of-ellipse.php
DO NOT TRY THIS. IT WILL ONLY END IN TEARS.

Basically, it leads to a really mean integral. I cheated by looking at the answer, and I still have no idea what happened. Most students leave public school knowing the formula for a circle's circumference: C=2πr. Now, try to think back if you've ever heard the formula for an ellipse's circumference. I can almost guarantee nobody has ever even mentioned it, and here's why:
C = 2\pi a \left[1 - \sum_{n=1}^\infty \left(\frac{(2n - 1)!!}{2^n n!}\right)^2 \frac{e^{2n}}{2n - 1}\right],
http://en.wikipedia.org/wiki/Ellipse#Circumference
Never-mind that this is the formula for just the circumference and I need pieces of this. I'm not even sure what coordinate system to use! Screw this. Bye bye. I need to start over.

When in doubt, solve the problem numerically
Lets define the spheroid as having semi-axes lengths of rx=1.00, ry=1.00, and rz=1.25 (these are kind of like radii). Now we just need to numerically determine u(z) and v(z) and plot the curve of v vs. u. to form the petal pattern which my wife will use to make the bee.

It'll be convenient to define r(z). This is the radius of the cross section of each slice along the z axis. Since the side view of the prolate spheroid is also an ellipse, we can use the following equation:
Made using the equation editor in Power Point.
From here, v(z) = 2π*r(z)/12 .
Determining u(z) is harder. First we have to define n=100 or so points so that z0=0 and zn=1.25 . We then use the Pythagorean Theorem to figure out the distances between each (ri, zi) and (ri+1, zi+1), and cumulatively add everything together:

We get the following curves:
r(z) vs. z and v(z) vs. u(z)
Now it's a simple manner of making a printout my wife can use:
Yaaaay! It's done.
So... that was interesting or as Finn would say:
adventure time animated GIF
http://media.giphy.com/media/9lMoyThpKynde/giphy.gif

Thursday, March 26, 2015

What is Algebra?

New article posted on LiveScience.com today!

What Is Algebra?

sanskrit
http://www.livescience.com/50258-algebra.html

I've been wanting to write about the history of Algebra since I first started with LS. In my first draft I made the mistake of trying to get an explanation out of the way so I could talk about the history. I have to thank my editor for his insistence that it didn't connect, and that we really had to pander to the average Googleing passerby. I kind of gave up on the topic, but to my surprise, after some time away, my editor expressed interest in picking it back up.

Somehow I'd forgotten how effective the technique of using a topic's history to explain it is. (See my articles on the transistor or quantum mechanics). I'm really happy with how this came out.

Friday, March 20, 2015

Why I tell the story of why an hour has 60 minutes

Next month I’m giving a talk on why an hour has 60 minutes. (Thanks @cafesciboston!) This story encompasses the progress of science from before the first writing, exploring contributions from Ancient Sumer, Babylonia, Greece, India, the Golden Age of Islam, and the early days of the Scientific Revolution in Europe. You can find the original article here:
I was told to be ready for questions on how I discovered this story and what drove me to tell it. What follows is a result of that prompt.

---------------------

This started with a series of dinner debates with a theologian friend. As a disclaimer on my atheism, let me say: whatever drives you to go out and look at the world, kudos. If it’s wanting to know the mind of God or wanting to understand God’s creation, I can’t criticize whatever motivation you have for driving the wheel of science forward.

I didn't always feel this way, and as a counter to my then opinion that religion makes people unscientific, my friend vehemently argued that the Christian Enlightenment was responsible for the Scientific Revolution in Europe. I was convinced there had to be more to it than this, and I was also sickened by the undercurrent that, when boiled down, amounted to little more than "because our civilization has the right god, we were the ones who got the bomb."

This was when I started researching the history of algebra (something I was curious about anyway). On page 4 of “An Introduction to the History of Algebra” by Jacques Sesiano, it reads:

"Several centuries of continuous [Mesopotamian] observations provided an invaluable body of data for the computation of planetary periods… As it would have been an overwhelming task to convert all these data into the decimal system… the Greeks maintained the sexagesimal system for astronomical measurements… This was also adopted by the Indians as early as antiquity… Then… it reached the Muslims who in turn transmitted this notation to medieval Christian Europe. The sexagesimal division… still used today is thus a living witness to the sexagesimal base once used by the Sumerians… in prehistoric times."

After picking my jaw up off the floor, I started looking at maps and reading about Babylonia, Greece, India, and Arabia. Later I took a course in science writing and decided to use the topic for my final paper.

Setting aside that this story is amazing… it is also incredibly useful! Firstly, because it encompasses the progress of science from before writing up until the Scientific Revolution. Knowing this story, I now have context when thinking about anything else from history.

The second reason concerns representation of ethnic minorities. This is a sensitive topic, so let me start with an image I saw the other day on Facebook.

Link here
Some commenters wanted to know about the quadratic formula. Since the ancient Babylonians, Greeks, Indians, and medieval Muslims all used some progressively advanced version of the quadratic formula, it’s hard to summarize in a list, and is thus not so great for going up on Facebook. Yet, something very dangerous has happened here!

This list goes from the 6th century BCE Europe to the 17th century CE Europe. Inadvertently (I hope), this creates the impression that mathematics is from Europe, and thus Europeans are somehow superior or representative of “true civilization.” If you don’t see this as a problem, allow me to point at (with a 10 foot pole) the unfathomable number of YouTube commenters who insist something along the lines of “white people invented science and other races should be thankful.”

An amazing set of circumstances triggered the scientific revolution in Europe, and it’s been fascinating looking to look at scholarly reasons why people think it happened there rather than India, Arabia, or China.

I view Newton as a linchpin for the following flood of applying mathematics to motion and the physical world. His Law of Gravitation has two halves: the terrestrial from Galileo for describing the parabolic motions of projectiles, and the celestial from Kepler and Brahe for describing the elliptical motions of the planets around the sun. Here are my favorite explanations for why these two halves happened in Europe:

The celestial half concerns mechanical clocks. According to David Landes, author of “Revolution in Time”, It gets cold in Europe, which is why Europeans went to the trouble to develop mechanical clocks that didn't rely on water. Clock chimes also set the rhythm of life for medieval Europeans, but this wasn't culturally important to the Chinese. The first mechanical clocks couldn't compete with water clocks in terms of accuracy, but their potential to do so was staggering. Tycho Brahe absolutely needed clocks accurate to within 4 seconds of the Earth's time to take the data upon which Johannes Kepler based his laws of planetary motion.


The terrestrial half concerns Renaissance art. According to Joseph W. Dauben, Professor of History at CUNY, it is because Renaissance artists became obsessed with accurately portraying reality in art that Galileo similarly became obsessed with accurately portraying reality using mathematics. I've not been able to answer if traditions in the medieval art of India, Arabia, or China would have been prohibitive of similarly inspiring great minds from those parts of the world… but it’s an interesting idea.

As a reply to my theologian friend (who has since lost interest in discussing this with me): you’d be correct to argue that Christianity is responsible for the culture of clock chimes and sponsoring great works of art… but this is far cry from the philosophy of the enlightenment being responsible for the scientific revolution.

Nationally, I’m American (US-ian, as my Canadian brother-in-law says), as were my mother and father. Ethnically, I’m half mixed European and half Chinese. While I do occasionally encounter blatant, personal racism, lately I've become more aware of the kind of insidious, subtle racism that comes with inadequately representing people of color. I think one of the best countermeasures to this problem is to tell stories like this. Such narratives eviscerate the idea that “white people invented science,” and they give context for other discoveries throughout history. It is my hope you’ll fall in love with the story too, and tell it to other people.

Monday, March 16, 2015

Ancient Babylonian Mathematics: Tablet VAT 8389

If the history of mathematics interests you, I highly recommend Jacques Sesiano's book “An Introduction to the History of Algebra“ (AMS, 2009). I particularly enjoyed the first chapter on ancient Babylonian algebra. There's one thing that I was really missing though... images of the original text. Digging just one of these images up sent me on something of a journey and sadly I can only use them for educational, non-profit purposes (which is why this is on my blog). The word "text" might give readers the wrong idea, so let me just throw this out there:
Photo credit: © Staatliche Museen zu Berlin – Museum of the Ancient Near East | Photo: Olaf M. Teßmer, May 2010 | USED WITH PERMISSION
What you're looking at is tablet VAT 8389 from the Museum of the Ancient Near East in Berlin, Germany. The tablet contains the solution to system of linear equations that today we'd write like this:
x + y = 1800
⅔∙x - ½∙y = 500
Such representation is anachronistic to say the least.

Today I'm going to pull out everything I know about this tablet: transliteration, literal and adapted translation, mathematical interpretation, and comparison with modern mathematics.

The Babylonians didn't have lined paper, so I've corrected that here:
Photo credit: © Staatliche Museen zu Berlin – Museum of the Ancient Near East | Photo: Olaf M. Teßmer, May 2010 | USED WITH PERMISSION | Edited by me in PowerPoint
Transliteration.
The following transliteration appears in Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin (Springer, 2013) by Jens Høyrup. I am no expert in ancient pronunciation of ancient Babylonian cuneiform (my background is in chemical engineering...). This was hard to type and I am not positive I got all the characters right.

I-1           i-na bùr 4 še.gur am-ku-us
I-2           i-na bùr ša-ni-im 3 še.gur am-ku-us
I-3           še-um ugu še-in 8,20 i-ter
I-4           garim ĝar.ĝar-ma 30
I-5           garim en.nam
I-6           30 bu-ra-am ĝar.ra 20 še-am ša im-ku-sú ĝar.ra
I-7           30 bu-ra-am ša-ni-am ĝar.ra
I-8           15 še-am ša im-ku-sú
I-9           8,20 ša še-um ugu še-im i-te-ru ĝar.ra
I-10        ù 30 ku-mur-ri a.šà garim.mes ĝar.ra-ma
I-11        30 ku-mur-ri a.šà garim.meš
I-12        a-na ši-na ḫe-pé-ma 15
I-13        15 ù 15 a-di si-ni-šu ĝar.ra-ma
I-14        igi 30 bu-ri-im pu-tur-ma 2
I-15        2 a-na 20 še ša im-ku-su
I-16        íl 40 še-um lul a-na 15 ša a-di ši-ni-šu
I-16a      ta-aš-ku-nu
I-17        íl 10 re-eš-ka li-ki-il
I-18        igi 30 bu-ri-im ša-ni-im pu-tur-ma 2
I-19        2 a-na 15 še-im ša im-ku-sú
I-20        íl 30 š-um lul a-na 15 ša a-di ši-ni-šu
I-20a      ta-aš-ku-nu íl 7,30
I-21        10 ša re-eš-ka ú-ka-lu
I-22        ugu 7,30 mi-nam i-ter 2,30 i-ter
I-23        2,30 ša i-te-ru i-na 8,20
I-24        ša še-um ugu še-im i-te-ru
II-1         ú-sú-uḫ-ma 5,50 te-zi-ib
II-2         5,50 ša te-zi-bu
II-3         re-eš-ka li-ki-il
II-4         40 ta-ki-ir-tam ù 30 ta-ki-ir-tam
II-5         ĝar.ĝar-ma 1,10 i-gi-a-am ú-ul i-de
II-6         mi-nam a-na 1,10 lu-uš-ku-un
II-7         ša 5,50 ša re-eš-ka ú-ka-lu i-na-di-nam
II-8         5 ĝar.ra 5 a-na 1,10 íl
II-9         5,50 it-ta-di-kum
II-10       5 ša ta-aš-ku-nu i-na 15 ša a-di ši-ni-šu
II-11       ta-aš-ku-nu i-na iš-te-en ú-sú-uḫ
II-12       a-na iš-te-en sí-im-ma
II-13       iš-te-en 20 ša-nu-um 10
II-14       20 a.šà garim iš-te-at 10 a.ša garim ša-ni-tim
II-15       šum-ma 20 a.šà garim iš-te-at
II-16       10 a.šà garim ša-ni-tim še-ú-ši-na en.nam
II-17       igi 30 bu-ri-im pu-tur-ma 2
II-18       2 a-na 20 še-im ša im-ku-sú
II-19       íl 40 a-na 20 a.šà garim iš-te-at
II-20       íl 13,20 še-um ša 20 a.šà garim
II-21       igi 30 bu-ri-im ša-ni-im pu-ṭur-ma 2
II-22       2 a-na 15 še-im ša im-ku-sú íl 30
II-23       30 a-na 10 a.šà garim ša-ni-tim
II-24       íl 5 še-um ša 10 a.šà garim ša-ni-tim
II-25       13,20 še-um a.šà garim iš-te-at
II-26       ugu 5 še-im a.šà garim ša-ni-tim
II-27       mi-nam i-ter 8,20 i-ter

Literal Translation
Next is a word-for-word translation. It's worth mentioning that Babylonian numbers were written in sexagesimal (base 60). For clarity I've included the place value of numerals which the Babylonians inferred by context.
               
I-1           From a bur 04. kur of grain I have collected.
I-2           from a second bur 03. kur of grain I have collected.
I-3           grain over grain, 08,20. it went beyond
I-4           My plots I have accumulated: 30,00.
I-5           My plots what?
I-6           30,00. the bur, posit. 20,00. , the grain which he has collected, posit.
I-7           30,00. , the second bur, posit.
I-8           15,00. , the grain which he has collected
I-9           08,20. which the grain over the grain went beyond
I-10        and 30,00. the accumulation of the surfaces of the plots posit:
I-11        30,00. the accumulation of the surfaces of the plots
I-12        to two break: 15,00.
I-13        15,00. and 15,00. until twice posit:
I-14        Inverse 30,00. , of the bur, detach 00.00,02
I-15        00.00,02 to 20,00. , the grain which he has collected
I-16        raise, 00.40 the false grain; to 15,00. which until twice
I-16a      you have posited,
I-17        raise, 10,00. may your head hold!
I-18        Inverse 30,00. , of the second bur, detach 00.00,02
I-19        00.00,02 to 15,00. , the grain which he has collected
I-20        raise, 00.30 the false grain; to 15,00. which until twice
I-20a      you have posited, raise 07,30. .
I-21        10,00. which your head holds
I-22        over 07,30. what goes beyond? 02,30. it goes beyond
I-23        02,30. which it goes beyond, from 08,20.
I-24        which the grain over the grain goes beyond
II-1         tear out: 05,50. you leave
II-2         05,50. which you have left
II-3         may your head hold!
II-4         00.40, the change, and 00.30 the change,
II-5         accumulate 01.10 . The inverse I do not know.
II-6         What to 01.10 may I posit?
II-7         which 05,50. which you head holds gives me?
II-8         05,00. posit. 05,00. to 01.10 raise.
II-9         05,50. it gives to you.
II-10       05,00. which you have posited, from 15,00. which until twice
II-11       you have posited, from one tear out,
II-12       to one append:
II-13       The first is 20,00. , the second is 10,00.
II-14       20,00. is the surface of the first plot, 10,00. is the surface of the second plot
II-15       If 20,00. is the surface of the first plot,
II-16       10,00. the surface of the second plot, their grains what?
II-17       Inverse 30,00. , of the bur, detach 00,00.02 .
II-18       00,00.02 to 20,00. , the grain which he has collected,
II-19       raise 00.40 to 20,00. . The surface of the first plot,
II-20       raise, 13,20. . The grain of 20,00. , the surface of the meadow
II-21       Inverse 30,00. , of  the second bur, detach 00.00,02
II-22       00.00,02 to 15,00. the grain which he has collect, raise, 00.30
II-23       00.30 to 10,00. , the surface of the second plot
II-24       raise, 05,00. the grain of the surface of the second plot.
II-25       13,20. the grain of the surface of the first plot
II-26       over 05,00. the grain of the surface of the second plot
II-27       what goes beyond? 08,20. it goes beyond.

Adapted Translation
Between the strange grammar, the use of sexagesimal, lack of units (which if written, would have been long outdated [says the guy from the US...]), and ancient methodology, I can't make much of this without differing expertise. The following adapted translation is based on that found in Jacques Sesiano's book: “An Introduction to the History of Algebra“ (AMS, 2009).

"Bur" and "sar" are units of field area. 1 bur = 1800 sar. A sar is about 36 square meters.
"Kur" and "sila" are units of grain volume. 1 kur = 300 sila. A sila is about 1 liter.
      
I-1           The 1st field gave 4 kur for each bur (2/3 sila for each sar)
I-2           The 2nd field gave 3 kur for each bur (1/2 sila for each sar)
I-3           The 1st field gave 500 more sila than the 2nd
I-4           The sum of the fields' areas is 1 bur (1800 sar)
I-5           What is the area of each field?
I-6           The 1st field gave 1200 sila for every 1800 sar
I-7           The 2nd field gave 900 sila for every 1800 sar
I-8           ↑
I-9           The 1st field gave 500 more sila than the 2nd
I-10        The total area of the two fields is 1800 sar
I-11        ↑
I-12        1800/2 = 900
I-13        You now have two values of 900
I-14        (1800)^(-1) = (1/1800)
I-15        (1/1800) × 1200 = (2/3) "false grain"
I-16        (2/3) × 900 = 600 "hold"
I-16a      ↑
I-17        ↑
I-18        (1800)^(-1) = (1/1800)
I-19        (1/1800) × 900 = (1/2) "false grain"
I-20        (1/2) × 900 = 450
I-20a      ↑
I-21        600 - 450 = 150
I-22        ↑
I-23        500 - 150 = 350 "hold"
I-24        ↑
II-1         ↑
II-2         ↑
II-3         ↑
II-4         (2/3) + (1/2) = (7/6)
II-5         (7/6)^(-1) = unknown
II-6         (7/6) × z = 350
II-7         ↑
II-8         300 × (7/6) = 350, z = 300
II-9         ↑
II-10       900 + 300 = 1200, 900 - 300 = 600
II-11       ↑
II-12       ↑
II-13       The 1st field has an area of 1200 sar, The 2nd field has an area of 600 sar
II-14       ↑
II-15       ↑
II-16       What are the yields of each field?
II-17       (1800)^(-1) = (1/1800)
II-18       (1/1800) × 1200 = (2/3)
II-19       (2/3) × 1200  = 800 "grain of the 1st field"
II-20       ↑
II-21       (1800)^(-1) = (1/1800)
II-22       (1/1800) × 900 = (1/2)
II-23       (1/2) × 600 = 300 "grain of 2nd field"
II-24       ↑
II-25       The 1st field produced 800 sila
II-26       The 2nd field produced 300 sila
II-27       800 - 300 = 500

Mathematical Interpretation
To make sense of this I've prepared the following graphic. I've done my best to identify the major events in each part of the procedure.

Image Credit: Self (Power Point)
The Babylonians solved this problem by figuring out how far the area of each field deviated (steps B-G) from half of the total area (step A). Step H has the answer, and steps I-K verify this answer is correct.

Analysis of Method
Working backward from the described method of verification, we can use modern methods to determine the ancient methods. Here we call the deviation from midpoint "z" and reverse engineer its method of determination. How did the Babylonians figure this out? I can speculate until the cows come home... but rigorous answers have been lost to the sands of time.

Image Credit: Self

Modern Method
Here's how the math looks today after filtering through 4000 years of advancement by civilization after civilization, specifically tenth-c. BCE Assyria, the seventh-c. BCE Chaldeans, sixth-c. BCE Persia, fourth-c. BCE Greece, first-c. CE Rome, fifth-c. CE India., ninth-c. CE Arabia, and 11th-c. CE Europe. By the beginning of the scientific revolution (17th-c. CE Europe), the notation and methodology would have been recognizable as something like this:
                x + y = 1800
                ⅔∙x - ½∙y = 500

                y = 1800 – x
                ⅔∙x - ½∙(1800 - x) = 500
                ⅔∙x - 900 + ½∙x = 500
                (7/6)∙x = 1400
                x = 1200

                (1200) + y = 1800
                y = 600

Is our modern method better? Yes. To be fair, it has the advantage of contributions of nearly every great civilization of Eurasia over the last 4000 years. Let us hope these methods survive even the fall of our own civilization.

Tuesday, March 3, 2015

Pretty Patterns!

LiveScience.com just posted my article about tessellation. I put a lot of work into making all the pretty graphics, so I'm really happy to see it up!

Go have a look
http://www.livescience.com/50027-tessellation-tiling.html