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From the Babylonian Mathematics website:
In this article we examine four Babylonian tablets which all have some connection with Pythagoras’s theorem. Certainly the Babylonians were familiar with Pythagoras’s theorem. A translation of a Babylonian tablet which is preserved in the British museum goes as follows:From the Counting in Babylon website:4 is the length and 5 the diagonal. What is the breadth ? Its size is not known.All the tablets we wish to consider in detail come from roughly the same period, namely that of the Old Babylonian Empire which flourished in Mesopotamia between 1900 BC and 1600 BC.
4 times 4 is 16.
5 times 5 is 25.
You take 16 from 25 and there remains 9.
What times what shall I take in order to get 9 ?
3 times 3 is 9.
3 is the breadth.
Some of the clay tablets discovered contain lists of triplets of numbers, starting with (3, 4, 5) and (5, 12, 13) which are the lengths of sides of right angled triangles, obeying Pythagoras’ “sums of squares” formula. In particular, one tablet, now in a collection at Yale, shows a picture of a square with the diagonals marked, and the lengths of the lines are marked on the figure: the side is marked <<< meaning thirty (fingers?) long, the diagonal is marked:For further interesting mathematical history, see the website of the University of St Andrews School of Mathematics and Statistics’ History of Mathematics Archive, which also has a Famous Curves Index, pages about Egyptian, Indian, Arabic, and Mayan mathematics, and histories of things like Zero, Indian Numerals (which are weird), Arabic Numerals (which rotated), and Pi. From this last I learn that the value of Pi was calculated as 3.162 by the Mesopotamians, 3.16 by the Egyptians, 3.0 by the builders of King Solomon’s Temple (must try harder), 3.1418 by Archimedes of Syracuse, 355/113 by Tsu Ch’ung Chi, and (sigh of relief) 3.1416 by Abu Ja’far Muhammad ibn Musa Al-Khwarizmi; also that Al-Khwarizmi is where we get the word “algorithm”, a word I’ve often wondered about at moments when I wasn’t able to look it up, and forgotten to look up when I could.<<<<11 <<11111 <<<11111. This translates to 42, 25, 35, meaning 42 + 25/60 + 35/3600. Using these figures, the ratio of the length of the diagonal to the length of the side of the square works out to be 1.414213… Now, if we use Pythagoras’ theorem, the diagonal of a square forms with two of the sides a right angled triangle, and if we take the sides to have length one, the length of the diagonal squared equals 1 + 1, so the length of the diagonal is the square root of 2. The figure on the clay tablet is incredibly accurate—the true value is 1.414214… Of course, this Babylonian value is far too accurate to have been found by measurement from an accurate drawing—it was clearly checked by arithmetic multiplication by itself, giving a number very close to two.
355/113 is actually a rather better approximation than 3.1416. In fact, that such an accurate approximation should have been discovered so early in human history AND be expressed in such an elegant pattern of just six digits is one of the minor miracles of mathematics.
One of my own favorite weirdnesses about ancient mathematics is the Egyptian system of fractions. For some reason they confined themselves to unit fractions (1/2, 1/3, 1/4, 1/5, and so on), with the inexplicable exception of 2/3, and all other fractions were expressed as sums of these -- and you weren't allowed to repeat one. So 3/5 was expressed not as 1/5 + 1/5 + 1/5, but as 1/2 + 1/10.
They developed methods of computation with fractions that got the job done but which seem incredibly roundabout to us now.
I am deeply interested in Hitory of Mathematics, and have quite a bit of it on my web domain.
For example, from my home page
magicdragon.com
if you click on "Timeline"
you get my Encyclopedia of History of Mathematics combined with History of Science Fiction and History of World Literature and History of Science and Technology.
Go to any Century that I cover. For most of them, I am in the top 5 worldwide according to Google, if you enter in Google: "3rd century",
"4th century", "5th century", and so forth. I have biographic and bibliographic data on major authors, philosophers, mathematicians, scientists, and anyone else who interests me. I have also posted the chronology of estimates of "pi" in the Midle East, Japan, China, India, and so forth.
You can read my chronology (including Math) also in "1st Millennium BC", "2nd Millennium BC", "3rd Millennium BC", "4th Millennium BC", "5th Millennium BC", "6th Millennium BC", and "Cosmic History" goes back into anthropology, biological and geological evolution, and cosmology.
We have another interest in common! And 12,000,000 hits per year on my domain shows that lots of other people do, also.
Don't forget that in the Old Testament, a large circular well is described as being "10 cubits" in diamter, and "30 cubits" in circumference, which make pi = 3.000000
As to Babylon, that friendly merger of Akkadian and Sumerian civilizations was amazingly high-tech. Did you know that one family kept as trade secret the process of electroplating, which they used to make jewelry for the King? We have carefully analyzed the crusted remnants of their wet-cell batteries.
Of course, Bablyon's creation myth was that extraterrestrials who breathed water, had space helmets, and lived for centuries came from a star and taught Earthlings about ceramics, metals, and mathematics... but that's another story.
And don't forget Carl Sagan's weird idea in the novel of "Contact" (started as film treatment) that trillions of digits deep in "pi" is a perfect digitized cicture of a circle, sort of a signature of God...
Sorry I've been out of touch for a month or two. Had to file yet another petition for review to the California Supreme Court, against the Hollwood Producers who ripped me off in 1994-1995, and against whom I won a loandmark Supreme Court case in August 2000. Also, I've been giving testimony to the Columbia Accident Investigation Board (there were SF fans who contributed to the Shutle disaster); attending a wedding at CBS studios with reception on "New York Street" outside set near the intersection of Newhart Street and Mary Tyler Moore Avenue, working on 3 appeals including one to save an historic theatre in Pasadena California, and ever so much more.
Bye for now...
I don’t think the Egyptian fraction system is much weirder than my high school math teachers’ insistence that I not leave roots in the denominators of my fractions.
It makes me wonder if the Egyptians had some set of physical counters — like a set of weights or measuring cups. It’s like a Microsoft interview problem, almost: given 1 cup, 1/2 cup, and 1/4 cup measuring cups, plus a tablespoon, how many of them can you get dirty while measuring out 1 3/8 cup of flour?
In one of the books wandering around somewhere in my house, is the claim that in Israel there is a prehistoric slab of cast glass which was one of two -- the other destroyed in recent times -- which were the largest in the world prior to the casting for the mirror for the Palomar telescope. IIRC the book has a picture of the slab.
The egyptian fractional method comes from the traditional splitting of land, esp., land that was given to the priests to support funerary rites. It was very common for a person to give one part of his land to make sure the priest took care of his parents shrine, then, when he died, his son would give another part -- you ended up with long strings of fractions. The 2/3rds part, apparently, comes from the fact that the normal tax in an ideal year (the level of the flood determined the tax rate) was 1/3 the normal predicted harvest, leaving 1/3 and 1/3 for the farmer. Since this was an incredibly common combination, it's not surprising that "1/3 and 1/3" got a symbol of it's own. However, reading it as "2/3" is risky -- the Egyptians had lots of shortcut symbols for common long phrases ("A voice offering which the kings gives" being another one) and the proper reading may well be "1/3 and 1/3", not "2/3."
And, of course, Egyptian teachers came up with problems that resulted in crazy long fractions as teaching aids.
Egyptian farm taxes were clever. The flood was measured by Nilometers (one still exists at Elphantine) and the farmers paid a set tax based on that mesurment and how much land he tilled. If a farmer worked hard, and had a better harvest because of it, his tax did not increase -- he was allowed to keep the excess as his own. Conversely, if he didn't work hard, and his harvest was less, he still paid the same harvest. There were many records of farmers pleading for relief from that years taxes, citing crop failures that they could not prevent, no matter how diligent.
We got lucky -- we found what was apparently a papyrus roll used in teaching mathematics, the Rhind Mathematical Papyrus. In short, we found a math book. Handy, that.
David Moles,
Mathematical convention is to leave roots only in the numerator of fractions. However, computation often requires the reverse. Ask any numerical analyst how to solve x^2 - 100 x + 1 = 0. The quadratic formula gives
(100 + \sqrt{9996}) / 2 and (100 - \sqrt{9996}) / 2
but the second one leads to horrible inaccuracies. The subtraction loses 4 significant digits. Fortunately,
2 / (100 + \sqrt{9996})
works fine.
Tell your high school teacher (or your kids' teacher).
