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April 26, 2006
"True beauty is to be found in natural forms. The more we magnify, and the closer we examine, the works of Artifice, the grosser and stupider they seem. But if we magnify the natural world it only becomes more intricate and excellent."
I know no one's probably interested in reading me ramble on about Quicksilver, but it's late and if I'm going to go for two straight months of daily posts, this is what it's going to be—so you'll just have to cope. =)
I just read some fascinating math I barely understand.
There's a weird conversation in Quicksilver between one of the protagonists, Daniel Waterhouse, and Isaac Newton—Newton's claiming that with calculus he's just rediscovering ancient geometrical secrets embedded in Solomonic architecture.
There's an annotation for the passage in the Quicksilver wiki from Neal Stephenson hisself, saying that Daniel's an unreliable narrator in that scene and we'll learn a lot more in the later volumes of the Baroque Cycle that'll make Newton seem a little less batshit insane. I haven't gotten further into the Cycle yet (although a pretty golden hardcover of The Confusion rests on my bed right now), so I can't really say.
This account may be justly criticized for making Newton seem a little wacky. In future volumes we will see more of Newton's side of the story and get a more balanced view of all this. In modern critical parlance, Daniel Waterhouse is an "unreliable narrator." We are seeing everything here through Daniel's eyes, but his eyes are clouded by his hostility towards Alchemy, and so the picture drawn here is biased and melodramatic.
Anyway, I decided to google, and found some amazing stuff.
Solomon, of course, hired the Phoenicians of Tyre to build his Temple. They'd learned architecture and math from the Egyptians.
There's this real-world papyrus (the Moscow papyrus) from the 1800s BC, which was discovered in the 1800s AD. It's a list of math exercises, one of only two documented examples of Egyptian math.
One of the problems in it is finding the volume of a frustum—a cone or prism with the top and bottom cut off.
The Moscow papyrus contains only about 25, mostly practical, examples. The author is unknown. It was purchased by V. S. Golenishchev (d. 1947) and sold to the Moscow Museum of Fine Art. Origin: 1700 BC. It is 15 feet long and about 3 inches wide.
Problem 14. Volume of a frustum. The scribe directs one to square the numbers two and four and to add to the sum of these squares the product of two and four. Multiply this by one third of six. "See, it is 56; your have found it correctly."
But see, there's this lingering issue:
Question. Speculate on how the Egyptians could have known the formula for a frustum, given that its derivation depends on the methods of modern calculus.
The prismoidal formula the Egyptians used is almost identical to what's now known as Simpson's Rule of Integration, a method of calculus used to find the integrals of polynomials of three or less degrees. But it can be used to bootstrap to formulas for integrating higher degree polynomials, the whole sequence of which is called the Newton-Cotes formulas. (Roger Cotes was one of Newton's assistants, and one of the historical models for the character of Daniel Waterhouse.)
And just to connect this even more to Quicksilver, the Egyptians claimed their mathematical knowledge came from Thoth, who merged over time with Mercury to become Hermes Trismegistus*...the crafter/trickster-god figure to whom alchemical mysteries are traditionally attributed.
The level of sophistication of this result is quite a bit higher than that of the rest of the papyrus (for example, the same papyrus gives incorrect formulas for some relatively simple plane areas), leading some people to suspect that either the Egyptians just stumbled into this particular formula, or else perhaps it was part of a more advanced body of mathematical results not generally reflected in the papyrus. Incidentally, according to tradition the god Thoth (Djhowtry), originally associated with time and the Moon, gave the calendar, astronomy, and mathematics ("reckoning") to the Egyptians. Thoth was later identified with the Greek god Hermes, who was later called Hermes Trismegistos (thrice great), the supposed author of the hermetic works revealing the secret knowledge of the ancients. Even as late as the 1600's this tradition was still influential in Europe. Isaac Newton, for example, was a devotee of hermetic studies, and actually seems to have believed that his own discoveries, such as calculus, universal gravitation, and much more, had been in the body of secret knowledge handed down from Thoth! Not surprisingly, Newton usually kept ideas like that to himself.
This is interesting because it is identical to what is known in calculus as Simpson's Rule of integration. In general if f(x) is any polynomial of degree less than or equal to 3, then we have [missing math] where m = (a+b)/2. It's easy to see why this is true for quadratic f(x), because it's essentially just the familiar integration rule for powers. For example, if we have f(x) = x2, then the indefinite integral of f(x) is (1/3)x3, which implies that the definite integral from a to b is [missing math]
This is identical to the rule described in the Moscow papyrus, bearing in mind that the area of a horizontal slice through a pyramid is proportional to the square of the distance from the (projected) apex of the pyramid, so we have A(x) = x2. (Notice that this applied to truncated pyramids whose bases have any shape, not just square as drawn above. Hence the expression is sometimes called the prismoidal formula.)
There's a bunch of math I'd love to copy, but can't seem to. Go read that last link. It's not some crank, it's serious math. It goes on into using the formula to derive integration formulas for higher-degree polynomials...
*I just checked Wikipedia to verify I was spelling Trismegistus right, and one of the first sentences in the entry for Thrice-Great Hermes is: "He has also been identified with Enoch." ("Enoch" is the first word of the novel—the name of a suspiciously long-lived character, Enoch Root, an enigmatic figure who seems to know everything and links the series with Stephenson's earlier novel, Cryptonomicon. I doubt Stephenson would go for anything so...incredible...as to have Enoch Root actually be the Biblical Enoch, but the connotations for the name are so rich.)
Posted by Jon Rubin at April 26, 2006 11:53 PM
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