Monday, 2 February 2009

Archimedes and the Square Root of Three

Snow here in my part of UK, so I stayed warm indoors reading other math blogs and some old journal articles...
Came across a couple of nice ones, but wanted to talk about one in particular.

Mark Dominus, atThe Universe of Discourse wrote a nice article (actually two of them) about Archimedes and how he might have obtained his fractional approximations to square roots, such as his use of 265/153 for the squre root of three that seem to confuse math historians.

He points out that Historians seem to treat it as if it must be some obscure and difficult approach that Archimedes had never revealed. W.W. Rouse Ball writes, "It would seem...that [Archimedes] had some (at present unknown) method of extracting the square root of numbers approximately." and Sir Thomas Heath writes, "the calculation [of π] starts from a greater and lesser limit to the value of √3, which Archimedes assumes without remark as known, namely 265/153 < √3 < 1351/780. How did Archimedes arrive at this particular approximation? No puzzle has exercised more fascination upon writers interested in the history of mathematics... "

So why wouldn't any of these great mathematicians think that probably someone in ancient Greece thought to make two lists and compare when they were close. One list of the squares (we know there are Babylonian tablets with columns of squares of numbers well back before Archimedes)..and one a list of them mutliplied by three (not a difficult task... In fact they could be prepared (as almost every Egyptian scribe must have known) by adding the sequential odd digits to get the squares, and three times these same odds to get the table of 3n2. Then you could just scroll down the list to find one in each list that was approximatly equal... Here is a list created quickly using Excel with two easy candidates for rough approximations marked in colored cells.

If 48, which is 3 (42) is about the same as 72. then by division we know that:

Mark also offers a second alternative. Perhaps a master of patterns like Archimedes noticed that as he looked down the list he saw that n2 was approximatly equal to 3p2 when for the following values of n and p;
n = 2....5.....7....19....26....71....97
p = 1....3....4....11....15....41....56

Do you see the pattern? Mark points out that 2+5=7 and 2x7 + 5 = 19... and then 19+7=26 and 2x26+19=71...
for the other string the same thing 1+3=4, and then 2x4+3=11. Each row continues by alternatly adding the last two numbers and then double the last plus the one before... A pattern that would be easily extended as far as he wished to take it; and the next fraction from the pattern are 265/153 which, when squared, gives 2.99991 (pretty close) but the next number is 362/209, which squares to approximatly 3.00002.... Now my question.. why didn't Archimedes use this for the upper bound instead of 1351/780... which follows the lower bound of 989/571 in this sequence

Perhaps the clever thing Archmedes did that baffled modern math historians is that he still had an eye for arithmetic.
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