10 votes

What is so mysterious about Archimedes' approximation of $\sqrt 3$?

There's a very simple geometrical response that provides a rounded solution which might reflect on this problem at http://www.gjbath.com/SQR3.htm. It's so much easier seeing it! This involves ...
Geoff Bath's user avatar
10 votes

What is so mysterious about Archimedes' approximation of $\sqrt 3$?

Archimedes might, of course, have used a better method. (He also produces the approximation $\frac{1351}{780}$, for which the foregoing is not obviously practical...) I disagree on the foregoing not ...
Peter Taylor's user avatar
9 votes
Accepted

How was the sum of squares formula discovered by Archimedes?

Although it is commonly said that Archimedes summed the geometric series in Quadrature of the Parabola, and summed squares of integers in On Conoids and Spheroids and On Spirals, that is not what he ...
Conifold's user avatar
  • 76k
7 votes
Accepted

Did Archimedes view fractions as "numbers"?

No, Archimedes, and ancient Greeks generally, did not see fractions as numbers, and they did not use fractions as we use them today, they did not use them at all. What they used was ratios of ...
Conifold's user avatar
  • 76k
6 votes
Accepted

What does this quote by Paul Halmos mean?

The meaning of the quote is more likely to be, that Archimedes' reasoning was on such a high level, that no one (example: the clever graduate student) could teach him to reason better because ...
terry-s's user avatar
  • 4,445
6 votes

What is so mysterious about Archimedes' approximation of $\sqrt 3$?

Some information can be found here 1 Using the secant method on the parabola $y = x^2$, a new estimate $x_{n+m}$ for $\sqrt{3}$ can be obtained from $x_n$ and $x_m$ as $$x_{n+m} = \frac{3 + x_n x_m}{...
Ignacio Rodriguez's user avatar
6 votes

How did Archimedes arrive at his principle in his time?

The bath is a later legend. The principle can be discovered and proved by the following argument which requires no calculation. Suppose that a body is submerged by water. The force with which water ...
Alexandre Eremenko's user avatar
4 votes

How was the sum of squares formula discovered by Archimedes?

You are right about the person (Archimedes), but not about the work. He stated it as proposition 10 of his text On spirals. In modern language, the statement is$$(\forall n\in\mathbb{N}):(n+1)n^2+(1+2+...
José Carlos Santos's user avatar
3 votes
Accepted

Did Archimedes make use of a Pappus chain according to the Book of Lemmas?

No, he did not. Wikipedia's editor was most likely misled by somewhat clumsy arrangement of the text in the Heath's 1897 translation of the Book of Lemmas. Right under the statement of Proposition 8 ...
Conifold's user avatar
  • 76k
3 votes

How did Archimedes's water clock work?

al-Jazari (and Ridwan too) explicitly says that Archimedes is the source for several of the components of his clock(s); he also says that one component of Archimedes' clock (the flow regulator) does ...
user2227111's user avatar
2 votes

How did Archimedes arrive at his principle in his time?

Fortunately there is some documentary evidence about Archimedes' discovery, in his work "On Floating Bodies", which survives in manuscript forms (https://en.wikipedia.org/wiki/On_Floating_Bodies). ...
terry-s's user avatar
  • 4,445
1 vote
Accepted

Looking for an online version of Archimedes' "The Method" (in Greek)

The question was answered by Viktor Blasjo in the comments: the Greek text is included in this article by Heiberg.
1 vote

What is so mysterious about Archimedes' approximation of $\sqrt 3$?

We don't know how Archimedes compute square root of 3 not because there is no known method, but because there are several methods that Archimedes could use, and we don't know which one he actually ...
Alexei Kopylov's user avatar

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