Talk:Archimedes Palimpsest

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To say that it was "discovered" in 1906 makes it sound as if no one knew of its existence before then. It is true that it was not generally known among either mathematicians or historians, but, if I understand correctly, neither was it something that had not been mentioned in print. Heiberg's translation was what made its contents well known, but its existence was hardly a secret. Michael Hardy 18:46 Apr 25, 2003 (UTC)

Yep, the second external link (my isisletter "Did Isaac Barrow read it?") contains an accurate chronology of the discovery compiled from various sources Arivero 15:28, 14 June 2006 (UTC).

I have read that the gold illumination was forged in the 10th century, not the twentieth. Can someone confirm this? —Muckapædia 5h38, 7e Août 2006 (EST)

Where was this read? --Wetman 11:13, 7 August 2006 (UTC)
http://www.abc.net.au/science/news/stories/s1707926.htm - erroneously says "Then 10th century forgers painted gold foil imagery onto the recycled pages in an effort to increase the manuscript's value." This is a typo, all other articles say 20th century, and it is also impossible, as the palimpsest was not created until the 13th century. MakeRocketGoNow 00:13, 8 August 2006 (UTC)

[edit] Unclear pagagraph

This is amusing because the collaboration on indivisibles between Galileo and Cavalieri—ranging between years 1626 to around 1635—has as a main argument the hull and pyramid of the n = ∞ dome. So in some sense it is true that the Method is only a theorem behind the modern infinitesimal theory.

I removed this because I don't understand the amusement, and the "a theorem behind" phrase. AxelBoldt 22:37, 29 March 2007 (UTC)

[edit] The Method

The article states: "Essentially then, the method consists in dividing the two areas or volumes in infinitely many stripes of infinitesimal width, and "weighing" the stripes of the first figure against those of the second". Really? So you think Archimedes actually succeeded in dividing the areas/volumes into infinitely many stripes of infinitesimal width? Wow. I am amazed at how stupid you are. Archimedes' method was one of approximation. It is very similar to natural integration by approximation (no trial and error involved). However, that Archimedes used infinitesimals is highly unlikely. Infinitesimal is an ill-defined concept that Archimedes knew nothing about. The term infinitesimal was only coined in the 16 or 17th century. Yet another factually incorrect article by Wikipedia Sysops/Administrators. 70.120.182.243 16:58, 4 July 2007 (UTC)

Archimedes himself said it's an ill-defined concept and therefore that the arguments he wrote that used them were not complete proofs. That didn't stop him from using them. See Archimedes' use of infinitesimals. Michael Hardy 22:01, 10 July 2007 (UTC)
How could he have used a concept he knew nothing about? Where did Archimedes make a reference to an 'infinitesimal' in the palimpsest? The Greek text nowhere mentions 'infinitesimal' or anything remotely similar. On what grounds do you arrive at your conclusion that equates the terminology Archimedes used with your interpretation of 'infinitesimal'? Do you read and understand ancient Greek? How could Archimedes have anticipated a concept he knew nothing about? 70.120.182.243 02:22, 11 July 2007 (UTC)

I don't think he used that word in this text. Nonetheless he used infinitesimals in this text. As for how he did it, just read this page; that tells you how. And he said in this text that these are not complete proofs. I think it was elsewhere that he in effect rejected any attempt to take infinitesimals literally, by stating a sort of Archimedean axiom, as we would now call it. Michael Hardy 02:32, 11 July 2007 (UTC)

I have read all the text you refer to. This page tells me how he used approximations but nothing about how Archimedes may have used infinitesimals (whatever these are). The link to how Archimedes found areas such as between the parabola and a secant also says nothing about how he might have used infinitesimals. All the articles claim he, Newton and certain others used infinitesimals, but there are no examples. What is an infinitesimal? Please don't refer me to your article on infinitesimal as it does not provide a satisfactory definition or anything that can be used in a calculation. Again, how do you perform calculations using a non-existent concept (infinitesimal)? 70.120.182.243 15:03, 11 July 2007 (UTC)
Archimedes did not have what we would call modern integral calculus. He did use summation to provide approximate answers to certain problems, and this is why he is sometimes credited with being the founder of integral calculus. However, the method of exhaustion was known to Eudoxus of Cnidus and approximation was known to the Babylonian mathematicians, so Archimedes built on this tradition.--Ianmacm 07:17, 11 July 2007 (UTC)
How is 'modern integral calculus' different from what Archimedes knew? As far as I know, there is no difference in any respect except in the few cases where one can apply the mean value theorem to evaluate integrals without approximation/exhaustion. Whether Eudoxus knew of exhaustion before or after Archimedes is actually irrelevant to the discussion since we are talking about the ill-defined 'infinitesimal' notion. 70.120.182.243 17:43, 15 July 2007 (UTC)

Modern integral calculus relies on the fundamental theorem of calculus, which says you can find an integral if you can find an antiderivative. Michael Hardy 17:49, 15 July 2007 (UTC)

The mean value theorem (mvt) is the fundamental theorem of calculus (ftoc). The ftoc follows in one step from the mvt. Just multiply both sides of the mvt by one average and you have the ftoc. In the case of single variable calculus, this average is the width. The following link illustrates this clearly: [[1]]. As for the Archimedean property (or Euxodian, it doesn't matter what it is called for this discussion), it is a result of the least upper bound property. It's 'complement' does not define an infinitesimal. 76.31.201.0 12:19, 16 July 2007 (UTC)
It's complement defines zero? 65.28.94.67 14:49, 7 September 2007 (UTC)
This discussion is becoming rather ill-tempered. The term "infinitesimal" has always been vague and hard to pin down, which is why it is rarely used nowadays. Integral calculus is used to provide approximate answers to certain types of problem involving area etc, and is not the same as a formal geometric proof. Archimedes knew how to do this although he did not have all the techniques of calculus that are available today.--Ianmacm 07:02, 16 July 2007 (UTC)

There are some specific contexts where it's not vague at all. See non-standard analysis. It's hardly true that it's rarely used in the present day. And as one with much experience teaching calculus, I think it ought to be used a lot more than it is in such freshman-level courses. Michael Hardy 11:45, 16 July 2007 (UTC)

When calculus was new in the 18th century, Bishop George Berkeley (after whom University of California, Berkeley is named) wrote The Analyst, in which he attacked the whole concept of infinitesimals or fluxions. Sir Isaac Newton seems to have realised that some people would not like calculus and argue that it lacked formal rigour. This may have been one of the reasons why he delayed publication of his work in this area. In today's mathematics textbooks calculus is regarded as mainstream, but at the time it was as controversial an non-Euclidean geometry during the early 19th century. Calculus is intended to provide approximations rather than exact proofs, and this is what may displease some of the purists. However, the success of the Apollo moon missions shows that approximations are all that is needed in most engineering situations. At one point during the Apollo 13 mission, the commander Jim Lovell remarked that Newton was in the driving seat. This is an impressive vindication of the uses of integral calculus, and tends to make arguments about the existence or non-existence of infinitesimals into a philosophical side issue. (see also:Oliver Heaviside)--Ianmacm 15:03, 16 July 2007 (UTC)