Archimedes and Calculus: A Lost Connection?

[Julian Solos]

Newton and Leibniz are generally credited with the invention of calculus, aren't they?

Last night I saw on TV that Archimedes actually knew some basic concept of calculus but his description of it was "lost" to the world until a recent "re-discovery."

What do you think the history of mankind would have been like if Archimedes's knowledge was followed up by others without interruption?

 

[uart]

Yeah Julian, I saw that same doco about 6 months ago, it was really quite facinating. Apparently he was working on problems like finding the area of an elipse and the mathematics he was using sounded very much like calculus indeed. I guess he just didn't have it formalized quite to the extent that Newton and Leibniz did, but it's almost certain that he had many of the fundamental concepts.

It's absolutley mind boggling how this kind of stuff can literally get lost for nearly two thousand years isn't it.

 

[master_coda]

Archimedes used concepts such as the method of exhaustion, where you would try to find the area of a shape by approximating it with other shapes whose area you already knew. You would then take better and better approximations. This is the same basic concept behind Riemann integration.

It's a little hard to comprehend these discoveries...I've been learning math all my life, and there are so many idea that now seem intuitively obvious to me that nobody would have known thousands of years ago. I sometimes think "how could they not know these concepts, they're so obvious" before remembering that these concepts are only obvious because others before me discovered them and helped make them obvious.

Standing on the shoulders of giants, indeed.

 

[Julian Solos]

Originally posted by master_coda
I sometimes think "how could they not know these concepts, they're so obvious" before remembering that these concepts are only obvious because others before me discovered them and helped make them obvious.

Your comment reminds me of the following quote:


"All truth passes through three stages. First, it is ridiculed. Second, it is violently opposed. Third, it is accepted as being self-evident."

Arthur Schopenhauer (1788 - 1860)



Kind of interesting quote, don't you think?

 

[master_coda]

Originally posted by Julian Solos
Your comment reminds me of the following quote:


"All truth passes through three stages. First, it is ridiculed. Second, it is violently opposed. Third, it is accepted as being self-evident."

Arthur Schopenhauer (1788 - 1860)



Kind of interesting quote, don't you think?

I've seen this quote before. Cranks think that since their theory is ridiculed, this quote proves that their theory is in fact self-evident.

It is an interesting quote, of course. And it turns out to be true more often than it should be. It's just irritating, because it's so easily abused.

 

[deda]

Archimedes is my favorite.
People should've started
counting time from his birthday!

I've a topic on Archimedes lever:
Archimedes and the three interactions

Newton simply has no chances over Archimedes!

 

[selfAdjoint]

Archimedes used a limiting process similar to integration in his approximation to $$\pi$$ and a process somewhat like differentiation in his work on spirals. He also had a mental trick for establishing areas of sections. He wrote this trick up in a publication calle "The Method" which was lost when the parchment it was written on was reused to write something else. Scholars recovered the text in the 20th century. This is the "lost" part of the story. The exhaustion and spirals work was saved and passed down to Europe by the Arab and Persian scholars of the middle ages.

Was it "calculus"? Well, mileage may differ. In my mind calculus is inseparable from analysis, that is from "algebraic" notation. And that was a product of the early modern European workers. No don't tell me about al-Khwarizmi. He didn't use notation. Everything he wrote about was already known to the Babylonians.

 

[phoenixthoth]

from what i understand, it was eudoxus who used exhaustion before archimedes but eudoxus was only the first one to write it down. what i find interesting is that those two were doing what is more similar to rigorous math, rather than Newton's infinitesimal approach. i don't know the details of how leibniz did it or whether he used infinitesimals. but Newton turned out to be rigorous in a certian light when robinson constructed the hyperreal extension of R in the 1960s.

Newton and/or leibniz should be given credit, i think, for something that as far as i know did escape the greeks: the fundamental theorem of calculus.

if only the libraries in alexandria were not burned! the theory of everything may have been completed 300 years ago!

 

[master_coda]

Originally posted by phoenixthoth
if only the libraries in alexandria were not burned! the theory of everything may have been completed 300 years ago!

Or perhaps humanity would have been destroyed in a nuclear war a thousand years ago.

 

[Tron3k]

Originally posted by master_coda
Or perhaps humanity would have been destroyed in a nuclear war a thousand years ago.

I see your glass is half empty.

 

[selfAdjoint]

Originally posted by phoenixthoth
from what i understand, it was eudoxus who used exhaustion before archimedes but eudoxus was only the first one to write it down. what i find interesting is that those two were doing what is more similar to rigorous math, rather than Newton's infinitesimal approach. i don't know the details of how leibniz did it or whether he used infinitesimals. but Newton turned out to be rigorous in a certian light when robinson constructed the hyperreal extension of R in the 1960s.

Newton and/or leibniz should be given credit, i think, for something that as far as i know did escape the greeks: the fundamental theorem of calculus.

if only the libraries in alexandria were not burned! the theory of everything may have been completed 300 years ago!

Yes, I agree about Eudoxus, who is my candidate for the greatest unremembered mathematician. He also introduced the theory of proportions, which is the forerunner of real arithmetic, and his "homocentric spheres" theory of the planets was the one that philosophers preferred to Hipparchos' epicycles.

IMHO what the Greek mathematicians lacked was a completeness axiom. You see this in their exhaustions, they have to approximate from both above and below, so that they squeeze the limit in between, because they have no way to assert that the limit exists from just one converging sequence.

This was to plague Nicole Oresme, the 14th century mathematician, in his construction of the exponential function (he called it a proportion between proportions). He wrote that he could do commensurable (rational) "exponents" of both commensurable and incommensurable (irrational) bases, but he couldn't do incommensurable exponents even of commensurable bases. What he needed was to be able to define a sequence of rational exponents going to his irrational as a limit. He was capable of doing that - he had summed simple convergent series - but had had no way of asserting that the limit existed.

 

[master_coda]

Originally posted by selfAdjoint
IMHO what the Greek mathematicians lacked was a completeness axiom. You see this in their exhaustions, they have to approximate from both above and below, so that they squeeze the limit in between, because they have no way to assert that the limit exists from just one converging sequence.

What does approximating from above and below have to do with completeness?

 

[Hurkyl]

What does approximating from above and below have to do with completeness?

To prove there's something in the middle.

(I guess it's no biggie, though, if you assume there's something in the middle before you start approximating)


If you strike out the completeness axiom, then you can't disprove things like "No rectangle can have the same area as a circle" or "A circle can have a point inside and outside another circle, but not intersect it".

 

[master_coda]

Originally posted by Hurkyl
To prove there's something in the middle.

(I guess it's no biggie, though, if you assume there's something in the middle before you start approximating)


If you strike out the completeness axiom, then you can't disprove things like "No rectangle can have the same area as a circle" or "A circle can have a point inside and outside another circle, but not intersect it".

I understand that. But the post I was quoting seemed to be saying that approximating from above and below somehow compensated for a lack of completeness.


The way I read selfAdjoints post, he was saying that the Greeks had to approximate from above and below because they lacked completeness, and so couldn't just use a single convergent sequence.

But if they don't have completeness, it doesn't matter if you use one sequence or two sequences.