What is the volume of a sphere?

[mathwonk]

huh?...

 

[tongos]

an increasing number is a number where all its digits are increasing from left to right (123, 234..) no digit being 0.

If i denote the first digit as 1, the second as 2, then i have (3,4,5,6,7,8,9) for the choices of the third digit. If i have the first digit as 1, the second as 3, then i have (4,5,6,7,8,9) to choose from. And so on. what i end up with is 7+6+5+4+3+2+1 combos for a three digit number with the first digit denoted as one. And (7+6+5+4+3+2+1)+(6+5+4+3+2+1)...+1 is the amount of combos for a three digit increasing number.
If all the digits are distinct, then there are six numbers to express with those three different digits. (123, 213, 321, ...) only one of those is increasing from left to right. so for all the digits to be distinct, we have nine choices for the first digit (1,2,...9), 8 choices for the second, 7 choices for the third. 9(8)(7) is the amount of combos for the three digit numbers which has all their digits different. and 9(8)(7)/6 for all the combinations of which the digits are increasing in a three digit number.
So...

9(8)(7)/6= (7+6+5+4+3+2+1)+(6+5+4+3+2+1)+(5+4+3+2+1)...+1 or

9*8*7/6= 7*1+6*2+5*3+4*4+3*5+2*6+1*7

 

[tongos]

Is there a relationship, yes!

because the squares go by a similar pattern...

(1)+(1+3)+(1+3+5)+(1+3+5+7)...

 

[dextercioby]

It's weird that a thread with a nonsensical title got so many replies,many of them even correct...

Daniel.

 

[Chronos]

Dang, isn't it easier to just anti-differentiate the formula for the area of a circle?

 

[dextercioby]

You mean the area of the sphere...?

Daniel.

P.S.$$\int_{0}^{R} 4\pi r^{2} dr=\frac{4\pi R^{3}}{3}$$

 

[vsage]

Speaking of sums of squares, I guess it was interesting when I was a HS freshman but if you integrate $$\frac{n(n+1)+1}{2}$$ and multiply it by 2 you obtain the formula for the sum of squares. Similarly you can integrate the formula for the sum of the squares and multiply by 3 to get the formula for the sum of cubes.

It's probably just a case of simple integration rules but it fascinates me as long as I'm too ignorant to see the connection!

 

[mathwonk]

By the way, "pleonastic" is a good one. I haven't heard that used since 1970, when Nat Hentoff, the jazz critic said that about Ringo Starr's drumming. Hentoff didn't like Jimmy Hendrix guitar playing either.

I'll use that in class today, if I really want to obfuscate something.

 

[saltydog]

assume your sphere has radius R, and you subdivide it into n equal parts each of length R/n, and circumscribe a stack of cylinders each of height R/n, inside the sphere.

Then by pythagoras, the radius r of the ith cylinder satisfies r^2 + [(i-1)R/n]^2 = R^2.

i.e. r^2 = R^2 - [(i-1)R/n]^2.

So the volume of the ith cylinder is pi r^2 (R/n) = pi (R/n) [R^2 - [(i-1)R/n]^2].

Hence the sum of all their volumes is the sum of these volumes as i goes from 1 to n.

which is pi R^3 - pi R^3 [1^2 + 2^2 +...+(n-1)^2]/n^3

now if we have the formula for the sum of these squares, namely (n^3)/3 +
terms of lower degree,

we see that as n goes to infinity, this quantity approaches pi R^3 - (1/3)(pi)R^3

= (2/3)pi R^3 = volume of a hemisphere (or semi - volume enclosed by a sphere).

thus indeed knowing the sum of the squares implies one can compute volume of a spherical solid.

Thanks Mathwonk. I've been trying to find out how he did that.

 

[saltydog]

Awareness

I attached a plot of a circle with 9 rectangles circumscribed in the upper half semi-circle representing 9 cylinders in the ball. The Archimedes analysis thus proceedes as follows:

Since:
$$x^2+y^2=R^2$$

then:
$$r_i^2=R^2-(\frac{iR}{n})^2$$
So that that volume of the i'th cylinder is:

$$V_i=\frac{\pi R}{n}[R^2-(\frac{iR}{n})^2]=\frac{\pi R^3}{n}-\frac{\pi R^3 i^2}{n^3}$$

Thus the total volume for the upper hemisphere with n partitions is:

$$V_{hem}=\sum_{i=1}^{n}[\frac{\pi R^3}{n}-\frac{\pi R^3 i^2}{n^3}]$$
$$V_{hem}=\pi R^3-\frac{\pi R^3}{n^3} \sum_{i=1}^{n}{i^2}$$

$$V_{hem}=\pi R^3-\frac{\pi R^3}{n^3}[\frac{n(n+1)(2n+1)}{6}]$$

$$V_{hem}=\pi R^3-\pi R^3[\frac{1}{3}+\frac{1}{2n^2}+\frac{1}{6n^3}]$$

Taking the limit of this process as n approaches infinity, obviously the last two terms in brackets go to zero leaving:

$$V_{hem}=\frac{2}{3}\pi R^3$$

Finally giving for the whole ball: $\frac{4}{3}\pi R^3$

You know, I spoke of "awareness" elsehere in the group. How nice to be "aware" of retracing the footsteps of a master. Now I understand why he's called "the Father of Integral Calculus".

(my thanks to MathWonk for explaining the process)

 

[dextercioby]

Thank you,i didn't know of this construction being assessed to Archimede's name.I liked the one with the area under of parabola.This is one is nice,too.

Daniel.

 

[mathwonk]

well i have never actually read archimedes. i have read that he discovered the sum of the first n squares, and that he could compute the area under a parabola, and the volume of a spherical ball.

having taught these subjects many times, and having always sought ways to explain and simplify them, I finally noticed these can all be derived by basically the same argument.

thus i surmised that this is the reason archimedes was able to do all of them.

so my attribution is that of a mathematical detective, deduced from evidence, not that of a historian working from actual sources.

still it seems very persuasive, does it not?

it seems a worthy goal is always to understand mathematical phenomena as clearly as possible, and as simply as possible. and to become freed from the canned presentations found in books, and only be guided by the underlying ideas of the masters.

once one knows the facts and tools available to the masters, one can be pretty confident that they would have successfully deduced any available consequences which follow from them.


so when i teach calc, I call this archimedes method, as opposoed to Newton's method, which uses the technique of antiderivatives to evaluate the same limit.

notice the different focus of mathematics in archimedes day, one of simply calculating an answer, rather than struggling with a precise definition of that answer, and questions such as whether or not area and volume actually
"exist".

 

[tongos]

do your calculus students see the beauty in mathematics? I mean, do they like it so much as to study it for fun?

 

[mathwonk]

well, some do.

many are afraid they will get a bad grade. and these have a fear of trying to understand the material. they think it is safer to memorize the answer to all possible problems, which of course is impossible.

our job includes at least showing them an example of someone who does love the subject.

 

[saltydog]

well i have never actually read archimedes. i have read that he discovered the sum of the first n squares, and that he could compute the area under a parabola, and the volume of a spherical ball.

having taught these subjects many times, and having always sought ways to explain and simplify them, I finally noticed these can all be derived by basically the same argument.

thus i surmised that this is the reason archimedes was able to do all of them.

so my attribution is that of a mathematical detective, deduced from evidence, not that of a historian working from actual sources.

still it seems very persuasive, does it not?

it seems a worthy goal is always to understand mathematical phenomena as clearly as possible, and as simply as possible. and to become freed from the canned presentations found in books, and only be guided by the underlying ideas of the masters.

once one knows the facts and tools available to the masters, one can be pretty confident that they would have successfully deduced any available consequences which follow from them.


so when i teach calc, I call this archimedes method, as opposoed to Newton's method, which uses the technique of antiderivatives to evaluate the same limit.

notice the different focus of mathematics in archimedes day, one of simply calculating an answer, rather than struggling with a precise definition of that answer, and questions such as whether or not area and volume actually
"exist".

Another words, maybe he didnt' do it that way. Sure took me a long time to figure it out even with your description. I can't imagine a better way to do it from "first principles" though. The leap is the limit!

 

[mathwonk]

yes, maybe he didn't, but I think I have made a conjectural case that he very likely did. It is hard to know how he did it since I understand some monk erased his works a few centuries ago, and the historians are only now trying to reconstruct it one symbol at a time, if you saw the tv show on it.

it seems well established for example that he did compute the area of a circle by approximating it by polygons with more and more sides, thus "exhausting" the circle in the limit. So it seems a small jump to exhaust a sphere by cylinders.

But yes, there is no guarantee he did it this way. What do you think?

The reason I think this, may sound strange if you are not a mathematician, but once the ideas are there, and all it takes is putting them together, then one often finds that all mathematicians do this independently in the same way.

I.e. if using these ideas that Archimedes had in his possession, I was able to construct these proofs, then it is not too much to expect that archimedes could certainly also do the same thing.

More bold perhaps, if Archimedes did it in some other way, then with my advantages of hindsight, I would also eventually succeed in doing it too. Since no one has suggested another way to deduce these results, probably Archimedes did not have one either.

Laypersons may believe for example that Fermat actually had a marvellous proof of his "last theorem" but I doubt any mathematician believes this. If such an elemetary argument has not been found in 350 years, then I think none exists.

All mathematicians share a grasp of logic, and an ability to reason by analogy. moreover the solution of a problem is most often not really created, but discovered, so if they are looking at it with the same tools, and in the same place, they will find it in the same way. That is why researchers hurry when they have made progress on a problem, because they know that anyone who hears what they have done, may be able to push it further in the same way as they are able.

But to be honest, since I am a mathematician and not a historian, I have no interest in doing research on Archimedes by reading parchments. I prefer to do it by thinking along what seem to be the same lines, and rediscovering the ideas myself.

It may seem odd, but i believe this is actualy more likely to lead to an understanding of what he did, than any other method available.

 

[saltydog]

But yes, there is no guarantee he did it this way. What do you think?

My only problem is the inscription on his tomb which was a SINGLE rectangle inside a circle. Seems if this was his most cherished discovery and he did so as you suggest (which I believe also), then would they not have inscribed the tomb with the same figure I drew in the attachement above (the pancakes in the circle)?

 

[arildno]

Here's, BTW, a link to the organization owning the palimpsest, where "The Method" is preserved:
http://www.thewalters.org/archimedes

 

[mathwonk]

saltydog, wasn't the inscription on his tomb rather a sphere inside a cylinder?

and when you say, arildno, that the palimpset "preserves" the method, have you seen the photos of a typical page of that document as it appears now? they showed it on tv , and it really is not readable. they are trying valiantly to eventually reconstruct some data from a parchment that was erased centuries ago.

I could be wrong, if say they were trying to show a particularly bad page. Maybe there are other pages with clearer writing, but I did not see anything you could learn from.

But I admit right off I am not a historian. I should not have said that Archimedes did this this way. I should have said, I believe he did.

Actually I would be very fascinated to read anything Archimedes wrote, if it is available. The palimpset seems not to be such. Is there a source for other works? I.e. a website for actual mathematical works?

By the way, although not a historian, I have of course read Plutarch's account of the siege of Syracuse by Marcellus, and the story of Archimedes inventions used in defense of the city, and of his death. I use this type of historical data to entertain my classes, and hopefully give some life to the subject.

 

[arildno]

Here's the relevant passage concerning the computation of the ball's volume:

"Archimedes is able to perform infinite sums: he takes a sphere, for instance, and calculates its volume as the infinite sum of the circles from which it is made... This was Archimedes' breakthrough, comparable to the modern integral calculus."

 

[mathwonk]

thanks. where does that come from?

by the way, if you understand "circle" to mean very small cylinder, then this is exactly the method I gave as his, and that saltydog illustrated with his pancakes.

 

[arildno]

Mathwonk: The palimpsest is in an awful state, so I think what is going on is a race to transcribe whatever can be retrieved from it, before the manuscript disintegrates completely.

 

[arildno]

thanks. where does that come from?

It comes from the link, in Dr. Naetz's comment there
This should be it:
http://www.thewalters.org/archimedes/frame.html

 

[mathwonk]

oh yes, and the sentence that he able to perform infinite sums also argues to me that he used limits.

Ok I checked that link, without however finding the quote you mention. THis link does not work so well onmy browser for some reason. I want to say however that these quotes found on this site do not have the force of historical reliability.

I.e. although I am not a historian I am more careful than the trnascribers of these statements. They quote as fact, statements which are written with considerably more caution in the original documents.

For example historians question the strict accuracy of the amazing descriptions of machines lifting ships from the water and so on, which occur merely as repeated stories in the original documents, not as strict historical fact.

Moreover the account of Archimedes death given on this website, is but one if several competing accounts. yet the website gives it as the truth.

so one should be careful about citing sentences found on some websites as correct. Many websites seem often to be much less reliable as sources of information than the original sources.

To get a better idea of Archimedes siege of Syracuse one should actually read Plutarch. And even then one is dependent on the translation, if one does not read Greek. Even then one is dependent on the accuracy of an old document which may or may not be genuine.

I.e I am not a historian, but I try to be a critical scholar.

 

[arildno]

oh yes, and the sentence that he able to perform infinite sums also argues to me that he used limits.

I "recently" saw a documentary on the palimpsest, where Netz said that some of the crucial passages Heiberg had been unable to transcribe had now yielded to modern reading devices.
the most interesting of these was precisely concerned with how Archimedes managed to compute infinite sums..

EDIT:
Unfortunately, both that program and the site are "popular" versions, I really would like a scholarly presentation of what they've found, but that is still lacking, I think.

 

[arildno]

thanks. where does that come from?

by the way, if you understand "circle" to mean very small cylinder, then this is exactly the method I gave as his, and that saltydog illustrated with his pancakes.

I agree, this must have been what Archimedes used.
In addition, the method of exhaustion would work quite nicely if he managed to derive an expression for the upper and lower finite sums used (which seems highly likely)

 

[mathwonk]

gee you said it so much better than I, and more briefly.

 

[mathwonk]

by the way, an interesting pooint to em is why he apparently did not deal with area and volumes of higher degree figures, such as cubics.

His method of exhaustion works just as well on them, and the formula for sums of cubes does not seem that much harder to us than the sum formula for squares.

maybe they just had no way to reporesent cubic figures. i.e. they lacked algebra, and so they met with objects that were defined more easily by geometry such as spheres.

but how did he come upon a parabola? how did the greeks describe a parabola?

Oh yes, I recall from field theory that all "constructible" lengths in geometry, i.e. lengths formed by intersecting lines and circles, are solutions of quadratic equations.

so maybe if eucldiean geometry is your main language and tool, you are restricted to quadratic objects.

 

[arildno]

oh yes, and the sentence that he able to perform infinite sums also argues to me that he used limits.

Ok I checked that link, without however finding the quote you mention. THis link does not work so well onmy browser for some reason. I want to say however that these quotes found on this site do not have the force of historical reliability.

I.e. although I am not a historian I am more careful than the trnascribers of these statements. They quote as fact, statements which are written with considerably more caution in the original documents.

For example historians question the strict accuracy of the amazing descriptions of machines lifting ships from the water and so on, which occur merely as repeated stories in the original documents, not as strict historical fact.

Moreover the account of Archimedes death given on this website, is but one if several competing accounts. yet the website gives it as the truth.

so one should be careful about citing sentences found on some websites as correct. Many websites seem often to be much less reliable as sources of information than the original sources.

To get a better idea of Archimedes siege of Syracuse one should actually read Plutarch. And even then one is dependent on the translation, if one does not read Greek. Even then one is dependent on the accuracy of an old document which may or may not be genuine.

I.e I am not a historian, but I try to be a critical scholar.

I agree with you that one should retain some scepticism as to whether the transcribers might have interpreted a bit too much into their findings.
We have virtually no documents from the ancient world which are older than, say 800-900 AD, that is, we only have copies of copies of..
However, my impression (from the show) was that Dr. Netz was a mathematician by education; the passage I quoted is quite far into his comment .

As for Archimedes' own work, maybe one can find them on the Gutenberg Project site

 

[arildno]

so maybe if eucldiean geometry is your main language and tool, you are restricted to quadratic objects.

Amen to that!

 

[mathwonk]

the gutenberg project does have one work.

In the introduction to it by a modern scholar, one finds justification for an opinion I stated earlier on this website that Euclid may not be a mathematician (in contradiction to statements on the palimpset website) as follows:

"It must
always be remembered that Archimedes was primarily a discoverer, and
not primarily a compiler as were Euclid, Apollonios, and Nicomachos."


I offer this for laypersons, who may have a different concept of what a mathematician does.

 

[mathwonk]

Wow! The paragraph following the one I just quoted is fantastic:

"Therefore to have him follow up his first communication of theorems to
Eratosthenes by a statement of his mental processes in reaching his
conclusions is not merely a contribution to mathematics but one to
education as well. Particularly is this true in the following
statement, which may well be kept in mind in the present day:

``l have
thought it well to analyse and lay down for you in this same book a
peculiar method by means of which it will be possible for you to
derive instruction as to how certain mathematical questions may be
investigated by means of mechanics.

And I am convinced that this is
equally profitable in demonstrating a proposition itself; for much
that was made evident to me through the medium of mechanics was later
proved by means of geometry, because the treatment by the former
method had not yet been established by way of a demonstration. For of
course it is easier to establish a proof if one has in this way
previously obtained a conception of the questions, than for him to
seek it without such a preliminary notion. . . .

Indeed I assume that
some one among the investigators of to-day or in the future will
discover by the method here set forth still other propositions which
have not yet occurred to us.''

Perhaps in all the history of
mathematics no such prophetic truth was ever put into words. It would
almost seem as if Archimedes must have seen as in a vision the methods
of Galileo, Cavalieri, Pascal, Newton, and many of the other great
makers of the mathematics of the Renaissance and the present time."


This reminds me of advice I once received from the outstanding Russian algebraic geometer, Boris Moishezon: "It is sometimes easier to find a proof, if you already know [the] answer."

 

[arildno]

Well, of all geniuses this world has seen, to me at least, Archimedes is still the greatest.
The gap between Archimedes and his contemporary researchers (by no means incompetent fellows) seems to have been so vast that they simply couldn't adjust themselves to his level (since he does not seem to have made lasting impact upon their thinking).
Since the Hellenic period in which Archimedes lived was the golden age of ancient technology (not only A. worked then), this fact is really remarkable.
This is quite different when comparing Newton, Gauss or Einstein to their ages; there were enough others who were able to appreciate their works.

 

[mathwonk]

My word! This is fantastic. Observe how Archimedes sums up many diificult calculations in a few words, which do contain the main ideas of the calculation:

"``After I had thus perceived that a
sphere is four times as large as the cone. . . it occurred to me that
the surface of a sphere is four times as great as its largest circle,
in which I proceeded from the idea that just as a circle is equal to a
triangle whose base is the periphery of the circle, and whose altitude
is equal to its radius, so a sphere is equal to a cone whose base is
the same as the surface of the sphere and whose altitude is equal to
the radius of the sphere.''

I.e. notice that the idea that a circle is merely a triangle whose base is the circumference of the circle, and whose height is the radius, is justified by approximating the cirfcle by polygons all having vertcies at thec enter, and bases on the circumference.

Then one takes the limit by allowing the number of sides of the polygon to increase without bound, and "Bob's your uncle!"

Similarly, the idea that a sphere ([ball]) is a cone whose base is the surface area, and whose height is the radius, is the same principle entirely.

holy smoke! I see this for the first time! i.e. you approximate a sphere's volume by that of a family of pyramids, each with vertex at the origin, nd base rectangles on the surface of the sphere. each has volume equal to (1/3) base area times height, whicha s you take more pyramids, approacjes (1/3) (area of sphere) (radius of sphere).

i.e. since the volume of a cone is (1/3) (area of base)(height), it follwos that the volume of a sphere is (1/3)(area of sphere)(radius of sphere).

but now you still have to get the volume some other way, since you do not know the area. but it shows that the area and volume of a sphere determine each other!

i.e.; to a modern student, the area of a sphere is the derivative of the volume, wrt radius, so either one determiens the other.

wow! young students take notice of how powerful it is to read the masters.

I now "see" (i.e. believe) I have been quite wrong (as apparently have others) to believe that Archimedes anticipated only integral calculus.

I.e. his calculation of the volume of a sphere, presumably by approximating slabs, pancakes, or cyl;inders, does indeed anticipate integral calculus.

But the deduction above of the area of the sphere from its volume, (to me at least) anticipates also differential calculus. I have never heard this said before.

By the way this answers my question in post 34 as to how he got the area of a sphere. Since my way of deducing that used what I consider the idea of differential calculus, and I did not think he had that idea, I could not see how he did it.

But I believe it now. What do you think Arildno?

By the way, the "idea" of differential calculus in this case is nothing but comparing the volume of a pancake with the area of its base. since archimedes had those components, and was a genius, he therefore must have seen the consequences.

 

[mathwonk]

Oh yes, another minor point perhaps, but relevant to understanding his work:

he did not have numbers and algebra, so all his calculations are ratios. I.e. he does not speak of formuals for voilume, but of the ratio of one volume to another or to an area.