What is the volume of a sphere?

[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.

 

[mathwonk]

Observe that all this arises from reading ONE SENTENCE by archimedes.

 

[arildno]

I hadn't seen his cone argument before; it is sheer brilliance.

And yes, Archimedes (and the other geometers) were always careful (we would say over-careful) with only comparing dimensionless numbers (i.e ratios) to each other;
for example, Archimedes' law of the lever is given in the form that in equilibrium, the ratios of the weights equals the inverse ratio of lever arms.
That is, the equality between moments about the fulcrum which we use was alien to Greek thought.

 

[arildno]

I wish to observe that I got all this from reading ONE SENTENCE by archimedes.

Some guys simply can't avoid being brilliant, huh?

 

[mathwonk]

i noticed in galileo that he reasons with real numbers also by considering a real number as a ratio of the lengths of two line segments. thus he draws pictures of real numbers as two segments. I always thought this was due to a lack of algebraic notation, as is implied in the footnotes of my translation. but maybe he was following a tradition of preferring geometry?

 

[arildno]

Yes, I would think so.
Irrationals, in their guise of incomensurable (was that the right word?) quantities, dates back to the Greeks, so I think Galileo was just following the conventional way of looking at this.

 

[mathwonk]

was descartes then a pioneer in marrying the traditions of algebra and geometry, which had existed separately for a long time?

the time line fits,a s galileo was born 1564 and descartes in 1596.

 

[arildno]

Yes, from what I've heard, Descartes is credited as the inventor of analytical geometry and showed how all geometrical propositions could be recast into algebraic equivalents.

 

[arildno]

I have to add to your previous comment, that it is quite striking how Archimedes derives the area of the sphere from its volume (calculated by the pan-cake method).
I've never heard of this derivation of his before (derivation in the double sense..)

 

[mathwonk]

Oh boy! Here is a quote from the introduction to the work of archimedes where it states explicitly, that archimedes found a volume of a certain section of a cylinder, by reducing it to the problem of the area of a parabola.

"Proposition XI is the interesting case of a segment of a right
cylinder cut off by a plane through the center of the lower base and
tangent to the upper one. He shows this to equal one-sixth of the
square prism that circumscribes the cylinder. This is well known to us
through the formula $v = 2r^2h/3$, the volume of the prism being
$4r^2h$, and requires a knowledge of the center of gravity of the

cylindric section in question. Archimedes is, so far as we know, the
first to state this result, and he obtains it by his usual method of
the skilful balancing of sections. There are several lacunae in the
demonstration, but enough of it remains to show the ingenuity of the
general plan. The culminating interest from the mathematical
standpoint lies in proposition XIII, where Archimedes reduces the
whole question to that of the quadrature of the parabola."


By the way, the famous work of Galileo in the 1600's of discovering that a moving projectile travels in the path of a parabola, and that the distances traveled by a falloing object, in succeeding units of time, stand to one another as the squares of the positibe integers, are also mathematical consequences of the work of archimedes.

this causes one to wonder why they were thought to be new in galileo's time, and why a genius like galileo did not realize they were corollaries of archimedes work.

of course the connection of the mathematics with the physics is in itself a significant discovery, but galileo seems to re-derive all the mathematics by geometry. this puzzles me.

 

[mathwonk]

i should be carefula s to my claim for archimedes. it seems from the sentence above that he connects the area and volume of a sphere, by a formula which to me resembles differentiation, i.e. by considering the area of the sphere as the base of a cone forming the sphere.

then to deduce the area from the volume requires fidnign the volume. I am assuming the reasonable fact that he found the volume by the usual method attritbuted to him, of approximation by cylinders.

I.e. what is visible to me in this sentence of his, is the connection between the two that to me anticipates differentiasl calculus. what i am assuming is the more widely accepted fact that he found the answer by methods that anticipate integral calculus.

i have not yet encountered this latter calculation in the manuscript.

 

[arildno]

Well, was the works of Archimedes actually accessible to Galileo?
Those copies we have today may have languished in monastery libraries, and their re-discovery happening after Galileo's time.
In any case, even if these were known to exist, it is probable that such works were preserved as one-of-a-kind documents, perhaps jealously guarded. Galileo might have been refused access to them, or he might have found a study journey too expensive.
(This is sheer speculation on my part, though..)

 

[arildno]

i should be carefula s to my claim for archimedes. it seems from the sentence above that he connects the area and volume of a sphere, by a formula which to me resembles differentiation, i.e. by considering the area of the sphere as the base of a cone forming the sphere.

then to deduce the area from the volume requires fidnign the volume. I am assuming the reasonable fact that he found the volume by the usual method attritbuted to him, of approximation by cylinders.

I.e. what is visible to me in this sentence of his, is the connection between the two that to me anticipates differentiasl calculus. what i am assuming is the more widely accepted fact that he found the answer by methods that anticipate integral calculus.

i have not yet encountered this latter calculation in the manuscript.

That calculation is what is claimed found in "The Method" (this has been accepted since Heiberg's edition in 1900 or so, I believe)

 

[mathwonk]

wow this was fun! thanks arildno. i definitely feel I learned something!

 

[mathwonk]

perhaps i should be more careful about maikng the link with derivatives. i.e. archimedes could have connected the area and volume of a sphere by as i said, approximating the spheres volumes by the volumes of a family of pryramids, whereas the differentiation method would seem to use instead a family of spheres, expanding their radii to that of the given sphere.

anyway i am tired now and will check out. thanks again.

 

[arildno]

As an after-thought, perhaps what Archimedes did should be thought of as devising two different volume computations; the pan-cake method, and V=Sr/3 (the cone method)

 

[arildno]

Seems that both of us got the same reservation here..

 

[mathwonk]

i guess what i need to know is how he found out the ratio between the volume and base area of a cone.

 

[arildno]

i guess what i need to know is how he found out the ratio between the volume and base area of a cone.

I would think he (or someone prior to him) used a clever "pan-cake" method.

I'm not sure, but I think the 1/3*base*height formula precedes Archimedes

 

[mathwonk]

oh yes, that would be the same as the other quadratic integral calculations today.

i.e. use similar triangles to express the radius r of the pancake as a proportion of the height.

i.e. let the cone have height H and base radius R, and consider the pancake at distance x from the top. then its radius r satisfies x/r = H/R, so r = Rx/H, so pi r^2

= pi (R/H)^2 x^2. so the volume of the pancake is this area times its height, i.e. times H/n. i.e. (pi) (R^2) (x^2)/(nH). I hope.

oh yes and the distance of the ith pancake from the top is i(H/n) = x,

so let's see the volume of the ith pancake is (pi) (R^2) (i^2)H/(n^3). hopefully


then add up as i goes from 1 to n, and get something like

(pi) (R^2 H)(1/n^3)( formula for sum of squares of i's)

= (pi) (R^2 H)(1/n^3)( n^3/3 + lower etrms),

and take limit as n gets larger,

getting ttata:

(pi) (R^2 H)(1/3). yep that's it. no derivatives needed. shoot. another great conjecture shot down by facts.

 

[mathwonk]

i have since read some archimedes, not in detail, nd other sources. it now seems likely that eh did the volume of a pyramid first, then approximated the volume of a sphere by pyramids with vertices at the center of the sphere, then took limits and obtained the volume as 1/3 the product of the surface area and the radius, in perfect analogy with the case of a circle. then he may showed the surface area of a sphere agreed with that of the lateral area of a cylinder, finishing it off.

the works of archimedes are highly recommended, in print from dover.

 

[arildno]

i have since read some archimedes, not in detail, nd other sources. it now seems likely that eh did the volume of a pyramid first, then approximated the volume of a sphere by pyramids with vertices at the center of the sphere, then took limits and obtained the volume as 1/3 the product of the surface area and the radius, in perfect analogy with the case of a circle. then he may showed the surface area of a sphere agreed with that of the lateral area of a cylinder, finishing it off.

the works of archimedes are highly recommended, in print from dover.

Thanks for the update!

 

[coomast]

I am not going into the discussion of the spere or ball or whatever the round thing is called, I'm not a native speaking English person, I make up from the context what is exactly meant :-)
However, one thing struck me in the following post, namely the second and the last sentence:

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.

I agree that this is true, but there are others who were also far ahead of their time. Evariste Galois for example. Don't misunderstand me, I am not seeking to open a discussion again, I completely agree with you Arildno. Taking this a bit further, what about Giordano Bruno?

 

[arildno]

Well, Abel would likely have understood some of Galois' work, if he had known it, and it didn't take more than 20-30 years after Galois' death before others recognized his importance.

However, it should be said that the manner in which Galois wrote his work, it was fairly illegible, and it had to be "cleaned up".
While there was rigour in his thinking, it was well hidden..
(Anyways, that's what I've heard about Galois)

 

[coomast]

Well, Abel would likely have understood some of Galois' work, if he had known it, and it didn't take more than 20-30 years after Galois' death before others recognized his importance.

However, it should be said that the manner in which Galois wrote his work, it was fairly illegible, and it had to be "cleaned up".
While there was rigour in his thinking, it was well hidden..
(Anyways, that's what I've heard about Galois)

This is certainly right. Galois' writing was difficult to read. He left out intermediate steps and didn't work systematically. However, if you can write papers and they are not recognized by the leading mathematical society, then you have the same "problem" as with Archimedes, you're ahead of your time, no?

Galois did read Lagrange and Abel, Cauchy rejected some of his papers, Poisson and Lacroix didn't come back on a memoir he wrote to them...

In Archimedes' time there were not so many people studying "science" compared to the time in which Galois lived so the chance of being understood was smaller, or is this incorrect? In Archimedes" case he was well recognized, fortunately.

Anyway, my admiration of both (and a lot of others) is the same.

An interesting book on Galois is "Galois Theory" written by Ian Stewart, ISBN 0 412 10800 3, with a small introduction on his life.

The remark on Giordano Bruno is a bit off post here. I only wanted to mention that things can go very bad if one is not understood. But in this case it is more related to religion and politics I believe.

 

[arildno]

Sure there were fewer scientists back in Archimedes' time, but that makes his insights all the more remarkable.

Consider how mathematicians work: They chat with each other all the time of various topics.
And so do all other scientists as well.

This is a positive feedback loop that spurs every one of them onto new research fields, and abandon worthless projects others have made them realize were worthless.

The lone genius is a very rare entity, mostly, gifted individuals without a social network of peers will degenerate into crackpots. Sad, but true..

Scientists need each other to stay on track and improve themselves.

 

[coomast]

Absolutely true, I can't agree more on this.

 

[mathwonk]

ok i have actually read more of archimedes and think i know how he found the volume of a sphere, or at least how he proved it. (he discovered it by setting up a lever and balancing the weights of different solids, knowing the centers of gravity of some of them, and deducing that of others.)

basic principles:
1) principle of parallel slices: two solids with equal areas for all plane slices parallel to a given plane, have equal volumes.
2) magnification principle: two pyramids with bases of equal area, have volumes in the same ratio as their heights.

these principles are proved by the method of approximation by blocks or cylinders, since solids with equal plane slices have equal approximating cylinders, and scaling the height merely scales the height of the approximating cylinders. then one proceeds as follows, first for pyramids and cones, then spheres.

step 1) right pyramids of height equal to base edge:
choose 2 opposite vertices on a cube, call them 1 and 2, and join them by a diagonal. choose a face having vertex 2 as a corner, and join every point of this face to vertex 1. this forms a right pyramid. the other two choices of faces having vertex 2 as corner, yield congruent pyramids, by rotation, and all three together make up the cube. thus the given right pyramid has volume 1/3 that of the cube, or 1/3 Bh, where B = area of base, and h = height.

step 2) using magnification principle, one extends the same formula to the case of arbitrary height in comparison to base edge, and using parallel slices one extends the same formula to pyramids which are not "right", but for which the angle to the vertex is arbitrary, since sliding a pyramid over at a new angle does not change the area of parallel slices.

step 3) approximating the base circle by polygons, hence approximating the cone by pyramids, gives the same formula for a cone, V = 1/3 Bh.

step 4) now circumscribe a cylinder about a sphere, and inscribe a double cone (vertex at center, bases at both top and bottom) in the same cylinder. then pythagoras shows that the area of a parallel slice of the cylinder has area equal to the sum of the parallel slices of the sphere and the cone.

Thus the volume of the cylinder equals the sum of the volumes of the cone and the sphere. in particular since the cone has 1/3 the volume of the cylinder, the sphere has 2/3 the volume of the circumscribing cylinder.

And that is how archimedes proved the volume of a sphere.

the by the argument above, viewing the sphere as a limit of pyramids with vertices at the center, he showed the surface area of the sphere, defined as the limit of the areas of the bases of the inscribed pyramids, was 3/R times the volume of the sphere, since tht is the formula for the base area of a pyramid in terms of the volume.

I.e. the volume of a sphere is 1/3 SR where S is the surface area and R is the radius.

and that's that! hooray for archimedes, who was obviously in almost complete command of the methods of purely integral calculus.

the only thing needing to be added, was the algebraic technique of antidifferentiating the algebraic formula for the area of the parallel slices and getting an algebraic formula for the moving volumes below each slice.

so as far as i know now it had nothing to do with ding up squares of integers at all, quite opposite to my original impression.

 

[mathwonk]

moreover archimedes said he could also compute that the volume of a bicylinder, intersection of two perpendicular cylinders, is 2/3 that of a circumscribing cube. his solution of this is lost, but you can guess it if you reflect that the horizontal slices of a bicylinder are intersections of horizontal slices of cylinders, i.e. intersections of rectangles, hence are squares.

thus you want to replace his prior use of a cone by some cone - like figure whose horizontal slices are squares. what do you guess? ...that's right! try it.

so this is archimedes actual work, and this i believe should be taught to every geometry student before attempting calculus. in fact harold jacobs' fine high school geometry book has this calculation of the volume of a sphere near the very end of his book.i also feel that this use of limits is not properly calculus, but that calculus is the combination of differentiation and integration, found in the fundamental theorem. i.e. i would preserve the term calculus for the use of antidifferentiation to compute the limits archimedes used to define volumes. of course this use of terminology is a matter of preference. note euler also declined to refer to the limits involved in infinite series, as calculus.

 

[arildno]

thanks mathwonk!
He becomes greater and greater, the more I get to know his work..

 

[dekyfineboy]

When the region between a and b of the function f(x) is rotated about the x-axis, the solid formed will have a volume

(pi)*(integration of f(x)^2). ----------------- 1

so we need the the formula of a circle so that we can put it into the formula

formula of a circle is given by r^2=x^2 + y^2 ---------------- 2
therefore making y the subject y^2=r^2 - x^2 ---------------- 3

put y=f(x) into the equation 1 and the formula for the volume of a sphere will be found.