What is the volume of a sphere?

[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.