What is the volume of a sphere?

[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.

 

[oscar_osa]

You are almost there,

When you integrate (r^2-x^2)dx you will have to fix the limits of integration, meaning, I can recommed from 0 to R, be careful, this is only half of the sphere, when you integrate the result is just r^2.R - R^3/3 this is iqual to R^3 - R^3/3 by just algebra this is equal to 2.R^3/3, as I told you this is just half of the sphere, double this number and you will obtain 4R^3/3, remember that pi was already out of the integration as constant. So to make the story short you have at the end 4.pi.R^3/3

I hope this helps

 

[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.

note: the integration will have to have its limits as -r and r since that si the boundries of
the circle

 

[counterpoint]

Yea, I know I'm slow. Anyway, here's the volume using a triple integral. And I didn't know either what Daniel meant about the volume being zero, and in fact it took me a while to figure it out even after Cepheid explained it.

In spherical coordinates, the problem can be defined as follows:

$$vol=8 \int_0^\frac{\pi}{2}\int_0^\frac{\pi}{2}\int_0^r \rho^2 \sin(\phi)d\rho d\theta d\phi$$

Beautiful isn't it!

So:

$$8 \int_0^\frac{\pi}{2}\int_0^\frac{\pi}{2} \sin(\phi)(\frac{\rho^3}{3}){|}_0^r d\theta d\phi$$

and then:

$$\frac{4r^3 \pi}{3}\int_0^\frac{\pi}{2}sin(\phi)d\phi$$

or:

$$-\frac{4r^3\pi}{3}[0-1]=\frac{4\pi r^3}{3}$$

Don't you just love Calculus!

good.

 

[rickcjmac]

r is a constant because if you have a sphere, then r is a specific number. For example, a sphere of radius 2, r=2. That's kind of tough to grasp when looking at it for the first time.

Counterpoint gave a triple integral to represent the sphere in the first octant, which is only an eighth of the sphere, which is why he multiplied by eight to get the final volume. His answer was correct, but here is the volume of a sphere in triple integrals without cutting it up:

$$\int_0^{2 \pi }\int_0^{ \pi }\int_0^r\rho^2d\rho d\phi d\theta$$

$$\left( -\cos \phi \Big|_0^{ \pi } \right) (2\pi)\left(\dfrac{r^3}{3}\right)$$

$$\dfrac{4}{3}\pi r^3$$​

 

[ayushojha]

excuse me Jameson there is only one variable in the integration given by you. the "r" is not a variable but is a constant which is actually the limit for the variable "x" (upper limit in this case). To integrate you need to put a trigonometric form of x i.e x=r cos(theta). then choose the limits for "theta" and "theta" becomes the variable instead of "x".