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mathwonk
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Observe that all this arises from reading ONE SENTENCE by archimedes.
Some guys simply can't avoid being brilliant, huh?mathwonk said:I wish to observe that I got all this from reading ONE SENTENCE by archimedes.
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 said: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.
I would think he (or someone prior to him) used a clever "pan-cake" method.mathwonk said:i guess what i need to know is how he found out the ratio between the volume and base area of a cone.
mathwonk said: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 said: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.
arildno said: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)
dekyfineboy said: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.
saltydog said: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:
[tex] 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 [/tex]
Beautiful isn't it!
So:
[tex] 8 \int_0^\frac{\pi}{2}\int_0^\frac{\pi}{2} \sin(\phi)(\frac{\rho^3}{3}){|}_0^r d\theta d\phi [/tex]
and then:
[tex] \frac{4r^3 \pi}{3}\int_0^\frac{\pi}{2}sin(\phi)d\phi [/tex]
or:
[tex] -\frac{4r^3\pi}{3}[0-1]=\frac{4\pi r^3}{3}[/tex]
Don't you just love Calculus!