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After discussing the dimensions of the Yam shel Shlomo, the gemara explains a verse in Divrei Hayamim that gives a different volume measurement of the Yam, by saying that that one is for dry measure. [The verse in Melachim says it contains 2000 bat, but the verse in Divrei Hayamim says 3000 bat.] Then the gemara says that you see from there that גודשא תלתא, dry measure is half again that of liquid measure (since you take into account the heaping up).
Since the piling up depends only on the shape of the top of the vessel, Rashi says this ratio is for a vessel where the height is half the horizontal dimensions, like the Yam shel Shlomo.
Okay, I have a number of questions.
1) Why would the author of Divrei Hayamim ever think of giving a dry measure for the Yam shel Shlomo - used to hold water?
2) I'm not much of an artist, but I tried to draw a picture:
These are the cases of a square or a round vessel (the filler material piles up above the top of the vessel.) In each case, the material being measured can pile up to a maximum height and volume. The maximum is, I guess, where the angle(s) shown in the diagrams equal what's called the Angle of Repose. Any steeper, the pile will slip. [Note that I drew the angle for the rectangular shape at its steepest slope, not along the diagonal. For the cone it won't matter; all directions are equally steep.]
Now the angle of repose depends on the material. Some things slip sooner than others. In the wikipedia article, it gives 45 degrees for flour, and 28 degrees for wheat kernels - both reasonable guesses for what the author of Divrei Hayamim had in mind. More than 45 degrees is quite unusual.
However, the rule of the gemara requires an angle of repose of 56 degrees! One calculates that using the volume rule for a pyramid, or a cone: 1/3 * base * height [not 1/2 like for a 2D triangle]. For that to equal half the volume of the rectangular/cylindrical region on the bottom, we're going to need a height of 3/4 of the whole horizontal width (remember that Rashi said the rectangular/cylinder was half as high as it was wide.) Then those angles in the pictures have a height of 3/4 width, and a base of 1/2 width, and the arctangent (arctan((3/4)/(1/2))) is 56.3 degrees. So the picture isn't drawn well, it's much steeper, and steeper than any likely dry filling material.
3) In any case, I don't see how this will work for the Yam shel Shlomo. The gemara concludes there that the base was partly square (bottom 3 amos) and partly round (top 2 amos). That's going to mean that we should be using my cone picture, but with a square bottom. That volume on the bottom (which the gemara said was actually part of the liquid volume measurement - not solid) is going to ruin the calculation of the gudsha being half of the volume of the base. And if it is true for the Yam shel Shlomo - by making the pile even steeper! - it would no longer be true for a standard rectangular or cylindrical vessel.
Since the piling up depends only on the shape of the top of the vessel, Rashi says this ratio is for a vessel where the height is half the horizontal dimensions, like the Yam shel Shlomo.
Okay, I have a number of questions.
1) Why would the author of Divrei Hayamim ever think of giving a dry measure for the Yam shel Shlomo - used to hold water?
2) I'm not much of an artist, but I tried to draw a picture:
These are the cases of a square or a round vessel (the filler material piles up above the top of the vessel.) In each case, the material being measured can pile up to a maximum height and volume. The maximum is, I guess, where the angle(s) shown in the diagrams equal what's called the Angle of Repose. Any steeper, the pile will slip. [Note that I drew the angle for the rectangular shape at its steepest slope, not along the diagonal. For the cone it won't matter; all directions are equally steep.]
Now the angle of repose depends on the material. Some things slip sooner than others. In the wikipedia article, it gives 45 degrees for flour, and 28 degrees for wheat kernels - both reasonable guesses for what the author of Divrei Hayamim had in mind. More than 45 degrees is quite unusual.
However, the rule of the gemara requires an angle of repose of 56 degrees! One calculates that using the volume rule for a pyramid, or a cone: 1/3 * base * height [not 1/2 like for a 2D triangle]. For that to equal half the volume of the rectangular/cylindrical region on the bottom, we're going to need a height of 3/4 of the whole horizontal width (remember that Rashi said the rectangular/cylinder was half as high as it was wide.) Then those angles in the pictures have a height of 3/4 width, and a base of 1/2 width, and the arctangent (arctan((3/4)/(1/2))) is 56.3 degrees. So the picture isn't drawn well, it's much steeper, and steeper than any likely dry filling material.
3) In any case, I don't see how this will work for the Yam shel Shlomo. The gemara concludes there that the base was partly square (bottom 3 amos) and partly round (top 2 amos). That's going to mean that we should be using my cone picture, but with a square bottom. That volume on the bottom (which the gemara said was actually part of the liquid volume measurement - not solid) is going to ruin the calculation of the gudsha being half of the volume of the base. And if it is true for the Yam shel Shlomo - by making the pile even steeper! - it would no longer be true for a standard rectangular or cylindrical vessel.
