Changing the load from wheat to barley: Yoni and Eitan make sense of a Talmudic discussion between Abaye and Rava

Since nothing really interesting happened today, (apart from my Valentine's Day ploy to photograph Marina for the blog, an attempt that was promptly foiled), I figured I'd write about an unusual experience I had in my night gmara. As I mentioned yesterday, we are learning about wheat and barley with respect to donkeys carrying it. The mishna in Bava Metzia states that a person can be liable if he changes the load on the donkey from wheat to barley, when the change is substantial enough. This is in part because of the fact that wheat is denser than barley, and in part because in changing the grain one is simultaneously changing two factors, volume and density. Basically (I think), one Talmudic sage, Abaye, says that changing the load means that you're influencing both volume and weight, which means that you have less leeway when changing grains. Rava, Abaye's foil and longtime nemesis, thinks that weight is the only significant factor, and therefore when you change grains, you can add more weight before becoming liable.

NOTE: If you are confused already, do not read on. It only gets more confusing.
NOTE: If you are confused already, yeshiva is not for you. Yeshiva, by definition, is confusing.

We (or should I say, Eitan) decided to turn the whole thing into a math problem. What really happened is that it turned into a Yoni problem, as the only math I've been doing in Israel is calculating the rate between dollars and shekels. This is the work we did; we created the densities of wheat and barley, making sure that the barley is less dense. Other than that, the math is real. Or so I'm told. I'm writing this down because I know that when I look back on this, I'll have no idea what it was. Come to think of it, as the minutes tick by, I am getting foggier and foggier as to what we did.

If you can make sense of the following, you're an Eitan, who is a Harvard genius. If you can't, no sweat, you're a Yoni, an average guy who just happens to be an award-winning blogger. Who Harvard rejected.

Density of Wheat = ⅔
Density of Barley = ½
w = weight
v = volume
L = load, as defined as weight + volume

v0 = v
w0 = ⅔v0
w’ = ½v’

L אביי= v+w
חייב= L0 < size="1">0 + v0 < size="1">0 + v0 < ½v’+ v’ 5/3 v0 < size="1">0 < size="1">אביי = v’- v0 = 1/9 v0

Lרבא= w
חייב= L0 < size="1">0 < size="1">0 < ½v’ v’ < size="1">0
Δvרבא = v’- v0 = 1/3 v0


Figure 1:1
Get it?