Rif Shabbat 3a

HIDE/SHOW IMAGE - there is a diagram
3a

{Shabbat 8a continued}
By way of explanation: any place 10 {handbreadths} high and 4 X 4 {handbreadths - the Bach emends and fixes the gender of the numbers} separates off a domain for itself, and is a private domain.

And therefore, someone who throws a {round} bin into the public domain is exempt {from liability}, for since it is 10 handbreadths high and 4 X 4 {handbreadths} wide, it separates off a domain for itself, and he is like one who has thrown from one private domain to another private domain, which is exempt.

And they only said that you need a width of 6 {handbreadths} because it {the bin} is circular, because it is impossible to inscribe {a square} 4 X 4 {handbreadths - gender of number is wrong} unless there is in the width {of the circle - that is the diameter} 6 X 6, as we say: for each cubit in the side of a square, there are 1 and 2/5 cubits in its diagonal. And it is a known fact that the width of a circle is equal to the length of the length of the diagonal of the square
inscribed in the circle, and this is its diagram, so that it is made clear by seeing it.
{It is in fact obvious, especially when you see the diagram. Click the link above to see the scan of the page, which has the diagram.}

And therefore we need to add in the width of the circle, which is the diagonal of the square, 2/5 of 4 handbreadths, for a total of 8/5th, that is 2/5 for each and every handbreadth, such that the length of the side of the square inscribed in it is 4 handbreadths. And we have further added an additional 2/5ths for the thickness of the walls of the bin, 1/5th here and 1/5th there, for a total of 10/5ths, which are = 2 handbreadths, such that we have 6 handbreadths, no less and no more.

{Using contemporary mathematics, rather than the fairly good estimate of 1:1.4 ratio for length of side:diagonal, we can get a slightly more accurate calculation. That is, the length of the diagonal of the inscribed square = the diameter of the circle, as we see in the Rif's diagram. We know the length of the square must be 4. What will be the length of the diagonal of the square, and thus the diameter of the circle?
We can use the Pythagorean theorem. For a right triangle with sides a and b, and a hypotenuse c:

a² + b² = c²

Now, since it is a square, a and b are the same. Therefore:

a² + a² = c²

or

2a² = c²

Now, a = 4, so a² = 16, and 2a² = 32;

So, c² = 32; Taking the square root of each side, c = square root (32) ~= 5.65685425. This means that the diameter of the circle, c, needs to be 5.65685425, which is less than 6. However, perhaps the gemara was dealing in whole units, and thus used 6 handbreadths, since 5 would not be sufficient. Or, like the Rif, we can divide the remaining width into the two walls of the bin. 6 - 5.65685425 = 0.34314575, which is even slightly less than the 2/5 = 0.4 that the Rif had in excess.

Update: To relate this to the Rif's calculation, consider:

2a² = c²

Therefore,

c = square root(2a²) = square root(2) X square root(a²) = a X square root(2)

Thus, to find the diagonal of a square, rather than multiplying a side a by 1.4, which is the good estimate of the Rif, one should multiply it by the square root of 2, which is approximately 1.414.
}

Rava said: Even if its width was not 6 X 6 he is exempt.
What is the reason?
It is impossible that the piece of cane not project above ten, so that it is a mekom petur.
If he overturns it {where it is less than 6 wide}, mouth downwards, [and throws it], then if it is a shade more than seven [in height] he is liable; if seven and a half, he is exempt.

{Because lavud makes it so that items within 3 handbreadths of the ground are considered resting on the ground. Therefore, when it is within 3, then if it is 7 handbreadths high, it is entirely within 10 handbreadths of the ground, and he is liable. The same for 7 and a tiny bit, since he needs to be within a tiny bit of the 3 for lavud to operate. However, if it is 7 1/2 handbreadths high, then when it first enters the 3 handbreadth space, part of it projects over 10 handbreadths, so that it is in a mekom petur.}

Rav Ashi said: Even if its height is 7 1/2 {handbreadths} he is also liable.
What is the reason?
The walls are made for their contents {to hold the contents, and not to create an imaginary wall downward}, and we cannot use it to make it higher than 10, and therefore he is liable.

Ulla said: If there is a column nine [handbreadths high] in the street, and the public rest and rearrange their burdens thereon, and one throws [an object] and it alights upon it, he is liable.
What is the reason?
Anything less than 3 {handbreads} the public tramples upon; from 3 until 9, the public does not trample upon, and they do not rearrange their burdens thereon; 9 they certainly rearrange their burdens thereon {Rashi: and since it is for the purpose of the public it is considered the public domain}.

Abaye said to Rav Yosef: What of a pit {which is 9 deep}?
He said to him: And so too a pit.
Rava said: A pit, no.
What is the reason?
He holds: Walking albeit with difficulty is considered walking; service {use by the public} with difficulty is not considered use.

{Shabbat 8b}
Rav Yehuda said: In the case of a zirza dekanei {a bundle of canes}: if one raises this end and lowers the other, then raises that end and lowers the other {thus transporting it head over heels}, he is not liable until he uproots it.
By way of explanation, a zirza dekanei is a bundle of canes, which is lying on the ground, and he lifts one end and the other end is resting on the ground. And he raises it until upright {perfectly vertical} and then casts it before him. And he returns and lifts up the other end which was resting on the ground, and raises it upright and then casts it before him. And he returns and lifts it up, etc., even the entire day, he is not liable, for he never uprooted it from the ground.

A man standing on a threshold may take [an object] from or give [it] to the master of the house, and may take an object] from or give [it] to the poor man.

This "threshold," what is it?
And we conclude that the "threshold" is a mekom petur, such that it does not have an area 4 X 4 {handbreadths} and like this {statement} that when Rav Dimi came, he cited Rabbi Yochanan that does not have 4 X 4 {handbreadths} it is permitted for the people in the public domain and the people in the private domain to rearrange their burdens thereon, so long as they do not switch.
{that is, someone from the public domain puts his burden in this place, and someone from the private domain takes it, or the opposite case}

{Shabbat 9a}
Others state: A threshold serves as two domains: if the door is open, it is as within; if the door is shut, it is as without.