Daf yud-dalet (Page fourteen). Today's daf is being studied le'ilui nishmat (for the elevation of the soul) Avraham ben Iser, lirefu'ah sheleimah (for a complete recovery) Dinah bat Sarah, Shoshana Mo'alem bat Sarah, and Yafah Sarah Elkeh Eidel Rivkah bat Rachel. We begin today's daf on yud-gimel amud bet. We start from the Mishnah.
Hakorah she'amru (the beam they mentioned): we're talking about the korah, the crossbeam over the mavoi (alleyway), rechavah kedei lekabel ariach (wide enough to receive a half-brick). An ariach is a shi'ur (measure) of half a brick. Rashi says, shi'ur domeh likvi'ut livnot aleha binyan (the measure is like a foundation on which to build), so you have to build level and put bricks upon it.
The Mishnah says, ve'ariach, how much is an ariach? Chatzi leveinah shel sheloshah tefachim: half of a brick that is three tefachim (handbreadths). Dayah lakorah shetehe rechavah tefach: it is enough for the korah itself to be a tefach (handbreadth) wide, kedei lekabel ariach lerochbo, so it can receive the brick across its width. As you see, the korah is a tefach wide, and you can put this ariach on it. Even though the ariach is a tefach vachetzi, it stays on it. There is a little overhang on both sides, a quarter of a tefach on each side, but it can hold it. It's sturdy enough.
Rechavah kedei lekabel ariach uvevri'ah: if it is wide enough to hold one of these bricks, it must also be strong enough to hold the ariach. See what the nafka mina is (practical difference). Rabbi Yehudah says, rechavah: no, it just has to be wide enough, af al pi she'ein beri'ah, even though it is not strong enough to actually receive it.
Therefore, haitah shel kash, if it was made of straw, veshel kanim, or reeds, which are not strong enough to hold the brick, ro'in otah ke'ilu hi shel matechet: we look at it as if it were metal, and therefore strong. However, akumah: if it is bent, so that the korah cannot rest on it, ro'in otah ke'ilu hi peshutah: we look at it as if it were straight. Agulah: if it is rounded, the brick will fall off, so ro'in otah ke'ilu hi meruba'at: we look at it as if it were squared.
Kol sheyesh beheikefo sheloshah tefachim (anything with a circumference of three handbreadths) yesh bo rochav tefach: its diameter will be one tefach. That's the way you make the calculation. Rashi explains that you take a string and go around it; it takes a string three tefachim long to go around it. If you straighten it out, its width will be a tefach. That's a general rule in mathematics.
The Gemara asks: Tefach? You told me the korah could be a tefach. What do you mean? Tefach umechtzah ba'i: seemingly, it should need to be a tefach and a half, because the ariach has to be a tefach and a half. So why did you say that a tefach is enough?
The Gemara says: keivan derachavah lekabel tefach, idach chatzi tefach melaben leih betina, mashehu mehai gisa umashehu mehai gisa, vekaima. As we said, there is a small overhang on each side. But melaben leih betina: you put some tar on the korah when you are building, place the brick there, and it stays. Even though there is a small overhang on each side, it will not fall off because something is holding it there. So as long as the korah is a tefach wide, you can fit a tefach and a half on it in an actual building situation.
Rabbah bar bar Chanah says, korah she'amru tzrichah shetehe beri'ah kedei lekabel ariach: the korah must actually be strong enough to receive an ariach, like the Tanna Kamma (first tanna) said in our Mishnah. Ume'amidei korah: the poles or pegs coming out of the wall, on which the korah rests, do not have to be strong enough to receive both a korah and an ariach. The korah has to be strong, but its supports do not.
Rav Chisda says, echad zeh ve'echad zeh tzrichim sheyihyu beri'im kedei lekabbel korah ve'ari'ach: both the korah itself and the pegs coming out of the wall have to be strong enough to receive the ariach and the korah.
Rav Sheshet says: hini'ach korah al gabei mavoi, he put a korah on top of the mavoi, uferas aleha machtzelet, and then put a mat over it, almost like a wall, to close the opening of the mavoi. Vehigbiha min hakarka sheloshah: he did it so that it was more than three tefachim off the ground. Korah ein kan, mechitzah ein kan: you have lost your korah. You put the korah there and then covered it with a sheet, trying to make a whole door at the opening of the mavoi. You lost the korah by covering it, and you have no mechitzah.
Korah ein kan deha mikasya: the whole purpose of the korah is heker. Once you cover it, you have lost the heker. Mechitzah ein kan deha havya lah mechitzah shegediyim bok'im bo: since it is three tefachim off the ground, what kind of mechitzah is that? The goats can get underneath. You might say it is not lavud, so it is not a mechitzah, because it does not serve its purpose; a mechitzah's purpose is to block, and this does not block.
The Gemara says: Tanu Rabbanan, korah hayotzah mikotel zeh ve'einah noga'at bekotel zeh: the korah does not reach the end of the area you are trying to cover. Vechen shtei korot, achat yotzah mikotel zeh ve'achat yotzah mikotel zeh ve'einan noga'ot zo bazo: similarly, there are two korot coming out from both sides, but they do not meet. Pachot misheloshah ein tzarich lehavi korah acheret: if the gap is less than three, you do not need to bring another korah. That is the lavud rule, as long as they are within three tefachim of each other or within three tefachim of the wall. Sheloshah tzarich lehavi korah acheret: if the gap is three, you need another korah.
Rabban Shimon ben Gamliel says, pachot me'arba'ah ein tzarich lehavi korah acheret, arba'ah tzarich lehavi korah acheret: if the gap is less than four, you do not need another korah; if it is four, you do. The famous machloket (dispute) between the Tanna Kamma and Rabban Gamliel is how much the shi'ur of lavud is: one says three, and one says four.
Vechen shtei korot hamat'imot: there are two korot placed next to each other. The case of shtei korot hamat'imot will be on page 102. Think of two train tracks: you made two korot right next to each other over the mavoi. Each one on its own is not wide enough to receive the ariach, but together they are wide enough. So im mekabbelot ari'ach lerochbo tefach, if, when you put them next to each other and place a brick on them, they can hold it, ein tzarich lehavi korah acheret. Ve'im lav, if they are not the shi'ur of a tefach, tzarich lehavi korah acheret. Basically, you can combine the two korot in order to put an ariach over them.
Rabban Shimon ben Gamliel says, im mekabbelet ari'ach le'orko sheloshah ein tzarich lehavi korah acheret ve'im lav tzarich lehavi korah acheret. If you place the ariach lengthwise, its length is three tefachim. If the two korot can receive the ariach lengthwise, ve'einan chazakot lekabbel ari'ach, you do not need another korah; otherwise, you do. The idea is that placing the ariach lengthwise gives you more room. Even if each korah is less than a tefach wide, it may still fit. The length of the ariach is three, while its width is one. Usually we place it widthwise, and the width is one tefach. Here, even if the two korot together are less than a tefach, as long as you can put the ariach lengthwise and it will hold, it works.
Hayu achat lema'lah ve'achat lematah: one is high and one is low. The two are staggered, with one a little lower and behind the other. Each is not the size of a tefach, so you cannot combine them because they are at different levels. Rabbi Yehudah says, ro'im et ha'elyonah ke'ilu hi lematah vehatachtonah ke'ilu hi lema'lah: we look at the higher one as if it were lower, and the lower one as if it were higher. Uvelvad shelo tehe elyonah lema'lah me'esrim amah vehatachtonah lematah me'asarah: this is as long as the higher one is not above twenty and the lower one is not below ten. If they are within the area between twenty and ten tefachim, we level them out, as it were, and then you can fit an ariach across them.
Rabbi Yehudah says, rechavah af al pi she'einah beriah. We saw that in the Mishnah: Rabbi Yehudah says that as long as it is wide enough, even if it is not strong enough, we look at it as if it were strong enough, and therefore it is acceptable.
Rav Yehudah taught the Mishnah to Chiyya bar Rav in front of Rav: rechavah af al pi she'einah beriah. Rav told Rav Yehudah to teach it as rechavah uveriah. Don't learn that Rabbi Yehudah is arguing with the Chachamim; he agrees with them. Don't teach rechavah af al pi she'einah beriah; teach your son rechavah uveriah, so Rabbi Yehudah is not arguing.
What, then, is the machloket? Did we skip a line? Rabbi Yehudah omer ufeliga aleh. Sorry. Abaye says, Rabbi Yosi berabbi Yehudah savar lah ke'avuh bechada: Rabbi Yosi ben Rabbi Yehudah holds like his father in one halachah (law), uveliga aleh bechada: and argues with him in one.
Let's go back. One is up and one is down. We said that we look at the one that is down as if it were up, and the one that is up as if it were down, making them equal, as long as they are within twenty and ten tefachim. So the Gemara says: Rabbi Yosi berabbi Yehudah savar lah ke'avuh bechada. He holds like his father in one point: de'it leh ro'im, he agrees with this use of ro'im to make a korah that is otherwise pasul acceptable. Rabbi Yehudah says ro'im: we imagine that the korah is strong, or that one is up and one is down; we consider these possible situations.
Ufeliga aleh bechada: he argues with him in one point. Rabbi Yehudah holds that even if the korah is above twenty, ro'im, we look at it as if it were below twenty, to make it acceptable. He permits it even above twenty. Rabbi Yosi ben Rabbi Yehudah only follows his father in one respect: he will level them out, but he will not say that something above twenty should be lowered, because above twenty is pasul. According to his father, even something above twenty is kesherah. So he agrees with him on one point and argues with him on another.
Now, Rabbi Yehudah in the Mishnah said, rechavah af al pi she'einah beriah: if it is wide, even if it is not strong, it is acceptable. Matni leh Rav Yehudah leChiyya bar Rav kameh deRav: Rav Yehudah was teaching Chiyya bar Rav in front of Rav. This is the Amora, not the Tanna: Rav Yehudah, not Rabbi Yehudah. He read the Mishnah to him, rechavah af al pi she'einah beriah. Rav told Rav Yehudah, atniyeh rechavah uveriah. The order has to be rechavah uveriah.
But Rabbi Elai says in the name of Rav, rechavah arba'ah af al pi she-einah beriah: Rav clearly says it can be four wide even though it is not strong. The Gemara answers: rechavah arba'ah shani. When it is four wide, it is different. What's the difference? Rashi says that the whole reason it has to look wide or be wide is so it will look significant, as though it is strong. Once it is four tefachim wide, it is significant enough to be considered a korah even though it is not strong.
But how does Rav understand our Mishnah, then? Is there no argument? Rav savar she-mishnateinu kulah keshita achat: Rav holds that our entire Mishnah is one position, rechavah uveriah, and Rabbi Yehudah does not disagree with that. Fine. Therefore, you would have to remove the cases of haytah shel kash and matechet, and the other examples, because we learned in the Mishnah that Rabbi Yehudah accepts potential cases: if it is straw, we look at it as metal, and so on. But according to this, we would have to throw that out.
The Gemara asks: Rav himself said it does not have to be strong. How do we know? He said that if it is four wide, it can be accepted even though it is not strong. Ah, four wide is already significant. The shi'ur is significant enough to be considered a korah even though it is not strong.
Now we return to the other position, which says there is a machloket in the Mishnah. If it is straw, we look at it as metal. Mai ka mashma lan? What is that teaching us? We already said ro'in. Why do we need an example? You told me Rabbi Yehudah says it is accepted even though it has the shi'ur and is not strong. Kash, ro'in otah ke'ilu shel matechet: straw is treated as if it were metal. That's the same law. What does this example add?
Mah hu deteima beminah amrinan, shelo beminah lo amrinan, ka mashma lan: you might have thought we say ro'in only within the same species, but not from one species to another. The teaching is that we say ro'in even from one species to another: we can look at straw as if it were metal. That is already a chiddush (novel insight). You might think straw can never be suitable, because how can we say ro'in that it is strong? There is no such thing as strong straw; straw is straw. The teaching is that you can say ro'in from one species to another.
Akumah, ro'in otah ke'ilu peshutah: if it is bent, we look at it as if it were straight. What is the problem with a bent korah? It cannot hold the brick. Why should we look at it as if it were straight? Peshita; it seems obvious, the same concept. The Gemara says this teaches us like Rabbi Zeira: hi betoch ha-mavoi ve-akmutehah le-mavoi or chutz la-mavoi. The case is where the end of the mavoi is here, and the korah goes across it, but the bend extends outside the mavoi. The korah itself is in the mavoi; the curve projects outside it.
Hi betoch esrim ve-akmutehah lema'lah me-esrim: the korah is within twenty, but the bend is above twenty. Or hi lema'lah me-asarah ve-akmutehah lematah me-asarah: the korah is above ten, which is acceptable, but the bend is below ten. Basically, the bend is in a place where it is not supposed to be: outside, too high, or too low.
The rule is: ro'in, kol she-ilu yinateil akmutehah, if you removed the bend, ve-ein bein zeh le-zeh sheloshah, and there were not three tefachim between the two areas, ein tzarich lehavi korah acheret. Ve'im lav, tzarich lehavi korah acheret. You look at the bend as if it were not there. If the space is less than three tefachim, it closes up and you are fine.
But isn't that obvious? What are you teaching me beyond lavud? Since it is less than three tefachim, of course we say lavud, just as we learned above about two korot next to each other: if there are fewer than three tefachim between them, they connect. The Gemara says no: ei betoch mavoi, lo, be'akmumita chutz lamavoi itztaricha leih. The chiddush is the first case, when the bend goes outside the mavoi. You might have thought that people in the mavoi would be drawn after the bend. The whole purpose is to make a heker, but when they see the korah extending outside, they might walk toward the bend and end up outside in the reshut harabim. The teaching is that we are not concerned about this.
Agulah, ro'im otah meruba'at: we said that if it is round, it cannot hold an ariach because the brick will fall off, but we look at it as if it were squared. What do you need that case for? The end of the Mishnah is needed for the calculation: kol sheyesh beheikifo sheloshah tefachim yesh bo rochav tefach. We need it to teach that anything with a circumference of three tefachim has a diameter one tefach wide.
Mena hanei milei? How do we know this? Rabbi Yochanan says, amar kra: it is in a pasuk. Unbelievable—the math is in a pasuk, in the account of Shelomoh Hamelech building the Beit Hamikdash and its pool. Vaya'as et hayam mutzak: he made this water structure out of certain types of metal. The pasuk says, eser ba'amah misefato ad sefato, ten amah from one edge to the other; agol saviv, it was circular; vechamesh ba'amah komato, it was five amah high; vekav sheloshim ba'amah yasov oto saviv, and a kav of thirty amah went around it.
What do we see? It was round, and its circumference was thirty amah, while its width was ten amah. So you see a three-to-one ratio. The Mishnah says three amah—or three tefachim—is one tefach. There were thirty amah around it, and its width was ten amah. Is that pi? 3.14? That's the math, the equation. That is the pi category.
The Gemara asks, veha'ika sefato: what about the rim? The rim means that the wall of the pool has to be included in the total width. If you add the wall to the ten amah of the interior, the total width is a little more than ten amah. But the circumference measured on the outside is only thirty amah. The ratio of the circumference to the width is therefore not exactly three to one. If that is so, for anything round with a circumference of three tefachim, the width would be greater than one tefach, not as the Mishnah says.
We are measuring thirty on the outside, and if we measure on the inside, we have ten plus the rim. So how did you tell me it is three to one? Is the thirty measured on the inside or the outside? Outside. The Gemara is assuming now that the thirty is on the outside. So the three-to-one ratio is not precise. The difficulty is that if thirty is measured on the outside, the interior circumference is less than thirty, after subtracting the rim.
Rav Pappa says, sfato, sfat perach shoshan ketiv bah. It is called perach shoshan. Dichtiv: ve'ovyo tefach, usfato kema'aseh sfat kos perach shoshan, alpayim bat yachil. Rashi says that the perach shoshan is very thin, like a lily blossom. What's perach shoshan? Petal of a rose. So the rim is very, very narrow and does not count. There is still a small amount, but ki ka chashiv, migavai ka chashiv: when we measure the circumference, we measure from the inside. We thought we were counting the outside and then measuring ten amah including the rim. No, we count the inside without the rim, and then it is exactly three to one.
They're still stuck on this three-to-one. They're calling it three to one. That's close enough for them. No, the original question was that it should have been smaller, not bigger. It's hard for us because we know from science that it is 3.14, and they are medakdek about all these details. But the Gemara is saying it is three to one because you count across without the rim. Without the rim, it is three to one, and that is exact.
What about the Vilna Gaon, who says it is a keri uchtiv (a read and written form)? The word is really spelled kav. It is pronounced kav, but it is spelled kaveh. So you divide kaveh by kav. Who learned that? 3.1415. The Gaon miVilna says that? It's famous. It's exactly pi. Say that chiddush; that's a great chiddush. The Gaon miVilna's teaching on the pasuk we just read is that it says, chamesh ba'amah komato vekav sheloshim ba'amah yasov oto saviv. You see the word kav: kuf-vav. It is written kuf-vav, but how is it written in the Tanach? There is a heh: kuf-vav-heh, but we do not read it kaveh.
How does he learn pi from there? Pi is 30, so he says kaveh divided by kav. If you divide 111 by 106, you get 1.04, and then the ratio of three to one—wait, 111 is the whole kaveh, and kav is 106. If you divide them, you get 1.04717, and then the ratio of three—three! So you get the extra 3.14 hinted at in the word kav, kaveh. Kaveh divided by kav. Amazing! What a Gaon! He knew the precise figure.
So you're saying the Gemara is not getting into the 3.14 here, but the pasuk itself knew it—Shelomoh knew pi. That's the point of kav, kaveh. You have to say that, but it's funny, because it sounds from the Gemara as though they are medakdek: what about the petal, and the thing, and the small amount? It's back to the 3.1. When it says three, it means 3.14. There was a little more than that. The petal does not count. We're not counting it at all.
Tanya, Rabbi Chiya: Yam she'asah Shelomoh. In the Beit Hamikdash, Shelomoh made this whole pool. Yam she'asah Shelomoh hayah machzik me'ah vachamishim mikveh taharah: it held one hundred and fifty mikvaot (ritual baths).
Mechdi, mikveh kamah? How much is a mikveh? Arba'im se'ah (forty seah measures). As the teaching says, verachatz bamayim, bemei mikveh, kol bamayim bemei mikveh: what is meant by mayim? Mikveh water. Kol besaro mayim shekol gufo oleh bahen: water in which his whole body can be immersed. And how much is that? Here is the volume. You do not measure a mikveh by the weight of the water; it is forty se'ah. You measure it by the volume that can hold forty se'ah.
What volume holds forty se'ah? Amah al amah berum shalosh amot: one amah by one amah, three amah deep. The Sages measured the water of a mikveh as forty se'ah. As we said, if you make that shape—one amah by one amah by three amah—it holds forty se'ah. The Torah does not give you the measure as forty se'ah; the Rabbis gave you an easier way to calculate it. If it holds forty se'ah, you have a mikveh, but volume-wise it is one by one by three.
Kamah havu lehu? How many amah were in Shelomoh's pool? It was eser al eser, ten by ten by five. Chamesh me'ah garmidei: that is five hundred cubic amah, because it is ten times ten times five. Now break it up. Letlat me'ah me'ah: from three hundred amah, you can make one hundred mikvaot. The measurement of a mikveh is three by one by one, so three hundred gives you one hundred. You have two hundred left. Ume'ah vachamshin chamshin: another one hundred and fifty lets you make fifty more. Be'arba me'ah vachamshin sagya: you only need four hundred and fifty cubic amah to make one hundred and fifty mikvaot.
Why does the pasuk say five hundred? What are you doing with the extra fifty? Hanei milei beribu'a: that is only when you make a square mikveh. Hayam she'asah Shelomoh agol hayah: the pool Shelomoh made was round, so it holds less than five hundred square amah. The pasuk says it was ten amah across, but that is only at its center; toward its edges, the measurement decreases as it goes around, so you lose the corners.
There is the round pool: at the bottom, the square section is one by one by three, and each of those sections is three high. The pool was round, and so it held one hundred and fifty mikvaot, but you needed the extra volume because it was not a perfect square. Uchevan she'iglo et shtei hakomot ha'elyonot chasarnu chamishim amot meruba'ot shel mayim: when they made the two upper sections round, we lost fifty square amah of water. You lost the square area here and the square area there. If he had built a square on top, it would have held more than one hundred and fifty mikvaot. Instead, he built a round pool.
I'll never forget this. Classic. We did this 7 1/2 years ago. Marvin, alav hashalom (peace be upon him), was sitting right in front of me here in the Torah Center, going through the calculations. He knew the math very well. He was saying, “The math isn't exact; what about this, the diagonal?” This line right here—he had it right. He had it right.
Mechdi, kamah meruba yeter al ha'agol? How much bigger is a square than a circle drawn inside it? Revi'a: a quarter. The square is a quarter more than the circle. So for four hundred, you take off one hundred and have three hundred left. That is how you get one hundred mikvaot: after taking off the quarter for the circle, you have three hundred, and the shi'ur of a mikveh is three by one by one.
But wait, let's get to the question first. We have not even said the question yet. All regions. We are not arguing that a square is bigger than a circle; let's get to the question. Are we talking about the circumference right now? No, we are dealing with how much space it takes—the little corners. The area. If you want to calculate it, take two by two: that gives four. Pi r squared, with the radius one, gives 3.14. So the area of the circle is 3.14, while the area of the square is four. The square is about a quarter larger. Now, what's your question?
Ume'ah ve'esrim vechamishah. The Gemara says, hanei me'ah ve'esrim vechamishah havu lehu. From one hundred amah, if you subtract a quarter, you have seventy-five. So from one hundred you take off twenty-five. The shi'ur was five hundred cubic amah: from four hundred you take off one hundred, and from the remaining one hundred you take off twenty-five. That leaves three hundred and seventy-five. Divide by three, the measure for a mikveh, and you have one hundred and twenty-five mikvaot. So how did you tell me it held one hundred and fifty? The Gemara knows the math; you can't get around it.
The Gemara says: tani Rami bar Yechezkel, yam she'asah Shelomoh shalosh amot tachtonot meruba'ot ushtayim elyonot agulot. That's the picture we are looking at: the lower three amah were square, and the upper two were round. Vecheivan she'igelu et shtei hakomot ha'elyonot, chasarnu chamishim amot meruba'ot shel mayim: since they made the two upper sections round, we lost fifty square amah of water.
Was there water underneath here? Yes, it was all water in here. So it was basically a square pool at the bottom and a round pool at the top. The bottom was square, not a perfect cylinder. The three lower amah were square, which gave one hundred mikvaot; the two upper amah were round, which gave fifty. We thought it was a perfect circle before, and we were stuck because it did not work. It works like this.
The Gemara says, ipcha lo matzit amart: you're right that you cannot say the bottom was round and the top square, because it says desfato agol ketiv. Ela eima chada: why not say only the top amah was circular? Why did you have to say two amah were circular? You could say the square went up another amah, and only the last amah was circular. Then it would hold more than one hundred and fifty.
Figure to the figure. No, why did Shelomoh do it that way? There is no magic number. We said that was his figure, that it came to one hundred and fifty, but why did he do it this way? It looks nice. By the way, why do we know that the circle is on top? Because it says desfato agol, and this is the safah.
The Gemara says, lo salka da'atach: we would not do it that way. Why? Because we know that it reaches one hundred and fifty, dichtiv alpayim bat yachil. It held alpayim. Bat kama havya? Shalosh sein: how much is a bat? Three se'ah. Dichtiv ma'asar habat min hachor: the measure of ma'aser taken from a chor is a bat. We know a chor is thirty se'ah, so ma'aser is three se'ah. Therefore, two thousand bat is six thousand se'ah. Dehavidu shita alfe girivei: six thousand girivei is one hundred and fifty mikvaot of forty se'ah. That's one hundred and fifty times forty.
Again, we know there were one hundred and fifty mikvaot in Shelomoh's pool; we did not guess that. The Gemara is not guessing. How do we know? The pasuk says that it held two thousand bat. A bat is three se'ah. How do we know? The Gemara says a bat is a tenth of a chor, and a chor is thirty, so a tenth is three. There were two thousand of these bat, making six thousand se'ah. There are forty se'ah in a mikveh, and six thousand divided by forty is one hundred and fifty. Very simple. We knew that already. Don't tell me that you put in more; it wasn't more.
Let's get to the Mishnah, and then we'll have a session this afternoon, Be'ezrat Hashem, before Shabbat. Baruch Hashem. We're not out of the woods yet.
Veha ketiv machzik batim sheloshet alafim: we have another pasuk that says, “3,000 bat it filled.” Three thousand bat is a problem because it is larger by one thousand bat, or three thousand se'ah. Hahu legudsha: that is for the gudsha, the heaping. We know that rule already. Water does not overflow; you fill water to the top. If you are filling this pool with wheat, you can pile it up. So we have a rule: an item that holds two thousand liquid can hold three thousand dry with the heaping. That is the chiddush.
Abaye says, shema minah hai gudsha tlata havei: we learn from this that the heaping is a third. A third of the total? You have six thousand, so do you add another amount? Is it a third of the total? You have to know how it works: is it five thousand to six thousand, or six thousand to—? Every time we get this rule, they say the relationship is two thousand to three thousand: fifty percent more. But if you go from three to two, it is a third. The Gemara is talking about an ordinary vessel. That's the ratio it means: three to two.
Tnan nami hachi: we have another proof that the gudsha is a shlish. Shidah: what is a shidah? It is the carriage the ladies used to ride on. Teivah umigdal are types of cupboards, aronot. Kaveret hakash vekaveret hakanim are baskets, either made of straw or reeds. U-vor is a cistern. Sefinah Aleksandrit is the boat that goes into salt water; they call it the boat of Alexandrit, a big boat.
Af al pi she-yesh lahem shulayim, even though these things have a base, ve-hen machzikot arba'im se'ah be-lach, and they can hold forty se'ah of liquid, she-hen korayim be-yavesh: they hold two kor of dry. How much is that? Each kor is thirty, so two kor is sixty. So you see forty and sixty: sixty to forty is a third.
Even though these vessels are so big, they are tehorim from mekabel-ing tumah (ritual impurity), because they are so big. We have a general rule that kelim can receive tumah, but a kli that is very large and cannot be moved or lifted is too big to receive tumah. We are not concerned with tumah here; we are concerned with the rule of sixty to forty. Baruch Hashem le'olam amen ve-amen.