Daf yud-dalet (page) is being studied le'ilui nishmat (for the elevation of the soul) Avraham ben Iser (Abraham son of Iser), and lirefu'ah sheleimah (for a complete recovery) Dinah bat Sarah (Dinah daughter of Sarah), Shoshana Mo'alem bat Sarah (Shoshana Mo'alem daughter of Sarah), and Yafah Sarah Elkeh Eidel Rivkah bat Rachel (Yafah Sarah Elkeh Eidel Rivkah daughter of Rachel). We begin today's daf on yud-gimel amud bet (folio side two).
We start from the Mishnah. Hakorah she'amru (the crossbeam they spoke of) refers to the korah (crossbeam) over the mavoi (alleyway): rechavah kedei lekabel ariach (wide enough to receive a half-brick). An ariach (half-brick) is a shi'ur (measure) of half a brick. Rashi says, shi'ur domeh likvi'ut livnot aleha binyan (a measure like a foundation on which to build): it is like a foundation that you are going to build something on, so you have to build it level and put bricks on it.
The Mishnah says that the half-brick is half of a brick that is three tefachim (handbreadth). Dayah lakorah shetehe rechavah tefach (it is enough for the crossbeam to be a handbreadth wide), kedei lekabel ariach lerochbo (so it can receive the half-brick across its width). As you see in the picture, the korah is a tefach wide, and you can put the ariach on it. The ariach is a tefach and a half, but it stays on the korah. There is a little overhang on both sides, a quarter of a tefach on each side, but it can hold it. It is sturdy enough.
Rechavah kedei lekabel ariach uvevri'ah (wide enough and strong enough to receive a half-brick): it has to be wide enough to hold one of these bricks, and it also has to be strong enough to hold the ariach. See what the nafka mina (practical difference) is. Rabbi Yehudah says rechavah (wide enough), even if it is not strong enough to actually receive the brick. Therefore, if it was made of straw or reeds, which are not really strong enough to hold the brick, ro'in otah ke'ilu hi shel matechet (we view it as if it were metal), and therefore as strong as metal.
However, akumah (bent): if the korah is bent and the brick cannot rest on it, ro'in otah ke'ilu hi peshutah (we view it as if it were straight). Agulah (round): since the brick would fall off a round korah, ro'in otah ke'ilu hi meruba'at (we view it as if it were square).
Kol sheyesh beheikefo sheloshah tefachim (anything whose circumference is three handbreadths), yesh bo rochav tefach (has a width of one handbreadth). The Mishnah gives a rule: whenever you have a round, cylindrical object whose circumference is three tefachim, its diameter will be one tefach. That is how you calculate it. Rashi explains that it takes a string three tefachim long to go around the object. If you straighten out the string, its length is the width of a tefach. That is a general rule in mathematics.
The Gemara asks: A tefach? You told me the korah can be a tefach. What do you mean? Tefach umechtzah ba'i (it needs a tefach and a half). Seemingly, it should need to be a tefach and a half, because the ariach is a tefach and a half. So why did you say it is enough if it is a tefach? The Gemara answers: keivan derachavah lekabel tefach, idach chatzi tefach melaben leih betina, mashehu mehai gisa umashehu mehai gisa, vekaima (since it is wide enough to receive a tefach, the remaining half-tefach is secured with tar, a little on each side, and it stays in place). As we said, there is a little overhang on each side. The text says melaben leih betina: if you were building, you would put some tar on the korah, put the brick there, and it would stay. Even though there is a little overhang on each side, it will not fall off because something holds it there. So, as long as the korah is a tefach, you can fit a tefach-and-a-half brick on it in an actual building situation.
Rabbah bar bar Chanah said that the korah has to be strong enough to receive an ariach, as the Tanna Kamma said in our Mishnah. Ume'amidei korah (the supports of the crossbeam)—sometimes the korah rests directly on the wall, and sometimes on poles or pegs coming out of the wall—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. The circular object in the picture is a tefach.
Rav Chisda says that both the korah and the pegs coming out of the wall have to be strong enough to receive the korah and the ariach. Rav Sheshet says: if someone put a korah on top of a mavoi and spread a mat over it, raising the mat more than three tefachim above the ground, korah ein kan, mechitzah ein kan (there is no crossbeam here and no partition here). You have lost your korah. You put a sheet over it because you want to make a whole doorway at the entrance to the mavoi, but by covering the korah you have lost it.
Korah ein kan deha mikasya (there is no crossbeam here, because it is covered). The whole purpose of the korah is heker (recognition), and once you cover it, you have lost the heker. Mechitzah ein kan deha havya lah mechitzah shegediyim bok'im bo (there is no partition here, because it is a partition that goats can pass under). Since it is three tefachim off the ground, what kind of partition is it? The goats can get underneath. You might say it is not lavud (joined across a gap), so it is not a partition: the purpose of a partition is to block, and this does not block.
The Rabbis taught: a korah that comes out of one wall but does not reach the other wall, and likewise two korot, one coming out of each wall, that do not meet each other: if the gap is less than three tefachim, there is no need to bring another korah. That is the lavud rule: as long as the gap is within three tefachim, either from the wall or between the two beams, you do not need another korah. If the gap is three, you need to bring another korah.
Rabban Shimon ben Gamliel says that if the gap is less than four, there is no need to bring another korah; if it is four, you need another korah. We know the famous machloket (disagreement) between the Tanna Kamma and Rabban Gamliel: how much is the shi'ur of lavud? One says three; one says four.
Vechen shtei korot hamat'imot (and likewise, two matching crossbeams): there are two korot placed next to each other. That case will come up on page 102. Think of two train tracks: you put two korot next to each other over the mavoi. Neither one is wide enough on its own to receive an ariach, but together they are wide enough. If, when you put them next to each other, they can hold an ariach across its width, there is no need to bring another korah. If not, you need another korah. Basically, you can combine the two korot to put an ariach on them.
Rabban Shimon ben Gamliel says that if together they can receive an ariach lengthwise, three tefachim long, there is no need to bring another korah; otherwise, you need another one. The Tanna Kamma says that together they must be a tefach. What does Rabban Shimon say? If you put the ariach lengthwise across the two beams, its length is three tefachim. Are you putting it down not across its width, but along its length? The picture shows the two tracks with the ariach lengthwise over them, so there is more space. It is longer, which gives you more flexibility. Even if the two beams together are less than a tefach, it can fit.
“Wait, I don’t understand. It’s three now.” “Well, its length is three, and its width is one.” “Right, so what?” “We are always putting it down widthwise, and widthwise is one tefach. So, even if the two beams together are less than a tefach, as long as you can fit the ariach lengthwise, it can hold it.” “No, it could still be less than a tefach.” “That’s the point. You’re saying they have to be a tefach between them. I’m telling you, no: even less than a tefach, as long as you can put it down in a way where it can still hold it.” “Right, it’ll hold it.”
Hayu achat lema'lah ve'achat lematah (if one was above and one below): one is higher, and the other is lower, a little behind it. What is the din (law) in that case? They obviously cannot hold an ariach together if they are at different levels. Each one is not the size of a tefach, and therefore you cannot combine them. Rabbi Yehudah says: ro'im et ha'elyonah ke'ilu hi lematah vehatachtonah ke'ilu hi lema'lah (we view the upper one as if it were lower and the lower one as if it were higher). They are treated as if they were level, provided the upper one is not above twenty amot (cubit) and the lower one is not below ten. As long as they are within the right zone, we level them out, and then you can fit an ariach across them.
Rabbi Yehudah says, rechavah af al pi she'einah beriah (wide, even though it is not strong). We saw that in the Mishnah: as long as it is wide enough, even if it is not strong enough, we view it as strong enough, and therefore it is acceptable.
Rav Yehudah was teaching the Mishnah to Chiyya bar Rav in front of Rav. He read: rechavah af al pi she'einah beriah (wide, even though it is not strong). Rav told him, “Teach it as rechavah uveriah (wide and strong).” Do not learn that Rabbi Yehudah is arguing with the Sages; he agrees with them. When you teach your son the Mishnah, teach him “wide and strong.”
So what is the disagreement, then? We skipped a line. Abaye says that Rabbi Yose berabbi Yehudah holds like his father in one matter and disagrees with him in one matter. Let’s go back. One is up and one is down. We said that we view the lower one as if it were up and the upper one as if it were down, making them equal, as long as they are within twenty and within ten tefachim.
Abaye says Rabbi Yose berabbi Yehudah holds like his father in one halachah: he accepts the principle of ro'im, viewing a korah that is invalid as if it were valid. Rabbi Yehudah said ro'im: we imagine that the korah is strong, or that the two beams are level. But he disagrees with his father in one matter. Rabbi Yehudah says that even if it is above twenty, we view it as if it were below twenty and permit it. Rabbi Yose berabbi Yehudah says that within twenty is acceptable, but above twenty is not. So Rabbi Yose berabbi Yehudah agrees with his father in one respect and disagrees with him in another. He will level the beams, but he will not use that principle to lower a beam that is above twenty. Above twenty is invalid according to him, whereas his father holds that even above twenty can be valid.
Now, in the Mishnah, Rabbi Yehudah says, rechavah af al pi she'einah beriah. Rav Yehudah was teaching this to Chiyya bar Rav in front of Rav. This is the Amora Rav Yehudah, not the Tanna Rabbi Yehudah. Rav tells Rav Yehudah to teach rechavah uveriah, “wide and strong.” But didn’t Rabbi Elai say in the name of Rav that it can be four wide even if it is not strong? The Gemara answers: rechavah arba'ah shani (four wide is different). Until now, we were talking about an ordinary case, say, a tefach. Rav understands that for a tefach, Rabbi Yehudah says it has to be strong. But Rav himself said that if it is four wide, it does not have to be strong; it only has to be fit for use.
Rashi explains that four tefachim is different. The reason it has to be wide is that it has to look important, as if it were strong. Once it is four tefachim wide, it is important enough to be considered a korah even though it is not strong.
But how does Rav understand our Mishnah? Then there is no disagreement: the Mishnah is all one view, and Rabbi Yehudah is not arguing. If that is so, we have to take out the cases of straw and metal and the other examples, because we learned in the Mishnah that Rabbi Yehudah accepts potential cases: if it is straw, we view it as metal, and so on. But according to this reading, those cases would have to be removed. The Gemara asks: what do you mean? 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 acceptable even though it is not strong. Four wide is important enough to be considered a korah even though it is not strong.
Now we are going back to the other view, which says that there is a disagreement in the Mishnah. If it was made of straw, we view it as if it were metal. What does that teach us? That we say ro'im. You already told me that Rabbi Yehudah says it is acceptable even though it is not strong. Why give me this example? The case is straw: ro'in otah ke'ilu shel matechet (we view it as if it were metal). The Gemara answers: you might have said that we apply ro'im only within the same species, but not across different species. It teaches us that we can say ro'im even from one species to another. I can view straw as if it were metal. That is the chiddush (novel point). You might say that straw can never be strong; there is no such thing as strong straw. The Mishnah teaches that we can view one species as another; that is the chiddush.
Akumah ro'in otah ke'ilu peshutah (if it is bent, we view it as if it were straight). What is the issue with a bent beam? It cannot hold the brick, so we view it as straight. That seems obvious; it is the same principle. The Gemara says that it teaches the case discussed by Rabbi Zeira: the korah is inside the mavoi, and its bend may extend inside or outside the mavoi. The picture shows the end of the mavoi and the korah running across it, with the curve extending out of the mavoi. That is the chiddush.
The case is also where the korah is within twenty, but its bend is above twenty; or the korah is above ten, but its bend is below ten. In each case, the bent part is somewhere it is not supposed to be: outside, too high, or too low. The rule is: ro'in, kol she'ilu yinateil akmutehah, ve'ein bein zeh le-zeh sheloshah (we view it as if its bend were removed, and if the gap is less than three), ein tzarich lehavi korah acheret; ve'im lav, tzarich lehavi korah acheret (there is no need to bring another crossbeam; otherwise, one is needed). You disregard the bent part. As long as the remaining gap is less than three tefachim, it closes up and you are all right.
But that too seems obvious. What is the chiddush in that as well? Since it is less than three tefachim, of course we would say lavud, just as we learned above about two korot next to each other: if the gap is less than three tefachim, they connect. The Gemara answers that the chiddush is when the bend extends outside the mavoi. You might have said that we should make a decree because people could be drawn after the bend and walk outside. The whole purpose is to create a heker. When people see the korah extending outside, they might follow the bend and end up in the reshut harabim (public domain). It teaches us that we are not concerned about that.
Agulah, ro'im otah meruba'at (if it is round, we view it as square). We said that if it is round, an ariach cannot rest on it and would fall off, but we view it as if it were square. Why do we need this case? The end of the Mishnah is needed for the rule that anything whose circumference is three tefachim has a width of one tefach.
How do we know this? Rabbi Yochanan says that a pasuk (verse) says so. The calculation is in a pasuk about Shelomoh Hamelech when he built the Beit Hamikdash (Holy Temple) and the mikveh (ritual bath). Vaya'as et hayam mutzak (he made the cast sea). It was made of certain types of metal. It says, eser ba'amah misefato ad sefato (ten cubits from one edge to the other), agol saviv (round all around), and chamesh ba'amah komato (five cubits high), and vekav sheloshim ba'amah yasov oto saviv (a line of thirty cubits went around it). What do we see from here? Its circumference was thirty amot, and its width was ten amot. So that is a three-to-one ratio. The Mishnah says three tefachim around is one tefach across. Here it was thirty amot around and ten amot across.
“Is that pi? 3.14.” “Shakai?” That is the pi discussion. The Gemara asks, veha'ika sefato (but there is its rim). What is the sefato? The edge, the rim. If you count the rim, the inside is a little more than ten, so it is not precisely a three-to-one ratio. The width, including the rim, would be more than ten amot, while the circumference measured on the outside is thirty. Therefore, the ratio of the circumference to the width is not exactly three to one. If a round object has a circumference of three tefachim, its width would be more than a tefach, unlike our Mishnah.
Rashi explains that “from one edge to the other” means the interior, not including the thickness of the walls, is ten amot. And it says that a line of thirty amot went around it, which indicates that for every amah of width, a round object has three amot of circumference. But what about the thickness of the rim? That would make the width more than ten. Even so, the line of thirty amot is measured around the whole thing. So the question is that the ratio would be a little less than three to one.
Rav Pappa answers: “Its rim is described as a perach shoshan (petal of a rose).” The verse says, “Its thickness was a tefach, and its rim was made like the rim of a cup, a perach shoshan; it could hold two thousand bat.” Rashi explains that the perach shoshan is very thin, like this rose. So it is a very narrow rim and does not count. “But there is still a little something.” The Gemara answers: ki ka chashiv, migavai ka chashiv (when it is measured, it is measured from the inside). When we measure the circumference of thirty amot, we measure from the inside of the rim. We thought the thirty was measured on the outside, and then the ten amot included the rim. The answer is that we measure on the inside, without the rim, so it is ten amot and exactly three to one.
Someone asks about the Vilna Gaon’s explanation of keri uchtiv (read and written forms): the word is read kav, but written kaveh. If you divide kaveh by kav, who learned that? Who says that? “The Gaon miVilna says that?” “Yeah, it's famous. It's exactly pi.” The pasuk says, “five cubits was its height, and a line of thirty cubits went around it.” You see the word kav, kuf-vav, but in the Tanach (Hebrew Bible) it is written kuf-vav-heh, kaveh, though we do not read it that way. How does he learn pi from there? The circumference is thirty, and the word kaveh divided by kav gives a ratio of 1.04. Then, with the ratio of three to one—wait, 111 is the whole kaveh, and 106 is kav. If you divide them, you get 1.04717. The extra 3.14 is hinted at in the words kav and kaveh. Amazing! What a gaon. He knew the precise figure.
So the Gemara is not getting into 3.14 here, but the pasuk itself knew it—Shelomoh knew pi. That is the point of kav and kaveh. But it is funny, because the Gemara sounds as if it is being precise: what about the petal, the rim, and the little something? The Gemara is saying that the ratio is three to one because you measure from the inside and do not count the rim.
Rabbi Chiya taught: “The sea that Shelomoh made held one hundred and fifty mikveh taharah.” Now the Gemara tells us how much a mikveh is. A mikveh is forty se'ah (biblical measure of volume), as it is taught: “And he shall bathe in water”—in the water of a mikveh. “All his flesh in water” means that all his body can be immersed in it. How much is that? Here is where we learn the volume. You do not measure a mikveh by the weight of the water, but by a volume that can hold forty se'ah. What volume can hold forty se'ah? Amah al amah berum shalosh amot (one cubit by one cubit and three cubits high). The Sages measured the water of a mikveh as forty se'ah. If you make that shape—one amah by one amah by three amot—it holds forty se'ah. The Torah does not say forty se'ah; the Rabbis gave you an easier way to calculate it. If it holds forty se'ah, you have a mikveh, but in terms of volume it is one by one by three.
How many amot were in Shelomoh’s pool? It was ten by ten by five: 10 by 10 by 5. That is 500 cubic amot. From 300 amot, you can make 100 mikva'ot, because the measurement of a mikveh is three by one by one. So 300 makes 100. You have 200 left, and now you need to get another 50. Another 150 lets you make 50. So 450 cubic amot is enough to make 150 mikva'ot. Why, then, does the pasuk say it was 500? What are you doing with the extra 50?
Hanei milei beribu'a (that is the measure when it is square). That is speaking about a square mikveh. The sea Shelomoh made was round. If it was round, it would hold less than 500 cubic amot, because the verse’s ten amot is measured through the middle, but at the sides its measure gets smaller. You lose the corners. The picture shows the round pool, with the square at the bottom indicating the three-by-one measure. The mikveh was round and held 150 mikva'ot, but it needed the extra volume because it was not a perfect square. When they rounded the two upper levels, they lost fifty square amot of water. If he had built a square on top, it would have held more than 150 mikva'ot. He built a round pool on top instead.
“I’ll never forget this. This is classic. We did this seven and a half years ago. Marvin, alav hashalom (peace be upon him), was sitting right in front of me in the Torah Center and going through the calculations. He knows the math very well. He was saying, ‘The math isn’t exact; what about the diagonal?’ He had it right.”
Mechdi, kamah meruba yeter al ha'agol (how much bigger is a square than a circle)? Revi'a (a quarter). If you draw a circle inside a square, the square is a quarter bigger. So from 400, a quarter is 100. After you take off the quarter for the circle, you have 300 left. We want to get to 300, because the measure of a mikveh is three by one by one, which makes 100 mikva'ot. You take 100 off 400. The calculation is 4r squared divided by pi r squared; the difference between the square and the circle is the part you take off.
“But wait, let’s get to the question first. We have not even said the question yet.” We are not arguing that a square is bigger than a circle. We are talking about the area, the corners. If the square is two by two, its area is four; pi r squared gives 3.14 for the area of the circle. That is about a quarter. Now, what is the question?
The Gemara says: “And one hundred and twenty-five.” How do we get 125? From 100, take away a quarter, and you have 75. So from 100 you take away 25. The measure was 500 cubic amot. From 400 you take away 100, and from 100 you take away 25. That means you have taken away 125, leaving 375. How many mikva'ot can you fit in 375? Divide by three, and you get 125. So how did you tell me it held 150 mikva'ot? The Gemara knows the math; you cannot get around it.
The Gemara says: “Rami bar Yechezkel taught that the sea Shelomoh made had three lower cubits square and two upper cubits round.” That is the picture we are looking at: a square base, with the two upper levels round. Since they rounded the two upper levels, they lost fifty square amot of water. There was water underneath, too. The pool was square at the bottom and round at the top. It was not a perfect cylinder. The bottom had 300, and the top had 200; take 25 percent off the top, and what remains is 150 mikva'ot: 100 on the bottom and 50 on top. We thought it was a perfect circle before, and we were stuck. This way it works.
The Gemara says you cannot say the reverse, that the bottom was round and the top square, because the verse says that its rim was round. The rim is the top. Could you say that only the top amah was round? Then the square would go up another amah, and only the last amah would be round, and it would hold more than 150. Why did he make it this way? “It looks nice.”
Why do we know the circle is on top? Because it says that its rim was round, and this is the rim. The Gemara says, lo salka da'atach (it would not occur to you to say that), because we know it reaches 150 from the verse alpayim bat yachil (it could hold two thousand bat). How much is a bat? Three se'ah, as it says, “a tenth of a bat from a kor.” A bat is a measurement. The amount of ma'aser (tithe) taken from a kor is a bat. We know a kor is thirty se'ah, so a tenth is three se'ah. Therefore, two thousand bat is six thousand se'ah, and six thousand se'ah is 150 mikva'ot of forty se'ah each. We thought that was just a kabbalah, and that the Gemara was guessing 150. We did not guess 150; the Gemara knew the pool held 150 mikva'ot because the pasuk says it held two thousand bat. A bat is three se'ah, and two thousand bat is six thousand se'ah. Divide that by forty se'ah per mikveh, and you get 150.
We are not out of the woods yet. The pasuk says, “It held three thousand bat.” That is a problem, because 3,000 is 1,000 bat more. That is another 3,000 se'ah. The Gemara answers: hahu legudsha (that was for the heap). Water does not overflow; you fill it to the top. If you fill the mikveh with wheat, you can pile it up. So we have a rule: an object that can hold 2,000 of liquid can hold 3,000 dry, because of the heaping. Abaye says, “Learn from this that the heap is a third.” Is it a third of the total? From 2,000 to 3,000 is 50 percent more; if you compare the extra 1,000 to the total of 3,000, it is a third. That is how I understand it.
We have another proof that the heap is a third: a shidah, the carriage that the ladies used to ride on; a teivah and migdal, cupboards and cabinets, a basket of straw and a basket of reeds, a cistern, and a sefinah Aleksandrit, a large boat that goes into salt water. Even though these things have a base, and they can hold forty se'ah of liquid, they hold two kor of dry material. Each kor is thirty, so the two kor are sixty. You see the ratio: 40 to 60, a third. Even though they are so large, they are tahor (ritually pure) with respect to receiving tumah (ritual impurity), because they are too big to be moved or lifted. We have a general rule that kelim (ritual vessels) receive tumah, but a very large kli that cannot be moved is too big to receive tumah. But the point here is not tumah; it is the ratio of 60 to 40.
Baruch Hashem le'olam amen ve-amen.