Steadying the storm in the kiddush cup
A solution to the halachic shiur crisis
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The difficulty with halachic shiurim for volume is well known and many different resolutions to the problem have been offered. While these resolutions differ in their premises and implications, they are all generically similar in that they start by assuming consistency between all the talmudic sources, and seek to solve the discrepancy that arises from applying these sources to observable reality by using forced arguments to modify one or another aspect of said observable reality. In this piece, I shall present a different way of resolving the problem, which both affirms traditional Jewish shiurim used until the modern era and avoids the need for forced arguments or untenable claims. Before doing so, I shall, at the risk of boring some readers, once more recapitulate the difficulty in calculating volume shiurim used in halacha and how the different approaches to addressing it arose. I shall take the shiur of the revi’it as my locus of the discussion because this is where the daily life of every Jew has been most affected.1
The problem and its discovery
There are two main ways of calculating the volume of the revi’it.2 The first, which was favoured by the Geonim, and was relied upon in practice by Jewish communities for at least a millennium, is to express it in terms of eggs. The Talmud Bavli in Eruvin 83a tells us that a se’ah is equivalent to 144 eggs. Since we know that there are 4 revi’ot in a log, 4 lugin in a kav, and 6 kabin in a se’ah, we can calculate the volume of a revi’it in terms of eggs through the sum 144 ÷ 6 ÷ 4 ÷ 4 = 1.5; it therefore follows that a revi’it is equivalent to the volume of one and a half eggs. Using a generous estimate of the egg as 50ml, we arrive at an estimate for the revi’it of 75ml.
The second way of calculating the revi’it is stated explicitly in B Pesahim 109a in the name of Rav Hisda:
אמר רב חסדא: רביעית של תורה אצבעים על אצבעים ברום אצבעים וחצי אצבע וחומש אצבע. כדתניא: ״ורחץ במים את כל בשרו״ — שלא יהא דבר חוצץ בין בשרו למים. ״במים״ — במי מקוה. ״את כל בשרו״ — מים שכל גופו עולה בהן, וכמה הן? אמה על אמה ברום שלש אמות, ושיערו חכמים שיעור מי מקוה ארבעים סאה.
Rav Hisda said: the revi’it of the Torah is two etzba’ot by two etzba’ot by 2.7 etzba’ot. As it was taught ‘and he will wash in water all of this flesh’ - [this shows that] there will not be anything separating his flesh from the water. ‘in water’ - [this means] in the water of a mikveh. ‘all of his flesh’ - [this means] water in which all of his flesh can get into them. And how much is that? 1 amah by 1 amah by 3 amot, and the sages measured the water of a mikveh as 40 se’ah.
Rav Hisda’s definition is based upon a tradition that states an equivalence between the minimum volume of a mikveh expressed as a cubic function of its dimensions measured in amot (cubits), and its volume as measured in the conventional large volume measurement, the se’ah. Rav Hisda’s mathematical argument is not spelled out in the gemara, but the gaps are filled in by the Behag, Rabeinu Chananel and the Rif.3 Though their calculations are without error, their methods for deriving the volume of a revi’it from a se’ah are extremely confusing for someone trained in modern mathematics. It is far easier for the modern reader to understand the proof as follows.
The minimum volume of a mikveh is 1 amah x 1 amah × 3 amot. There are 6 tefahim (fists) in the standard halachic amah and 4 etzba’ot (fingers) in a tefah. There are therefore 6 × 4 = 24 etzba’ot in an amah. The cubic volume of a mikveh can therefore be defined as (24 × 1) × (24 × 1) x (24 × 3) = 41,472 cubic etzba’ot in a mikveh. There are 40 se’ah in a mikveh, 6 kabin in a se’ah, 4 lugin in a kav and 4 revi’yot in a log. We can therefore divide the volume of a mikveh using the sum 41,472 ÷ 40 ÷ 6 ÷ 4 ÷ 4 to show that a revi'it 10.8 cubic etzba’ot. This is expressed by Rav Hisda as 2 x 2 x 2.7 (=10.8).
This formula is not only explicitly stated in the gemara, but also canonised as practical halacha by Geonim and Rishonim including the Behag, Rif, Rambam, Rosh and the Tur.4 All the above authorities, with the exception of the first, explicitly stipulate that the etzba in question is actually a godel, that is to say a thumb. The rule that an etzba used in measurement means a thumb is explicitly stated by Rav Papa in Menahot 41b, and also follows from a baraita in B Bechorot 39b, which states that etzba always refers to a quarter of a tefach, something that can only be true of a thumb.
However, while a measurement in terms of cubic etzba’ot is an apparently very precise way of defining a revi’it, it was not, before the modern age, a very practical way of doing it either for the ordinary person or for a scholar. It was therefore not for nearly a thousand years - and perhaps not coincidentally shortly after the invention of the metric system - that the poskim came to appreciate a very serious problem. Human thumbs vary in width, but a reasonable average is 2.3cm. If we plug that into Rav Hisda’s formula, we get (2 × 2.3) × (2 × 2.3) × (2.7 × 2.3) = 131 ml, close to double the volume of the revi’it calculated in terms of eggs.
This general problem was observed already in the period of the Rishonim by the Tashbetz, who, writing about the volume of the mikveh, observed that 3 cubic amot is a great deal larger than 5760 (= 144 × 40) eggs, and therefore, endorsed using the larger measure to avoid the possibility of the mikveh being invalid.5 However, in practice, mikveot had always been built using cubic amot measurements anyway, and those who noticed that there was a problem did not apply their observation to the smaller shiurim of the revi’it and the shiur for hallah, but instead stuck with traditional calculation based on eggs. The first to raise this problem as a practical issue was the Noda BiYehuda, R.Yehezkel Landau זצ’’ל, who made two vessels for an asirit haephah, the minimum shiur for hallah, one using an egg-based measurement derived from Eruvin 83b, and the other from cubic dimensions derived from Pesahim 109b. His thumb seems to have been around 2.4cm, and, accordingly, he found that the latter vessel was fully twice the size of the former. This discovery threw the halachic system of shiurim into conceptual crisis, one that demanded, and still demands, a resolution.
Solutions to the Problem
The Noda BiYehuda himself was the first to offer a resolution. Since the problem essentially boils down to an irreconcilability between the measure based on thumbs and the measure based on eggs, there must, he reasoned, be something wrong either with our estimate of the width of a thumb or the volume of an egg. The Noda BiYehuda opted for the latter, and concluded that, in the talmudic era, eggs must have been twice the size they were in his own day. The practical ramification was that all shiurim that had hitherto been based on the volume of eggs (which was, in fact, all volume shiurim except for mikveot) now had to be doubled.6
The Noda BiYehudah’s solution to the problem has been accepted, at least lechatchila, by the bulk of Orthodox Jewry, though later than most people likely assume, partly because he was the first to provide an answer, partly because of his justified reputation as a top-tier posek and talmudist, and partly because it was endorsed, though, again, more tepidly than most people likely realise, by the Mishnah Berurah and Chazon Ish.7 However, the fact is that this is not a possible solution to the problem. The largest eggs that can be produced commercially today after generations of intensive breeding for egg size have a volume of not much more than 60 ml. Occasionally, eggs significantly larger than this are laid, so it is strictly biologically possible, but there is precisely no evidence to indicate that in the time of the gemara, the period of the Rishonim, or at any other time, chickens were ever routinely laying eggs with a volume of 90 or 100ml. To the contrary, numerous different strands of evidence from archaeology and the Rabbinic sources all point to the conclusion that chicken eggs a thousand or two thousand years ago were slightly smaller, and, at any rate, certainly no larger, than those today. The arguments have been made at length by others, so it will suffice here to say that the resolution of a contradiction in the gemara cannot justify such a far-reaching and unlikely claim about the world around us with so much contrary, and a total lack of supporting, evidence.
If we reject the option of adjusting up the size of eggs, what remains is to adjust down the size of thumbs. This option was taken by R. Chaim Naeh זצ’’ל, who worked backwards from the volume of a revi’it to find the length of a talmudic etzba. R. Naeh calculated the revi’it as 86 ml, by measuring the amount of water that weighed 27 dirhams, a coin used in the Muslim world, employing a method for practically measuring a revi’it found in Rambam’s commentary on the Mishnah (Eduyot 1:2). Working backwards from this definition, R. Naeh estimated the size of a Rabbinic etzba as 2 cm [(2 × 2) × (2 × 2) × (2 × 2.7) = 86.4ml]. His estimate of revi’it is widely cited today as the ‘lenient shiur’ for a revi’it that can be relied upon in cases where an obligation is d’rabanan or extenuating circumstances apply.
The first problem with R. Naeh’s shiur for the revi’it is that, while it is much smaller than the Chazon Ish shiur, it is still too big. This is so, first, because it implies an egg size of 57ml, which is bigger than the average size of an egg today, and even more so than an egg in the time of Hazal. Perhaps more importantly, though, it has subsequently been shown that the dirham used by R. Naeh to calculate a revi’it was larger than those in circulation during the era of the Rambam. As Rav Beinush Finkel זצ’’ל found, if we use the coin Rambam was referring to, we arrive at a shiur of 75ml, more or less exactly what we could calculate using modern eggs. Though the figure of 86ml is still widely quoted as the lenient opinion, it is generally recognised by those who understand the sugya, that the actual shiur according to the logic of R. Naeh’s way of reconstructing the sources is 75ml.8
This, however, intensifies the second, more serious, problem with R. Naeh’s shiur. If we plug 75ml into Rav Hisda's formula and work backwards, we arrive at an estimate for the etzba of 1.9cm. 2cm is at the lower boundary for the width of an adult thumb, and 1.9cm is below it. Neither is a plausible estimate of what Hazal intended to be understood by the measure etzba, which, as we have seen, is defined by the width of the thumb, and is equal to ¼ of a tefach.
R. Naeh addressed this issue by pointing to the difficulty of accurately measuring the width of the thumb, which varies from person to person, and relied upon the testimony of the Rambam who claimed to have calculated the revi'it himself using thumbs and arrived at a volume equal to 1.5 eggs.9 Neither argument, however, is convincing. It is true that, if pressed, we would have trouble, based on the sources in the gemara, in determining whether 2.2cm, 2.3cm or 2.4cm was a better estimate of an etzba, but we can be confident of the range of plausible answers. 2cm and, still more so, 1.9cm are simply outside that range. The Rambam's testimony certainly requires analysis, but it cannot change the width of the human thumb or fist. To put the matter in Yeshivish terms, it's a kasha on the Rambam, not a ra'aya for a definition of the etzba.
Various answers have been given to this problem by defenders of traditional shiurim and what we might loosely call rationalists. One popular argument is that human hands used to be smaller because of worse nutrition.10 This argument is often paired with the claim that the Noda BiYehuda was unusually tall and long limbed.11 Neither of these are true to the extent necessary to solve the problem. More serious arguments include the claim that the etzba should be calculated by the thickness, not the width, of the thumb,12 and that ancient workmen pressed their thumbs very closely together when measuring.13 Aside from the inherent implausibility of such claims, they all founder on the rock of modern archaeology. Careful studies of pots from ancient Israel have found that a tefach measure was used with great consistency and that it was between 8.85 and 8.97cm. If we take the middle of this small range, and divide by 4, we get an estimate of 2.2275cm for an etzba, which, using Rav Hisda's formula, yields a revi'it of 119ml, smaller, it is true, than the Chazon Ish shiur, but far too large to be 1.5 eggs.14
If we accept the evidence of biology, archaeology, and our own eyes, we are left with no choice but to admit that the incompatibility between the two measurements of the revi'it cannot be explained away by reference to larger eggs or smaller thumbs, but rather exists within the gemara itself. This suggests a totally different approach to addressing this problem. Sometimes, it is true, an apparent difficulty in a rabbinic text can be resolved by reference to context or realia known to the author but not immediately apparent to the reader. Our usual approach, however, to a talmudic problem is to delve further into the sources and see if additional information in the gemara itself or close textual analysis can shed new light. I will now attempt such an analysis.
A New Approach to the Problem
Once we approach this issue as a contradiction within the gemara itself, rather than a contradiction between the gemara and observable reality, we need to start by defining precisely where the difficulty in the gemara is located. Our first step, then, is to note that the gemara does not define the volume of a revi’it in terms of eggs, but rather this definition is inferred from what gemara says explicitly about the volume of a se'ah. The next observation is that, while the gemara does explicitly provide a measurement for the revi'it in terms of cubic etzaba’ot, this is an amoraic derivation from a primary source that, once again, describes the volume of a se'ah, or, more precisely, forty of them, this time in terms of cubic amot.
We can therefore re-describe the problem as follows. A baraita that appears in four places in the Talmud Bavli, as well as Sifra,15 defines the minimum volume of a mikveh as 3 amot by 1 amah by 1 amah and states that this is equivalent to 40 se’ah. The Talmud Bavli also states that the volume of one se’ah is 144 eggs. However, these two formulae do not appear to match. An amah measured in centimetres is 6 [tefahim in an amah] × 4 [etzba’ot in a tefah] × 2.2275 [the width of a thumb] = 53.46, which results in the formula for a mikveh being 53.46 × 53.46 × 53.46 × 3 = 458,362ml or 458.3 litres. However, if we divide this figure using the second formula we get 458,362 [ml in a mikveh] ÷ 40 [se’ah in a mikveh] ÷ 144 [eggs in a se’ah], yielding an impossible volume of an egg as 79.6 ml.
This leaves us in apparently exactly the same position as we started, but now with different options. When the problem was expressed in terms of the volume of a revi’it (or, as it was by the Noda BiYehuda, the shiur for hallah), the variables were eggs and thumbs, and so the only means for solving the problem were to look outside of the gemara and redefine the size of either eggs or thumbs, both of which prove to be untenable. However, when we focus on the problem as it actually manifests itself in the gemara, we introduce two different variables, the length of the amah, and the volume of the se’ah, both of which allow us to look within rabbinic sources for a resolution to the problem. We shall now look at each option in turn.
The first method involves revisiting our assumption about the length of the amah. Our calculation of the volume of a mikveh above started with the assumption of all modern shittot that the amah contains 6 tefahim. However, two amot were in use by ancient Jews, one five tefahim long and the other six.16 The length of the amah used to construct King Hizkiyahu's tunnel that brought water to Jerusalem has been shown to be 44.4cm, corresponding to a 5-tefah amah, based on an etzba of 2.22 cm. If we find an amah measurement in the Talmud, there are certainly strong grounds for considering the possibility that the amah referred to was made up of 5, not 6, tefahim.
If we do so for the baraita which states that the minimum shiur of a mikveh is 3 cubic amot we get the following calculation: (2.2275 [cm in an etzba] × 4 [etzbaot in a tefach] × 5 [tefahim in an amah])3 × 3 = 265,255ml. If we divide that volume by 40 [se'ah in a mikveh] and 144 [eggs in a se'ah], we get a volume of 46ml, which fits with estimates of average egg size at the time of Hazal.17 The contradiction between the primary sources can therefore be completely resolved by assuming that the amah used to calculate the minimum size of a mikveh is the smaller amah of 5 tefahim.
This is not, however, the only possible approach to addressing the problem, because, in addition to the amah, there are also multiple definitions of the se'ah found in the gemara:
הא סאה דהיכא אי דמדברית מאה ארבעים וארבע הויא ואי דירושלמית מאה שבעים ושלש הויא ואי דציפורית מאתים ושבע הויין … תנו רבנן סאה ירושלמית יתירה על מדברית שתות. ושל ציפורית יתירה על ירושלמית שתות.
Which se’ah is referred to [in the preceding discussion]? If it is a midbar se’ah, that is 144 eggs. If it is a Yerushalmi se’ah, that is 173 eggs. If it is a Tzipporean se’ah, that is 207 eggs … It was taught, the Yerushalmi se’ah is bigger than the midbar se’ah by a sixth, the Tzipporean se’ah is bigger than the Yerushalmi se’ah by a sixth.
The system of volume measurements used by halacha was not static. On two occasions, an additional ⅕18 was added to the original midbar measures, resulting first in the Yerushalmi measures and, then, in the Tzipporean system. In the latter of these, a se'ah is defined as 207 eggs, which, with a 50ml egg, gives us a Tzipporean se'ah of 10,350ml. Calculating a se'ah using cubic etzba’ot of 2.2275cm and amot of 6 tefahim gives us 11,519ml, a discrepancy of just over 10%. The equivalents for the revi'it are 107.8ml and 120ml, respectively.
In order to resolve this discrepancy, we are once again forced to resort to the solution of larger eggs and/or smaller thumbs. However, we can do so within the realm of the plausible. An egg of 52ml gives us a Tzipporean revi'it of 112ml, and an etzba of 2.125cm - the minimum value found by archaeologists investigating the tefach - with an amah of 6 tefahim gives us 117ml. I leave it to the discretion of the reader to judge how far it is reasonable and necessary to go in pushing these figures closer together.
Choosing between our two options
It is now time to state clearly what the perceptive reader will likely have already observed. Either of these solutions to the problem of the incompatibility of the two definitions of se’ah necessarily implies that Rav Hisda’s way of deriving the volume of a revi’it shel Torah is mistaken. To be specific, either Rav Hisda correctly assumed that the definition of a mikveh as 40 se’ah refers to the original se’ah of the midbar, but incorrectly assumed that the definition of three cubic amot referred to a 6-amah tefah, or he correctly assumed that the amah referred to was 6 tefahim long, but incorrectly assumed that that the 40 se’ah referred to the smaller se’ah of the midbar. In the first case, his formula does not correspond to any recognised measurement, except perhaps incidentally; in the second, it is a way of deriving a Tzipporean revi’it. If we accept that Rav Hisda’s formula must be mistaken in some way, we have to determine which specific mistake is more likely. This substantially hinges on two questions:
Is it more likely that the primary tannaitic source cited in the Talmud Bavli and Sifra refer to midbar measurements and a 5-cubit amah, or to Tzipporean measurements and a 6-tefah amah?
Is it more likely that Rav Hisda made a mistake about what amah was being referred to or what se’ah was referred to?
My personal answers to these questions are as follows:
It would seem logical that a tannaitic source referring to a se’ah without further specification intends the Tzipporean measure, since this was the measure in use during the period.19 It is also more likely that a source that refers to an amah without specification refers to an 6-tefach amah, since this rule is stated clearly in Eruvin 3b (even though, according to Abaye, it applies only to amot in hilchot kilayim)
It is clear that Rav Hisda was trying to calculate the midbar revi’it, to the exclusion of the Tzipporean revi’it.20 It seems logical that he would be careful to ensure that the source he was using as a basis for his measurements would be a midbar measure, but he would be more likely to make a mistaken assumption about the meaning of the term amah within the source. In addition, since there was apparently a tendency over the course of the amoraic period to make the 6-tefah amah the normative default amah, it is plausible that Rav Hisda would mistakenly interpret an earlier source used when the 5-tefach amah was more common.
In other words, a priori logic here points in opposite directions. However, one way of resolving the problem is much more convincing from the mathematical perspective. A 5-tefah amah and a midbar se’ah correspond to an egg volume that matches neatly to archaeological evidence. Conversely, a Tzipporean se’ah and a 6-tefah amah force us to push both the volume of an egg and the width of a thumb to the limits of what is plausible, unless we are willing to entertain the possibility of substantial inaccuracy in the primary halachic sources.
Concluding Remarks
The most convincing resolution to the shiur crisis is that the baraita that describes the minimum dimensions of a mikveh refers to amot of five tefahim, and that, therefore, the correct measure of a revi’it in terms of cubic etzba’ot is not 10.8 (= 2 × 2 × 2.7), but rather 6.25,21 corresponding to a volume of around 71ml. This solution maintains the absolute validity and accuracy of the tannaitic sources, and also the geonic tradition for calculating shiurim by world Jewry until recently, while corresponding perfectly with archeology, biology, history and basic rationality. However, there is no avoiding the fact that it entails asserting that one of the greatest Amoraim made a mistake, that this mistake was given authoritative status by the editors of the gemara, and that this mistake was then ratified by the greatest Rishonim and poskim, until, eventually, this codified halacha became paramount over the authentic traditional measure.
This is not the solution I was looking for when I began researching the topic, and I suspect most readers will feel similarly. While the thesis can be stated less baldly than I have above, the probability that, however presented, it will be accepted by any significant proportion of Orthodox Jewry is vanishingly small. Most will simply reject the possibility that Rav Hisda's formula is based on an error out of hand. Others will argue that regardless of its origin, its codification in classic sifrei halacha makes it authoritative. For practical purposes, it is probably wise for supporters of the traditional authentic shiurim to rely on a mixture of mesorah, the authority of R. Chaim Naeh, and the lack of popular appreciation of the insurmountable difficulties of the smaller-thumbs hypothesis. However, the truth also has, if not a veto, at least a voice with a right to be heard by those who seek it.22
Many readers will object that a better candidate is the kezayit, but, here, the doubling of shiurim in the modern era is only relevant as a side effect of the more fundamental development of defining olives in terms of eggs.
Many MSS of the gemara do provide a step by step calculation, but this appears to be a later interpolation.
משנה תורה, הלכות תפילה ט''ו:ד; טור, א''ח תעב; בה''ג סוף הלכות חלה/כלאים; רי''ף פרק ערבי פסחים כג ב, ורא''ש פרק ערבי פסחים כג.
חידושי הצל''ח, פסחים קטז ב
See the famous remarks of Prof. Hayyim Soloveitchik in Rupture and Reconstruction, pp. 323-5
For example, פניני הלכה and Rav Kaganoff
שיעורי תורה, ו:א-ב
This has been extensively and ably argued by Prof. Avraham Greenfeld
A different type of solution was suggested by J.J. Ajdler, who proposed that the gemara’s definitions of the three types of se’ah in terms of eggs referred not to multiples of the volume of a standard egg, but rather how many actual eggs could be placed in these measures. Because eggs are egg-shaped, a vessel that can hold 144 eggs has a liquid volume equivalent to around 275 eggs. The Geonim, on this argument, simply misunderstood the gemara and the contradiction between the volume of eggs and the lengths of thumbs never existed. Filling a large basket with eggs sounds like quite a bad, not to say messy, way of measuring its volume. The conclusive refutation, however, of this claim comes again from archaeology. In Eruvin 83a, after giving the measurements of the three types of se’ah in terms of eggs, the gemara also describes Rabi as measuring a certain vessel, the מודיא, as 217 eggs. The modius was 8.73 litres, which divided by 217 gives us 40.2ml. This clearly proves the gemara is not talking about how many eggs could be fitted into the vessel, but rather how many multiples of the volume of a standard egg it contained. The most likely method used to make this calculation was simply to crack eggs into a se’ah measure until it was full. This explains why it is only the se’ah that is calculated in such a way, because for a small measure, like a revi’t, trying to do so with any accuracy would be impossible. The other arguments made by J.J. Ajdler for larger measurements can be explained simply by the observation that the Mishnah, and early sources generally, by default employ Tzipporean volume measurements, also using Yerushalmi measures in some cases.
See מנחות צז:א. Note that, in the wider Middle East, there were two cubits, the standard one, and the longer ‘royal’ cubit. It is widely stated that the first was six, and the second seven, fists, but there does not appear to be much evidence for this, and their estimated lengths actually correspond to five and six fists, retrospectively. Bizarrely, Wikipedia cites Leonardo Da Vinci’s drawing of Vitruvian man and a book by apparent crank, Steven Skinner, as well as [sic!] ‘many other sources’.
See recent research by Prof. Zohar Amar. In fact, 46ml is still slightly too big, though it is very close to the 45ml that has been found by measuring the amphora.
The baraita, as is common in Rabbinic sources, refers to it as a sixth, because a fifth of the original total is a sixth of the increased total.
See, for example, עדויות א:ב, which uses the Tzipporean kav without explanation or warning. Of course, this question hinges on one's views about the antiquity of the baraita. If it is either much older than the Mishnah or much younger the probability that it is using Tzipporean measurements decreases. In cases such as this, however, source criticism is not much more useful than guessing.
The words רביעית של תורה appear in all MSS of the gemara.
The volume of 40 se’ah in terms of etzba’ot (4 [etzba’ot in a tefah] × 5 [tefahim in an amah])3 × 3 = 24,000. If we divide this by 40 [se’ah in a mikveh], 6 [kabin in a se’ah], 4 [lugin in a kav] and 4 [revi’yot in a log] we get 6.25
What does MSS mean in footnote 3?