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We see various math issues that show up in Chazal, but I don't usually see calculus. Thus, I was surprised to see the same topic in two places in Maseches Mikvaos, and in both I think you need calculus to solve it. And there is actually a third place that a discrete version of the same problem appears.

Mikvaos 3:3

בּוֹר שֶׁהוּא מָלֵא מַיִם שְׁאוּבִין וְהָאַמָּה נִכְנֶסֶת לוֹ וְיוֹצְאָה מִמֶּנּוּ, לְעוֹלָם הוּא בִפְסוּלוֹ, עַד שֶׁיִּתְחַשֵּׁב שֶׁלֹּא נִשְׁתַּיֵּר מִן הָרִאשׁוֹנִים שְׁלשָׁה לֻגִּין.

If a cistern is full of drawn water and a channel leads into it and out of it, it continues to be invalid until it can be reckoned that there does not remain in it three logs of the former [water].

The Tosefos Yom Tov points out that the language "reckoned" sounds like you need a calculation, אלא לפי חשבון המים שהיו בבור, והמים היורדים בתוכו הם יוצאים. It's hard, because: As kosher water flows in, initially almost all the water leaving is she'uvim. But as more kosher water flows in, assuming it's all mixed up, the percentage gradually changes; when it's half and half, half the water leaving is kosher water and only half is she'uvim - etc.
It's not a hard problem, first-year calculus, and the answer is going to be that there is an exponential decay, getting slower and slower with time but eventually getting down to 3 לוגין. Does anyone before Isaac Newton להבדיל do this calculation?

Then here it is again, in the fifth perek (5:1):

הֶעֱבִירוֹ עַל גַּבֵּי בְרֵכָה וְהִפְסִיקוֹ, הֲרֵי הוּא כְמִקְוֶה. חָזַר וְהִמְשִׁיכוֹ, פָּסוּל לַזָּבִים וְלַמְצֹרָעִים וּלְקַדֵּשׁ מֵהֶן מֵי חַטָּאת, עַד שֶׁיֵּדַע שֶׁיָּצְאוּ הָרִאשׁוֹנִים

If [a spring] is made to pass over into a pool and then is stopped, ... If it is made to flow again, it is invalid for zavim... until it is known that the former [water] is gone.

The Tosefos Yom Tov there points us to what he said on the earlier mishnah, that it has to be done through a calculation. That would mean another exponential decrease, as the non-maayan water is gradually diminished by the new water flowing in.
However, I don't know how to do it here; over there it gave an endpoint, 3 לוגין of שאובים remaining. Here it just says, Until the former water is gone. But it will never be completely gone.

The third case starts in the Mishnah 7:2:

אֲבָל שְׁאָר הַמַּשְׁקִין, וּמֵי פֵרוֹת, כו' פְּעָמִים מַעֲלִין וּפְעָמִים שֶׁאֵינָן מַעֲלִין. כֵּיצַד. כו' הָיוּ בוֹ אַרְבָּעִים סְאָה, נָתַן סְאָה וְנָטַל סְאָה, הֲרֵי זֶה כָשֵׁר

But other liquids, and the juice of fruits, ... sometimes raise it up to [the required quantity] and sometimes do not raise it up. ... But if the mikveh contained forty seah and a se'ah of any of these was put in and one seah was removed, the mikveh is still valid.

Yevamos 82b goes further: You can keep repeating this, until you no longer have a majority of the original water.

This is also an exponential decay, but it is discrete instead of continuous. Each time you add a seah (bringing the total up to 41 seah) and remove one, the percentage of the original water is multiplied by 40/41 - until it gets down to 50%. Someone could do that without calculus. But did they?

The kadmonim say you need a calculation; do any of them discuss how to do it?

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  • 1
    It is worth noting that it is generally assumed that Isaac Newton invented calculus in the 17th century
    – אילפא
    Feb 20, 2023 at 18:09
  • I think should change title to "logarithmic calculations" - where's the calculus in this?
    – Rabbi Kaii
    Feb 20, 2023 at 20:11
  • I think you need calculus to understand the idea of a continuous exponential decay, or see how the rate of decay can be calculated.
    – MichoelR
    Feb 20, 2023 at 20:58
  • Why assume it's a mathematical equation? It would seem just likely if not more so to assume we should guesstimate based on on our logic when the change would have occurred. Similar to when we assume how long it takes for blios to be cooked out of utensils when kashering.
    – user6591
    Feb 20, 2023 at 22:19
  • Related: judaism.stackexchange.com/a/75703
    – Fred
    Feb 21, 2023 at 0:05

1 Answer 1

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There is a Mirkeves HaMishneh on Rambam Hilchot T'rumot 13:5 that discusses your latter case and goes through the calculation, which is a discrete calculation of repeated mixings. It's a difficult one with seeming mistakes. It seems clear that generally speaking we do not have much information on how to do these calculations from the kadmonim. For the discrete cases, twenty 40ths of mixing would be assumed to switch the majority, based on a simple calculation based on even mixing. Rabbi Akiva Eiger on SH YD 201:24 states that the Shulchan Aruch wasn't so much concerned with mathematics as it was certain halachic considerations.

All of the above can be borrowed to the continuous cases as well, by estimating how long it takes a se'ah to flow in, and waiting the number of those units of time until the number of se'ahs matches half the se'ahs in the mikva, plus one.

My friend has recommended "Approaching Infinity" by Rabbeinu Shlomo of Chelme who goes through the interesting mathematics of Chazal, and from which I got information for this answer.

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  • Twenty twentieths isn't right? Unless you oversimplify by assuming the percentage doesn't change on the way, which is maybe was what you are saying. But the discrete case is the easy one: just multiply 40/41 by itself till it goes below 1/2!
    – MichoelR
    Feb 20, 2023 at 21:02
  • Excellent post. Do you have a reference for that R' Akiva Eiger - I don't see it?
    – MichoelR
    Feb 20, 2023 at 21:17
  • @MichoelR Oops yeah I meant fortieths. Re: R' Akiva Eiger, the reference in my source isn't clear, I believe it might be s'k 6 but I haven't had time to go through it properly. He brings halachic considerations about not presuming they properly mix etc, rather than mathematical ones
    – Rabbi Kaii
    Feb 22, 2023 at 17:28
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    Yeah, in real life I also wouldn't assume it mixes perfectly! And if it doesn't you really would need R' Akiva Eiger to figure out what to do.
    – MichoelR
    Feb 23, 2023 at 11:03
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    @MichoelR depends on the ink! Coffee will mix completely faster than my eye can see, as will a lot of inks. I might imagine water is faster at mixing with fellow water, but you are right, we can't just rely on our assumptions
    – Rabbi Kaii
    Feb 23, 2023 at 11:56

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