# Chazal using calculus

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?

• It is worth noting that it is generally assumed that Isaac Newton invented calculus in the 17th century Feb 20, 2023 at 18:09
• I think should change title to "logarithmic calculations" - where's the calculus in this? 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. 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. Feb 20, 2023 at 22:19
• – Fred
Feb 21, 2023 at 0:05