22

Try Pesachim 109a-b where the Gemara (and more elaborately in Rashi and Tosfot) tries to work out the volume of a Reviit in Etzba^3 based on its knowledge of the volume of a Mikva in Amah^3 (ie lots of basic algebra and unit conversion).


15

It's approximating π, as is clear from the g'mara (Eruvin 14:1). The problem is that that g'mara seems to be saying that it's a pretty precise approximation, and we know it's not. (Tosafos there raise this question and offer no answer.) But to answer your question, whether it's an approximation of π or a miracle, it's the former.


15

Pi can be derived from the Torah by rolling up a Torah scroll and measuring the end's diameter and circumference. The ratio between them will be pi.


13

I remember doing the gemara on Sukkah 8a in high school while I was also in a geometry class in the afternoons. It's pretty basic high-school geometry stuff. Squares and circles. It's the Tosfos there, though, that go all out. It's particularly ingenious how Tosfos (bottom of the page) demonstrates that the ratio of the diagonal of a square to its side (...


12

According to this (p. 9) the Gra never said this, it was actually proposed by Max Munk and was wrongly attributed to the former, This solution is attributed to Eliyahu the Gaon of Vilna (18th century), known as the Gra. The Gra was not only a Gadol in Torah, but also an accomplished mathematician. However, there is evidence that this solution was ...


11

You might want to show him Rambam Hil. Kiddush HaChodesh and the diagrams in the back, as well as Chazon Ish on Kiddush HaChodesh and the attendant illustrations. At the very end the Chazon Ish even includes a handy sine table! In R' Chaim Kanievsky's Shekel HaKodesh there is an appendix that explains the trigonometric underpinnings of the numbers given by ...


11

In Hakirah vol. 14, they published an article called "'Learning' Mathematics" which includes examples of different mathematical applications in classical Jewish literature.


11

The GR"A points out the following: The word circumference (kav) is spelled קוה but pronounced קו. The gematria of the former is 111 and the latter is 106. The ratio of 111 to 106, multiplied by the approximation of 3, gives you: (111 / 106) * 3 = 3.1415 Perhaps pi to five digits is a better approximation than 3?


10

That is why we say "yesterday was the fourth day" before counting. You do not want to say "today is" in any way because even in an indirect manner you are counting today. Once you have said "today is" then however you say the number, that is still a count. "Code of Jewish Law Ganzfried - Goldin, volume 3 page 52 chapter 120 number 3 (translation of Kitzur ...


10

Wikipedia has a set of answers in their article on Approximations of pi. That links to a terrific article on rabbinic approximations of π by Boaz Tsaban and David Garber. Tsaban and Garber summarize as follows (pp. 10-11): The rational-religious approach of Maimonides holds that, since we cannot know the exact values, the Bible tells us that we do ...


8

This might simply have been a confusion of units. Sizes.com claims that a Russian verst is equal to 500 sazheni, but a Moscovy verst is 1000 sazheni. The sazhen was fixed at 7 English feet (2.134m) in both systems way back during the 1700s in Peter the Great's rule. In 397:1, the Aruch HaShulchan says that specifically, he is talking about the Russian ...


8

In Rava's scenario the entire square is the numerator, so it's not (green/orange), but ((green+orange+gray)/orange). So for your second diagram, the formula should be: BigArea/MigrashArea = ((2(r+1000+1000))^2)/(3(r+1000)^2 - 3r^2) or 3.6x10^7 / 9x10^6 = 4 Abaye, on the other hand, doesn't include the city, so for him it's ((green+orange)/orange). Since ...


8

I think I may have discovered the answer. After mulling over the problem for a bit, I realized that not all cycles are created equally. The 19-year cycle just determines which years are leap years; it's the 247-year cycle which determines the lengths of Cheshvan and Kislev. To test out my theory, I built an Excel spreadsheet, then went, column by column, ...


7

The number of people in a generation is (N/2) * x where N is the number of people in the previous generation and x is the number of children each couple has. If N_0 = 70 and x = 6, after 10 generations, there would be over 4 million children. And that's assuming everyone in all previous generations had died.


7

I think that the point is being missed here. There are not that many places where there is a difference between the written word (k'siv) and the way the word is pronounced (kri). This is especially true where the written word would be pronounced the same way. The reason is generally that neither is quite correct. The "real" word should be some combination. ...


7

The gemara (Y'vamos 82b) discusses iteratively replacing mikva water with fruit juice.1 The case involves adding one se'ah of fruit juice to a 40 se'ah mikva, and then removing one se'ah of the solution. R' Yochanan rules that this iterative process may be repeated so long as more than 50 percent (or at least 50 percent) of the solution remains water. Rashi ...


7

I'd say if an average human, using an average human brain and eye, can't figure out the pattern, that's good enough. Rabbi Asher Weiss discusses the criteria of "does an unaided human notice it?" with regards to triggering some change deep instead some semiconductor someplace that I'd never notice. He points out that I may not drag a heavy bench on shabbos ...


7

The sefer Shmiras Shabbos K'Hilchasah (16:33) paskens without reservation that dice games for recreation (not gambling) are fully permitted on Shabbos. No qualification is made for fear of writing by accident. A strict opinion (which not all hold like as Halachah) is brought by the Chayey Adam (Shabbos 11:38). He holds that any game which usually involves ...


7

There seems to be three mainstream opinions in the Rishonim and Acharonim. Only if you're not planning on cheating someone The Sefer HaChinuch (Mitzvah 258) says explicitly that one may not rely on such estimations by Choshen Mishpat. In his words: ותזכר...שלא אמרו זכרונם לברכה על הכיוון כי אם בקירוב, ולכן אל תסמוך בזה בחלוקת הדברים בין בני אדם. ואל ...


7

The rules try to keep the date for requesting rain in the late fall. They are not perfect. No rule would be except adding the ever-changing long decimal tropical year length. If we were to take a simple rule and start requesting rain always 365 days after the previous time, we'd end up shifting a day earlier every four years. If we were to do that, but ...


6

"ריש חזית" means "the beginning of מדרש חזית". The Midrash Rabbah on Shir Hashirim and Kohelles is often called מדרש חזית because it begins with derashos on the pasuk "חָזִיתָ אִישׁ, מָהִיר בִּמְלַאכְתּוֹ" (Mishlei 22:29). The author of the introduction you are reading begins by quoting Shir Hashirim Rabbah (1:8).


6

A couple examples from Eruvin come to mind: Eruvin 14a brings proof that for halachic matters, pi = 3, and then continues to explain the dimensions of Solomon's pool. This contains some prealgebra and simple geometry. Eruvin 23b discusses the difference between 7.666 and sqrt(5000), with Rashi (s.v. "אלא אמרה תורה טול חמשים") giving an interesting way of ...


6

This might refer to Deut. 4:2. There is the advice not add or remove commands. http://ohr.edu/this_week/its_not_quite_that_simple/5926 So associating this advice with maths and numbers might actually remove the context. Or in other words: This might add somthing by substracting.


6

The Rambam here is giving an easy way to calculate the moment of conjunction for future months. A (lunar synodic) month, as you stated, is 29 days 12 hours and 793/1080 hours. Since 28 days is exactly 4 weeks (bringing us back to the same point in the week), the next conjunction will occur 1 day 12 hours and 793/1080 hours later in the week relative to the ...


5

Amazing what you can see when you look. Irv Bromberg at the University of Toronto discusses an adjustment. Currently there are 13 months in 7 years out of every 19; the new formula would involve 130 leap years out of every 353. As he clarifies, witnesses only determined when exactly the new month would start; the Sanhedrin could decide whether to make it a ...


5

The discussion is indeed Dr. Aumann's discussion on Kesuvos 93a regarding the man who dies with insufficient funds to pay for his three wives' Kesubos. He claims to be the first to come up with this theory, but I think Rashi figured it out first. He theorizes as follows: First, award equally amongst all parties until one receives half of his claim. Then ...


4

Since your need seems to be to figure out "when to start saying ותן טל ומטר in diaspora", I'll just focus on that (and ignore the mention of the tekufah in the title): To simplify things, start with the date November 21. You will then need to adjust based on two rules: If the following year is a Gregorian leap year, push the day off by one (i.e. November ...


4

You may be interested in my Sefer which includes the Sugyos Eruvim Daf 14,57,76, Pesachim 109, Succos 7,8. It explains Gemara, Rashi, Tosfos, Marsha, Maharom, Gra, others, and presents every step in equation form and diagrams; it also makes corrections and clarifications on diagrams found in the Shas. It includes an Appendix on Basic Algrebra, 21 definitions ...


4

R. Abarbanel (in his Zevach Pesach) explains: The magicians had tried to replicate the first three plagues, but were unsuccessful with the third, lice, which proved to them retroactively that all three had indeed been God's finger, as would any remaining plagues be. That's why they didn't bother trying to replicate the remaining plagues, and why Pharaoh didn'...


4

That is one reason that many people calculate the gross ma'aser as after withholding and then count any income tax refund as "income" for the purpose of ma'aser after the tax has been paid. Thus, the recursion that you suggest does not occur. Halachically, the ma'aser owed is calculated at particular times (even if you keep track each month). If on the ...


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