Friday, July 19, 2024

Counting the Judges


In the end of Parshas Balak, (pasuk 25:5), Moshe passes on HaShem's command to carry out justice upon those who worshipped Ba'al Pe'or.

Rashi states that there were 88,000 "Dayanei Yisroel" and cites the gemara at the end of the first perek of Sanhedrin. However, the gemara over there clearly calculates the number of Dayanei Yisroel to be 78,6001. Of course the obvious easy way out would be to say that there is an error in our version of Rashi, which would require only the replacing of the word shemonas with the word shiv'as to get close enough to the true number. This, in fact, seems to be the version of Rashi that the Ramban had. However, whenever this is avoidable it is best not to rely on such an answer and to justify our reading of Rashi. But is it avoidable in this instance?

Perhaps, it is possible that Rashi is not referring to the actual number given in Sanhedrin but to the calculation done there. Just as when B'nei Yisroel were 603,550 the number of dayanim was 78,600 the number of dayanim based on B'nei Yisroel's current population would be around 88,000. The next step, then, is to calculate what population would require 88,000 dayanim. Being that the dayanim included judges of groups of 1000, 100, 50 and 10 the following equation is given:
88,000

= (x/1000) + (x/100) + (x/50) + (x/10)
(where x = total population)
= (100x + 20x + 10x + x)/1000
(multiplying by 1000/1000, i.e. 1)
= 131x/1000
(131 judges per 1000 citizens)
x= 88,000,000/131
x

= 671,755
(rounded down)


A population of close to 672,000 is needed to necessitate 88,000 dayanim. This is an odd number since none of the recorded censuses rendered a number anywhere near this. But as clearly shown in the very Rashi in question, there was to be a large population decrease before B'nei Yisroel would reach the figure of 601,730 given in Pinchas. To justify this figure of 671,755 we must account for 70,000 lost lives. The only definite casualty count we are given is the 24,000 who perished in the plague following the worship of Ba'al Pe'or. That still leaves 46,00 lives unaccounted for. Starting from Beha'aloscha there were a number of catastrophic incidents recorded in which many fell from B'nei B'nei Yisroel. However, many of these may not be considered in this particular calculation. If B'nei Yisroel did in fact reach a population as large as we are suggesting then it must have happened gradually from the time of the census in Bemidbar to the time that they began their decline to the figure given in Pinchas. Therefore, since the individual plagues from Beha'aloscha to Korach were still in the second year and, for the most part, immediately after the census in Bemidbar we may not consider them in the decline of the population toward the figure of 601,730.


We are left, then, with only three incidents to consider. The first is the episode following the death of Aharon when B'nei Yisroel began to return toward Mitzrayim and B'nei Levi ran after them (see Rashi 26:13). The Yerushalmi records that only eight (or seven, see Rashi and the Yerushalmi inside) families were wiped out from Yisroel in that incident. This would seem to represent a significant loss but perhaps not 46,000.


The second is the episode with the snakes (Bemidbar 21) where, as recorded in the Mishna (Rosh HaShanah 29a), those who were bitten and did not have appropriate kavana when shown the copper snake perished. Here, too, there is no significant loss recorded but only that proper kavana was required for the snake to cure you. Before the cure, however, the pasuk states (pasuk 6) that a great multitude perished from Yisroel. We are not given any further information, though, on the number of casualties2.


Finally, there is the plague of Ba'al Pe'or. That leads into a discussion as to what in fact transpired besides the loss of 24,000 in the plague. From the fact that the Torah refers to a magefa, it is doubtful that this refers to those killed by the shoftim. So how many people did the shoftim kill? The Ramban (on this pasuk) quotes a Yerushalmi, the simple reading of which implies that each shofet killed two men as commanded by Moshe. This would render well over 150,000 casualties. Ramban, however, concludes that the figure given by the Yerushalmi is just referring to how many would have been killed had Moshe's command been carried out but in the end the shoftim never had a chance to do so and they didn't kill a soul. Perhaps it is possible to take this idea of the Ramban that the shoftim were interrupted before having a chance to complete their mission, but to suggest that they had already begun to carry it out when they were interrupted. With or without such a supposition, one could suggest that in some way, these three incidents combined for a grand total of 46,000 fatalities. The other 24,000 died in the magefa. Now we have accounted for all 70,000 lives and all the figures work out.


Nevertheless, it is doubtful that Rashi actually wrote 88,000 and had this convoluted calculation in mind. Rather, it is more reasonable to assume that this version of Rashi is a mistake and that he originally wrote 78,000, particularly because the Ramban had such a reading of Rashi. However, I am not the first to try and justify the figure of 88,000. The Margaliyos HaYam on the gemara in Sanhedrin cites the sefer Techeles Mordechai who offers a calculation based on a population of 603,550:

603shoftim over a thousand
6,035shoftim over a hundred
12,071shoftim over fifty
60,355shoftim over ten
70zekeinim
276(12x23 small Sanhedrin for each tribe)
72(12x6 Nesi'im for each tribe)
8,580Levi'im
=88,002

There are a number of details involved in this figure that may be questionable. Firstly, we have previously determined that before the sin of Ba'al Pe'or the population was at least 625,000. Also, there is no source that indicates that all these different parties were included in the term "Dayanei Yisroel." The gemara in Sanhedrin certainly did not include them. Then Rashi's comment on this pasuk would have absolutely nothing to do with the gemara and from the text of Rashi it seems that Rashi himself cited the gemara in Sanhedrin. So, the conclusion remains that the proper reading of Rashi is most likely 78,000 rather than 88,000.


1Although it is not the subject of this piece, it is interesting to note the various discussions concerning the calculation in Sanhedrin. The Yad Ramah and one opinion in Tosafos state that the shoftim were all over 60 and were not part of the general population. Another opinion in Tosafos states that the shoftim of 50 were taken from the shoftim of ten, the shoftim of 100 were taken from the shoftim of 50, etc. This, however, is not in accordance with the Yerushalmi quoted here by the Ramban. See also Margaliyos HaYam who raises a number of interesting questions regarding the figure given there. It bothered me, though, that the calculation is based on 600,000 not 603,550 which would render a different total.
2The Zohar in Parshas Balak cites an opinion that this pasuk is referring to Tzelafchad alone for he was the leader of his tribe (Source: Ta'ma D'Kra, R' Chaim Kunyevsky)

Friday, May 17, 2024

Omer Counting in Different Bases

My father-in-law showed me a ספר that discusses whether or not you can fulfill your obligation to count the עומר using other base systems besides decimal. This a good case where the question is far more interesting than the answer. Surely, one should not do that. However, it was a very interesting concept I had never thought of before. So, I added a widget on the blog's sidebar which will display the day of the Omer in various relevant bases.

On the insistence of reader Pi (that's his shorter name), I have added base 7 as well as a pick-your-own-base section.
Here is the excerpt from the ספר.

Thursday, March 14, 2024

Happy π Day

We wish you all a happy Pi Day, today being March 14th which, in the US anyway, is expressed as 3-14. Pi day was first observed in the year 1593. Ok, I'm just making that up (and rounding.) Just to give this some semblance of a Torah flavour, here is our post on Pi in the Torah 

In European countries where the day is written before the month, Pi Day is observed on April 31. For information on that, you would have needed to contact me this past Sunday morning at 2:30 am.
והמבין יבין.

Here are 10 ways to celebrate Pi Day, including this young chap who memorized 2,552 digits (eat your heart out, Brodsky.)


Friday, February 17, 2023

10,000 Kikars

Someone just sent me a brilliant piece explaining a very difficult תוספות regarding the 10,000 ככר of silver which המן gave for the extermination of the Jews. Check it out!


Wednesday, February 3, 2021

טומאה under a beam

In the משנה יומית, we recently encountered a משנה I had long been awaiting. Just as a background, טומאה which is inside of a room will generally make anything else in the room טמא. In order for the טומאה to spread, however, it needs to be inside a space which is at least a cubic טפח. So, the משנה אהלות י"ב:ז discusses how big a cylindrical beam would have to be in order to be certain that there is a cubic טפח underneath it. The משנה gives the figure of a 24 טפח circumference. Let us explore the math behind this:

The key to figuring this out is actually simple geometric fact mentioned in the .גמרא סוכה ח
כל אמתא בריבועא אמתא ותרי חומשא באלכסונא
An isosceles, right-angle triangle whose two sides are a טפח will have a diagonal of 1.4 טפחים. Or, more simply, if you draw a line from one corner of a 1x1 טפח square to the other, the length will be 1.4 טפחים. This is actually a simple (and rounded) version of the Pythagorean Theorem. The square of the hypotenuse is equal to the sum of the squares of the other two sides. 12+12=2 so the hypotenuse should be √ 2 which is more like 1.414. Let's be as precise as we can for this.


Above you will find a rough sketch of the problem we are trying to solve. We are going to solve for the radius of the beam. So in order for the space to be a cubic טפח, the diagonal has to be √ 2 טפח. If we draw a line from the centre of the beam to the corner of the square which surrounds the beam we can form a triangle which I have highlighted in faint green. It is a right-angle triangle where the two sides are of length x and the hypotenuse would be x+√ 2. But using our rule from סוכה (and Pythagoras) we know that the length of the hypotenuse can also be expressed as √ 2x. Therefore, we have
√ 2x = √ 2 + x
√ 2x - x = √ 2
(√ 2 - 1)x = √ 2
x = √ 2 / (√ 2 - 1)
x ≈ 3.41
Double that to get the diameter and multiply by π to get the circumference
c ≈ 21.45 טפחים
Now we know that the תלמוד does not necessarily use 100% precise figures. The aforementioned גמרא in סוכה clearly uses a round figure of 3 for π. But how would the משנה have come to a figure of 24 for this calculation?

רע"ב explains: A circumference of 24 means a diameter of 8 (when using the talmudic π of 3) If the sides of the large square above are 8, the diagonal, from one corner of that square to the other, which includes the diameter and two cubes, would be 8 * 1 2/5. Basically, that's 16/5 extra, 8/5 diagonal on either side. But for a טפח x טפח cube, we were looking for a diagonal of just 7/5. רע"ב explains that משנה was not concerned with the minuscule margin of the extra 1/5. The troubling issue is that if you go through all those calculations with a circumference of 21, a diameter of 7, you get exactly 7/5 on the dot! I saw explained according to one source that since a beam would likely be sunken into the ground slightly, the משנה did not want to give the exact measurement which would end up being overly stringent.

To be continued...

Friday, March 11, 2016

Frequency of חזק's and סיומי דפ יומי

This coming שבת, the סיום דף יומי מסכת גיטין coincides with the חזק of finishing ספר שמות. See this fascinating article by R' Dovid Heber of the Star-K. In it, he shows that since the beginning of דף יומי there have in fact been the same number of סיומים as חזק's and goes through the numbers to show exactly why that is. עיין שם,

I decided to take this matter a little further and investigate how many times this has actually happened or will happen, assuming the exact structure of דף יומי remains the same. This is what I found, going back to the beginning in 1923 all the way to the year 2050:

Beitzah 40 1/16/2044 16 Teves 5804 Vayechi
Avodah Zarah 76 7/7/2040 26 Tammuz 5800 Matos Masei
Moed Katan 29 7/14/2029 2 Av 5789 Matos Masei
Gitin 90 3/12/2016 2 Adar II 5776 Pekudei
Shabbos 157 3/9/2013 27 Adar 5773 Vayakhel Pekudei
Kesubos 112 12/22/2007 13 Teves 5768 Vayechi
Yevamos 122 12/24/1977 14 Teves 5738 Vayechi
Temurah 34 7/7/1945 26 Tammuz 5705 Matos Masei

It is interesting to note that it does not occur on בחוקתי at all in this time, nor does it ever land on שמחת תורה. In fact, looking even further ahead, the only time it will ever land on שמחת תורה between now and the year 2200 will be the following on שמחת תורה in ארץ ישראל:

Horiyos 14 Thu Oct 26, 2062 22 Tishrei, 5823
For anyone curious as to how I was able to figure all this out, I wrote a very short and simple Java application and made use of the brilliant Zmanim API by KosherJava.

And if you're even more curious, here's my code:


import java.util.Calendar;
import java.util.Date;

import net.sourceforge.zmanim.hebrewcalendar.HebrewDateFormatter;
import net.sourceforge.zmanim.hebrewcalendar.JewishCalendar;


public class Daf {

      public static void main(String[] args) {
            int daf;
            HebrewDateFormatter hdf = new HebrewDateFormatter();
            Calendar day = Calendar.getInstance();
            day.set(Calendar.YEAR,2050);
            //System.out.println(day);
            boolean siyum=false;
            // While loop to expose all siyumim that happen on Shabbos. I have to go in manually afterwards to identify the chazaks.
            // I could have possibly used getParshaIndex() and played with that to expose just the chazaks but that would have been clunkier
            while (day.get(Calendar.YEAR) > 1922){
                  JewishCalendar jewCal = new JewishCalendar(day);           
                  try{
                      daf = jewCal.getDafYomiBavli().getDaf();
                  } catch (IllegalArgumentException iae){
                      break;
                  }
                  if (siyum){
                        if (day.get(Calendar.DAY_OF_WEEK) == Calendar.SATURDAY){
                       
                        System.out.println(jewCal.getDafYomiBavli().getMasechtaTransliterated() + " " +
                                    daf + ", " + day.getTime() + "," +  jewCal + "," + jewCal.getParshaIndex());
                        }
                        siyum=false;
                  }
                  if (daf == 2) siyum = true; //If today is the beginning of the masechta, yesterday was the siyum of the previous
                  day.add(Calendar.DAY_OF_YEAR, -1);
            }
            day.set(Calendar.YEAR,2200);
            //Separate loop to identify Simchas Torah since it isn't usually on Shabbos
            while (day.get(Calendar.YEAR) > 1922){
                  JewishCalendar jewCal = new JewishCalendar(day);           
                  try{
                      daf = jewCal.getDafYomiBavli().getDaf();
                  } catch (IllegalArgumentException iae){
                      break;
                  }
                  if (siyum){
                        // If the 22nd or 23rd day of Tishrei
                        if (jewCal.getJewishMonth() == 7 && (jewCal.getJewishDayOfMonth() == 22 || jewCal.getJewishDayOfMonth() == 23)){
                        System.out.println(jewCal.getDafYomiBavli().getMasechtaTransliterated() + " " +
                                    daf + ", " + day.getTime() + "," +  jewCal + "," + hdf.formatParsha(jewCal));
                        }
                        siyum=false;
                  }
                  if (daf == 2) siyum = true;
                  day.add(Calendar.DAY_OF_YEAR, -1);
            }           
      }
}


Thursday, February 20, 2014

Pi in the Torah

This week, the הפטרה is read from מלכים א פרק ז. There is a difference of opinion as to which part of the פרק is read. The ספרדים and some אשכנזי shuls will read the part that contains a reference to the figure of π. פסוק כג states that the pool was 10 אמות in diameter and 30 אמות in circumference. The obvious difficulty is that π is more than 3. Even if the פסוק were to round to the nearest whole number, it would be 31.

There is a famous approach to this problem which is said over in the name of the גר"א although I am not sure there is any written evidence of it. The word used for the circumference is וקוה. However, it is read וקו. If you drop the leading ו which is simply a וי"ו החיבור, you are left with קו and קוה. The respective גמטריות of those words are 111 and 106. If you divide 111/106 and multiply that by the apparent figure of π used - 3 - then the result is 3.1415094! That is correct to 3 decimal places!

I once heard an interesting addendum to this. There needs to be some meaning behind the figure of 30 אמות. The thickness of the walls was 1/4 אמה (I'm not sure where we know that from.) Therefore, the inner diameter was 9 1/2 אמות. If you multiply that by π you come awfully close to 30. I thought heard this in the name of the מלבי"ם but I looked it up and he definitely did not say that.

Wednesday, April 10, 2013

BlackBerry® App

I am happy to announce that the Al Pi Cheshbon BlackBerry® App is now available, compatible with all devices, including the new BB10 devices.


Apps for other WeeklyShtikle blogs are available as well.

Monday, March 18, 2013

A Pythagorean Question

A math-oriented cross-post from the Daily Leaf. I guess we should be expecting more of those over the course of מסכת עירובין.

Monday, June 4, 2012

Jewish Pi Day

Today is Jewish Pi Day: 3rd month (Sivan), day 14. Shall we eat pi(e) at 1:59 shaot zmaniyot into the day?