חמושים

When B'nei Yisrael leave Mitzrayim, the pasuk says (13:18) that they left "chamushim." Rashi says that the literal meaning is that they left armed. But he brings a Midrash from Mechilta that only 1/5 of B'nei Yisroel came out of Eretz Mitzrayim and the rest died during the plague of darkness because they did not want to leave. In the Mechilta itself there are two other opinions, that 1/50 or 1/500 of B'nei Yisroel came out. This is indeed a disturbing Midrash. It would mean that 2.4 million, or 24 million, or 240 million of B'nei Yisrael died during the plague of darkness. That would make it a far greater blow than everything brought upon the Mitzrim put together. R' Shimon Schwab in Ma'ayan Beis HaSho'eva asks, as well, that the point of the B'nei Yisrael dying during the darkness was so that the Egyptians would not see them dying. It's kind of hard not to notice at least 2.4 million people missing. 

 R' Schwab answers, therefore, that really only a few people died at that time. The discussion in the Midrash is concerning the long term effects of it and how to gauge it. The first opinion looks only so many generations down the line and sees that at that point, 2.4 million Jews that would have been born never were. The other two opinions simply look farther down the time line to see the greater long term effects of this loss. This would answer all the questions. But what I find very bothersome about this answer is that the wording of the Midrash simply does not seem to lend itself to such interpretation. The number of deaths is not given at all but rather the fraction of B'nei Yisroel that left Eretz Mitzrayim. This fraction would not change over the generations. In other words, let's say 600,000 went out but it could have been 3 million. If generations down the line, we reached a population of 6 million, the projection should dictate that we could have been would be 30 million, still a 1:5 ratio. I don't see how the fraction can be interpreted the way R' Schwab did. Nevertheless, it is the only answer I've seen to these difficulties with the Midrash.

Ironic Observation

Well, I guess it's more an observation of irony. We are rapidly approaching ספר במדבר which is known as the book of Numbers. And for good reason. But it is interesting to note that we find numbers and counting as a recurring theme in the פרשיות leading up to במדבר. First, we cover ספירת העומר in אמור. Actually, we already began dealing with counting at the end of מצורע with the הלכות of זב and זבה. Then, בהר begins with the counting for שמיטה and יובל. The ensuing הלכות all involve calculations based on the proximity to יובל. We end of ספר ויקרא discussing the laws of ערכין which involve considerable calculation as well as a brief mention of מעשר which also involves numbers.

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!

The Weight of the Teiva

If I were such a prolific author that I would have a magnum opus, I suppose this would be it. To this day, there are still people who identify me as "that guy who wrote the thing on the teiva."

It is told that one year, on a 12th Grade chumash test, Rav Moshe Heinemann שליט"א asked his students how to calculate the weight of the Noah's Ark. He did not ask for an answer, he simply asked how one would go about figuring it out. These are the calculations. And the answers:

Later on in the Parsha, (8:4), Rashi calculates based on the rate at which the waters of the flood receded, that the ark was submerged 11 amos in the water. A variety of commentaries deal with the calculation cited by Rashi and its validity, most notably the Ramban. The Sifsei Chachamim quotes the Nali"t as saying that the figure of 11 amos is only a minimum but it could have been more. There are a number of problems raised with different aspects of the calculation, some of which will be dealt with later on. Nevertheless, if the words of Rashi are taken at face value, they hold within them the key to unlocking this mystery. With the application of a single principle, the weight of the ark can be calculated. The law required for this calculation is Archimedes' Principle which states that the weight of a body floating in water is equal to the weight of the water it displaces. The ark's virtually cubic structure (according to Rashi) makes the measurement of water displacement easy to achieve. The ark was 300x50x30 amos³ in volume (Breishis 6:15). Therefore, the water displaced by the ark was 300x50x11 = 165,000 amos³. 

The next step, of course, is to convert the figure of cubic amos into conventional measures. Unfotunately, we are unsure as to the exact measure of the amah. There are three primary opinions amongst the contemporary poskim as to the actual length of the amah: Chazon Ish, R' Moshe Feinstein and GRA"CH Noeh. Because of this disagreement, they will differ on the measure of the ark's water displacement and therefore, the final figure for the weight of the ark will be different according to each. The following is a chart calculating the water displacement in cm³ based upon each of the opinions.

Metric to Imperial conversion table below

	Chazon Ish	R' Moshe Feinstein	GRA"CH Noeh
Length of amah	57.66 cm.	53.98 cm.	48 cm.
Volume of cubic amah (length/100)3	0.192 m3	0.157 m3	0.111 m3
Calculation	= 165000 x 0.192 ≈ 31630	= 165000 x 0.157 ≈ 25950	= 165000 x 0.111 ≈ 18250
Water Displacement	31630 m3	25950 m3	18250 m3

Now that we have determined the amount of water displaced by the ark, all we have to do is calculate how much that water weighed. Then by Archimedes' Principle we can assume that the ark weighed the same amount. This, however, is not necessarily so simple. The density of sea water is slightly more than that of regular water at approximately 1025 kg/m³. This figure usually remains about the same, without significant deviation, regardless of the exact temperature. The only drastic changes are observed when the water reaches extreme conditions such as freezing or boiling.

The first difficulty encountered is that during the initial 40 days of the flood, the waters were boiling hot (Rosh HaShanah 12a). This would change the density of the water substantially and consequently interfere with the calculation. However, it is important to note that Rashi's calculation is based on the rate at which the water receded after the 150 days which followed the 40 days of destruction. By that time, the waters had calmed down and most probably dropped to a more moderate temperature. Therefore, it can be assumed that the temperature of the water is a negligible factor in the calculation of the water density. However, what does seem problematic is that Rashi brings in the figure of 11 amos in 7:17 when the waters were at their highest intensity. It is almost certain that the density of the water at this point was much less than it was 190 days later. If the ark was calculated to have been submerged 11 amos by a calculation based on cooler waters, that figure should presumably be greater at the time of the actual flood.

The next issue of question in this calculation is the fact that the water was not necessarily pure sea water. It is suggested in Rashi (6:14) that the water contained sulfur. The presence of this sulfur and whatever other solvents in solution with the water could change the density of the water and affect the accuracy of the calculation greatly. This is only a problem, of course, if the words of Rashi are taken literally. The Sifsei Chachamim seem to suggest that what Rashi means is that the sulfur caused the heating of the water. Even if the interpretation is as originally perceived, it is possible that the ratio of solute to solvent was such that it would not have affected the density anyway. Therefore, for the purposes of this calculation I have chosen to ignore whatever effects the sulfur could have had on the water density and thus we are left with approximate figure of 1025 kg/m³. Based on this figure, these are the final calculations of the weight of the ark according to the three aforementioned opinions:

Chazon Ish	R' Moshe Feinstein	GRA"CH Noeh
31630 m3	25950 m3	18450 m3
x 1025 kg/m3
32420750 kg	26598750 kg	18706250 kg

In conclusion, considering the relevant opinions, it would appear that the ark weighed somewhere between 18 and 33 thousand metric tons. In comparison with other famous ships, the Queen Mary weighed 73,850 tons. It was 309 m long, about twice as long as the ark. The Titanic weighed approximately 42,000 tons. Of course, this refers to the weight of those vessels without anyone inside whereas the above calculation for the teiva included the inhabitants.

Table of Metric Conversions
57.66 cm	=	22.7 in.
53.98 cm	=	21.25 in.
48 cm.	=	18.9 in.
31630 m3	=	1117003 ft3
25950 m3	=	916416 ft3
18450 m3	=	651556 ft3
25o C	=	77o F
1025 kg/m3	=	2260 lb/61024 in3(35.3 ft3)
32420750 kg	=	71475519 lb = 35737.8 tons
26598750 kg	=	58640206 lb = 29320.1 tons
18706250 kg	=	41240222 lb = 20620.1 tons
6372500 m	=	20907152 ft
53 cm	=	20.87 in.

The Constant Rate of Recession

No, this has nothing to do with the American economy. There is another difficulty with the calculation that Rashi uses to conclude that the teiva was submerged 11 amos. How could Rashi base his calculation on the depth of the water decreasing at a constant rate. One can generally assume that when water decreases, it does so at a constant rate of volume. However, mathematically, if the volume of a sphere decreases at a constant rate, the rate of change of the depth will increase as the waters become shallower. The shallower the water gets, the faster it will decrease depth-wise. How then could Rashi assume that the depth decreased at a constant rate? This is the question posed by מהרי"ל דיסקין. He gives his own answers to this question. One, for instance, is that the waters receded, the ground became more saturated which slowed down the overall receding process and hence balanced out the constant rate of change of depth. But a rebbe of mine from Yeshivas Ohr Yerushalayim posed this question of none other than Nobel Prize winner Yisrael Aumann. He answered simply that mathematically, none of this is needed. True, the rate of change of depth is not directly proportional to the rate of change of volume. However, considering the size of the globe, the difference between the two within the scope with which we are dealing is negligible and would not affect Rashi's calculation. Is this true? The short answer is "Yes". The longer answer requires a little Calculus.

The radius of Earth is 6372500 m. To make things simple we will convert this to amos. Instead of using three separate measures of the amah, we will keep things neat and use an average figure of 53 cm. (6372500 ÷ 0.53 = 12023585) That translates to 12023585 amos. To make things simpler, we will round it off to 12000000 amos. This will have little effect on the final outcome. This figure will be called r_w.

The standard equation for volume:

V_w= 4/3πr_w³

Through implicit differentiation:

ΔV= 4πr_w² Δr,

where ΔV is the rate of change of volume and Δr is the rate of change of radius. We have already set r_w to be 12000000 and Δr is ¼ (amos/day according to Rashi). Therefore,

ΔV= 4π (12000000)² ¼

ΔV= 4.524 x 10¹⁴ (constant)

The goal of these calculations is to see whether or not Δr changes significantly over the course of the decreasing of the water. To see how much Δr changes, we must switch around the equation to define Δr and instead of using the figure of 12000000 for the radius, we will use the new radius when the top of the mountains became visible, 11999985.

As stated before, ΔV= 4πr² Δr

Therefore, Δr₂ = ΔV/ 4πr_new²

Δr₂ = 4.524 x 10¹⁴/ 4π(11999985)²

Δr₂ = 0.2500006250012

This means that if the waters were receding at a rate of change of depth of 0.25 amos per day when they began receding, then 60 days later they were receding only 0.0000006250012 amos/day faster, a rather negligible amount indeed.

QED

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...

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);

            }           

      }

}

The Search for Worthy ... Humans

In the end of פרק ו, Koheles is in search of the worthy man, free of sin. פסוק כ"ח says: "אדם אחד מאלף מצאתי ואשה בכל אלה לא מצאתי." I have found one man in a thousand. And a women in all of these I have not found. I'm not quite sure how to understand this and right now, I'm not going to try. But R' Chaim Kanievsky has a startling ha'ara on this pasuk which vastly changes how we approach it mathematically.

If we were to look at this pasuk statistically, one would say that the ratio of worthy men is 1:1000 and of worthy women is 0. Not so, says R' Chaim. In no place do we ever find the word "אדם" referring only to males. It refers to Man as a species. Therefore, we must view the 1000 as being a mixture of men and women, presumably an even mixture of 500 and 500. Koheles therefore tells us that one in a thousand human beings are worthy. And within this thousandth, he found none to be women. Although this doesn't change the state of the women, it does change the ratio of worthiness for the men. Instead of a 1:1000 worthiness ratio, according to R' Chaim Kanievsky's interpretation of the pasuk, it is 1:500.

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.

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.