Tuesday, March 11, 2014
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!
Labels:
מגילה
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.
Friday, January 31, 2014
עמודי החצר
In the end of Parshas Terumah, (pesukim 27:9-19), the Torah describes the beams that held up the curtain that surrounded the courtyard of the Mishkan. Pasuk 10 discusses the beams on the southern side of the courtyard:
In Rashi's seemingly innocent comment on the pasuk, there is a grave arithmetic difficulty which is the subject of much discussion amongst the commentators on Rashi. If there are five amos between each beam and 20 beams, that would provide only 19 spaces of five amos. That would yield only 95 of the 100 amos that the pasuk tells us make up the length of the courtyard. Of course, the first notion is that the space does not include the width of the beams. Therefore, there may have been 95 amos of space and five amos of beams to complete the 100 amos. This is in fact the suggestion of the Riva, in the name of his rebbe and is also the opinion of the Abarbanel. The 20 beams on the north and south sides added up to five amos on either side. This would make each beam one quarter amah (1½ tefachim). This interpretation would avoid all our problems from the outset. However, R' Eliyahu Mizrachi takes issue with this interpretation on two accounts. Firstly, he sees no reason why there should be such a large difference between the thickness of the beams of the courtyard and that of the planks of the Mishkan itself (nine tefachim). His second objection is that within the beams themselves you would have some of different thickness than others. On the east and west sides, there are only 10 beams needed to make up five amos. (The nine spaces between the ten beams make up 45 of the 50 amos width of the courtyard.) Therefore, each beam would be three tefachim, twice the width of those on the north and south sides. The lack of symmetry involved in this understanding of Rashi causes the Mizrachi to disregard it and give his own interpretation.
Firstly, the Mizrachi suggests that the five amos referred to by Rashi are not five amos of space but rather five amos from the beginning of one beam to the beginning of the next.. This view is generally accepted amongst all those who deal with this problem with the obvious exception of the aforementioned Riva and Abarbanel. In pasuk 18, the Mizrachi infers from Rashi that the beams were one amah thick. Therefore, the actual space between each beam would be four amos and the thickness of the beam would complete the five amos. However, we have now only accounted for 95 amos. Therefore, the Mizrachi suggests that the north and south sides actually had 21 beams and the east and west had 11 but that the seemingly extra beam on each side belonged to the set of of beams of the side perpendicular to it. For instance, 21 beams were placed on the southern side of the courtyard. The beam in the southwest corner, though, was officially part of the western side. So, too, the beam in the northwest corner was not counted as part of the western beams but as part of the northern beams and so on. See illustration. With this arrangement another space of five amos is added to complete the 100 amos referred to in the pasuk.
In pasuk 18, the Mizrachi suggests that the 100 amah measurement of the courtyard was in fact a measurement from within the beams and the one amah taken up by the beams is not included. This reasoning was given in order to justify Rashi's calculation of 20 amos distance between the Mishkan and the curtains of the courtyard on the north, south and west sides. The Gur Aryeh objects to this with the claim that the pesukim (9,11,12,13) clearly state that the curtains were exactly 100 amos long on the north and south sides and 50 amos long on the east and west sides. But according to the Mizrachi's interpretation, the outer perimeter of the courtyard would be 102 amos by 52 amos. He offers a defence for the Mizrachi that perhaps the only purpose of the curtains was to cover up the open spaces and they did not need to cover the corners (as illustrated on page 3). However, in his own opinion, the Gur Aryeh suggests that the 100 amah measurement is in fact referring to the outer perimeter of the courtyard. He then was required to justify Rashi's measurement in pasuk 18 in a different manner.
The Levush HaOrah, another commentator on Rashi is unhappy with both the Mizrachi and the Gur Aryeh's explanations of Rashi in regards to the placement of the beams. From the fact that Rashi mentions the measurement of five amos between each beam more than just once, he infers that Rashi meant for this to be consistent throughout the entire perimeter of the courtyard. According to the Mizrachi the length of the north side, for instance, was really 102 amos and according to the Gur Aryeh it was 100. However, if you add up 21 beams each of one amah thickness and 20 spaces of four amos each, we are given 101 amos. So, too, on the east and west sides we would end up with 51 amos instead of 50 or 52. He concludes that the only way for the Mizrachi's figures to work out would be to say that one space on each of the four sides was actually one amah bigger. For the Gur Aryeh's figure to work one space would have to be one amah smaller. The Levush does not accept that such a lack of symmetry was present in the building of the Mishkan and offers a rather unique arrangement of the beams. Each of the beams were circular on the bottom for one amah and were inserted into circular holes in the copper sockets that held the beams in place. The beam itself was a semi-cylinder whose diameter was one amah. On each of the corners was placed a quarter-cylinder beam so that the curtain could wrap around it. See illustration. The thickness of this beam was only one half amah on either side. This removes one half amah one either end of each side of the courtyard. With this arrangement, the spaces between all of the beams were all four amos wide without any exception and the perimeter of the courtyard was exactly 100 amos by 50 amos as stated in the pesukim. Amongst all the interpretations mentioned thus far, this is by far the most symmetric and arithmetically accurate.
Finally, the sefer Ma'ase Choshev offers another possible arrangement of the beams which matches that of the Levush's in symmetry and arithmetic correctness. He suggests that there were no beams in the corners. The curtains were suspended from wooden bars. On these bars were placed the hooks that were used to hang the curtains from the beams. Each of these bars was five amos long. The north and south sides had twenty such bars and the east and west sides had ten. These wooden bars would allowed the curtains to change direction at the corners without the need to wrap it around a beam. See illustration. Once again the figure of five amosrefers to the distance from the beginning of one beam to the beginning of the next. With this arrangement the thickness of the beams becomes irrelevant. All of the figures mentioned in the pesukim work out perfectly as well. One advantage of this arrangement over that of the Levush's is that all of the beams are the exact same shape.(The illustration assumes the beams to be one amah thick.)
The arrangement of the Ma'ase Choshev is the one quoted in the seforim Meleches HaMishkan and Tavnis HaMishkan (etc.). The sefer Lifshuto Shel Rashi, however, is content with the opinion of the Riva and the Abarbanel. Whatever the true arrangement of the beams was, it is clear that when Rashi said that there were five amos between each beam, he had some logical calculation in mind. The only question that remains is "Which?".
On a Related Topic
The Mishkan was covered by three layers of material(*). The first covering described by the Torah (26:1-6) was made of twisted linen, turquoise, purple and scarlet wool. The covering was made up of 10 panels of 4x28 amos2. This yields a total area of 40x28 amos2. The Mishkan was 30x10 amos2. The beams that made up the walls of the Mishkan were 1 amah thick. Thus, the Mishkan required 32x12 amos2 of roofing.
The beams were 10 amos tall. The covering was 28 amos wide and 12 amos covered the roof of the Mishkan. That leaves 16 amos for the two sides which is 8 amos on each side. So the wool/linen would reach two amos from the ground. There is a dispute as to whether or not the front beams were covered. We will go with the opinion of the gemara (Shabbos 98b) that they were uncovered as Rashi (26:5) notes that the pesukim seem to indicate as such. Therefore, 31 amos of the covering's width provided roofing, leaving 9 amos to hang from the back. The second covering was a covering of goat hair. This covering was wider and longer than the wool/linen layer and covered it fully on all sides.
Rashi (26:13) notes that the Torah teaches us a lesson that one should show compassion for valuable objects. The twisted linen and assorted wools were very precious and thus, as Rabbeinu Bachya explains, it was made not to drag on the ground so that it would not be soiled by dirt and rain and was protected fully by the goat hair. This lesson is easily understood considering the measurements mentioned thus far. However, there is one simple question to be asked. What about the corners? As the accompanying diagram shows, if a piece of material hangs only 8 amos off one side and 9 amos off the other, simple Pythagorean geometry dictates that the corners will hang down more than 12 amos! (This effect is well demonstrated by the corners of a rectangular tablecloth hanging from the table.)This is hardly an efficient way to care for valuables.
This problem seems far too obvious to have been overlooked by Chazal in teaching us this lesson. However, finding the answer was not easy. But finally, an answer was found in R' Chaim Kunyevsky's elucidation of Braisa diMleches haMishkan. There he asks exactly this question. He answers that the corners of the coverings were folded against the back of the Mishkan as illustrated. The Ritv"a (Shabbos 98b) apparently provides the same answer in the name of Braisa diMleches haMishkan but our versions show no evidence of any such discussion. One of the books on the Mishkan actually show such an arrangement but there is no discussion as to any source or reason for it.
*This and a number of other facts discussed on this page are actually subject to a large-scale dispute between R' Yehudah and R' Nechemiah. For our purposes, all figures are according to R' Yehudah.
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תרומה
Friday, January 10, 2014
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 Yisroel 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 Yisroel 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 ration. 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.
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בשלח
Friday, December 6, 2013
Can you Count to 70?
Question: How many males are counted as coming to Mitzrayim with Yaakov? One thing is for sure, it wasn't 70. I still have not been able to figure out how all the numbers worked - who were the 66 mentioned in 46:26 and the 70 in 46:27? 66+3 = 69, the last time I checked. If you add up all of the children and grandchildren, it does come out to 70 but then it should have been 67 and then 70. All that aside, it was not only males who were counted. Dinah is counted along with her brothers which is understandable. Serach bas Asher is counted as well which is slightly more puzzling. One must assume she was not the only granddaughter. From Rashi (46:27) it seems Yocheved was somehow part of the 70 as well.
While I was not able to find anything explaining why these particular women figured in the count as opposed to others, I did see an interesting insight into the pesukim in consideration of that fact. Tzeror HaMor and Emes L'Yaakov both point out a discrepency in the per-wife tallies found in the pesukim. The numbers for Rachel ("arba'ah asar") and Bilhah ("shiv'ah") are of the masculin form. The numbers for Leah ("sheloshim veshalosh") and Zilpah ("sheish esreih") are feminine. They both explain that Leah and Zilpah both had women counted among their offspring - Dinah from Leah and Serach from Zilpah. Therefore their numbers are delivered in feminine. Rachel and Bilhah had no feminine offspring counted and thus their numbers are in masculine.
One might wonder why this is so, considering that the generic plural is usually masculine by default. However, Emes L'Yaakov points out that the word "nefesh" which the number is qualifying is feminine. So the default gender of the number for "nefesh" should be feminine. Rachel and Bilhah were the exceptions.
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ויגש
Friday, November 15, 2013
Goats and Amicable Numbers
Goats and Amicable Numbers
In this week’s parasha, we find Yaakov preparing for his encounter with his twin brother Esav in several ways. Among other preparations, Yaakov sends him gifts consisting of various different kinds of animals. The Torah tells us (Bereishit 32:14–16) how many of each kind of animal Yaakov sent: 200 female goats and 20 male goats; 200 female sheep (ewes) and 20 male sheep (rams); 30 nursing camels with their young; 40 female cows and 10 bulls; 20 female donkeys and 10 male donkeys. What is the significance of these numbers?
In his ספר בעלי ברית אברם, R’ Avraham Azulai provides an explanation for the number of goats, which he attributes to R’ Nachshon Gaon of the 9th century. The total number of goats is 200 + 20 = 220. What significant property does the number 220 have?
Consider the factors of 220, that is, numbers that multiply together to give the product 220. We can factor the number 220 in the following ways:
220 = 1 × 220
220 = 2 × 110
220 = 4 × 55
220 = 5 × 44
220 = 10 × 22
220 = 11 × 20
Now consider only the “proper factors” of 220 – that is, all the factors in the above list, excluding the number 220 itself – and add them up:
1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284
So the proper factors of 220 add up to 284.
We now repeat the process, considering the factors of 284. We can factor the number 284 in the following ways:
284 = 1 × 284
284 = 2 × 142
284 = 4 × 71
Again, we consider only the proper factors of 284 – all the factors in the above list, excluding the number 284 itself – and add them up:
1 + 2 + 4 + 71 + 142 = 220
So the proper factors of 284 add up to 220. Does this number look familiar?
As we have just shown, the numbers 220 and 284 have the property that the proper factors of each number add up to the other number. A pair of numbers with this property is known as a pair of amicable numbers, or according to R’ Nachshon Gaon, מנין נאהב. Apparently it was known to the ancients that in order to gain the love of kings and princes, a person would give one of a pair of amicable numbers as a present, keeping the other number for himself. This is so that the factors of the number given add up to the number kept, and the factors of the number kept add up to the number given. So this is what Yaakov did. He sent Esav 220 goats, and kept 284 for himself.
Wait a minute: The Torah tells us that Yaakov gave Esav 220 goats, but where do we see in the Torah that he kept 284 for himself? Several pesukim later, as Yaakov gives instructions to the servants carrying the gifts, the Torah records (32:21), “כי־אמר אכפרה פניו במנחה ההולכת לפני” – “for he said: I will win him over with the gifts that are being sent ahead.” R’ Nachshon Gaon explains that this sentence contains a hint to the number 284, in the following way. The word אכפרה can be divided in two parts: אכ פרה. When the Torah uses the word אך, it is generally interpreted by the Rabbis to indicate exclusion or reduction. Calculating the numerical value of the second part of the word, פרה, we get: 80 (פ) + 200 (ר) + 5 (ה) = 285. Applying a reduction (indicated by אך) to the value 285 (given by פרה), we obtain a value of 284. This represents the number of goats that Yaakov kept for himself, according to R’ Nachshon Gaon.
Special thanks to Daniel Levenstein for bringing this insight to my attention.
References:
Leonard Eugene Dickson, History of the Theory of Numbers, Volume I: Divisibility and Primality, Carnegie Institute of Washington: Washington, 1919, p. 39, available at:
http://www.archive.org/stream/historyoftheoryo01dick#page/38/
ר' אברהם ב"ר מרדכי אזולאי, ספר בעלי ברית אברם, published 1873 but existed in manuscript for 300 years previously; pp. 48–49, available beginning at:
http://www.hebrewbooks.org/pdfpager.aspx?req=3997&pgnum=47
הנאהבים והנעימים - על רעות אצל מספרים, in Michlalah Jerusalem College's mathematical journal אלף אפס (ℵ₀):
http://alefefes.macam.ac.il/article/article.asp?n=15
(may not work in all browsers)
(Thanks to Yaaqov Loewinger for this link via Hebrew Wikipedia)
In this week’s parasha, we find Yaakov preparing for his encounter with his twin brother Esav in several ways. Among other preparations, Yaakov sends him gifts consisting of various different kinds of animals. The Torah tells us (Bereishit 32:14–16) how many of each kind of animal Yaakov sent: 200 female goats and 20 male goats; 200 female sheep (ewes) and 20 male sheep (rams); 30 nursing camels with their young; 40 female cows and 10 bulls; 20 female donkeys and 10 male donkeys. What is the significance of these numbers?
In his ספר בעלי ברית אברם, R’ Avraham Azulai provides an explanation for the number of goats, which he attributes to R’ Nachshon Gaon of the 9th century. The total number of goats is 200 + 20 = 220. What significant property does the number 220 have?
Consider the factors of 220, that is, numbers that multiply together to give the product 220. We can factor the number 220 in the following ways:
220 = 1 × 220
220 = 2 × 110
220 = 4 × 55
220 = 5 × 44
220 = 10 × 22
220 = 11 × 20
Now consider only the “proper factors” of 220 – that is, all the factors in the above list, excluding the number 220 itself – and add them up:
1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284
So the proper factors of 220 add up to 284.
We now repeat the process, considering the factors of 284. We can factor the number 284 in the following ways:
284 = 1 × 284
284 = 2 × 142
284 = 4 × 71
Again, we consider only the proper factors of 284 – all the factors in the above list, excluding the number 284 itself – and add them up:
1 + 2 + 4 + 71 + 142 = 220
So the proper factors of 284 add up to 220. Does this number look familiar?
As we have just shown, the numbers 220 and 284 have the property that the proper factors of each number add up to the other number. A pair of numbers with this property is known as a pair of amicable numbers, or according to R’ Nachshon Gaon, מנין נאהב. Apparently it was known to the ancients that in order to gain the love of kings and princes, a person would give one of a pair of amicable numbers as a present, keeping the other number for himself. This is so that the factors of the number given add up to the number kept, and the factors of the number kept add up to the number given. So this is what Yaakov did. He sent Esav 220 goats, and kept 284 for himself.
Wait a minute: The Torah tells us that Yaakov gave Esav 220 goats, but where do we see in the Torah that he kept 284 for himself? Several pesukim later, as Yaakov gives instructions to the servants carrying the gifts, the Torah records (32:21), “כי־אמר אכפרה פניו במנחה ההולכת לפני” – “for he said: I will win him over with the gifts that are being sent ahead.” R’ Nachshon Gaon explains that this sentence contains a hint to the number 284, in the following way. The word אכפרה can be divided in two parts: אכ פרה. When the Torah uses the word אך, it is generally interpreted by the Rabbis to indicate exclusion or reduction. Calculating the numerical value of the second part of the word, פרה, we get: 80 (פ) + 200 (ר) + 5 (ה) = 285. Applying a reduction (indicated by אך) to the value 285 (given by פרה), we obtain a value of 284. This represents the number of goats that Yaakov kept for himself, according to R’ Nachshon Gaon.
Special thanks to Daniel Levenstein for bringing this insight to my attention.
References:
Leonard Eugene Dickson, History of the Theory of Numbers, Volume I: Divisibility and Primality, Carnegie Institute of Washington: Washington, 1919, p. 39, available at:
http://www.archive.org/stream/historyoftheoryo01dick#page/38/
ר' אברהם ב"ר מרדכי אזולאי, ספר בעלי ברית אברם, published 1873 but existed in manuscript for 300 years previously; pp. 48–49, available beginning at:
http://www.hebrewbooks.org/pdfpager.aspx?req=3997&pgnum=47
הנאהבים והנעימים - על רעות אצל מספרים, in Michlalah Jerusalem College's mathematical journal אלף אפס (ℵ₀):
http://alefefes.macam.ac.il/article/article.asp?n=15
(may not work in all browsers)
(Thanks to Yaaqov Loewinger for this link via Hebrew Wikipedia)
Labels:
number theory,
וישלח
Thursday, October 3, 2013
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 amos^{3} in volume (Breishis 6:15). Therefore, the water displaced by the ark was 300x50x11 = 165,000 amos^{3}.
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 cm^{3} 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 m^{3} | 0.157 m^{3} | 0.111 m^{3} |
Calculation | = 165000 x 0.192 ≈ 31630 | = 165000 x 0.157 ≈ 25950 | = 165000 x 0.111 ≈ 18250 |
Water Displacement | 31630 m^{3} | 25950 m^{3} | 18250 m^{3} |
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/m^{3}. 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/m^{3}. 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 m^{3} | 25950 m^{3} | 18450 m^{3} |
x 1025 kg/m^{3} | ||
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 m^{3} | = | 1117003 ft^{3} |
25950 m^{3} | = | 916416 ft^{3} |
18450 m^{3} | = | 651556 ft^{3} |
25^{o} C | = | 77^{o} F |
1025 kg/m^{3} | = | 2260 lb/61024 in^{3}(35.3 ft^{3}) |
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. |
Wednesday, October 2, 2013
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 depthwise. 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 Rebbie 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_{w}^{3}
Through implicit differentiation:
ΔV= 4πr_{w}^{2} 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 10^{14} (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= 4πr_{}^{2} Δr
Therefore, Δr_{2} = ΔV/ 4πr_{new}^{2}
Δr_{2} = 4.524 x 10^{14}/ 4π(11999985)^{2}
Δr_{2} = 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.
Tuesday, September 17, 2013
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.
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קהלת
How many בקשות in יעלה ויבוא
Since we are about to say יעלה ויבוא numerous times (perhaps as many as 40 times) I figured it would be a good time to explore this:
Wow, 347 בקשות packed into one small תפילה!
A friend once approached with an interesting project - to calculate the total number of בקשות in יעלה ויבוא. Now, of course, that's not as simple as it seems. It is not simple addition. It involves a lot of multiplication.
So let's dive into it:
יעלה | ויבוא | ויגיע | ויראה | וירצה | וישמע | ויפקד | ויזכר | = 8 |
(Keep in mind that all the above verbs will apply to all of the following nouns:) | |
זכרוננו | ופקדוננו | וזיכרון אבותינו | וזיכרון משיח בן דוד עבדך | וזיכרון ירושלים עיר קדשך | וזיכרון כל עמך בית ישראל | x 6 |
= 48 | |
(And then the following modify all of the above:) | |
|לפניך לפליטה | לטובה | לחן | ולחסד | ולרחמים | לחיים ולשלום | x 7 |
So the entire first section | = 336 |
זכרנו ה' אלוקינו בו לטובה | ופקדנו בו לברכה | והושיענו בו לחיים טובים | + 3 |
= 339 | |
Now we add the final portion: | |
ובדבר ישועה | ורחמים | = 2 |
חוס | וחננו | ורחם עלינו | והושיענו | x 4 |
= 8 | |
So the final count is 339 + 8 | = 347 |
Wow, 347 בקשות packed into one small תפילה!
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תפילה
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