Parashat Noah 5766/ November 5, 2005
the weekly Torah reading by the faculty of
Prof. Ido Kantor
Department of Physics
The dimensions of the ark were dictated to Noah by G-d with great precision: “The length of the ark shall be three hundred cubits, its width fifty cubits, and its height thirty cubits” (Gen. 6:15). Rashi, commenting on the verse “in the seventh month, on the seventeenth day of the month, the ark came to rest on the mountains of Ararat” (Gen. 8:4), added the following interesting detail: “Hence, one learns that the ark was sunk in the water 11 cubits.”
1) How did Rashi reach the conclusion that the ark was 11 cubits deep in the water? 2) Can one deduce the mass of Noah’s ark (its weight when fully loaded) from the dimensions given for the ark? 3) Are there people who disagree with Rashi’s approach, and on what do they base their different opinions?
Rashi deduced that the ark was sunk 11 cubits deep in the water from two basic data that appear in this week’s reading:
1. At the peak of the flood the water came 15 cubits over the tops of the mountains.
2. The water receded 15 cubits over a period of 60 days, from the first of Sivan until the first of Ab. Hence Rashi concludes that the water receded one cubit every four days. Therefore, if the Torah notes that the ark came to rest on the mountains of Ararat on the seventeenth of Nisan (Gen. 8:4), sixteen days from the point at which the water began to recede, then the height of the flood-waters on this day must have been 11 cubits.
Rashi calculated an intermediate value by drawing a straight line between two known points. In scientific terminology this sort of assumption is called linear interpolation. Before examining whether other Bible commentators have made different assumptions, let us briefly look at the significance of Rashi’s deduction.
We shall use Archimedes’ law, which states that a body immersed in water is buoyed up by a force (buoyant force) equal to the weight of the water displaced by the body. Rashi also assumed that the shape of the ark and its dimensions were maintained in its lower part (Gen. 6:16). Therefore, for the purposes of computation if we take a cubit to be 0.5 meters, the volume of water displaced by the ark was 5.5x25x150 = 20,635 cubic meters, whose weight exceeds 20,000 tons. This is equivalent to the mass of an average ship today. 
It would be interesting to consider how the weight was apportioned between the weight of the ark itself and the weight of its contents (people, animals, food). To answer this question we must make an assumption regarding the ratio between the volume of the space contained within the ark and the volume of its sides and partitions. If we assume that the volume of the exterior walls and partitions was approximately 15% of the overall volume, and that the ark was made mostly of wood, whose specific gravity is close to the specific gravity of water, we can calculate that Noah’s ark was laden with approximately 17,000 tons, or, in concrete terms, the weight of 250,000 people. Clearly such an ark could have carried several tens of thousands of varieties of mammals, birds, and reptiles. 
Considering the third question, we note that some commentators have taken issue with Rashi. Nahmanides, in his commentary on the words, “in the seventh month ... the ark came to rest” (Gen. 8:4), cautiously challenged Rashi’s linear interpolation. We cite part of what he said:
Rashi wrote that we can deduce from this that the ark was immersed 11 cubits in the water... But since in other instances Rashi is critical of Midrashic homilies and takes pains to explain the plain sense of Scripture, he has in effect allowed us to do likewise, for there are seventy faces to the Torah... I say that the computation given does not fit what is written in Scripture ... in that it argues that the ark was immersed a certain amount in the water on the basis of positing an equal decrease in water level for each day – four cubits per day. But it is known that when water recedes in a large river, at first it might recede one cubit in four days, and by the end, four cubits in one day. According to such a computation, on the first of Ab the tops of the high mountains became visible, and on the first of Tishre the water had entirely dried up; thus in sixty days the water had receded the entire height of the high mountains, which is several thousand cubits.
In other words, Nahmanides challenged Rashi’s assumption of a linear interpolation, arguing, on the basis of his familiarity with similar processes, that the most likely process is actually not linear. If we assume that over the course of sixty days the water receded approximately 10,000 cubits, which is the height of the mountains of Ararat, then each day the water receded an average of 170 cubits. According to Nahmanides’ approach, the forty days of rain are included in the 150 days during which the water level rose. On the 17th of Heshvan the rain began, and on the 17th of Nisan, 150 days later, the water level began to recede, and on the same day the ark also came to rest on the mountains of Ararat. On the day that the rain ceased, the 17th of Nisan, G-d caused a strong wind to blow on the water and the water level decreased about 12 or 13 cubits in a single day. Therefore, the ark was immersed only 2 to 3 cubits. Over the following 72 days, from the 17th of Nisan until the first of Tammuz, on which day the tops of the mountains became visible, the water level dropped only another 2 to 3 cubits.
Underlying the difference of opinion between Rashi and Nahmanides is the question whether, on the basis of the partial data given in this week’s reading, one should make a linear or a non-linear interpolation. It seems that the fundamental disagreement rests on the question whether the condition of the planet earth prior to and during the time of the flood was similar to its condition today, and whether phenomenon with which we are familiar today also existed then.
Rashi apparently held that at the time of the flood things were totally different from what we know today, and therefore it was not valid to make any inferences from reality as we know it today to reality in the time of the flood. Since it is impossible for us to define reality at the time of the flood from the few details which we have available, in such a situation the most reasonable assumption would be the most simplistic one – a linear function. Any assumption that diverges from a linear function must necessarily posit a model having a variable rate of change in the water level.  Rashi’s assumption that the condition of the planet was different from what we know today stands out most clearly in his commentary on chapter 8, verse 22: “So long as the earth endures ... day and night shall not cease.” He wrote there: “Insofar as they did cease throughout the days of the flood, for the planetary system did not function, so that there was no distinction between day and night.”
Malbim, who lived in the 19th century and was exposed to modern science, tried to go one step further in his commentary. He constructed a theory that explains the momentous changes that occurred during the flood, writing on as follows:
“So long as the earth endures”: It has become clear to scientists studying nature that the planet earth changed its location at a certain point ... Our Sages explained that prior to the flood the sun’s angle did not fluctuate from 23° north to 23° south of the equator in the course of its annual cycle,  as it does today; its annual course then was over the equator or close to it, and therefore people lived longer, for they did not suffer the hardships of the seasons and changes of weather. 
Malbim also provided a scientific explanation for the appearance of the rainbow, in his commentary on verse 9:13.
Nahmanides, in contrast, assumed that one can draw inferences from phenomena that we know today regarding the time of the flood. Therefore, when one makes an interpolation one should take into account non-linear functions, as they are known to us today. In his commentary on 8:4 he wrote:
If it was immersed 11 cubits, which is more than one third of its height, it would sink, since it was wider on the bottom and became smaller as it went up, the opposite of what we find in ships today, and hence was very heavy... The rain ceased after forty days, but the fountains of the great deep and the floodgates of the sky remained open, so that the air was extremely humid and the entire earth was filled with water that did not drain and never dried up. The water remained at full height until 150 days after the rain, at which time the Lord caused a strong wind to blow in the sky and the fountains the deep were stopped up. The water that had been issuing from them returned to their place until the water in the deep was restored as it had been before, and the openings of its fountains closed and the floodgates of the heavens closed... The water receded greatly on that day, and the ark came to rest, being immersed in the water two or three cubits. 
We conclude with a question to which we have found no clear answer. Archimedes’ law of buoyancy was discovered and became generally known over two thousand years ago; halahkic literature through the ages is replete with detailed investigations and computations about all sorts of strange cases and realities. It seems that Archimedes’ law did not find its way into many halakhic discussions, and the question remains why this was so?
 Prof. Moshe Kaveh, Parasha Page for Parashat Noah 5762, no. 415 (Hebrew).
 It should be noted that finding the most probable model on the basis of partial information is one of the questions currently in the forefront of many fields of research, such as bio-informatics.
 This refers to the inclination of the orbit plane to the ecliptic, i.e., to the plane of the earth’s orbit.
 One could offer a playful interpretation based on the numerical value of the first letters in the latter part of the verse (zera ve-katzir ve-kor ve-hom ve-kayitz ve-horef ve-yom ve-lailah) plus 2 (the fewest number that could be said to be “days” in the plural), showing them to total 365 days in a year.
 The ark being immersed two to three cubits in the water was primarily the result of its own weight, notwithstanding the fact that Nahmanides believed the ark was wider at the bottom and narrower at the top. Therefore, according to his theory the ark’s load (the weight of the animals and their food) was far smaller that what Rashi hypothesized, and possibly the number of species was also much more limited according to his approach.