Chronology of the Flood

Exegetical Approaches

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150 Days Which Includes the First 40 Days

The two mentions of 150 days refer to the same time period and encompass the forty days of rain. The ark landed at the conclusion of these 150 days.  This approach subdivides regarding how the commentators understand the chapter's dating:

Months from Creation

All the months mentioned in the verses are calculated according to the yearly calendar, from creation.

Sources:Qumran Scroll 4Q252Fragment 1 Columns 1-2About the Qumran Scrolls, the KaraitesYefet b. Eli Bereshit 8:4Aharon b. Yosef Bereshit 8:3-8Aharon b. Eliyahu Bereshit 8:3-6, rejected approach in Lekach TovBereshit 7:24About R. Toviah b. Eliezer, Ibn EzraBereshit 8:3-5Additional Commentary Bereshit 7:11Additional Commentary Bereshit 8:14Iggeret HaShabbatAbout R. Avraham ibn Ezra, R. Yosef Bekhor ShorBereshit 7:24Bereshit 8:3-10About R. Yosef Bekhor Shor, RidBereshit 7:12Bereshit 8:4About R. Yeshayah of Trani, approach in ChizkuniBereshit 7:17Bereshit 7:24Bereshit 8:3-13About R. Chizkiyah b. Manoach, RambanBereshit 8:4-5About R. Moshe b. Nachman #2,¹ Akeidat YitzchakBereshit 13About R. Yitzchak Arama, AbarbanelBereshit 7:24About R. Yitzchak Abarbanel, SfornoBereshit 7:24About R. Ovadyah Sforno, ShadalBereshit 7:24Bereshit 8:2Bereshit 8:4About R. Shemuel David Luzzatto, Hoil MosheBereshit 7:11About R. Moshe Yitzchak Ashkenazi, R. D"Z HoffmannBereshit 7:21-24About R. David Zvi Hoffmann, U. CassutoBereshit Introduction to the FloodAbout Prof. Umberto Cassuto

Months – These commentators assume that all the months mentioned in the account of the flood are numbered from the beginning of the calendar year.  They disagree, though, regarding whether the first month is Tishrei or Nissan.²

150 and 40 Overlap – This position assumes that the forty days of rain are subsumed in the 150 day period, reading the final verse of Chapter 7 "וַיִּגְבְּרוּ הַמַּיִם עַל הָאָרֶץ חֲמִשִּׁים וּמְאַת יוֹם", as a summary statement of the previously described events rather than a successive event.³

One set of 150 – These sources similarly assume that the statement "וַיַּחְסְרוּ הַמַּיִם מִקְצֵה חֲמִשִּׁים וּמְאַת יוֹם" does not mean that the water receded after a second set of 150 days, but rather that it decreased after (or the reduction was first noticeable at the end of) the 150 days previously mentioned.

"וַיִּגְבְּרוּ הַמַּיִם" and "וַיַּחְסְרוּ הַמַּיִם" – Though all these sources agree that there was but one set of 150 days, they disagree in how they understand the relationship between the periods of increasing and receding waters:⁴

• Water first decreased after 150 days ­ - The author of the Qumran Scroll, Yefet the Karaite, Ibn Ezra,⁵ Ramban, and Sforno all understand the phrase "וַיַּחְסְרוּ הַמַּיִם מִקְצֵה חֲמִשִּׁים וּמְאַת יוֹם" to mean that the water only began to decrease at the end of the 150 days. Until that point the water level was either still rising, or at least, maintaining its height. Sforno suggests that even though the rain stopped after 40 days, the underground sources of water were still open,⁶ causing continued water pressure during this entire period.⁷ Ramban, in contrast, seems to maintain that though the waters did not continue to increase after the fortieth day, due to the great humidity⁸ they retained their height throughout this period.
• Water decreased after the forty days of rain – Most of the other commentators in this approach⁹ assume that the water started to recede right after the rain stopped. The phrase " וַיִּגְבְּרוּ הַמַּיִם עַל הָאָרֶץ חֲמִשִּׁים וּמְאַת יוֹם" simply means that the waters were is a state of strength (relative to normal water heights), not that they were increasing, or even maintaining their level.  R"Y Bekhor Shor, R. Hoffmann and U. Cassuto explain the verse "וַיַּחְסְרוּ הַמַּיִם מִקְצֵה חֲמִשִּׁים וּמְאַת יוֹם"   to mean not that the water first began to recede at this point but that the decrease was first noticeable to Noach after this period, when the ark landed.¹⁰

Chronology

• Chronological –Those who maintain that the waters maintained their strength throughout the 150 days, read the verses chronologically, with Chapter 7 describing the rising  flood and the opening  verses of Chapter 8 detailing the decreasing waters at the end of the 150 days.¹¹
• Achronological – The other commentators, though, assume that verses 8:2-4 (the sending of the wind and closing of the heavenly windows) occurred in the midst of the 150 days that are mentioned in the preceding chapter and that there is an element of achronology in the verses.  Akeidat Yitzchak and Shadal explain that the Torah purposely separated the images of destruction and salvation, detailing the world's collapse in Chapter 7 before describing its rebuilding in Chapter 8.

How do 150 days fit in five months? According to the fixed lunar calendar, months generally alternate between having 29 and 30 days,¹² which means that there can be at most 147-149 days in any 5 month period.¹³  As such, this position must explain how the 150 days fit into the five months between 2/17 when the rain began and 7/17 when the ark rested.   The commentators resolve this issue in different ways:

• Different calendar – Several sources reject the assumption that the dates in the story are based on the Hebrew lunar calendar:
• Solar Calendar –  Some commentators¹⁴ assert that the months mentioned relate to the solar calendar. According to this reckoning there are 152 days in five months and the 150 day period ended a couple of days before the ark rested.¹⁵
• Persian/Egyptian – Ibn Ezra suggests, in contrast, that Noach was using a calendar like that of the Egyptians or Persians who would intercalate their year by adding 5 days to a certain month. The total days in the five month period can thus equal or surpass 150.¹⁶
• Lunar calendar based on eyewitnesses – Yefet and Anan the Karaites maintain that Noach did use a lunar calendar, but one in which the months were determined by eyewitnesses rather than calculations.¹⁷ They assume that if no one can see the moon (as was the case when Noach was in the Ark)¹⁸ the default length of a month is thirty days.¹⁹  This allows for exactly 150 days between 2/17 and 7/17.²⁰

• Fixed Hebrew Lunar Calendar – A last group of sources attempt to uphold the idea that the Torah is employing the Rabbinic calendar:
• Round Numbers - The Rid and U. Cassuto solve the problem by simply suggesting that 150 days is a round number. Though, in reality there were fewer days in the five month period, the Torah rounded up to the the next ten.²¹
• Full Leap Year - R. Saadia suggests that one can reach 150 days using a regular fixed lunar calendar if the year was a "full" leap year. In such a year, 4 of the 5 months( Marcheshvan, Kislev, Shevat and Adar I) would all have 30 days. As such 7/17 is the 150th day.²²

How did the ark land so soon? These commentators differ in how they understand how the ark managed to land right after the 150 days of strong waters, if the mountain tops were still covered  by fifteen cubits of water:²³

• Receded over 110 days – According to those who say that the water started to decrease already after the 40 days of rain, this is not an issue,²⁴ and is, in fact, one of the factors that motivate them to explain the verses as they do. Shadal even maintains that by this point the higher mountain tops were totally uncovered, and that Bereshit 8:5 describes only the revealing of the shorter mountains.

• Receded over 2 days – According to those who maintain that the waters retained their strength throughout, but that a solar calendar was used and thus the 150 period ended by the 15th of the seventh month, there was a period of two days in which the water lessened, allowing the ark to land.
• Miraculous intervention – Ramban, in contrast, is forced to assert that Hashem sent a miraculous wind which decreased the waters substantially on the 150th day itself.²⁵ 

Shifting water levels – A simple reading of the verses suggests that the waters decreased at an extremely varied pace:

• If one posits that the water first decreased after the 150 days, then it went down only 15 cubits in the 2 1/2 months from 7/17 until 10/1 compared to thousands of cubits in the 3 months from then until the land dried on 1/1.²⁶  Ramban explains that there is no reason to think that the water needed to decrease at a steady pace. In addition, according to him, Hashem's intervention on the first day led to a vast decrease in the water, ²⁷ minimizing the difference in the rate of reduction thereafter slightly.²⁸
• Those who posit that the water began to recede after the rain stopped have an even bigger problem, since according to them, initially the water receded only 15 cubits in over 180 days!²⁹   Shadal minimizes the problem by asserting that by the time the ark landed on 7/17, more than 15 cubits of water had already receded, since Mt. Ararat was not the tallest, but rather one of the shorter mountains.³⁰  There would, nonetheless, still seem to be a significant increase in the rate of water reduction in the last 3 months, assuming that even the shorter mountains were a few thousand cubits high.

"וַיְהִי מִקֵּץ אַרְבָּעִים יוֹם" and the sending of doves

• From the revealing of the mountain tops – Most of these commentators count the 40 days from the preceding date mentioned in the verses, the first of the tenth month, when the mountain tops were visible. This leaves about a month between the landing of the dove and opening of the ark's cover when the land dried on the first of the first month.³¹ It is not clear, though, why Noach waited.³²
• From the landing of the ark – The Rid questions the dating of the above approach, wondering why the dove could not find a resting place if the land had already been visible for over a month.³³  This prompts him to instead count the forty days from the landing of the ark on the seventeenth of the seventh month.  He assumes that Noach waited for ten days to see how the raven was faring, then sent the first dove 7 days later and the final dove 14 days after that, reaching the first of the tenth month. It was only then, when the mountaintops were finally revealed, that the dove could rest.³⁴

Varied Dating

Some months are counted to creation and some are dated in reference to key points during the flood.

Sources:Ramban Bereshit 8:4-5About R. Moshe b. Nachman# 1,³⁵ R. Eliezer AshkenaziMa'asei Hashem Bereshit 26About R. Eliezer Ashkenazi (Ma'asei Hashem) #2

Months – According to Ramban and R. Ashkenazi, the beginning and end of the flood (2/17 and 2/27) are dated to the beginning of the calendar year, but the months mentioned in the middle of the story, the landing of the ark on 7/17 and the revealing of the mountain tops on 10/1, are both dated to the start of the flood, setting them at 8/17 and 11/1 respectively.³⁶ The last date, though, when Noach opens the ark's cover on 1/1, is once again dated to the beginning of the year.  This approach is motivated by a desire to make sense of the various time-frames in the chapter³⁷ but its obvious disadvantage is its inconsistency.³⁸

40 and 150 overlap – Like the above approach, these commentators also view the phrase "וַיִּגְבְּרוּ הַמַּיִם עַל הָאָרֶץ חֲמִשִּׁים וּמְאַת יוֹם", as a summary statement of the previously described events, and assume that the 40 days are included therein.

One set of 150 – Both Ramban and R. Ashkenazi maintain that the verse "וַיַּחְסְרוּ הַמַּיִם מִקְצֵה חֲמִשִּׁים וּמְאַת יוֹם" does not refer to a new period of 150 days, but to the end of the previously mentioned one.

"וַיִּגְבְּרוּ הַמַּיִם" and "וַיַּחְסְרוּ הַמַּיִם" – These commentators assert that the waters maintained their strength throughout the 150 days, and only began to abate afterwards.³⁹

Chronology – This position does not need to assume any achronology in the verses describing the increasing and decreasing waters.

How do 150 days fit in five months? According to this approach there are really six months from the start of the flood until the resting of the ark.⁴⁰  This easily fits the period of 150 days of increasing waters, even leaving a month over afterwards.

How did the ark land so soon? According to this position, there was a month between the 150 days and the resting of the ark, enough time for the water to recede sufficiently to allow the ark to find a  resting place.

Shifting water levels – As this position reduces the time between the revealing of the mountain tops and the drying of the land to two months,⁴¹ it exacerbates the problem of the changing rate of reduction.  It has the waters reducing 15 cubits in about 2.5 months, while they are lowered thousands of cubits in 2 months!

"וַיְהִי מִקֵּץ אַרְבָּעִים יוֹם... וַיְשַׁלַּח אֶת הָעֹרֵב" – R. Ashkenazi dates this from the previously mentioned event, the revealing of the mountain tops, which this position maintains occurred on the first of the eleventh month (the tenth month being counted from the start of the flood).⁴²  As such, the sending of the final dove fell on the first of the first month,⁴³ when the land dried.⁴⁴ This explains why it was only now, that it did not return.

150 Days Which Do Not Include the First 40 Days

The two mentions of 150 days are one unit of time, while the forty days of rain are a distinct unit.  As such, there was a 190 day period before the waters began to decrease.

Sources:Seder Olam Rabbah4About Seder Olam Rabbah, Bereshit Rabbah33:7About Bereshit Rabbah, Sefer HeArukhs.v. קלAbout R. Natan b. Yechiel, RashiBereshit 7:12Bereshit 8:3-14Ms. Leipzig 1 Bereshit 8:5-6About R. Shelomo Yitzchaki, Lekach TovBereshit 7:24Bereshit 8:3-6Bereshit 8:10-14About R. Toviah b. Eliezer, Rashbamfrom Ms. Oxford 970About R. Shemuel b. Meir, ChizkuniBereshit 7:24Bereshit 8:3-13About R. Chizkiyah b. Manoach, R. Chaim PaltielBereshit 8:6About R. Chaim Paltiel, Ba'alei HaTosafotMS JTS 793Tosafot HaShalem Bereshit 8:4:8Tosafot HaShalem Bereshit 8:5:5)About Ba'alei HaTosafot, Minchat YehudaBereshit 8:6About R. Yehuda b. Elazar, Daat ZekeinimBereshit 8:3About Daat Zekeinim, first approach of R. Eliezer AshkenaziMa'asei Hashem Bereshit 26About R. Eliezer Ashkenazi (Ma'asei Hashem)

Distinct 40 and 150 days – This approach assumes that there is no overlap between the description of the flood and the 150 days of strong waters.  The verses follow sequentially, with the 150 days following the forty days of rain.

One Set of 150 – This position, nonetheless, agrees that there is only one set of 150 days and that the waters receded at the end of the previously mentioned 150 days, and not after a new set.

"וַיִּגְבְּרוּ הַמַּיִם" and "וַיַּחְסְרוּ הַמַּיִם" – Though all these commentators maintain that the water started to abate only after the 150 days (on the first of the ninth month), they differ regarding what happened during the initial period itself:

• Waters retained strength – According to Seder Olam Rabbah and the Arukh, the water stopped increasing after the forty days of rain, but maintained their level throughout.⁴⁵ According to this, the phrase "וַיִּגְבְּרוּ הַמַּיִם" does not mean that the waters increased but that they stayed in their strength.
• Waters increased – According to Ba'alei HaTosafot, in contrast, even after the rain stopped, the water continued to increase throughout the 150 days. He might maintain that the waters continued to emerge from underground.⁴⁶

How do 190 days fit into 5 months? The verses state that the ark rested on the 17th of the seventh month.   According to this position, though, at this point the waters were still at their height (and would be through the 29th of the eighth month)!⁴⁷  This forces these commentators to reinterpret this date and to assume that it is dated to an event in the flood narrative rather than the beginning of the calendar year:

• End of rain – Following Seder Olam Rabbah, most of these commentators⁴⁸ assert that the 17th of the seventh month is being counted from the month in which the rain stopped.  As such, it is really equivalent to to the 17th of the ninth month,⁴⁹ which gives ample time for the waters to decrease before the ark landed.⁵⁰
• Beginning of rain – The Arukh, Lekach Tov and R. Yitzchak⁵¹ get to the same date but through a different calculation. They assert that one needs to count seven full months from the date the rain began. Thus, 2/17 plus 7 months brings one to 9/17.⁵²

"בָּעֲשִׂירִי בְּאֶחָד לַחֹדֶשׁ נִרְאוּ רָאשֵׁי הֶהָרִים" – These sources suggest that this date, too, should not be counted from the beginning of the year and almost unanimously date it to the start of the flood, reaching a date of 11/1.⁵³ This, of course, is inconsistent both with the dates of the start and end of the flood (dated to the beginning of the calendar year), and with the way most of these commentators dated the landing of the ark, which was to the end of the rain, not the beginning.⁵⁴ They are motivated to do so, though, by their understanding of the verses regarding the sending of the doves - see point below.

"וַיְהִי מִקֵּץ אַרְבָּעִים יוֹם" and the sending of the doves – The verse does not say explicitly from which event the forty days are to be counted.  These commentators offer three different possibilities:

• From when the waters began to abate – According to Seder Olam Rabbah, Rashbam, and one version of R. Tam⁵⁵ the 40 days are counted from the first of the ninth month, when the waters began to decrease.  The motivation for their interpretation is the assumption that the first two doves would have found a resting place if the mountaintops had been visible.  Thus, their chronological reconstruction have the third and final dove being sent on the day the mountain tops appear.⁵⁶  This reading, though, necessitates dating the appearance of the mountains to the first of the eleventh month,⁵⁷ forcing them to adopt the notion mentioned above that the date mentioned in the verses, the first of tenth month, is really dated to the beginning of the rain (and not the year).⁵⁸
• From the revealing of the mountain tops – The Arukh and Rashi,⁵⁹ in contrast, maintain more simply that the forty days are counted from the last date mentioned in the verses, the revealing of the mountain tops.⁶⁰ They assume that the final sending of the dove must have coincided with the drying of the land on the first of the first month,⁶¹ as only this would explain why it did not return.⁶² Thus, working backwards from this date (1/1), they, too, need to suggest that the mountain tops appeared on the first of the eleventh month,⁶³ and that the date mentioned in the verses (10/1) is from the beginning of the rain (and not the year).⁶⁴
• From the resting of the ark – R. Tam, as brought by R. Chayim Paltiel,⁶⁵ and R. Eliezer Ashkenazi assert that the forty days began with the landing of the ark, which they date to the seventeenth of the ninth month.⁶⁶  R. Tam rejects Seder Olam's possibility above, asserting that Noach would have been unaware of the day in which the water started decreasing,⁶⁷ and so he must have counted the forty days from an event he was cognizant of,⁶⁸ such as the landing.⁶⁹ 

Shifting water levels – Seder Olam Rabbah, and many of these sources in its wake, assume that the water receded at a steady pace from the first of the ninth month until the mountain tops were seen two months later.  As such they calculate that they receded at a rate of one cubit  every four days.⁷⁰  This would mean that afterwards, though, there was a drastic change in pace, since in the remaining 3 1/2 months, the water would have to go down the height of an entire mountain!  Chizkuni attempts to suggest that there was lot more volume to the water that was above the mountain tops since there were no land forms intervening.

Two sets of 150 Days Which Include the Second 40 Days

The verses speak of two different sets of 150 days, one in which the waters rose and one in which they receded. The forty days of rain are not included in the first unit, but the forty day wait before the sending of birds is subsumed in the second set.

Sources:RadakBereshit 7:24Bereshit 8:3-14About R. David Kimchi

Two sets of 150 – Radak assumes that the verses speak of two distinct periods of 150 days, one in which the waters rose and one during which they fell.  When the verse says, "וַיַּחְסְרוּ הַמַּיִם מִקְצֵה חֲמִשִּׁים וּמְאַת יוֹם", it is not referring to the previously mentioned days of rising waters described in Chapter 7, but rather a new unit of time depicted in Chapter 8.

40 days of rain – According to Radak, Chapter 7 is sequential and the forty days of rain do not overlap with the first set of 150 days.  Thus, between the two there were 190 days, ending on the first of the ninth month.

40 day wait – Radak assumes that the forty days before the sending of the birds occurred at some point during the 150 days in which the waters decreased.

How do 190 days fit into 5 months – In order to fit the 190 days of rain and rising waters into the period before the landing of the ark, Radak must explain like Seder Olam Rabbah and others above, that the date given for the landing (7/17) is not counted from the beginning of the calendar year but rather from the end of the 40 days of rain.  It is, thus, equivalent to to the 17th of the ninth month, 16 days after the water began to decrease.

"בָּעֲשִׂירִי בְּאֶחָד לַחֹדֶשׁ נִרְאוּ רָאשֵׁי הֶהָרִים" – Radak follows Seder Olam Rabbah in counting the tenth month, when the mountains became visible, to the beginning of the flood.  Thus, according to his reckoning they appeared on the first of the eleventh month.

"וַיְהִי מִקֵּץ אַרְבָּעִים יוֹם" and the sending of the doves – Radak is inconsistent in the way he understand the timing of these events:

• At the end of the 150 days – In his comments to 8:3, Radak suggests that the 21 days worth of sending of the birds occurred after the second set of 150 days.  He seems to be motivated by a desire to have the times mentioned in the text reach a year: 40+ 150+ 150 + 21=361, or about 12 months of 30 days each.  Such a reckoning, though, is very difficult as it assumes that the birds were sent in the second month of the second year, after Noach had removed the ark's covering and saw that the waters had already dried!⁷¹
• After the ark landed – In his explanation to 8:6, Radak writes that the forty days be counted from when the ark landed (on 9/17),⁷² assuming that the starting point had to be an event that Noach was aware of.⁷³ This would mean that the first dove was sent after the mountains were already visible,⁷⁴ making it difficult to understand why it could not find a resting place.⁷⁵  One of the Ba'alei HaTosafot answers that perhaps "וְלֹא מָצְאָה הַיּוֹנָה מָנוֹחַ" means that she did did not find anything to serve as a sign for Noach and was, thus, not at peace for she did not fulfill her mission.⁷⁶  
•  After the mountain tops appeared – A few verses later, Radak suggests that the last dove found a place to land on the first of the first month,⁷⁷ which would mean that the forty days were counted from the appearance of the mountain tops, two months before.⁷⁸ This possibility works best with the chronology of the verses.

¹ Ramban raises two possible understandings of the chapter.  This approach represents his second option; see below for his first possibility.

² The Rid, Ramban, Akeidat Yitzchak, Abarbanel and R. David Zvi Hoffmann all date them to Tishrei while Ibn Ezra, R"Y Bekhor Shor, and Shadal date them to Nissan.  The difference of opinion relates to the controversy in Bavli Rosh HaShanah 10b-11a regarding whether the world was created in Tishrei or Nissan.

³ They do regard the later mention of forty days as its own independent unit.  There the text writes, "וַיְהִי מִקֵּץ אַרְבָּעִים יוֹם" but it is hard to suggests that this refers to the previously mentioned 40 days of rain, since there are several other time markers in between.  See point below.

⁴ The various opinions relate to a number of other issues raised by the text, including the landing of the ark, chronology of the verses, and the shifting pace of water reduction, which will be discussed in the bullets below.

⁵ Ibn Ezra does not refer to this phrase explicitly but also agrees that the waters first abated at the end of 150 days.

⁶ Chapter 7:17 speaks of the flood lasting for 40 days, but it is only in 8:2 that the text mentions the closing of the heavenly windows and subterranean sources of water.  This leads Sforno (and Ramban) to suggest that there were two stages.  However, the continuation of 8:2 reads, "וַיִּכָּלֵא הַגֶּשֶׁם מִן הַשָּׁמָיִם" which leads others to suggest that the two events happened simultaneously.

⁷ His words are slightly ambiguous, and can be interpreted to mean either that the subterranean waters caused an increase in the water throughout this period or that they simply caused a continuous pressure, preventing the water from standing still or decreasing.

⁸ This was caused by the fact that the heavenly windows and subterranean water sources were still open.

⁹ See R"Y Bekhor Shor, Akeidat Yitzchak, Abarbanel,, Shadal, R. D"Z Hoffmann, and U. Cassuto.

¹⁰ Until then, when the ark was still moving and Noach could only see water and sky he had no way of knowing if the water level was increasing, staying the same or decreasing.

¹¹ Ramban might have been motivated to read the verses in this manner due to his general tendency to posit that the Torah is written in chronological order.

¹² There are two potential exceptions to this rule. In a "regular" year, Marcheshvan has 29 days and Kislev has 30 days, but there are some "full" years in which both have 30 days, and some "missing" years in which both have only 29 days. In addition, in a leap year, Adar I precedes the regular Adar and has 30 days.

¹³ See the introduction and notes there for elaboration.

¹⁴ See one of the possibilities raised by R. Saadia Gaon in his arguments against the Karaitic position regarding the establishing of new months, the opinion rejected by Ibn Ezra, and the opinion of R. Yehuda HaParsi (quoted in Ibn Ezra's Iggeret HaShabbat).  The Qumran sect also use a solar calendar  which is divided into four quarters, in each of which two out of every three months has thirty days and the third month has thirty one days. They thus calculate that the 150 days lasted through the 14th of the seventh month, several days before the ark rested on 7/17. 

¹⁵ According to this approach, the 150 day period mentioned is not equivalent to the five month period and ends a couple of days before it. The Karaites, Yefet and Anan, reject this possibility for this very reason. [See Ibn Ezra as well.] They assume that there have to be exactly 150 days between 2/17 and 7/17, asserting that only the receding waters, which began after the 150 days, could possibly explain why the ark was able to finally rest. [They assume that had the water decreased earlier, the ark would have rested earlier.] See below, though, for other explanations of how and when the water level fell and how this impacted the landing of the ark.
See also R. Ashkenazi below, who makes an opposite argument, claiming that if the 150 days were equivalent to the five month period, it would be superfluous for the Torah to give both time markers.

¹⁶ Based on the math, one would think that such a calendar would allow for more than 150 days in the five month period.  Yet, Ibn Ezra himself seems to assume that such a calendar would equal exactly 150 days, since right beforehand and in his comments elsewhere, he rejects the possibility that Noach used a solar calendar because that would add two days to the 150 given in the text.

¹⁷ In their arguments with the Rabbis, they even point to this story as proof of their position that moths are determined by eyewitnesses, claiming that if one uses calculations the dating in the story does not work.

¹⁸ R. Saadia questions this assumption, suggesting that after the rain ceased on the fortieth day, the moon should have been visible.  Moreover, there is no reason to assume that the months mentioned were calculated by Noach, rather than Hashem, who does not need the moon to calculate time.

¹⁹ R. Yosef Bekhor Shor, Sforno, and Hoil Moshe also maintain that Noach was using a thirty day per month calendar, but do not elaborate regarding how they reach this conclusion.  Hoil Moshe suggests that in the time of Noach people might not have been very adept at mathematical and astronomical calculations, leading to an inexact calendar.

²⁰ This count does not include the end date, 7/17. Alternatively, according to Aharon b. Yosef and Aharon b. Eliyahu the Karaites, though 7/17 is included and there are really 151 days, the Torah was not bothered by the extra day and rounded it to 150.

²¹ U. Cassuto points out that this is common usage.  Often, a person might speak of a three month period as 90 days, even though it is not an exact calculation.  The Rid points to other examples in Torah, such as the punishment of lashes which is referred to as 40 lashes, even though there are only 39. 

²² This end date in included in the count. Cf. R D"Z Hoffmann who counts the days similarly.

²³ Since the ark was only 30 cubits high, this would mean that half of it was submerged in water!

²⁴ There was a full 110 days for the water to recede enough for the ark to land. According to Lekach Tov and R. Yosef Bekhor Shor, the water decreased at an even pace (of 1 cubit every 12 days) over the 180 days until the mountain tops were seen on 10/1. As such, in the 110 days before the ark landed, the water decreased nine cubits, allowing the ark to rest while partially submerged in the remaining 6 cubits of water.

²⁵ Ramban compares this to Yam Suf where it says: "וַיּוֹלֶךְ י"י אֶת הַיָּם בְּרוּחַ קָדִים עַזָּה כׇּל הַלַּיְלָה וַיָּשֶׂם אֶת הַיָּם לֶחָרָבָה וַיִּבָּקְעוּ הַמָּיִם".

²⁶ In these last three month the water would have to have receded the height of an entire mountain!

²⁷ In his comments to 8:4, he suggests that the water fell by about 9 cubits on the first day, and assumes that the ark was submerged in 6 cubits when it rested on the mountain.  In his comments to 8:5, however, he offers a variation of the approach and asserts that since Mt. Ararat was one of the shorter mountain, the reduction in water level need not be limited to fifteen cubits and was much more than that.  He suggests that many of the taller mountain tops would have already been revealed, and had Noach been in their vicinity, he might have landed earlier.  It was only the shorter mountains in the region of Ararat that were first uncovered on the first of the tenth month.

²⁸ As the water needed to decrease the height of an entire mountain in the period from 10/1 to 1/1, while it needed to decrease just the difference in height between the taller and shorter mountains in the period between 7/17 and 10/1, there would still seem to have been a vast increase in pace at the end.

²⁹ There were 110 days until the ark rested on the 17th of the seventh month and then almost 2 1/2 months until the mountain tops were revealed on the first of the tenth month.  See note above that R"Y Bekhor Shor assumes that for these 180 days there was a steady rate of reduction, 1 cubit every 12 days.  The next 90 days (until the land dried on the first of the first month), though would have to see a reduction of at least 100 cubits per day, if a mountain's height of water still needed to be lowered!

³⁰ In other areas, the tops of the tallest mountains would have already been revealed.  It was only the shorter mountain tops that were first revealed on 10/1. See Ramban in the note above who explains similarly.

³¹ The exact time relates to how one reads the time-frame of the sending of the birds.  The verses do not say how long Noach waited between sending the ravenand the first dove.  If the dove was sent immediately, then the text allots fourteen days for the bird's mission.  Before sending the second dove, though, the verse writes, "וַיָּחֶל עוֹד שִׁבְעַת יָמִים", suggesting that before the first dove had departed Noach has also waited a week.  This reading assumes that the events took place over 21 days.

³² It is possible that the the doves indicated an in between phase in the drying of the land - the drying of treetops, but not the ground.  Thus the verse says that Noach realised "כִּי קַלּוּ הַמַּיִם" when the dove returned, but it was only a month later that he saw that "הִנֵּה חָרְבוּ פְּנֵי הָאֲדָמָה".

³³ They explain that perhaps the dove could not find a resting place either because it preferred the valleys or the air was too moist.

³⁴ This position must posit that the verses describing the sending of the doves are not written in chronological order.  As they could easily have been placed between verses 4-5 (and then the phrase "at the end of forty days" would much more obviously refer back to the landing of the ark) the placement is difficult.

³⁵ Ramban offers two different ways of understanding the chapters.  This position represents his first possibility; see above for his second option.

³⁶ R. Saadia Gaon brings support for the idea that one event can be dated to another, rather than to the beginning of the year, from Yirmeyahu 28:17. There Yirmeyahu prophesies that Chananyah will die by the end of the year, but he dies in the "seventh month", which would already be the new year. R. Saadia, therefore concludes that the seventh month is dated from Yirmeyahu's prophecy. This prooftext is very questionable, though, as one could easily posit Yirmeyahu counted the new year from whatever he called the first month, not the seventh.

³⁷ See below how such dating easily fits 150 days into the period between 2/17 and 7/17 when the ark landed, and explains how the ark could land on the mountain tops when otherwise they would seem to have still been covered by 15 cubits of water.

³⁸ R. Asheknazi attempts to defend the approach, pointing out that the opening of the flood must be dated to the calendar year, since there is no other event to date it to.  Since the final date of the flood occurred in the same month, it would be confusing to the reader to refer to it differently, and so Torah maintains the calendar month for that date as well.  Though the drying of the land on the first of the first month could conceivably have been dated to the flood, the Torah opted not to, since it is Rosh HaShanah, and calling it the eleventh month would be strange.

³⁹ R. Ashkenazi even posits that the rain continued (albeit sporadically) throughout this 150 day period.

⁴⁰ See first point above, that both dates, 7/17 and 10/1, are dated to the beginning of the flood and not the calendar year.

⁴¹ See above that they maintain that  "the first of the tenth month" is dated to the beginning of the flood, making it really the first of the eleventh calendar month. At the same time, the day of the drying of the land, 1/1 is dated to the calendar year, leaving but two months between the two dates.

⁴² As discussed regarding the above approach, he could maintain that despite the fact that the mountains had already been visible for more than a month when the dove is sent, it could not find a resting place since it preferred the valley.

⁴³ This date, unlike the previous one, is dated to the calendar year.

⁴⁴ R. Ashkenazi asserts that the forty days ended on the tenth of the twelfth month and that "21 days after he sent the dove, the land dried on the first of the first month."  This sounds as if there was 21 days between when Noach sent the first dove and the last, but the verses actually only allow for 14 days.  (Noach waited a week in between each sending.)  It is possible that R. Ashkenazi means "21 days after sending the raven" and is including  a seven day waiting period afterwards.  Otherwise, R. Ashkenazi is not trying to say that the final dove was sent on the same day as the drying of the land, but a week before.

⁴⁵ The Arukh explains that the verse, "וַיָּשֻׁבוּ הַמַּיִם מֵעַל הָאָרֶץ הָלוֹךְ וָשׁוֹב" mans that after the rain stopped, the waters went back and forth, neither increasing nor decreasing during this period. Seder Olam Rabbah explains that during these 150 days the wicked were each being sentenced for their actions.

⁴⁶ Cf. Sforno above.

⁴⁷ The rain lasted through the 27th of the third month. 150 days from there, in a regular year in which months alternate between 29 and 30 days, would conclude on the 29th of the eighth month, more than a month after the ark landed.
Seder Olam Rabbah writes, "עד מתי חמשים ומאת יום עד א' בסיון" leading many to assumes that he calculated the end of the 150 days to the first of the ninth month rather than the 29th of the eighth.  It is possible, though, that he simply means that the 150 days ended before the first of the month (עד א' בסיו is not inclusive).
Rashi apparently tries to solve the problem emerging from Seder Olam Rabbah in two different ways in his commentary, leading to an internal contradiction in Rashi.  In his comments to 7:12, he posits that the last day of the 150 really was the first of the ninth month, because the first day of rain was not included.  In his comments to 8:3, though, he asserts (as per the discussion above) that the first of the month is the day after the 150.

⁴⁸ See Seder Olam Rabbah, Bereshit Rabbah, Rashi, Ba'alei HaTosafot and R. Ashkenazi.

⁴⁹ Since the rain stopped at the end of the third month, one would count seven months starting from the third month (Kislev, Tevet, Shevat, Adar, Nissan, Iyyar, Sivan). 

⁵⁰ Seder Olam Rabbah assumes that the waters receded at a steady pace of 1 cubit per four days until the mountain tops were revealed. As such, by the time the ark landed they had receded 6 cubits, allowing the ark to land while submerged in 11 cubits of water.  Ramban questions that if 1/3rd of the ark was under water, it would probably have sunk.

⁵¹ See his gloss found in Ms. Leipzig 1 of Rashi's commentary.

⁵² A rejected opinion in Baalei haTosafot also counts from the beginning of the rain, but uses the method above, counting seven months from the second month, thereby reaching a date of 8/17.  As this is before the waters began to decrease, one must posit (as R. Saadia suggests) that the ark did not really land but rather somehow got stuck, despite the high water level.  If so, though, it was still half submerged, making this improbable.  In addition, it is not clear why the Torah would share such information if the event did not mark a difference in the flood levels. 
This opinion would work just as well if one did not recalculate the months (one could just as easily say that on 7/17 the ark got stuck, despite the high water levels), and is only motivated by a desire to have both of the middle dates of the flood (the resting of the ark and revealing of the mountain tops) be dated to the same point.  See the point below regarding the date of 10/1.

⁵³ One exception is R. Ashkenazi, who dates it to the end of the rain, getting to the first of the twelfth month.  He is thus able to maintain consistency in the dating, having both the landing of the ark and revealing of the mountains, dated to the same event.
The second exception is one of the Ba'alei HaTosafot who also counts both events to the end of the rain, but suggests that it was a leap year. Thus, 7/17=9/17 is not the 17th of Sivan but of Iyyar and 10/1=12/1 is not the first of Elul, but the first of Av.

⁵⁴ The Arukh, Lekach Tov and R. Yitzchak who dated 7/17 also to the beginning of the rain were, in fact, only motivated to do so in order to be consistent with this dating.  They lose much of their advantage, though, by using different methods of counting.  While they reckon 7/17 to be 9/17 by adding seven full months to the date of the begininng of the flood (2/17+ 7 = 9/17), they reckon 10/1 to be 11/1 by counting ten partial months from the month in which the flood began (from month 2 to month 11, there are 10 months).

⁵⁵ This is the way Ba'alei HaTosafot recount his position.  R. Paltiel, though, brings a different version of R. Tam, according to which he maintains that the forty days began with the landing of the ark. See below.

⁵⁶ One might question this approach on several grounds:
1) One of the Ba'alei HaTosafot asks how the second dove could bring back a leaf from an olive tree, if it was sent before the mountain tops were revealed. He answers simply that the trees, being on the mountain tops, must have appeared before the tops of the mountain themselves, but he questions why this fact is not mentioned as a marker of the decreasing flood waters. [see Daat Zekenim as well.]
2) This position is also somewhat difficult as it suggests that the entire passage regarding the birds (8:6-11) is achronological.  It is not clear why this would be necessary, as the verses could have easily been written in their proper place. 
3) Finally, R. Paltiel further questions why, according to this approach, it took Noach two more months before he removed the cover of the ark.

⁵⁷ The verses speak of a forty day period before sending the raven and then 21 days worth of sending doves.  Thus, if the waters began to abate on the first of the ninth month, the last dove was sent sixty days later, on the first of the eleventh month.  [This assumes that the dove was sent on the fortieth day itself, and that the end dates, 9/1 and 11/1 are included in the count.]

⁵⁸ This, then, also explains the inconsistency in the dating of the landing and appearance of mountains (one to the end of the rain and the other to the start).  These sources could not date the resting of the ark from the beginning of the rain since the 190 days would not fit in the allotted time frame. Conversely, they could not date the revealing of the mountain tops from the end of the flood, for that would mean that the 61 days mentioned in the verse ended before the tops were revealed, making it impossible for the dove to land.

⁵⁹ The explanation given here follows the printed version of Rashi's commentary.  In the Leipzig manuscript of Rashi, though, this explanation is written and then crossed out, replaced in the margins of the manuscript with an explanation that matches that of Seder Olam Rabbah. The scribe writes that this is how he found the text in the manuscript of R. Shemayah, the student/scribe of Rashi. It is unclear, though, if R. Shemayah made the change by himself or under Rashi's instructions.  See also Daat Zekeinim which brings this explanation of Rashi, questions it, and then writes "לכך פירש"י" and brings the version which is similar to Seder Olam Rabbah.  For a fuller discussion of the issue, and an image of the manuscript, see here.

⁶⁰ As opposed to Seder Olam Rabbah, they can thus preserve the chronology of the chapter.

⁶¹ Thus, 8:12 and 8:13 overlap in time, and Noach's opening of the cover of the ark is a direct response to the dove's not returning.

⁶² R. Tam questions this assumption, pointing out that it is not clear why the first dove could not find a resting place, if the tops of the mountains had appeared over forty days before dove was sent. [This is what motivates his alternative suggestion above.]  Chizkuni and R. Elyakim suggest that doves prefer valleys, and would not nest on the mountains tops, while one of the Ba'alei HaTosafot posits that the phrase "וְלֹא מָצְאָה הַיּוֹנָה מָנוֹחַ" means that the dove did not find any food.

⁶³ R. Tam questions this calculation, pointing out that, from the appearance of the mountains on the first of the 11th month until the end of that month is only 29 days, and that the twelfth month, being a "missing" month, is also only 29 days. Thus, 61 days from the appearance of the mountains would bring one to the third of the first month, not the first. 
Rashi, though, was likely assuming that the dove was sent on the fortieth day itself, adding up to a 60 (not 61) day period.  Thus, if one includes both the first and last day (11/1 and 1/1) in the count, there are exactly 60 days from the revealing of the mountains to the drying of the land. See Rashi's language, "הרי ששים יום משנראו ראשי ההרים עד שחרבו פני האדמה".

⁶⁴ This necessitates the above mentioned inconsistency in the method of dating. Rashi could not date the revealing of the mountain tops to the end of the rain (as he had regarding the resting of the ark), for that would date it to the first of the twelfth month, and would only leave a month until the drying of the land, instead of the 60 days needed according to the verses.
One of the Ba'alei HaTosafot tries to get around the problem by maintaining that the year was a leap year, giving an extra month's worth of time for all the events to occur. As such, he is able to date both the landing of the ark and the revealing of the mountains to the end of the rain, reaching dates of the 17th of Iyyar (9/17 due to the extra Adar) and the 1st of Av (12/1 due to the extra Adar). He, thus, has the same amount of days pass until the landing (with ample room for the 190 days), but because there are 13 months in the year, he still has two months after the appearance of the mountain tops to send the doves.

⁶⁵ The Ba'alei Ha Tosafot bring a different version of his position, which fits with Seder Olam Rabbah above.

⁶⁶ See discussion above.

⁶⁷ R. Ashkenazi's motivations are different. Desiring to count both the date of the ark's landing and the appearance of the mountains to the same event, he assumes that both are counted from the end of the rain. As such, he posits that the mountains appeared on the first of the 12th month. This means that he can't count the forty days from the mountain's appearance (as it would place the sending of the doves a couple of weeks after the drying of the land), leading him instead to the previously mentioned date, the landing of the ark. His, position, though, becomes extremely difficult, since according to his calculations, even the last dove is sent more than a week before the land appears, making it unclear how it found a place to rest. The position would only work, if he posits, like the Rid above, that there was an extended waiting period between the sending of the raven and the sending of the doves.

⁶⁸ One could disagree, though, for there is no need to posit that Noach reckoned the forty days himself. It is possible that the Torah is simply providing the time frame of the events for the reader.

⁶⁹ According to this approach, the first dove was sent a few days after the mountain tops appeared on 11/1.  As this is still very soon afterwards, they can easily suggest that despite the land being visible, it was still too wet to find a resting place.  A week late, though, the second dove, though, could already find a leaf, and by the third week, it was dry enough to land.

⁷⁰ Seder Olam thus posits that when the ark landed it was still submerged in 11 cubits of water.  Ramban questions that if this were true, the ark should have sunk, as 1/3 of it was under water.

⁷¹ In addition to the fact that there would seem to be no point is sending the doves on a fact finding mission when Noach could see the land on his own, it is also not clear why the doves would not have been able to find a place to land.  The words of the text are difficult as well, since they assume that only beacuse the dove returned with the olive leaf, did Noach know "כִּי קַלּוּ הַמַּיִם".  Finally, this dating does not work with the chornology of the verses which places the event before the drying o fthe water son the first of the first month.

⁷² See above that Radak assumes that the date given in the verses, 7/17, is dated to the end of the rain.

⁷³ He, thus, dismisses the approach that suggests that the forty day count began when the waters stated to wane, as Noach would have had no way of knowing that.  See R. Tam above who explains the same.

⁷⁴ According to Radak they appeared on the first of the eleventh month and the first dove was sent on the fourth of that month.

⁷⁵ Another disadvantage of this reading is that it needs to assume that the verses are achronological.

⁷⁶ He could have also explained that though the mountain tops were visible, it was still too wet to land.

⁷⁷ He writes, "אז ידע נח כי חרבו פני האדמה והיונה מצאה מנוח בארץ ובעצים הרבה לפיכך לא שבה אליו, וזה היה באחת ושש מאות שנה לנח כמו שאמר ראשון באחד לחדש".

⁷⁸ See above that he maintians that they appeared on the first of the eleventh month.  This reading of the sending of the doves matches that of Rashi above.