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The Significance of the Number 40
1. The Shulchan Aruch (Code of Jewish Law) brings the customs of the month of Elul

(א) נוהגים לקום באשמורת לומר סליחות ותחנונים מראש חדש אלול ואילך עד יום הכפורים:

הגה: ומנהג בני אשכנז אינו כן אלא מראש חודש ואילך מתחילין לתקוע אחר התפלה שחרית ויש מקומות שתוקעין ג"כ ערבית ועומדים באשמורת לומר סליחות ביום א' שלפני ר"ה...

(1) We have the custom to get up at the break of dawn to recite Selichot and tachanunim from Rosh Chodesh Elul until Yom Kippur.

REMA: The minhag of Ashkenazi Jews is not like that. Instead from Rosh Chodesh on we blow the Shofar after Shacharith, and there are some places where they also blow the Shofar after Arvit. We get up at dawn to recite Selichot from the Sunday before Rosh HaShanah...

2. The Mishnah Berura (1884) by Rabbi Yisrael Meir Kagan, better known as the Chafetz Chaim

(א) ואיכא אסמכתא מקרא אני לדודי ודודי לי ר"ת אלול וס"ת עולה מ' כנגד ארבעים יום מר"ח אלול עד יוה"כ כי באלו ארבעים יום התשובה מקובלת להיות לבו קרוב אל דודו בתשובה ואז דודו קרוב לו לקבל תשובתו מאהבה ועוד סמך מקרא ומל ד' אלקיך את לבבך ואת לבב זרעך ר"ת אלול:

(1) There is a hint in the verse “I am for my Beloved and my Beloved is for me” (Ani L’Dodi v’Dodi Li - Shir Hashirim) that the first letters spell Elul and the last letters have a numerical value of 40. This alludes to the forty days from the beginning of Elul until Yom Kippur for during these forty days repentance is [more readily] accepted so a person should bring their heart near to their Beloved [G-d] with repentance, and then the Beloved will be close to them to accept the repentance with love.

3. The verse quoted above from the Song of Songs

(ב) דּוֹדִי֙ יָרַ֣ד לְגַנּ֔וֹ לַעֲרֻג֖וֹת הַבֹּ֑שֶׂם...

(ג) אֲנִ֤י לְדוֹדִי֙ וְדוֹדִ֣י לִ֔י הָרֹעֶ֖ה בַּשּׁוֹשַׁנִּֽים׃

(2) My beloved has gone down to his garden, to the beds of spices, to browse in the gardens...

(3) I am my beloved’s And my beloved is mine; He browses among the lilies.

4. The Kitzur Shulchan Aruch or Shortened Code of Jewish Law (1864), written by Rabbi Shlomo Ganzfried

(א) מֵרֹאשׁ חֹדֶשׁ אֱלוּל עַד אַחַר יוֹם הַכִּפּוּרִים, יָמִים אֵלּוּ מֻבְחָרִים יוֹתֵר וּמְזֻמָּנִים לִתְשׁוּבָה, לִהְיֹתָם יְמֵי רַחֲמִים וִימֵי רָצוֹן, כִּי בְּרֹאשׁ חֹדֶשׁ אֱלוּל, עָלָה משֶׁה אֶל הַר סִינַי לְקַבֵּל לוּחוֹת שְׁנִיּיּם, וְנִשְׁתֵּהָה שָׁם אַרְבָּעִים יוֹם, וְיָרַד בַּעֲשָֹרָה בְּתִשְׁרִי, שֶׁהָיָה אָז גְּמַר כַּפָּרָה. וּמִן אָז הֻקְדְּשׁוּ יָמִים אֵלּוּ לִימֵי רָצוֹן, וְיוֹם עֲשִׂירִי בְּתִֹשְרִי לְיוֹם הַכִּפּוּרִים.

(1) [The days] from Rosh Chodesh Elul until after Yom Kippur, are unexcelled and most suitable for repentance, because they are days of mercy and favor. On Rosh Chodesh Elul, Moshe went up Mount Sinai to receive the Second Tablets; he remained there for forty days, and came down on the tenth day of Tishrei when the atonement was completed. From then on these days have been designated as days of Divine favor [and acceptance,] and the tenth day of Tishrei as Yom Kippur [Day of Atonement].

5. When did Moses go up to Mount Sinai for 40 days?
(ט) בַּעֲלֹתִ֣י הָהָ֗רָה לָקַ֜חַת לוּחֹ֤ת הָֽאֲבָנִים֙ לוּחֹ֣ת הַבְּרִ֔ית אֲשֶׁר־כָּרַ֥ת יְהֹוָ֖ה עִמָּכֶ֑ם וָאֵשֵׁ֣ב בָּהָ֗ר אַרְבָּעִ֥ים יוֹם֙ וְאַרְבָּעִ֣ים לַ֔יְלָה לֶ֚חֶם לֹ֣א אָכַ֔לְתִּי וּמַ֖יִם לֹ֥א שָׁתִֽיתִי׃
(9) I had ascended the mountain to receive the tablets of stone, the Tablets of the Covenant that the LORD had made with you, and I stayed on the mountain forty days and forty nights, eating no bread and drinking no water.
(יח) וָֽאֶתְנַפַּל֩ לִפְנֵ֨י יְהֹוָ֜ה כָּרִאשֹׁנָ֗ה אַרְבָּעִ֥ים יוֹם֙ וְאַרְבָּעִ֣ים לַ֔יְלָה לֶ֚חֶם לֹ֣א אָכַ֔לְתִּי וּמַ֖יִם לֹ֣א שָׁתִ֑יתִי עַ֤ל כׇּל־חַטַּאתְכֶם֙ אֲשֶׁ֣ר חֲטָאתֶ֔ם לַעֲשׂ֥וֹת הָרַ֛ע בְּעֵינֵ֥י יְהֹוָ֖ה לְהַכְעִיסֽוֹ׃
(18) I threw myself down before the LORD—eating no bread and drinking no water forty days and forty nights, as before—because of the great wrong you had committed, doing what displeased the LORD and vexing Him.
(יח) וַיָּבֹ֥א מֹשֶׁ֛ה בְּת֥וֹךְ הֶעָנָ֖ן וַיַּ֣עַל אֶל־הָהָ֑ר וַיְהִ֤י מֹשֶׁה֙ בָּהָ֔ר אַרְבָּעִ֣ים י֔וֹם וְאַרְבָּעִ֖ים לָֽיְלָה׃ {פ}

(18) Moses went inside the cloud and ascended the mountain; and Moses remained on the mountain forty days and forty nights.

6. When is the first time the number 40 appears in the Torah?
(ד) כִּי֩ לְיָמִ֨ים ע֜וֹד שִׁבְעָ֗ה אָֽנֹכִי֙ מַמְטִ֣יר עַל־הָאָ֔רֶץ אַרְבָּעִ֣ים י֔וֹם וְאַרְבָּעִ֖ים לָ֑יְלָה וּמָחִ֗יתִי אֶֽת־כׇּל־הַיְקוּם֙ אֲשֶׁ֣ר עָשִׂ֔יתִי מֵעַ֖ל פְּנֵ֥י הָֽאֲדָמָֽה׃

(4) "For in seven days’ time I will make it rain upon the earth, forty days and forty nights, and I will blot out from the earth all existence that I created."

(יב) וַֽיְהִ֥י הַגֶּ֖שֶׁם עַל־הָאָ֑רֶץ אַרְבָּעִ֣ים י֔וֹם וְאַרְבָּעִ֖ים לָֽיְלָה׃

(12) The rain fell on the earth forty days and forty nights.

(א) הסבה באריכות ימיהם כי אדם הראשון מעשה ידיו של הקב"ה נעשה בתכלית השלימות בנוי בכח בקומה וגם אחרי שנקנס עליו שיהיה בן מות היה בטבעו לחיות זמן רב וכאשר בא המבול על הארץ נתקלקל עליהם האויר והלכו ימותם הלוך וחסור כי עד המבול היו ימיהם בארך ההוא ויש מהם שחיו יותר מאדם...

(1) The reason for their longevity is that the first man, the handiwork of the Holy One, blessed be He, was made in absolute perfection as regards beauty, strength, and height. Even after it was decreed upon him that he be mortal, it was in his nature to live a long time. But when the flood came upon the earth, the atmosphere became tainted, and as a result their days kept on decreasing. Until the flood, their days were about the length of Adam’s; some even lived longer than Adam...

7. The number 40 appears twice in the story of the spies
(כה) וַיָּשֻׁ֖בוּ מִתּ֣וּר הָאָ֑רֶץ מִקֵּ֖ץ אַרְבָּעִ֥ים יֽוֹם׃
(25) At the end of forty days they returned from scouting the land.
(לד) בְּמִסְפַּ֨ר הַיָּמִ֜ים אֲשֶׁר־תַּרְתֶּ֣ם אֶת־הָאָ֘רֶץ֮ אַרְבָּעִ֣ים יוֹם֒ י֣וֹם לַשָּׁנָ֞ה י֣וֹם לַשָּׁנָ֗ה תִּשְׂאוּ֙ אֶת־עֲוֺנֹ֣תֵיכֶ֔ם אַרְבָּעִ֖ים שָׁנָ֑ה וִֽידַעְתֶּ֖ם אֶת־תְּנוּאָתִֽי׃
(34) You shall bear your punishment for forty years, corresponding to the number of days—forty days—that you scouted the land: a year for each day. Thus you shall know what it means to thwart Me.
8. Is it forty or thirty nine lashes?
(ג) אַרְבָּעִ֥ים יַכֶּ֖נּוּ לֹ֣א יֹסִ֑יף פֶּן־יֹסִ֨יף לְהַכֹּת֤וֹ עַל־אֵ֙לֶּה֙ מַכָּ֣ה רַבָּ֔ה וְנִקְלָ֥ה אָחִ֖יךָ לְעֵינֶֽיךָ׃
(3) He may be given up to forty lashes, but not more, lest being flogged further, to excess, your brother be degraded before your eyes.

(ד) וְלָמָּה חִיְּבַתּוּ תּוֹרָה לִתֵּן לוֹ מַלְקוֹת אַרְבָּעִים, לְפִי שֶׁעָבַר עַל הַתּוֹרָה שֶׁנִּתְּנָה לְאַרְבָּעִים יוֹם, וְגָרַם מִיתָה לְעַצְמוֹ שֶׁנּוֹצָר בְּאַרְבָּעִים יוֹם, יִלְקֶה אַרְבָּעִים וְיֵצֵא יְדֵי עָנְשׁוֹ. כְּשֵׁם שֶׁנַּעֲשָׂה לְאָדָם הָרִאשׁוֹן שֶׁחָטָא וְנִתְחַיֵּב מִיתָה וְנִלְקָה אַרְבָּעִים, שֶׁנִּתְקַלֵּל הָעוֹלָם בְּחֶטְאוֹ אַרְבָּעִים קְלָלוֹת (עֶשֶׂר לְאָדָם, וְעֶשֶׂר לְחַוָּה, וְעֶשֶׂר לַנָּחָשׁ וְעֶשֶׂר עַל הָאָרֶץ)...

(4) And why did the Torah require him to receive 40 lashes? Because he transgressed the Torah which was given in 40 days, caused (spiritual) death to himself who was formed in forty days therefore he receives 40 lashes and he has been punished. This is similar to Adam who sinned and was required to die and the world was smitten with 40 i.e. 40 curses due to his sin (10 to Adam, 10 to Chava, 10 to the snake and 10 to the ground)...

9. There are 'forty minus one' forbidden categories of activity on Shabbat

(ב) אֲבוֹת מְלָאכוֹת אַרְבָּעִים חָסֵר אֶחָת. הַזּוֹרֵעַ. וְהַחוֹרֵשׁ ... הַמּוֹצִיא מֵרְשׁוּת לִרְשׁוּת. הֲרֵי אֵלּוּ אֲבוֹת מְלָאכוֹת אַרְבָּעִים חָסֵר אֶחָת:

(2) This fundamental mishna enumerates those who perform the primary categories of labor prohibited on Shabbat, which number forty-less-one. One who sows, and one who plows ... and one who carries out an object from domain to domain. All these are primary categories of labor, and they number forty-less-one.

10. Forty is the age of wisdom

(כא) הוּא הָיָה אוֹמֵר ...בֶּן אַרְבָּעִים לַבִּינָה.

(21) He used to say ....At forty for wisdom.

Indeed, the sages say one shouldn't learn kabbalah until the age of 40. Only at 40 years of age does the mind and soul develop in a higher understanding of 'Binah', where one can start to understand and grasp the lofty and esoteric concepts of kaballah.

(א) ...יש שכתבו שלא ללמוד קבלה עד שיהא בן ארבעים שנה כמ"ש בן ארבעים לבינה בשגם שצריך קדושה וטהרה וזריזות ונקיות לזה ורוב המתפרצים לעלות בחכמה זו קודם הזמן הראוי קומטו בלא עת...

(1) ...Some write that one should not study Kabbalah until they are 40 years old as it says (Avot 5:21) 'at forty for wisdom' not least because one needs holiness, purity, speediness and cleanliness for this. Most of those who move to study this insight before the time are grabbing it prematurely...

We also find that Moses only told the Jewish nation that they understand him a full 40 years after they started learning Torah from him.

(דברים כט, ג) ולא נתן ה' לכם לב וגו' אמר רבה ש"מ לא קאי איניש אדעתיה דרביה עד ארבעין שנין

Moses told the Jewish nation: “But the Lord has not given you a heart to know, and eyes to see, and ears to hear, until this day” (Deuteronomy 29:3). Rabba said: A person does not understand the opinion of his teacher until after forty years, as Moses said this to the Jewish people only after forty years of accepting the Torah.

11. After 40 days a fetus is Halachically considered a child (in some cases)

(ז) הַמַּפֶּלֶת לְיוֹם אַרְבָּעִים, אֵינָהּ חוֹשֶׁשֶׁת לְוָלָד. לְיוֹם אַרְבָּעִים וְאֶחָד, תֵּשֵׁב לְזָכָר וְלִנְקֵבָה וּלְנִדָּה.

(7) A woman who has a miscarriage on the fortieth day need not be concerned that it might have been an offspring and she became impure. But on the forty-first day after immersion, there is concern that perhaps it was an offspring and she shall observe the period of impurity for a male and a female; and for niddah.

הָיְתָה אִשְׁתּוֹ מְעוּבֶּרֶת וְאָמַר: ״יְהִי רָצוֹן שֶׁתֵּלֵד אִשְׁתִּי זָכָר״ — הֲרֵי זוֹ תְּפִלַּת שָׁוְא. וְלָא מַהֲנֵי רַחֲמֵי? ... אֵין מַזְכִּירִין מַעֲשֵׂה נִסִּים. וְאִיבָּעֵית אֵימָא: מַעֲשֶׂה דְּלֵאָה בְּתוֹךְ אַרְבָּעִים יוֹם הֲוָה.

We learned in the mishna: One whose wife was pregnant and he said: May it be God’s will that my wife will give birth to a male child, it is a vain prayer. Is a prayer in that case ineffective? ... [The Talmud quotes the story of Leah] Answer 1: One does not mention miraculous acts to teach general halakha. Answer 2: And if you wish, say instead that the story of Leah and her prayer with regard to the fetus was within forty days of conception.

12. A Mikva (ritual immersion bath) requires 40 Se'ah (around 575 litres or 152 Gallons).

״וְרָחַץ אֶת כָּל בְּשָׂרוֹ (בַּמַּיִם)״, שֶׁלֹּא יְהֵא דָּבָר חוֹצֵץ בֵּין בְּשָׂרוֹ לַמַּיִם. ״בַּמַּיִם״ — בְּמֵי מִקְוֶה. ״כׇּל בְּשָׂרוֹ״ — מַיִם שֶׁכׇּל גּוּפוֹ עוֹלֶה בָּהֶן, וְכַמָּה הֵן — אַמָּה עַל אַמָּה בְּרוּם שָׁלֹשׁ אַמּוֹת. וְשִׁיעֲרוּ חֲכָמִים מֵי מִקְוֶה אַרְבָּעִים סְאָה.

It is written in the Torah: “And he shall bathe all his flesh in the water” (Leviticus 15:16), and the Sages derived that nothing should intervene between his flesh and the water. The definite article in the phrase “in the water” , i.e., specifically in the water of a ritual bath. The phrase “all his flesh” indicates that it must be in water into which all of his body can enter. And how much water is that? It is a cubit by a cubit by the height of three cubits. And the Sages calculated that the waters of a ritual bath measure forty se’a.

(123) Mikveh: A ritual immersion pool. In Jewish law, some changes of status, such as conversion, repentance, and ritual purification require ritual immersion in a pool of 40 seah, approximately 200 gallons, of pure water.

Chart of Biblical measurements:
13. Our forefather Abraham recognised God at 40, the famed sages Rabbi Yochanan and Rabbi Akiva (and Hillel) became learned at the age of 40.

(ג) וּבֶן אַרְבָּעִים שָׁנָה הִכִּיר אַבְרָהָם אֶת בּוֹרְאוֹ...

(3) And, when Abraham was forty years old he recognized his Creator...

מה היה תחלתו של רבי עקיבא. אמרו בן ארבעים שנה היה ולא שנה כלום.

What were the origins of Rabbi Akiva? They say that he was forty years old and had still not learned anything.

...תניא כל שנותיו של רבי יוחנן בן זכאי מאה ועשרים שנה ארבעים שנה עסק בפרקמטיא מ' שנה למד ארבעים שנה לימד.

It is taught in a baraita: All the years of Rabbi Yoḥanan ben Zakkai were 120 years. For forty of those years he dealt in business, for forty of those years he studied, and for forty of those years he taught and guided the Jewish people.