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After sleeping, when one sleeps 60 breaths or more, they have to do netilat yadayim. Is 60 breaths a span of time, or a calculated amount of breaths one takes while sleeping?

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It is a combination of both.

In Shulchan Aruch, Orach Chaim 4:16 it relates that this was the practise of Dovid HaMelech. It writes there:

דוד היה נזהר שלא לישן שיתין נשמין (פירוש ששים נשימות) כדי שלא יטעום טעם מיתה: הגה ובגמרא פ' הישן משמע דדוקא ביום היה נזהר: (דברי עצמו ועיין בב"י):

David (the King) was careful to not sleep sixty breaths in order to not feel the taste of death. [And the Gemara states in the chapter ofהישׁן that David was only careful to do this during the day.

The Mishna Berurah there explains it as meaning:

שיתין נשמי - ר"ל רצופין אלא היה מתנמנם כמה פעמים פחות משיתין נשמי עד חצות לילה ומחצות ואילך היה מתגבר כארי

60 breaths - Meaning, [60] continuous [breaths]. Instead, he would nap several times, [each one] less than "60 breaths", until mid-night, and from mid-night and on he would "strengthen himself like a lion"....

And then the Biur Halacha helps to contextualise this into an actual timeframe:

רבו בו הדעות בשיעור זה י"א דהוא ג' שעות וראיה מהא דהאר"י ז"ל היה ישן בשבת ב' וג' שעות ויש דוחין דשאני ת"ח בשבת דמצוה לענג השבת ועוד אינו מוכרח כלל דשמא היה ניעור כמה פעמים בתוך שינתו ולא היה ישן ס' נשמין בפעם אחת וי"א דהוא יותר מחצי שעה וי"א דהוא שיעור מעט יותר משלשה מינוט ע"כ בעל נפש יחמיר לפי כחו

Therefore some hold that it equates to three hours whilst other understand it to mean only half an hour, whilst others say even a timeframe of a little more than three minutes.

Kitzur Shulchan Aruch agrees with the half an hour view:

אִם הִשְׁכִּים וְנָטַל יָדָיו בְּעוֹד לַיְלָה כְּדִינוֹ וְהָיָה עֵר עַד אוֹר הַיּוֹם, אוֹ שֶׁהָיָה יָשֵׁן אַחַר כָּךְ שֵׁנִית בְּעוֹד לַיְלָה, וְכֵן הַיָּשֵׁן בַּיּוֹם שִׁתִּין נִשְׁמִין (שֶׁהוּא לְעֵרֶךְ חֲצִי שָׁעָה), וְכֵן הַנֵּעוֹר כָּל הַלַּיְלָה [וְלֹא יָשַׁן שִׁתִּין נִשְׁמִין], בְּכָל אֵלּוּ יֵשׁ סָפֵק אִם צְרִיכִים נְטִילַת יָדַיִם אוֹ לֹא. לָכֵן יִטֹּל יָדָיו שָׁלֹשׁ פְּעָמִים בְּסֵרוּגִים כְּדִלְעֵיל סָעִיף ג', אֲבָל לֹא יְבָרֵךְ עֲלֵיהֶם.

If you awakened and washed your hands while it is still night, as required, and remained awake until daybreak, or if you slept a second time while it was still night, or if you slept during the day [long enough] for sixty breaths (which is about one half hour), or if you are awake all night [without sleeping long enough for sixty breaths,] in all of these cases it is doubtful if there is a requirement to wash the hands or not. You should, therefore, wash three times in the alternate manner (as mentioned in paragraph 3) but without reciting the berachah.


UPDATE

I found here a full breakdown:

How much sleep is “sixty breaths”?[24]

Some Poskim[25] rule that sixty breaths is the equivalent of more than three hours. Others[26] rule it is slightly more than three minutes. Others[27] rule it is less than a minute. Others[28] rule it is slightly more than thirty minutes. Practically if one slept for over a half hour he is to wash his hands with a blessing upon awakening [past midnight].[29] If one slept for less than a half hour he is to wash hands without a blessing.[30]


[24]Shaareiy Teshuvah 4/10; Ketzos Hashulchan 2 footnote 1; Kaf Hachaim 4/55

Opinion of Admur: From Admur in the Siddur it is evident that 60 breaths is a short amount of time being that he defines a nap [temporary sleep] as being more than 60 breaths.

[25]Tiferes Tzevi [brought in Shaareiy Teshuvah ibid] based on that the Arizal slept 3 hours during the day despite being careful not to sleep for 60 breaths.

[26]Rameh Mepuna [Alfasi Zuta Perek Hayashein]

[27]Chayeh Adam 7/8 writes it is equivalent to the amount of time it takes to walk 100 Amos. This is approximately 1/67 of an hour. [Ketzos Hashulchan ibid]

[28]Machazik Bracha in Kuntrus Achron brought in Shaareiy Teshuvah ibid; As the 60 breaths refer to the 60 breaths of a horse which takes approximately 30 minutes.

[29]So concludes Ketzos Hashulchan ibid; Kitzur SH”A 2/8; As this is the average opinion brought in the Shaareiy Teshuvah.

[30]As it is proper to be stringent like all opinions. [Ketzos Hashulchan ibid]

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