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The degree of contact required between the mikveh and the other body of water varies, as reflected in the following halachot.

The size of this measure is mentioned in Halachah 6. Since these cavities are outside the mikveh per se, it is necessary that they share a significant point of contact. In his Commentary to the Mishnah (Mikveot 5:6), the Rambam states that even if the height of the water that connects the two bodies of water is as thin as a garlic peel, the two are connected. To bring that statement in accord with the ruling here, we must say that the width must be the size of the mouthpiece of a drinking pouch, but there is no minimum measure for the height of the water, as stated in Halachah 5 (Beit Yosef, Yoreh De’ah 201; note the Rama, Yoreh De’ah 201:52, who rules more stringently).

I.e., they may used when immersing small utensils (ibid.), for the entire body of the object being immersed must be immersed at the same time (Siftei Cohen 201:127).

Although Rambam uses the term ma’arah which literally means “cave,” he is merely borrowing the wording of the Mishnah (Mikveot 6:1). As he states in his Commentary to the Mishnah, the intent is a mikveh in a cave. The term “cave” is used because most of the mikveot of the Talmudic era were frequently in caves (Rav Ovadiah of Bartenura)

Since they are within the pool of mikveh itself, greater leniency is shown.

I.e., it is considered to have been immersed in the mikveh, because the water of the mikveh flowed over it. When quoting this law, the Rama, Yoreh De’ah 201:57, emphasizes that one must be careful not to uproot the wave from the body of the mikveh. If one does so, the immersion is unacceptable. See Chapter 9, Halachot 17-18.

Our translation is based on the Rambam’s Commentary to the Mishnah (loc. cit.).

I.e., vertically. This represents the Rambam’s interpretation of term shti, the woof, used by the mishnah (ibid. 6:9). Here, even a crack of a very small size is sufficient to cause the mikveot to be considered as joined. (It must be emphasized that Rabbenu Asher and others define shti as the warp, i.e., a horizontal line.)

This represents the Rambam’s interpretation of term airev, the warp, used by the mishnah (ibid.).

The larger measure is necessary in this instance because since the opening is horizontal, it is more likely to become filled with debris or the like (Rambam LeAm). Alternatively, when the wall is split vertically, it is as if it has been destroyed entirely and there is no divider between the two mikvaos. When, however, it is split horizontally, the bottom of the wall is still a viable divider. Hence it is necessary that there be a point of contact with a hole of the required measure to connect them (Turei Zahav 201:64).
The Siftei Cohen 201:118 writes that since there is a difference of opinion among the Rabbis whether it is a vertical or a horizontal split that requires a hole the size of the mouthpiece of a drinking pouch, we rule stringently and do not consider the mikveh acceptable unless there is such a hole regardless of whether it is cracked vertically or horizontally.

Since the water covers the wall and faces the open space, the mikveot are considered as joined even though the height of the water is minimal (Turei Zahav 201:65).

Between 4 and 5.6 centimeters in contemporary measure.

Which is usually the intent when the terms “finger” or “fingerbreadth” are mentioned in the Talmud.

The obstructing entity must be removed or the hole widened for the mikveh to be acceptable.

E. g., algae and living beings which come into existence from the water [the Rambam’s Commentary to the Mishnah (Mikveot 6:7)]. Although that mishnah mentions an opinion that maintains that anything created from water is considered as water, that view is not accepted as halachah. See also Halachah 11.

The bracketed addition is taken from the Rambam’s Commentary to the Mishnah (Mikveot, op. cit.).

As stated in Chapter 1, Halachah 2.

As Eruvin 4a states, the required measures for the fulfillment of the mitzvot were not written in the Torah, but communicated to Moses at Sinai as part of the Oral Tradition.

Here the term “purified” means made acceptable for use. As the Rambam proceeds to explain, when there is one acceptable mikveh and one pool filled with drawn water, the pool filled with drawn water can become an acceptable mikveh by being connected to the acceptable water. Indeed, this is the most common method used for mikveot today. There is a pool of rainwater and next to it, a pool filled with tap water. There is a hole in the connecting wall that can be plugged. After the pool of tap water is filled, the plug is opened, connecting the mikveot and causing the pool of tap water to be considered as an acceptable mikveh.
In the instance described in this halachah, this concept is extended and the Rambam illustrates how the connection between the pools can be made even when the pools are far from each other.

These are just examples. A pipe does not disqualify a mikveh regardless of the substance from which it is made [the Rambam’s Commentary to the Mishnah (Mikveot 6:8)]. See Chapter 6, Halachah 2 above.

I.e., a pipe is not a k’li, a container [the Rambam’s Commentary to the Mishnah (Mikvaos 6:8)]. Hence, the fact that water passes through it does not cause the water to be deemed as “drawn” and unacceptable for a mikveh.

I.e., plugs the outer end of the pipe with his hand so that the water will not flow out.

I.e., as long as the two pools are even slightly in contact, the water from the kosher mikveh makes the mikveh of drawn water acceptable. There is no need for the point of contact to be the size of the mouthpiece of a drinking pouch. Such contact is required only when making a mikveh that contains less than 40 se’ah acceptable. If a pool contains 40 se’ah of water, but is unacceptable for immersion because the water was drawn, even the slightest contact is sufficient to render it acceptable. The rationale is that, according to Scriptural Law, a pool of drawn water is acceptable for immersion (Chapter 4, Halachah 1). It is invalid only because of Rabbinic decree. Hence the Sages were lenient regarding the measure of the point of contact (Rabbenu Shimshon, as quoted by Rav Yosef Caro in his Kessef Mishneh).
Rav Yosef Caro also rules in this manner in his Shulchan Aruch, Yoreh De’ah 201:53. Nevertheless, the Tur and the Rama require that the point of contact be at least the size of the mouthpiece of a drinking pouch in such an instance as well. This is the practice followed throughout the Jewish community at present. See note 2 above.

Where water collects.

Unlike a natural stream, rainwater is acceptable for immersion only when it is collected in a reservoir and not when it is flowing.

For it alone contains the minimum measure for a mikveh.

Since the rainwater is not acceptable for immersion when flowing, it also cannot connect two pools together. Were the current of natural water to come from a natural stream, the mikveot would be considered as joined (Siftei Cohen 201:129).

I.e., when calculating whether it contains the minimum measure of 40 se’ah. Note a similar ruling in Hilchot Bi’at HaMikdash 5:12.

Even if the feet of the person or the article enter the mud, the immersion is considered acceptable. The rationale is that they came in contact with the water first and since the mud is soft, the water from the mud is still considered as in contact with the water of the mikveh.

Even though the mud is considered as part of the 40 se’ah of the mikveh (Shulchan Aruch, Yoreh De’ah 201:32).

I.e., can be included in the measure of 40 se'ah necessary to comprise a mikveh. Significantly, the Shulchan Aruch, Orach Chayim 160:10, states that such worms are also considered as water with regard to washing one's hands. There, however, the Rama rules that they must be crushed before they are considered acceptable.

I.e., if the eye of a giant fish decomposes, it can be considered as a pool for immersion (Kessef Mishneh).
In his Beit Yosef, Yoreh De’ah 201, Rav Yosef Caro notes that this ruling and one in Halachah 6 are seemingly contradictory, for they both have their source in the teachings of Rabbi Shimon ben Gamliel (see Zevachim 22a) and seemingly, depend on the same principle: that an entity that came into being from water is still considered as water. Nevertheless, Rav Yosef Caro finds a way to reconcile both rulings. Therefore, in his Kessef Mishneh, he does not question the Rambam’s acceptance of the two seemingly opposite positions, and in his Shulchan Aruch, Yoreh De’ah 201:33 and 52, he rules like the Rambam does. See, however, the gloss of Tosafot Yom Tov (Mikveot 6:7) which questions how both rulings can coexist.
See the Magen Avraham 162:17 who explains that it is possible to immerse in fluid coming from the living beings of the water only when the beings themselves are crushed or decomposed. When their form is intact, they are considered significant. (He notes that this is somewhat problematic, because snow is acceptable for immersion even when it has not melted.)

I.e., his immersion is considered as invalid.

A certain amount of water remained on the body and hair of the first person who immersed. Thus there will not be 40 se’ah in the mikveh when the second person immerses [the Rambam’s Commentary to the Mishnah (Mikveot 7:6)].

Thus it could be said that the water on his body was not entirely removed from the mikveh.

I.e., an object that will certainly absorb water.

I.e., one that contained only 40 se’ah and thus the absorption of the water by the sponge will present a problem.

Without removing it, even briefly from the mikveh. If he would remove it, the water in the sponge could no longer be considered as part of the mikveh and, indeed, could disqualify it as drawn water does.

The Kessef Mishneh quotes the Rivosh who explains that this clause differs from the preceding one, because in the preceding instance, the water on a person's body is on the surface, while the water in a sponge is absorbed within it. The Kessef Mishneh himself does not accept that interpretation; he explains that the water in the sponge is all connected and is connected with the water in the mikveh. The water on the person's body, by contrast, is not all connected, because certain portions of his body may be dry.

I.e., long articles which cannot easily fit in the mikveh at the same time.

Unlike loose mud (see Halachah 10), thick mud is generally not considered as part of a mikveh.

In this instance, since the feet of the bed will be covered with water before they enter the mud, they are still considered as within the mikveh.

For taking the pot out of the mikveh with the water inside would cause that water to be considered as “drawn.” Even when it is taken out of the mikveh upside-down, there will still be some water that drips into the mikveh, but it will be far less than three lugim and will not disqualify the mikveh.

When quoting this law, the Shulchan Aruch, Yoreh De’ah 201:66 says that one may also use stones, but that it is forbidden to employ containers for this purpose. The Turei Zahav 201:80 explains the rationale: the mikveh should not be formed by objects that are susceptible to impurity. The Siftei Cohen 201:140 gives a different rationale: the space in the bundles of straw and reeds is larger than the size of the mouthpiece of a drinking pouch and thus the water is always connected.

Here and in Hilchot Tum’at Ochalin 15:1, the Rambam speaks of drawn water. In his Commentary to the Mishnah (Mikveot 1:1), however, he states that this applies only to natural water.

Hilchot Tum’at Ochalin, op. cit. Note the objection of the Ra’avad and the resolution offered by the Kessef Mishneh there.

Or to wash utensils used for challah [the Rambam’s Commentary to the Mishnah (op. cit.:5).

Implied by the Rambam’s words is that immersion in such water is unacceptable. The Ra’avad states that if the bodies of water do not contain drawn water, one may immerse his hands even if there are less than 40 se’ah of water, provided the water is not drawn. In his Kessef Mishneh and his Beit Yosef, Orach Chayim 159, Rabbi Yosef Caro states that the immersion of hands is acceptable only in a mikveh containing 40 se’ah of water that was not drawn or in a spring.

Hilchot Berachot 6:5.

In this, it possesses an advantage over the water in the first category.

Impure water can be purified by being brought into contact with a larger quantity of pure water. This concept — that hashakah is effective when the parent body of water is less than 40 se’ah — is a new idea (Ma’aseh Rokeach).

As described in the previous halachah.

The Kessef Mishneh explains that since the water from a sea and a river are not considered as “living water,” they are not in the same category as a spring. Hence, the laws applying to them differ from those applying to a spring as obvious from the following halachah.

This is also speaking about an instance when there are less than 40 se’ah in the pit that was dug.

And is acceptable for immersion even if it does not contain 40 se’ah, as stated in Halachah 8.

Who must immerse in “living water;” see Halachah 9, and Chapter 1, Halachah 5.

See Chapter 11.

Hilchot Sha’ar Avot HaTum’ah 8:7; 12:12.

This applies even if there is more drawn water than the water originally present in the spring (Shulchan Aruch, Yoreh De’ah 201:15). See also Siftei Cohen 201:42. See Halachah 11 where this law is illustrated. The Rama (Yoreh De’ah 201:9) mentions more stringent views which he recommends to follow as an initial preference. Nevertheless, he also fundamentally accepts the Rambam’s view.

Unlike a spring whose water imparts purity even when flowing.

In contrast to a mikveh whose water does not impart purity unless there are 40 se’ah.

From the Rambam’s wording [here, Halachah 8, and his Commentary to the Mishnah (Mikveot 1:7)], it can be inferred that a natural spring can impart purity to a human being even when it contains less than forty se’ah (Kessef Mishneh). Now, a man of ordinary size cannot immerse in less than forty se’ah of water. Nevertheless, a woman or a child could immerse in a smaller amount. There are other authorities who do not accept this principle. Their view is accepted by the Tur and the Shulchan Aruch, Yoreh De’ah 201:1. Nevertheless, all authorities agree that implements can be immersed in a spring even though it contains less than forty se’ah.

Since this water is not fit to drink, the spring is not categorized as “living water.”

When speaking of similar springs in Hilchot Parah 6:12, the Rambam also mentions “hot springs.”

The term used by the Rambam (and his source, Mikveot 1:8, et al.) literally means “crawling.” See Deuteronomy 32:24 which speaks of animals “crawling on the earth.”

I.e., the water sprinkled on a person afflicted with tzara’at must be living water. See Hilchot Tum’at Tzara’at 11:1.

See Hilchot Parah 6:1.

Needless to say, the entire body of the person or the entire article being immersed must be submerged in the water of the spring at the same time (Kessef Mishneh).

The Rambam’s wording implies that even if water is flowing out of the upper portion of a mikveh, if one immerses in the water that is collected, he or she is pure. See Shulchan Aruch, Yoreh De’ah 201:50.

This refers to a receptacle that is considered as a k'li, an independent container. If it was hewn into the stone itself, it does not disqualify the water as stated in Chapter 6, Halachah 3.

For water contained in a k’li or that had been contained in a k’li is considered as drawn. Even though the water is in contact with the natural spring, as long as it is in the k’li, it is unacceptable.

And thus did not enter the receptacle and become disqualified.

The Siftei Cohen 201:33 concludes from the words of the Rama that he would postulate that the water is acceptable for immersion only if it is collected in a reservoir. While it is flowing, it is unacceptable. Nevertheless, the Siftei Cohen himself maintains that it is acceptable for immersion even when flowing.

I.e., makes water fit for immersion.

In contrast to a mikveh where, in certain situations, it is necessary that the point of contact be the size of the mouthpiece of a drinking pouch.

Whether drawn water or water that is acceptable for a mikveh.

Because contact with the natural spring made its water fit for immersion, as in the previous clause.

For these require “living water.”

From the Rambam’s wording, it can be inferred that one must calculate how much water entered the pool and how much of that water flowed out with the water that was originally there. See Chapter 5, Halachah 6, and the commentaries to Shulchan Aruch, Yoreh De’ah 201:20.

In his Commentary to the Mishnah (Mikveot 5:2), the Rambam states that this law also applies when the water of the spring flows over the outer surface of a receptacle.

And it only purifies when collected in a reservoir and when there are 40 se’ah. Since it passes over the articles, its status declines.
The Turei Zahav 201:23 and the Siftei Cohen 201:35 clarify that this is speaking about a situation where all the water from the spring passes over the article and comes in contact with it. If some does not, it purifies it, as stated in the previous halachah. The Turei Zahav also explains the reason the water is disqualified even though it was never in a container and thus cannot be considered as “drawn” in the simple sense: Since all of the water passes over the article, the article is considered as having intervened between the spring and the water beyond the article. Thus that water is no longer considered as in contact with the spring that is its source.

Previously, it was stated that one may immerse in a chest or cabinet submerged in water. It is possible to distinguish between that instance and the law here as follows. There we are speaking about a container totally submerged in a body of water. Here, by contrast, we are speaking about a situation where the implement separates between the spring and the ground.
Therefore, the situation is comparable to immersing in a wave, as mentioned in Halachah 18.

There was not a large amount of water flowing from the spring and it did not have the power to flow forcefully into the ditches until other water was added to it.

Since the drawn water comes in contact with the water of the spring, it is considered like it, as stated in Halachah 6 [the Rambam’s Commentary to the Mishnah (Mikveot 5:3)].
The Ra’avad differs with the Rambam and maintains that the water of the spring does not purify entities immersed in it when flowing in those areas where its flow would not have reached had the water not been added. The Shulchan Aruch, Yoreh De’ah 201:2 cites the Rambam’s ruling, but the Siftei Cohen 201:11 states that most authorities follow the Ra’avad’s view.

I.e., in the previous clause, the water was not flowing powerfully, but it was flowing. In this clause, it was not flowing at all.

See Halachah 6 which describes this situation in the abstract.

This applies only when the majority of the water comes from the spring. If the majority is drawn water, it is unacceptable unless there are 40 se'ah collected in the reservoir. See Halachah 13.

See also Halachah 17.

For the water of a sea is not considered “living water.”
The commentaries note that in Hilchot Parah 6:10 (based on the Mishnah, Parah 8:8), the Rambam writes:
The Great Sea is considered like a mikveh and not like a spring. Therefore water to be sanctified may not be drawn from it. [Water from] all of the rivers is unacceptable to be sanctified to [serve as] the water for the [ashes of] the red heifer. Other seas or lakes are like springs.
The Kessef Mishneh resolves the contradiction in the Rambam’s wording as follows:
The Great Sea (the Mediterranean Sea or the ocean) is never considered like a spring with regard to these three purposes. Other seas and lakes are considered as natural springs and may be used for the three purposes mentioned above. Nevertheless, unlike a spring, they may be used for these purposes only when the water in those bodies is stationary; not when it is flowing.
(In his gloss to Halachah 18, the Kessef Mishneh explains that even the Mediterranean is considered as a spring with regard to ordinary immersion and purifies a person who immerses in it even when the water is flowing. See also Halachah 7, which states that a spring of salty water is not considered as “living water.” Seemingly, this would apply to an entire body of salt water.)

The bracketed addition is based on the Rambam’s Commentary to the Mishnah (Mikveot 5:5) which states that this refers to rivers and streams that flow from natural springs.

It imparts purity while flowing even when there are less than 40 se’ah and it is acceptable for situations that require “living water.”

As would happen when there were heavy rains and even more so in areas where there was snow which melted and flowed into rivers.

This law is derived from the conduct of the father of the Sage, Shmuel, who would make enclosures of mats to enable his daughters to immerse in the fall when the rain would flow into the rivers to the extent that there was more rainwater than water from the river’s natural source (Nidah 65b). Significantly, in Chapter 1, Halachah 11, the Rambam learnt a different concept from that passage.
The Rambam’s ruling is quoted as halachah by the Shulchan Aruch, Yoreh De’ah 201:2. The Rama rules that it is proper to be stringent and follow this ruling. Nevertheless, he does find room for leniency if one immerses in a river even in the rainy season, since he maintains that it is unlikely that the amount of rainwater will be more than the water that flows from the river’s source.

Which as stated in Hilchot Keilim 1:8 is not susceptible to ritual impurity.

Although the Rambam uses the term mikveh here, his source (Mikveot 5:5) speaks of water that is dripping from a natural spring.

For a mikveh should be brought into being only by articles that are not susceptible to ritual impurity (ibid.).

Our translation is based on the Rambam’s Commentary to the Mishnah (Eduyot 7:4).

The person placed the nut shell there so that the water will flow onto the nut shell and from there to a different place (ibid.).

I.e., were the nut shell to be considered as a container, the water would be considered as “drawn.” Nevertheless, since the nut shell imparts its color to the objects with which it comes in contact, it is not fit to be used as a container and, hence, does not disqualify the water.

Because it is considered as if a reservoir has formed.
The commentaries question this ruling, because it appears that even though the containers are susceptible to ritual impurity, it is possible to construct a mikveh from them and several halachot, including the previous one, appear to run contrary to that conclusion. Indeed, on this basis, Rav Ovadiah of Bartenura explains that a similar statement in Mikveot 5:6, the Rambam’s source, follows the perspective of the School of Shammai and is not accepted as halachah.
The Rambam, however, does not consider the mikveh to have been formed by an entity that is susceptible to ritual impurity. The rationale is that, in this instance, the containers that are susceptible to ritual impurity are not bringing the water to the mikveh. They are merely preventing it from flowing away. See Halachah 13 above.

Because they were never totally submerged in the water. Only their outer edge served as a wall for it (the gloss of Rav Ovadiah of Bartenura to Mikveot, op. cit.).

But not for the second tithe, terumah, or sacrificial foods. Such an immersion is acceptable for a woman who seeks to emerge from the nidah state.

Even though the wave is “flowing,” it still imparts purity, because, as stated in Halachah 12, seas are considered as natural springs and impart purity even when flowing. The question is asked: If seas are like natural springs, why are 40 se’ah required? A natural spring imparts purity regardless of the amount of water it contains.
Some commentaries (Turei Zahav 201:6) explain that 40 se’ah are required only for a person, because a smaller quantity of water will not cover a person’s body entirely. Others explain (Siftei Cohen 201:20) that since the wave has flowed out of the basin of the sea proper, greater stringency is required.

Hilchot Sha’ar Avot HaTum’ah 13:2; see also Chapter 1, Halachah 8, of the present halachot.

To partake of the second tithe, terumah, or sacrificial foods, or to take part in the purification process involving the ashes of the red heifer.

The Rambam is employing a kal vechomer, a fortiori reasoning. If water flowing on the earth is unacceptable, certainly water in the air is unacceptable. The commentaries have, however, raised questions regarding the inference, because it is acceptable to immerse in the flowing water of a sea (see Halachah 12 and the previous halachah).
In his Kessef Mishneh, Rav Yosef Caro states that, based on the interpretation offered in the notes there, that one may immerse in the flowing waters of other seas, but not in the flowing water of the Mediterranean Sea (or the ocean), it is possible to say that this halachah is speaking about the Mediterranean Sea. Nevertheless, if that was the Rambam's intent, he should have stated so explicitly. Furthermore, from the concluding clause in the halachah, it appears that water that is flowing is acceptable for immersion. Therefore he explains that the Rambam is speaking in general terms. Just like a spring which drips is unacceptable, so too, a wave in the air is unacceptable.
The Leshad HaShemen offers a different explanation: Since the wave was "separated" from the sea, the leniency of purifying while flowing does not apply and it is only when it is at rest that one may immerse in it.
(Most commentaries accept Rashi’s explanation (Chagigah 19a) that since purity is attained through immersion in a mikveh or a natural spring, any water used for immersion must resemble them and, like them, be in contact with the earth.)

Our translation follows the authoritative manuscripts and early printings of the Mishneh Torah. The standard published text contains a slightly different version.

See Hilchot Sha’ar Avot HaTum’ah 14:1.

As explained above, if three lugim of drawn water fall into a mikveh containing less than 40 se’ah of acceptable water, they disqualify it. If, however, there are less than three lugim or more than 40 se’ah of acceptable water, the mikveh remains acceptable. As a preface to the entire chapter which deals with questions of uncertainty, the Rambam clarifies that as long as one is not certain that the mikveh has been disqualified, it is considered acceptable.
According to the Rambam, the reason for the leniency is that the concept that drawn water disqualifies a mikveh is a Rabbinic decree and we follow the general principle: “Whenever there is a doubt regarding a question of Rabbinic Law, we rule leniently” (see Turei Zahav 201:81).

We assume that the drawn water fell into the mikveh that contained the required measure and thus would not be disqualified by the drawn water. Hence the other mikveh is also acceptable. Although there is no empirical basis for this assumption, since the doubt about the acceptability of the mikveh involves a question of Rabbinic Law and there is a possibility that the water fell into the acceptable measure, we follow this approach. Note parallels in Hilchot Terumot 13:14.

Hence, since there is no way of determining which mikveh the water fell into, they are both disqualified, for there is no reason to favor one over the other and one is certainly invalid (Siftei Cohen 201:142).

Needless to say, this law also applies when the mikveh contained less than 40 se’ah of acceptable water. See the conclusion of the following halachah.

The Kessef Mishneh maintains that the rationale for accepting the mikveh here is different than that in Halachah 1. The concept stated there — that when there is a doubt whether water is drawn or not, we rule leniently — applies only in an instance when there is already an existing mikveh. If the mikveh was empty, it is considered acceptable because of a different rationale: that when there is no factor that indicates that it is likely to have been filled in an unacceptable manner, we operate under the presumption that any mikveh that is discovered is acceptable (Tosefta, Mikveot 2:1). The Shulchan Aruch, Yoreh De’ah 201:69 accepts the Rambam’s ruling. The Rama, however, maintains that a mikveh of drawn water is unacceptable according to Scriptural Law. Therefore, he maintains that the present situation is comparable to a doubt arising with regard to a point of Scriptural Law, in which instance, a stringent ruling is called for.

Which is acceptable, because a pipe is not a container and it does not disqualify the water. See Chapter 6, Halachah 2.

I.e., a mill turned by water. The mill is considered as a k’li and disqualifies the water that passes through it.

In the previous halachah, there was a possibility that the mikveh had been filled by drawn water, but there is no evidence that this was what had happened. Here, when the disqualifying factor is next to the mikveh and there is a likelihood that the water came from it, the Sages were not willing to allow the leniency. The Siftei Cohen 201:144 explains that this is speaking about an instance when it is more likely that the water came from the mill, e.g., the mill is closer to the mikveh. Hence, that possibility is given more weight.
The Rabbis have questioned this ruling, because although there is a possibility that the water came from the mill, there is no evidence of such. It is just as possible that it came from the pipe. Hence, since the matter is one of Rabbinic Law, there is room for leniency. The Turei Zahav 201:83 explains that the rationale mentioned by the Kessel Mishneh, that we operate under the presumption that the mikveh is acceptable does not apply in this instance, since a factor that could disqualify the mikveh is present.

I.e., more than 20 se’ah.

This law applies even according to the authorities who maintain that a mikveh filled with drawn water is unacceptable according to Scriptural Law. In this instance, since there already is the majority of an acceptable mikveh, according to Scriptural Law, the drawn water is considered as insignificant. Hence, when it is not certain that drawn water was in fact added to the mikveh, we rule leniently.

In the Talmudic era — and in the Rambam’s time as well — the term mikvaot did not refer to a special ritual bath, as it does today. Instead, it referred to any pool or collected body of water.

I.e., if the status of a pool of water is unknown, we rule stringently. If, however, a mikveh is known to be acceptable — as the mikveot, ritual baths, used today — there is no problem in relying upon it [see the Rambam’s Commentary to the Mishnah (Mikveot 8:1)].

And may be used by a woman to purify herself from nidah impurity. There is no need to verify the status of the mikveh (ibid.).
The Kessel Mishneh states that this ruling applied only in the Talmudic era when Eretz Yisrael was inhabited primarily by Torah observant Jews. In eras when that is not the case, there is no difference in this regard between Eretz Yisrael and the Diaspora.

For in previous eras, it is unlikely to have pools of water in outlying areas that did not come from rain water or a natural stream. Indeed, the Siftei Cohen 201:150 states that even in the Diaspora, if it is apparent that a pool came about through natural means, it is acceptable for immersion.

I.e., he immersed, but he was unsure if his entire body was immersed as required.

The Turei Zahav 201:85 suggests that this law applies even when it is known that the mikveh once contained the required measure.

Since the person is known to be impure, he cannot change his status unless he immerses in a manner that leaves no doubt that he is pure.

With regard to contracting ritual impurity, there is a difference between the public domain and a private domain, as explained in Hilchot Sha’ar Avot HaTum’ah 14:1, 16:1-2. Nevertheless, with regard to purification, no such distinction is made.

The rationale is the same as in the first clause. Since the articles were known to be impure, their status does not change until it is known with certainty that they were immersed in an acceptable mikveh. In this instance, since the mikveh was discovered to be unacceptable, the status of all the articles immersed in it from the last time that it was measured is thus deemed questionable.

The intent is a person who contracted impurity through contact with a source of impurity. This includes even a source of impurity of Rabbinic origin (Ra’avad, Kessef Mishneh, based on Taharot 4:7, 11).

See Hilchot Sha’ar Avot HaTum’ah 8:10.

Ibid. 9:1.

For this is considered an ordinary instance of a doubt regarding a Rabbinic injunction.

The Merkevet HaMishneh states that this — like the final clause of the halachah — is speaking about a mikveh that is unacceptable because its water was drawn. Others maintain that this is talking about a mikveh that does not contain 40 se’ah.

Either terumah or sacrificial foods.

Since it is possible that his immersion was unacceptable, his status might not have changed and thus he would impart impurity to the articles that he touched. On the other hand, since he is not definitely impure, the status of the articles is questionable. They are not considered as impure and consigned to being burnt (because it is forbidden to burn terumah or sacrificial foods that are acceptable), nor are they considered pure and eaten. Instead, they are left until they definitely contract impurity or are disqualified for other reasons.

Since he immersed in both mikvaot, he obviously purified himself. Moreover, he is not considered impure, because he immersed his body in drawn water. Since the impurity of drawn water is merely a Rabbinic stringency and it is not known with certainty that he immersed in it after purifying himself, it is assumed that he immersed in the acceptable mikveh last.

The same ruling that applied to them originally still applies. Even if the person immersed in the acceptable mikveh originally and then in the mikveh filled with drawn water, the foods that he touched originally do not become impure because of the contact with the second group of foods. Although the second group becomes impure, their impurity does not impart impurity to other foods, because it is only a Rabbinic safeguard.

They are definitely impure. If the first immersion was in the acceptable mikveh, the second immersion was in drawn water and the person became impure. He then imparts impurity to the foods he touches. If the first immersion was in the invalid mikveh, the articles he touched afterwards also become impure. Moreover, since their impurity stems from contact with a severe type of impurity, they also impart impurity to other foods with which they come in contact. Therefore they impart impurity to the second batch of foods.

In Halachah 6.

I.e., the fact that he contracted a light impurity and immersed does not change any of the logic mentioned in note 30.

For the reasons explained in Halachah 6.

Because as in the first situation mentioned in this halachah, it is not known whether or not the person’s immersion was acceptable.

This follows the same rationale as the ruling in the first clause. The first batch of foods should be burnt because if the first immersion was acceptable, the second was not and the foods touched afterwards contracted impurity and imparted that impurity to the first batch. And if the first immersion was unacceptable, the person became impure due to the immersion in drawn water and he imparts that impurity to the foods he touches.

For the reasons stated in note 27.

Because it is a questionable situation regarding a Rabbinic safeguard, as stated in Halachah 6.

In the first clause above.

Which is merely a Rabbinic safeguard. Hence, when a doubt arises in such a situation, we rule leniently.

As explained with regard to the last clause of the previous halachah, this is a situation where a doubt has arisen regarding whether a person entered drawn water or not. Hence he is considered as pure.

Since he entered both pools. Since the acceptable pool did not contain 40 se'ah, there is no possibility that he was purified after entering the pool of drawn water.