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One cannot use drawn water to fill a mikveh. The Mishnah (2:7) records a debate whether one can use rainwater that collected in jugs that were left on the roof to dry out. Since one did not intend to collect the water, it would seem, that that which collected is not considered drawn water. Everyone agrees in the Mishnah that if one picked the jugs up to pour them into the mikveh, it would be invalid as they would then be considered drawn water. Smashing the jugs could possibly be one solution. R' Eliezer however argues that if the jugs were placed on the roof when rain clouds had already formed, then smashing the jugs would only work, if a majority of valid rainwater had already collected in the mikveh. R' Yehoshua however argues that having rainwater in the mikveh first is unnecessary, and the jugs can either be smashed or tipped over for the water to run into the mikveh below.
The Rishonim explain the basis of the debate differently. The Bartenura understands that when breaking the vessel, the water that runs into the mikveh is valid based on the principle of hamshachah. To explain, drawn water can be used to fill a mikveh provided it is poured at a distance from the mikveh and flows into the mikveh. The Bartenura therefore understand that in this case the water that collected is considered drawn water. The Mishnah Achrona explains that when everyone agrees that if the jugs are picked up and poured into the mikveh that it is invalid, it must mean that it is poured directly into the mikveh and not by way of hamshacha.
Note however that the Rambam rules (Mikvaot 4:8) that hamshachah would only work as long as there is a majority of valid water in the mikvah. Explaining the Mishnah in this way, it appears that the Rambam rules like R' Eliezer who requires rainwater to already be in the mikveh. The Rambam (Mikvaot 4:4) however rules like R' Yehoshua in the Mishneh Torah. From a simple reading of R' Yehoshua it appears that one could fill the mikveh entirely with the water jugs if there was sufficient water. That is indeed how the Rambam explains R' Yehoshua in his commentary to the Mishnah. The Tosfot Yom Tov notes this difficulty in understanding the Rambam that it appears to contradict his ruling that hamshacha only works if the drawn water is minority of the water in the mikveh.
The Tosfot R' Akiva Eiger however does not understand the question from the outset. The limit of hamshacha, for there to be a majority of rainwater in the mikveh, is if one is doing hamshacha with drawn water. The Tosfot R'Akiva Eiger suggests that the Rambam understands that in this case, if the jugs are smashed or tipped over (but not picked up and poured out) the water is not considered drawn water at all. That being the case, the entire mikveh could be filled with that water.1
In a similar vein, the Mishnah Achrona explains that the Rambam understands the Mishnah like the Rosh. The Rosh understands that the Mishnah is not based on hamshacha. The Mishnah is addressing whether the water is considered drawn water or not (as suggested in the opening to this article). If it is smashed, then it is not and if it is picked up and poured out then it is.
If we explain the Mishnah like the Rosh, how do we understand R' Eliezer? If it was at time when there would possibly be rain, why does he require the mikveh to already have valid water inside it? The Mishnah Achrona explains that in that context the water appears to be intentional collected and therefore appears to be considered drawn water. R' Eliezer therefore rule stringently out of concern that someone may misunderstand the situation and thereby allow drawn water to be used to a fill a mikveh. The requirement of having valid water already in the mikveh in that situation makes it clear that the water on its own, if it was indeed drawn, is invalid.
1 See also the Rashash. R' Akiva Eiger however continues that the question is still relevant to the Bartenura who explains that R' Yehoshua maintains that hamshacha could be used to fill an entire mikveh with drawn water.
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