Where is Ein Eitam?

  The Gemara says (Yoma 31a) that the mikveh on top of Sha'ar Hamayim that the Kohen Gadol used on Yom Kippur was from a spring called Ein Eitam, which was 23 amos higher than the Azarah.
     The Gemara in Zevachim (54b) says:
ה " Shmuel and David said: It is written: “Then you shall arise, and you shall go up to the place which the Hashem shall choose” (Devarim 17:8). This teaches that the Beis Hamikdash is higher than all of Eretz Yisroel... They did not know where the highest place in Eretz Yisroel was.
They brought the book of Yehoshua. With regard to all of the borders of the tribes it is written: “And went down” (see Yehoshua 15:10, 16:3, 17:9), and it is written: “And the border went up" (Yehoshua 15:6), and: “And the border was drawn from the top of the mountain to the fountain of the waters of Naftoach” (Yehoshua 15:9). The verse uses different terms to describe the borders of the portion of each shevet. And with regard to the shevet of Benyamin it is written only: “And went up” (Yehoshua 18:11), but it is not written: “And went down.” They said: Conclude from here that here is its place.  ​

 They thought to build it at Ein Eitam, which is higher than any other place in the portion of Benyamin. They said: Let us lower it a bit, as it is written: "and He dwells between his shoulders” (Devarim 33:12). This indicates that the Beis Hamikdash is slightly lower.
And if you wish, say instead that it is learned as a tradition that the Sanhedrin is to be located in the portion of Yehuda, and the Shechina is to be located in the portion of Benyamin. And if we raise it and place it near Ein Eitam it will be too distant from the portion of Yehuda. It is preferable that we lower it a bit, as it is written: “and He dwells between his shoulders” (Devarim 33:12)." 
    Rashi there writes, that Ein Eitam is Mayan Mey Naftoach, which is the highest point mentioned in Yehoshua. (Click here for the full sugya with Rashi and Maharsha.)
     
    The researchers nowadays say that Ein Eitam is the springs by Solomon's pools, to the south of  Beis Lechem, from which water goes to Har Habayis, because near there is a spring called Eitan. But this cannot be, because that is in middle of Yehuda's part of Eretz Yisroel, and the Gemara clearly says it is by Binyamin's part! (like Rabbi Dov Levanoni points out in his book The Temple In Jerusalem, page 166-167.)  And also, Rashi on the Gemara, as well as the meforshim in Yehoshua (Metzudas David 18:16, and Radak 15:9), say it is Mey Naftoach, on the border! The researchers say Mey Naftoach is by Lifta, but that cannot be, because it is lower than Har Habayis is! So, we need to figure out where Ein Eitam-Mey Naftoach really is. 
     The Tavnis Heichal (2:7:48 in the Hagaha) writes, from Josephus: This spring flowed from an orchard that King Shlomo had, called Eitam, which was four thousand five hundred amos from Yerushalayim. The researchers say this was by Solomon's pools, but there is no proof from the text that that is where it actually was. So, based on this Tavnis Heichal, we should measure around 4,500 amos from Yerushalayim, and that is where Ein Eitam-Mey Naftoach is. In Sefer Yehoshua it says that Mey Naftoach is to the west of Yerushalayim, so we need to measure in that direction.   
    I measured around 4,500 amos to the west from Migdal David, which was by the western wall of Yerushalayim. I then looked around to see where is the highest point in that area. I noticed that the ground slopes slowly up, reaching its highest point, 830 meters above sea level, around one thousand amos more to the northeast, (right near Allenby square, where Hamon street meets HaAdrichal street)   so that is probably where Ein Eitam is. Although this is a thousand amos longer than what the Tavnis Heichal writes, his number could be not so exact, and also, it could be that Shlomo's orchard started around 4,500 amos away, but Ein Eitam was at the other end of the orchard.       

general map showing the location of Ein Eitam and the Beis Hamikdash

Map of the location of Ein Eitam

   However, there is a big problem, because that is around two hundred amos higher than Har Habayis is now, not twenty three! However, there is anyways a big problem, because Har Habayis is not even the second to highest mountain in the area, Har Hazeisim is higher, Har Tziyon is higher, etc. In a future post, I will Im"h attempt to answer this question. ​