In my last post I suggested that the correct position for C397 (UMME 139) was at 37.085266° N, 22.055458° E
In the same paragraph from Simpson which I quoted earlier we read this:
"About 500 m. to SW and considerably below the site just described, we noted prehistoric sherds about 2.50 m. down from the surface in the sides of a sunken track that leads up in a cleft between two hills to the convent of the Panayia. The spot is only about 250 m. E of the outskirts of Thouria village. There is a new concrete cistern about 30 m. to E, with a sluggish spring slightly higher to NE. Among the most distinctive sherds are a flat ribbon handle of fine pink clay, a horizontal pinched handle of soft reddish brown fabric, and some plain rims and bases of darkish clay. Several large pieces of thick pithoi might indicate that this was a cemetery. The pottery seems to belong mainly to MH, but some may be LH.
The material collected on the habitation site is not closely datable but confirms its use in prehistoric times. It could have been associated with the "cemetery" below it."
Also in Simpson and Dickinson  this:
"... About 500 m. SW, some MH sherds and others which may be LH were noted about 2.5 m. down from the surface in the sides of a sunken track between two hills on the way to the Panayia convent. Some large pithos fragments suggest that the track may have cut through a prehistoric cemetery."
So. From the previously located site move 500 m. to SW (225°) and we should be in the vicinity of the CEM. Let's look at the map.
This is what it looks like when I apply all of Simpson's constraints. The large yellow circle is centered on C397 and has a radius of 500 m. The azimuth line of 225° (SW) is included for reference.
Near where the azimuth line meets the circle I noticed two things. The first is what appears to be a cistern and 30 m. distant to the left (W) is a road. Simpson et al. clearly state that their finds were at the side of a road in the (visible) strata and about 2.5 m. from the top.
Here I've zoomed in on that location. The circle is 30 m. in radius and it is centered on what I consider to be a cistern. Another meter or two and we're on the roadside. I propose this as the approximate location for C5407, our supposed CEM.
All of this fits precisely with Simpson's directions - except for one thing. Where is the convent of the Panayia and does it fit Simpson's text? This poses a problem because I can find no religious establishment with that specific name. The two best candidates might be 1) Ayios Vlasios at 37.079550° N, 22.055526° E and 2) another unnamed church at 37.082563 N, 22.058166 E. The two candidates can be seen in the next photo:
Ayios Vlasios is to the S and connected to C5407 by a driveable field trail. But it looks like it might be too small to be more than a simple country church. Here it is in close-up:
The second church complex looks like this:
In this photo I think we have two things. One is the church itself and the other seems nothing more than tin-roofed farm buildings. Truthfully these two church complexes both appear to be simple country chapels.
Nonetheless I slightly prefer the second church, the unnamed one, simply because it is reached through a path between hills. In the next photo I have exaggerated the vertical by a factor of 2 in order to bring this idea out.
This picture shows the entire area. Our proposed CEM (C5407) is at the lower left. Ayios Vlasios is at the lower right. The large complex right in the center looks to me like a private home. That leaves 'Church 2' at the upper right as the most likely candidate for the convent but - I am not sure. The path in red that I have labelled 'trail' is taken from Topoguide. It is here to emphasize that it is possible to walk from C5407 to 'Church 2' by means of a trail that goes between two hills.
If anyone has more information about these several structures then I would very much like to hear. For right now I propose 37.081622° N, 22.052162° E (C5407) as the most likely position for Simpson's MH/LH CEM.
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 Messenia III, p. 160, no 79B. 'Ayios Athanasios (Thouria)'.
 Simpson and Dickinson : 164.
McDonald and Rapp : McDonald, William A. and George R. Rapp, Jr., The Minnesota Messenia Expedition: Reconstructing a Bronze Age Regional Environment, University of Minnesota Press, Minneapolis, Minnesota. USA. 1972
Messenia III: McDonald, William A. and Richard Hope Simpson. 1969. 'Further Explorations in Southwestern Peloponnese: 1964-1968'. American Journal of Archaeology. (73:2), pp. 123-177.
Simpson, Richard Hope and O.T.P.K. Dickinson, A Gazetteer of Aegean Civilization in the Bronze Age, Vol. I: The Mainland and the Islands, Paul Åströms Förlag, Goteborg. 1979.