1) SUMMARY: HOW TO TAKE INTO ACCOUNT A VALLEY WHEN MEASURING THE SHABBOS TECHUM
The Mishnah (57b) presents the basic guidelines for measuring the 2000-Amah Techum of Shabbos. The method for measuring level ground is straightforward. How, though, does one measure the Techum over hills and through valleys?
There are four different ways to measure the 2000-Amah Techum:
(a) "Medidah Yafah," straight measurement, is used to measure level ground, flat land at the bottom of an extremely steep cliff, and slight inclines (where the slope does not rise 10 Tefachim before a distance of 5 Amos).
(b) If the hill is steeper than that, then the methods of "Mavli'a v'Koder" are applied. How steep the slope must be to apply these more lenient ways of measuring depends on the two versions of Rava's statement in the Gemara.
(c) If the slope rises to a height of 10 Tefachim before a distance of 4 Amos, then the area atop the slope is measured by "Omed" -- by estimating the distance, without including the rise in the measurement.
(d) If it is a steep cliff that does not protrude 4 Amos on bottom no matter how deep or high it is, then the method of "Medidah Yafah" applies, and one measures only the flat part on the bottom and not the slope.
The ROSH says that the leniency of estimation ((c) above) applies only to a rise (or to a wall) that is difficult to climb. If it is a drop, though, such as a valley, which is not more difficult to measure (when less than 50 Amos) than any other valley or level ground, then the leniency of estimation does not apply, and the valley must be measured with the method of being "Mavli'a" with a 50-Amah rope.
In addition, the Rosh says that the method of "Koder" ((b) above), which applies when the distance from one side of the valley to the other is greater than 50 Amos, is used not only for a valley which is very steep (for which we do not permit estimating), but even for a hill or a rise which is very steep (and we do not permit estimating). The only time that the Chachamim permit estimating is when there is a rise that is less than 50 Amos, and it is too steep to be "Mavli'a." "Koder" applies whether it is a height that one is climbing or a depth.