Before us is Mishnah 4 in the fifth chapter of Maseches Eruvin. The Mishnah deals with two matters: in its first part - how to measure the Shabbos boundary in a uniform way, so that when a city sets up markers at the point where the boundary ends, all the measurements will be done in one and the same manner; and in its second part - what the law of the measurement is when one reaches a place where it is not physically easy to carry out the measurement.
"Ein moddin ela b'chevel shel chamishim amah":
The Mishnah opens: "Ein moddin ela b'chevel shel chamishim amah, lo pachos v'lo yoser" - one measures with no other instrument except a rope exactly fifty amos long. At the root of this are two concerns:
A rope shorter than fifty amos: those measuring tend to pull it taut with greater force, and the result is that the length of the boundary comes out longer than it should.
A rope longer than fifty amos: the tendency is not to pull it taut enough, and then a sag is created in the line of measurement, and in the end the boundary comes out shorter.
"V'lo yimdod ela k'neged libo":
The manner of holding the rope also has to be uniform: those measuring hold the rope opposite their heart, and since they are at roughly the same height, the rope is stretched in a straight line and does not run at an angle - for an angle takes up part of the distance and distorts the measurement.
Measuring on a slope - "mavli'o":
What is the law of measuring when "hayah moded v'higia l'gai" - he came to a steep valley, "o gader"? Literally 'gader' means a wall, but in practice it refers to a wall that has fallen, which is like a heap of stones forming a slope or something hill-shaped. Further on the Mishnah also mentions a mountain. And in all of these: "mavli'o v'chozer l'midaso" - he spans the place and continues his measurement.
What is havla'ah, spanning? They would take that same rope and tie it to tall poles, and at least where a heap or a place with height is involved, they would measure with the rope above the heap, and from there continue measuring opposite the chest. Where a valley is involved, this can be done even without poles, and it is enough to hold the rope opposite the chest. The point of the Mishnah is that the slope, whether of the valley or of the heap of stones, is not taken into account in the measurement of the two thousand amos.
And so the Mishnah continues: "Higia l'har - mavli'o v'chozer l'midaso". A mountain has the same law as the heap of stones: they use poles in order to pass over the mountain and measure that area, and from there they continue the measurement.
"U'vilvad shelo yeitzei chutz la'techum":
If the width of the valley or the mountain exceeds fifty amos, it cannot be spanned along that line, so the one measuring moves along the line of the hill - southward or northward - until he finds a place where there are not fifty amos. This is permitted, but on condition "shelo yeitzei chutz la'techum": the permission to move off the original line in order to span and to find a width of less than fifty amos exists only as long as there is no need to go outside the boundary. But if in order to find such a place he must go outside the boundary, it is not permitted. Why? Because those who see the measuring being done there will think that the boundary reaches that far, and that is not so - the measuring is being done only in order to find a way to measure the boundary at the original place, and the one measuring will return to that place and continue his measurement there; it is only that those who see him measuring lower down will think that the boundary passes there.
"Shem'kadrin b'harim":
And what is the law if no place at all can be found within the boundary where the width of the valley or the mountain is less than fifty amos? On this the Mishnah says: "Ein yachol l'havli'o - b'zeh amar Rabbi Dostai bar Rabbi Yannai mishum Rabbi Meir: shamati shem'kadrin b'harim". Rabbi Dostai, the son of Rabbi Yannai, said in the name of Rabbi Meir that he has a tradition that one pierces through mountains.
The Gemara explains that kidur, piercing the mountains, is a method of measuring: when going up a slope or down into a valley, they would take a short rope of four amos. The one standing below would hold the rope opposite his heart, and the one going up the mountain above him would measure opposite his feet at the bottom. In this way they shortened the distance, that is, they removed the distance of the slope from the measurement, and thereby obtained a close to accurate reading of the distance as it would have been measured in a straight line through the mountain, without taking the slope into account.
The conditions for spanning and for kidur:
A very important point: the whole matter of havla'ah - spanning the slope with the help of poles - and likewise the method of "m'kadrin b'harim", were stated only where the slope, going up or going down, is ten tefachim for every five amos of forward progress; that is, for every five amos on the horizontal plane there is a vertical movement of ten tefachim. But if the movement of ten tefachim upward or downward occurs within only four amos - which is a sharper angle - the Chachamim said that one may be lenient, and estimate the measure of the place and proceed according to that estimate.
In summary: in this Mishnah we learned the rules of uniformity in measuring - a rope of exactly fifty amos, no less and no more, and holding it opposite the heart; the law of spanning a valley, a fallen wall and a mountain, that the slope is not taken into account in the measurement of the two thousand amos; the condition that one does not go outside the boundary in order to span, because of those who see; and the method of "m'kadrin b'harim" with a rope of four amos when spanning is impossible, together with its conditions and the degrees of slope in which it applies.