1)

(a)One squares a round town in the directions of the world (i.e. so that the north side of the town faces due north, the east side, due east etc.

(b)Taurus is the sign by which it is possible to gauge due north, and Scorpio due south.

(c)When Rebbi Yossi refers to ...

1. ... 'Mehalchasan' - he means the Mazal's course by day.

2. ... 'Mesavevasan' - he means its course by night.

(d)According to Rebbi Yossi, one gauges ...

1. ... the north side of the town - by drawing a line from the point where the sun rises in mid-summer (on the north east) to where it sets (on the north-west).

2. ... the south side - by drawing the line from the point from where the sun rises in mid-winter (on the south east) to where it sets (on the south west).

3. ... the east side - by noting where the sun rises and sets in Tekufas Nisan (the spring equinox) and Tishrei (the autumn equinox), because then it rises and sets exactly in the middle of the east and west sides respectively.

2)

(a)A full solar year lasts three hundred and sixty five and a quarter days.

(b)The luminaries were placed in the sky - at the beginning of the first Tuesday night following the creation of the world, at the beginning of Nisan (it is not clear on which date).

(c)In the following year ...

1. ... Tekufas Nisan will have fallen one a quarter days after that, at midnight of the first Wednesday at the beginning of Nisan.

2. And the year after that, it will fall at sunrise of Friday morning at the beginning of Nisan.

3)

(a)Tekufas Tamuz (the summer solstice) will always fall either at one and a half hours or at seven and a half hours, either after sunrise or after sunset.

(b)Tekufas Teves (the winter solstice) will fall either at four and a half or at ten and a half hours after sunrise or sunset.

4)

(a)When the Gemara says 'Ein Tekufah Mosheches me'Chaverta Ela Chatzi Sha'ah' - it refers to the difference in time between the rising of the Mazal from one Tekufah to the next. This works out in the following manner: There are seven Mazalos (constellations) - Kochav, Levanah, Shabsi, Tzedek, Ma'adim, Chamah, Nogah, each of which dominates from the time it appears until the next Mazal appears exactly one hour later. Each Tekufah lasts ninety one days plus seven and a half hours. Now since the ninety one days are divisible by seven, and so are the seven hours, it transpires that if Kochav (for example) ruled at the beginning of Tekufas Nisan, then Tekufas Tamuz would arrive when Kochav was half way through her 'reign', and it will be Levanah who will rule at the beginning of Tekufas Tishri and so on. Each Tekufah will find the Mazal half an hour in advance of where it was at the previous Tekufah.

(b)When the Tekufah of ...

1. ... Nisan falls in the Mazal of Tzedek - then it causes the trees to become broken.

2. ... Teves falls in the Mazal of Tzedek - it causes seeds to dry up.

(c)This speaks when Tzedek dominates at the time of the Molad (the birth of the new moon).

5)

(a)When the Beraisa says that, after one has squared the town, one squares it again - it means that, by a town of two thousand Amos, after one has squared the circle (to form a square of two thousand Amos by two thousand Amos), one measures the Techum by forming four squares of two thousand Amos square (one on each side of the square of the town), completing the large square by filling in the four corners.

(b)One gains by each corner ...

1. ... of the town - four hundred Amos (the length of the diagonal that the square is in excess - of the circle).

2. ... of the Techum another four hundred Amos.

6)

(a)One thousand Amos surrounding the Levi'im's town - had to remain Migrash (an unused plot of land which served to beautify the town).

(b)The outer thousand Amos - was designated for fields and vineyards (for the Levi'im's use).

(c)'Meshucha'a' means - the land-measurer.

(d)According to Bar Ada, who establishes the Beraisa (which gives the Migrash as a quarter of the area surrounding the Levi'im's cities) by a town of two thousand Amos by two thousand Amos, we can divide the entire two thousand by two thousand Amos area surrounding the town into two times sixteen blocks (one for the area adjacent to the town, the other to fill in the corners), each block a thousand by a thousand Amos. Of these blocks, twelve will surround the town to serve as the Migrash, leaving twenty two for fields and vineyards. - But twelve out of thirty-two is more than a third, asks the Gemara, so how can the Beraisa refer to the Migrash as a quarter?

7)

(a)The Gemara raises the fraction from more than a third Migrash to exactly a third - by adding the four blocks of a thousand by a thousand Amos of the town itself to the field and vineyard section of the entire area, making it twelve out of thirty-six blocks (exactly one third).

(b)We arrive at the stated quarter - by establishing the case by a circular town, in which case the Migrash too, will be circular (instead of the square town which we discussed until now). Since a circle is a quarter less that the square that surrounds it, we will have reduced the sixteen blocks of a thousand by a thousand Amos (which previously took up the town together with the Migrash), to twelve - three quarters (nine) of which are ascribed to the Migrash. Nine out of thirty-six (i.e. the unchanged square of two thousand Amos square that surrounds the town) is a quarter.

(c)According to Abaye too, we are speaking about a round town - and the six blocks of a thousand Amos which comprise the Migrash, make up one quarter of the twenty four blocks of the total area surrounding the town. The advantage of Abaye's explanation is - that it is not necessary, according to him, to include the town in the reckoning.