The solution:

 3 8 7 2 0 5 4 9 1 4 4 0 2 3 8 4 1 1 1 0 7 2 1 9 1 8 9 7 9 2 2 7 1 6 9

An explanation:

• Since 8 horizontal and 11 horizontal are years, both should start with a 1.
• 15 horizontal ("the walking speed (in miles per hour) of farmer Dunk, to the power of three") can only be 27 (33) or 64 (43). However, if 15 horizontal would be 64, then 16 horizontal ("15 horizontal minus 9 vertical") would end on a 3, and 7 vertical ("the square of the width (in yards) of Dog's Mead") would also end on a 3, which is impossible. Therefore, 15 horizontal is 27.
• Since 16 horizontal equals "15 horizontal minus 9 vertical", 16 horizontal must be 16 and 9 vertical must be 11.
• Since 9 vertical equals "10 vertical divided by 10 horizontal", 10 horizontal must end on a 2.
• 8 vertical ("the number minutes in which farmer Dunk walks 11/3 times around Dog's Mead") is therefore 12, the number of minutes in which farmer Dunk walks 1 time around Dog's Mead is 9, and since from 15 horizontal it follows that farmer Dunk walks 3 miles per hour, 14 horizontal ("the circumference of Dog's Mead (in yards)") is 792.
• 12 vertical ("the sum of the digits of 10 vertical plus 1") can now only be 19.
• 11 horizontal ("the year in which Mary was born") must start with 191, and since the current year is 1939, 3 vertical ("the age of Mary, the daughter of farmer Dunk") must start with a 2.
• Since 6 horizontal ("the difference between the length and width of Dog's Mead (in yards)") is at least 10 and at most 99, and the sum of the length and width of Dog's Mead equals 396 (because of 14 horizontal), the minimum value for the width is 149 and the maximum value is 193. Because of 7 vertical ("the square of the width (in yards) of Dog's Mead"), the width must end on a 4 or a 6. Therefore, only eight possible values remain: 154, 156, 164, 166, 174, 176, 184, and 186. Of these values, only the value 176 has a square that ends on 976. Now we can calculate that the length of Dog's Mead is 220, that 7 vertical ("the square of the width (in yards) of Dog's Mead") is 30976, that 1 horizontal ("the area of Dog's Mead (in square yards)") is 38720, that 6 horizontal ("the difference between the length and width of Dog's Mead (in yards)") is 44, and that 7 horizontal ("the number of roods in Dog's Mead multiplied by 8 vertical") is 384.
• Since 6 vertical ("the current age of Ted, son of farmer Dunk, who will be in 1945 twice as old as his sister Mary will be in that year") is 48, we can calculate that 3 vertical ("the age of Mary, the daughter of farmer Dunk") is 21 and that 11 horizontal ("the year in which Mary was born") is 1918.
• 13 vertical ("the number of years that Dog's Mead is owned by the Dunk family") can now only be 829, and therefore 8 horizontal ("the year in which the Dunk family became owner of Dog's Mead") is 1110.
• Because of 9 vertical ("10 vertical divided by 10 horizontal") and 12 vertical ("the sum of the digits of 10 vertical plus 1"), 10 horizontal ("the age of farmer Dunk") can only be 72 and 10 vertical ("see 9 vertical") can only be 792.
• Since 2 vertical ("the square of the age of the mother-in-law of farmer Dunk") is a number of four digits, starting with a 7 and ending on a 1, the age of the mother-in-law of farmer Dunk is at least 84 and at most 89. Of these numbers, only the value 89 has a square that ends on a 1. Therefore, 2 vertical is 7921.
• From the description of 1 vertical ("the value of Dog's Mead (in shillings per rood)") we can deduce that the price per rood is at least 15 and at most 19 pounds. Since the area of Dog's Mead is 32 roods (see 1 horizontal), and because of 4 vertical ("the value of Dog's Mead (in pounds)"), the price per rood can only be 17 pounds, and we find that 1 vertical is 340 and 4 vertical is 544.

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