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1. A circular fish-pond is to be dug in a garden, that shall take up just half an acre; what must be the length of the cord that strikes the circle? Ans. 27.75 yards.
2. Two sides of a triangle are 20 and 40 perches respectively; required the third side, so that the content may be just an acre.
Ans. Either 23.099 or 58.876 perches.
3. In 110 acres of statute measure, in which the pole is 5.5 yards, how many Cheshire acres, where the customary pole is 6 yards, and how many of Ireland where the pole in use is 7 yards?
Ans. 92A. 1R. 29P. Cheshire; 67A. 3R. 25P. Irish.
4. The ellipse in Grosvenor square, London, measures 840 links the longer way, and 612 the shorter, within the rails; now the wall being 14 inches thick, it is required to find what quantity of ground it encloses, and how much it stands upon.
Ans. It encloses 4A. OR. 6P. and stands on 1760 square feet.
5. Required the dimensions of an elliptical acre, with the greater and less diameters in the ratio of 3 to 2.
6. The three sides of a triangular field, containing 6A. 1R. 12P. are in the ratio of the three numbers 9, 8, 6, respectively; required the sides.
Ans. 59.029, 52.47, and 39.353 perches.
7. In a pentangular field, beginning with the south side and measuring round towards the east, the first or south side is 27.35 ch., the second 31.15 ch., the third 23.70 ch., the fourth 29.25 ch., and the fifth 22.20 ch.; also the diagonal from the first angle to the third is 38 ch.. and that from the third to the fifth 40.10 ch.; required the area of the field. Ans. 117A. 2R. 39 P.
8. Required the dimensions of an oblong garden, containing three acres, and bounded by 104 perches of pale fence.* Ans. 10 perches by 12.
9. How many acres are contained in a square meadow, the diagonal of which is 20 perches longer than either of its sides? Ans. 14A. 2R. 11P.
10. A gentleman has a garden 100 feet long and 80 broad, and a gravel walk is to be made of equal width half round it; what must be the width of the walk, so that it may take up just one-fourth of the ground?
Ans. 11.8975 feet.
11. A person has a circular yard that is 150 feet in diameter, and wishes a walk of equal width made round it within the fence; required the width of the walk, so that it may occupy a fifth part of the ground.
Ans. 7.918 feet.
* This question may be neatly constructed by 28.6 Playfair's Geometry. It may not be improper also to observe, that the 2nd question, and all those fol
12. From a point within a triangular field, the sides of which were equal, I measured the distances to the three angles, and found them 12.5, 10, and 7.5 chains, respectively; required the area. Ans. 12A. 1R. 22P.
13. On examining the field-notes of a lot of ground of which I wished to know the content, I found them as follow; 1st. S. 72° W. 24 per., 2nd. North, 38 per., 3rd. N 82° E. 41 per., 4th., 20 per., 5th. S. 80° E. 11.5 per., 6th. S. 26° W. 22 per., and 7th. 37 per., to the place of beginning. The bearings of the 4th and 7th boundary lines were illegible; but the data remaining being sufficient, the area is required.
Ans. 12A. 3R. 2P.
14. It is required to lay out 44 acres of land in a triangular form, so that the length of one side may be 15 chains, and the lengths of the other sides in the ratio of 2 to 3; what must be the lengths of those sides?
Ans 7.7914 and 11.6871 chains; or 29.58536 and 44 37804 chains.
15. It is required to lay out five acres of ground in a triangular form, to be bounded by 135 perches of fence; the length of one side is to be 50 perches; what must be the lengths of the other sides?
Ans. 33.3785 and 51.6215 perches.
16. The area of a rectangular field is 7 acres, and the length of the diagonal 50 perches: required the sides. Ans. 30 and 40 perches.
17. In a rectangular tract of land, containing 58 A. 3R. 8P. the difference of the lengths of the sides is just
equal to the difference between the lengths of the longer side and the diagonal; hence the sides are required. Ans. 21 and 28 chains.
18. The boundaries of a tract of land are as follow; 1st. N. 14° W. 15.20 ch. 2nd. N. 70° E. 20.43 ch. 3rd. S. 6° E. 22.79 ch. 4th. N. 86° W. 18 ch. to the place of beginning; within the tract there is a spring, the bearing and distance of which, from the 2nd corner is S. 75° E. 7.90 ch. It is required to cut off 10 acres from the west side of this tract by a straight line running through the spring; what must be the distance of the division line from the 1st corner, measured on the fourth side?
Ans. 4.6357 chains.
19. The boundaries of a quadrilateral tract of land are as follow; 1st. N. 35° E. 23 ch. 2nd. N. 75 E. 30.50 ch. 3rd. S. 3° E. 46.49 ch., and 4th. N. 66° W. 49.64 ch., to the place of beginning. This tract is to be divided into four equal parts by two straight lines, one of which is to run parallel to the 3rd side; required the distance of the parallel division line from the first corner, measured on the 4th side; also the bearing of the other division line and its distance from the same corner, measured on the 1st side.
Ans. Distance of the parallel division from the 1st corner 32.50 chains, the bearing of the other, S. 88° 22′ E. and its distance from the same corner 6 chains.