Building-ground may sometimes be very elegantly laid out in a square or a rectangle. When this is the case, the houses are built on the margin or outside of the square; and in the middle is left an open area, which is generally ornamented with grass-plots, gravel walks, trees, &c.

If the ground will admit, it is very desirable for each house to have a garden laid out in the front; which must, of course, be sold with the house-stead. The open area may be divided into as many equal parts, as there are house-steads in the square; one part may be sold with each house-stead; and the respective purchasers may occupy the whole as joint property.

Ground laid out in this manner, generally fetches a good price, as most persons think it more pleasant and healthful living in squares than in streets.

After every thing has been properly and judiciously arranged upon the plan, such dimensions must be taken by the scale as will enable the Surveyor to stake out all the streets, squares, lots, house-steads, &c. &c. in the field. This being done, the ground may then be considered as ready for inspection and sale.

NOTE 1. House-steads must be laid out of different sizes, according to the respectability of the intended buildings. A room 14 by 15 feet will be found quite large enough for any cottage; and these dimensions may be increased at pleasure, to 20 or 24 feet, according to situation and circumstances.

2. A plan of ground or buildings, intended for sale, is generally left for inspection, at the office of the surveyor or solicitor, employed on the occasion, from the time of advertising to the time of sale. Also, the special conditions of the sale, not specified in the advertisement, may commonly be known at those offices; or by applying to the proprietor, or to his agent, previously to the day of sale.

3. Building-ground is generally sold by auction; and if it be divided into small lots, it will tend much to promote the sale; as many persons may be desirous of purchasing a single house-stead, who would not find it convenient to purchase a lot containing two or three house-steads.

4. The price of building-ground varies from sixpence to upwards of two guineas per square yard, according to the eligibility of the situation.

5. The method of laying out building-ground so as to form straight streets at right-angles to each other, is exemplified in the Plan of a new Town, Plate VII. This town bears a considerable resemblance to Somers-town, and to Pentonville, near London.

SECTION III.

Miscellaneous Questions relating to Surveying, Laying-out, Partingoff, and Dividing Land.

1. THE base of the largest Egyptian pyramid is a square, whose side is 693 feet; how many acres of ground does it cover?

Ans. 11a. Or. 4p.

2. Required the side of a square garden that cost 3£. 18s. 1d. trenching at 1d. per square yard. Ans. 25 yards. 3. Required the area of a rectangle whose length is 1275, and breadth 675 links. Ans. 8a. 2r. 17p. 4. The area of a rectangular field is 14a. 2r. 11p.; what is its length, its breadth being 925 links ? Ans. 1575 links. 5. A rectangular allotment upon a common, cost 78£. 1s. 101d. digging and levelling, at 7£. 10s. per acre; what will be the expense of fencing it half round, at 5s. 6d. per rood; its length being 1225 links? Ans. 17£. 18s. 8d.

6. Measuring along the base of a field in the form of a rhomboides, I found the perpendicular to rise at 678, and its length 1264 links; the remainder of the base measured 2435 links; what is the area of the field? Ans. 39a. 1r. 153p.

7. A grass-plot, in a gentleman's pleasure-ground, cost 3£. 14s. Id. making, at 4d. per square yard; what is the length of the base, the perpendicular being 40 feet, and the figure a rhombus ?

Ans. 50 feet.

8. What is the area of a triangular field, the base of which measures 3568 links, the perpendicular 1589 links, and the distance between one end of the base and the place of the perpendicular 1495 links? Ans. 28a. 1r. 151p.

9. After measuring along the base of a triangle, 895 links, I found the place of the perpendicular, and the perpendicular itself = 994 links; the whole base measured 1958 links; what is the area of the triangle? Ans. 9a. 2r. 37p.

of

10. The area of a triangle is 6 acres, 2 roods, and 8 perches, and its perpendicular measures 826 links; what will be the expense making a ditch, the whole length of its base, at 2s. 6d. per rood? Ans. 6£. 4s. 74d.

11. What is the area of a triangle whose 3 sides measure 15, 20, and 25 chains respectively? Ans. 15 acres.

12. Required the area of a grass-plot in the form of an equilateral triangle, whose side is 36 feet? Ans. 561.18446 feet.

13. What is the area of a triangular field whose 3 sides measure 2564, 2345, and 2139 links? Ans. 23a. 2r. Op.

14. The 3 sides of a triangular fish-pond measure 293, 239, and 185 yards; what did the ground which it occupies cost, at 185£. per acre? Ans. 843£. 7s. 8d. 15. How many square yards of paving are there in a trapezium whose diagonal is found to measure 126 feet 3 inches, and perpendiculars 58 feet 6 inches, and 65 feet 9 inches?

Ans. 871.47569 yards.

16. In taking the dimensions of a trapezium, I found the first perpendicular to rise at 568, and to measure 835 links; the second at 1865, and to measure 915 links; the whole diagonal measured 2543 links; what is the area of the trapezium? Ans. 22a. 1r. Op.

17. Lay down a trapezium, and find its area from the following dimensions; namely, the side A B measures 345, BC 156, CD 323, D A 192, and the diagonal AC 438 feet.

Ans. 52330.33406 feet.

18. What is the area of a trapezoid whose parallel sides measure 25 and 18 feet; and the perpendicular distance between them, 38 feet? Ans. 1197 feet.

19. The parallel sides of a piece of ground measure 856 and 684 links, and their perpendicular distance 985 links; what is its area? Ans. 7a. 2r. 131p.

20. If the parallel sides of a garden be 65 feet 6 inches, and 49 feet 3 inches, and their perpendicular distance 56 feet 9 inches; what did it cost, at £325. 10s. per acre?

Ans. 24£. 6s. 74d.

21. It is required to lay down a pentangular field, and find its annual value, at 2£. 5s. per acre, the first side measuring 926, the second 536, the third 835, the fourth 628, and the fifth 587 links; and the diagonal from the first angle to the third 1194, and that from the third to the fifth 1223 links ? Ans. 18£. 10s. 71⁄2d.

22. The diameters of an elliptical piece of ground are 330 and 220 feet; how many quicks will plant the fence forming the circumference, supposing them to be set 5 inches asunder?

Ans. 2073.

23. Given the lengths of 7 equidistant ordinates of an irregular piece of ground, as follows; 15, 19, 20, 23, 25, 30, and 33 feet; and the length of the base 72 feet; required the plan and area?

Ans. 1704 feet.

24. What must be the length of a chord which will strike the circumference of a circular plantation that shall contain just an acre and a half of ground? Ans. 48.072 yards.

25. The annual rent of a triangular field is 43£. 15s., its base measures 25, and perpendicular 14 chains; what is it let for

per acre? Ans. 2£. 10s.

26. The transverse diameter of the ellipse in Grosvenor-square measures 840 links, and the conjugate 612, within the wall; the wall is 14 inches thick; what quantity of ground does it inclose, and how much does it occupy? Ans. the wall incloses 4a. Or. 6p. and occupies 1760.531 square feet.

27. Two sides of an obtuse-angled triangle are 5 and 10 chains; what must be the length of the third side, that the triangle may contain just two acres of ground? Ans. 8.06225 or 13.60147 chains.

28. What is the area of an isosceles triangle inscribed in a circle whose diameter is 24; the angle included by the equal sides of the triangle being 30 degrees? Ans. 134.3538.

29. The side A B of a triangular field is 40, B C 30, and C A 25 chains; required the sides of a triangle parted off by a division-fence made parallel to A B, and proceeding from a point in C A, at the distance of 9 chains from the angle A.

Ans. 16, 19.2, and 25.6 chains.

30. A field in the form of a right-angled triangle is to be divided between 2 persons, by a fence made from the right angle, meeting the hypothenuse perpendicularly, at the distance of 880 links from one end; required the area of each person's share, the length of the division-fence being 660 links?

Ans. 2a. 3r. 241p. and 1a. 2r. 211p. 31. It is required to part from a triangular field whose 3 sides measure 1200, 1000, and 800 links respectively, 1 acre, 2 roods, and 16 perches, by a line parallel to the longest side.

Ans. The sides of the remaining triangle are

927, 772, and 618 links respectively.

32. The base of a field, in the form of a trapezoid, is 30, and the 2 perpendiculars are 28 and 16 chains respectively; it is required to divide it equally between 2 persons, by a fence parallel to the perpendiculars. Ans. the division-fence is 22.8035 chains, and it divides the base into two parts, whose lengths are 17.0087 and 12.9913 chains respectively.

33. A gentleman a garden had,

Five score feet long and four score broad;

A walk of equal width half round

He made, that took up half the ground:

Ye skilful in geometry,

Tell us how wide the walk must be.

Ans. 25.96876 feet.

NOTE 1. If the sum of the two diameters of an ellipse be multiplied by 1.5708, the product will be the circumference, exact enough for most practical purposes. (See Question 22.)

2. All the foregoing Questions are taken from the Author's Treatise on Practical Mensuration; consequently, their Solutions may be found in the Key to that Work.

X

PART THE EIGHTH.

PLANE TRIGONOMETRY.

THE object of Plane Trigonometry is to point out methods of finding unknown parts of a plane triangle, from having other parts of it given. These methods are derived from the relation existing between the radius of the circle and lines depending on its arcs, or parts of its circumference.

DEFINITIONS.

DHEI is the circumference of a circle, C its centre, C A its radius, A D an arc, A C D an angle, DCE and HCI two diameters perpendicular to each other, and dividing the circumference in four equal parts, called quadrants; and since the whole circumference is usually divided into 360 degrees, one of these four quadrants, as D H, contains one fourth part of 360°, or 90°, which is also the measure of the right angle DC H, and the arc AD may be.. assumed to contain 56° 18′ 37", which is the measure of the angle A CD.

The complement of an arc is what it wants of 90°.

The supplement of an arc is what it wants of 180°. Thus the arcs AH and AE are the complement and supplement of the arc AD; the angles ACH and ACE are also called the complement and supplement of the angle AC D.

The sine of an arc is a straight line drawn from one end of the arc, perpendicular to the diameter drawn to the other end: thus AF is the sine of the arc A D, and of its supplement A E.

The tangent of an arc is a straight line E drawn from one end of the arc perpendicular to the extremity of the diameter, and terminated by the prolongation of the radius through the other end of the arc: thus DG is the tangent of the arc A D, and of its supplement A E.

G

H

L

K

A

D

F

B

The secant of an arc is a line drawn from the centre of the circle, through the end of the arc to meet its tangent: thus C G is the secant of the arc A D, and of its supplement A E.