multiply the square of the circumference by 0.07958, and the product will be the area.

RULE 6. The area being given to find the diameter: divide the area by 0.7854, and the quotient will be the square of the diameter; from this extract the square root, and you will have the diameter.

RULE 7. The area being given to find the circumference: divide the area by 0.07958, and the quotient will be the square of the circumference; from this extract the square root, and you will have the circumference.

SECTION II.

The following CASES teach the most usual methods of taking the survey of fields; also, how to protract or draw a plot of them, and to calculate their area.

NOTE. The field-book is a register containing the length of the sides of a field, as found by measuring them with a chain; also the bearings or courses of the sides, or the quantity of the several angles, as found by a compass or other instrument for that purpose; together with such remarks as the surveyor thinks proper to make in the field.

CASE I.

TO SURVEY A TRIANGULar field.

Measure the sides of the field with a chain, and enter their several lengths in a FIELD BOOK, protract the field on paper, and then find the area by PROB. IX. Rule 1. Or, without plotting the field, calculate the area by prob. IX. rule 3.

NOTE. When there are ciphers at the right hand of the links, they may be rejected; remembering to cut off a proper number of fig. ures according to decimal rules.

Observe, That in measuring with a chain, slant or inclined surfaces, as the sides of hills, should he measured horizontally, and not on the plane or surface of the hill; otherwise, a survey cannot be accurately taken. To effect this, the lower end of the chain must be raised from the ground, so as to have the whole in a horizontal line; and the end thus raised must be directly over the point where the chain begins or ends, according as you are ascending or descending a hill; which point may be ascertained by a plummet and line.

CASE II.

TO SURVEY A FIELD IN THE FORM OF A TRAPEZIUM.

Measure the several sides, and a diagonal between two op. posite angles; protract the field, and find the area by PROBLEM XI. Or, without protracting the field, calculate the area according to the note at the end of that PROBLEM.

To PROTRACT THIS TRAPEZIUM.

Draw the side AB the given length; with the diagonal AC 28 and the side BC 11.70 describe cross arcs as at C, from A and B as centres; and the point of intersection will represent that corner of the field: then, with the side CD 21.50 and the side AD 14.70, describe cross arcs as at D, from A and Cas centres; and the point of intersection will represent that corner of the field.

NOTE. The perpendiculars need not be actually drawn; their length may be obtained as follows: From the angle opposite the diagonal open the dividers so as when one foot is in the angular point, as at B, the other, being moved backwards and forwards, may just touch the diagonal at A, and neither go the least above or below it; that distance in the dividers being measured on the scale will give the length of the perpendicular.

CASE III.

TO SURVEY A FIELD WHICH HAS MORE THAN FOUR SIDES, BY THE CHAIN ONLY.

Measure the several sides, and from some one of the angles

them; draw a plot of the field, and find the area by PROBLEM XII.

FIELD BOOK. See Fig. 52.

22.40

20.40

Fig. 52.

Diagonals.

Ch. L. 45.

35.

24.20

TO PROTRACT THIS FIELD.

Draw the side AB, making it the given length 30.60; with the diagonal AC 45 and the side BC 20.40, describe cross arcs as at C, from the points A and B as centres, and the point of intersection will represent that corner of the field; draw the side BC and the dotted diagonal AC; with the diagonal AD 35 and the side CD 22.40, describe cross arcs as at D, from the Points A and C, and draw the Side CD and the dotted diagonal AD. Proceed in this manner till all the sides and diagonals are drawn.

TO FIND THE AREA.

The field being plotted, may be divided into one trapezium and two triangles; the area of which is calculated as fol

As each of the sides of the several triangles, into which the preceding plot of a field is divided, is known from the field book, the area of the field may be calculated arithmetically, by finding the area of each triangle, according to PROB. IX. Rule 3; and then adding the whole together. This method, though it may require more time, is preferable to the other, because more accurate. Indeed it is always better to calcu. late the area of a field arithmetically than geometrically; for in the former no two persons can differ in their calcula tions; whereas, according to the latter, which is the common method of casting the contents of a field, it is hardly to be expected that any two persons will perfectly agree. The inaccuracy of scales, and the difficulty of determining with precision the length of sides and perpendiculars with a scale and dividers, render it almost if not quite impossible to obtain the exact area of a field, in the method commonly practised, even if the surveyor has measured it accurately in the first place. Other methods of taking the survey of field by the chain only are mentioned in some treatises on this subject, but they