ber corresponding to one half the sum of those logarithms will be the area of the triangle.

The half sum, 31

1.491362

0.845098

1.041393

RULE 4. When two sides of a triangle and their contained angle, that is, the angle made by those sides, are given, the area may be found as follows:

Add together the logarithms of the two sides and the logarithmic sine of the angle; from their sum subtract the lo garithm of radius, the remainder will be the logarithm of double the area.

EXAMPLE. Suppose a triangle one of whose sides is 105 rods and another 85, and the angle contained between them 28 5'. Demanded the area.

NOTE. Radius may be subtracted by cancelling the left-hand figure of the index, or subtracting 10, without the trouble of setting down the ciphers.

BY NATURAL SINES.

Multiply the two given sides into each other, and that product by the natural sine of the given angle; the last product will be double the area of the triangle.

Nat. sine of the angle 28° 5′ 0.47076.

105×85-8925, and 8925×0.47076-4201 the double area of the triangle.

RULE. Multiply half the sum of the two parallel sides by the perpendicular distance between them, or the sum of the two parallel sides by half the perpendicular distance, the product will be the area.

PROBLEM XI. To find the area of a trapezium, or irregular four sided figure.

RULE. Draw a diagonal between two opposite angles, which will divide the trapezium into two triangles. Find the area of each triangle and add them together. Or, multiply the diagonal by half the sum of the two perpendiculars let fall upon it, or the sum of the two perpendiculars by half the diagonal, the product will be the area.

NOTE. Where the length of the four sides and of the diagonal is known, the area of the two triangles, into which the trapezium is divided, may be calculated arithmetically, according to PROB. IX. Rule 3.

PROBLEM XII. To find the area of a figure containing more than four sides.

RULE. Divide the figure into triangles, and trapezia, by drawing as many diagonals as are necessary, which diagonals must be so drawn as not to intersect each other; then find the area of each of the several triangles or trapezia, and add them together; the sum will be the area of the whole figure.

NOTE. A little practice will suggest the most convenient way of drawing the diagonals; but whichever way they are drawn, provided they do not intersect each other, the whole area will be found the

same.

PROBLEM XIII. Respecting circles.

RULE 1. If the diameter be given the circumference may be found by one of the following proportions: as 7 is to 22, or more exactly, as 113 is to 355, or in decimals, as 1 is to 3.14159, so is the diameter to the circumference.

RULE 2. If the circumference be given the diameter may be found by one of the following proportions: as 22 is to 7, or as 355 is to 113, or as 1 is to 0.31831, so is the circumference to the diameter.

RULE 3. The diameter and circumference being known, multiply half the one into half the other, and the product will be the area.

RULE 4. From the diameter only, to find the area: multiply the square of the diameter by 0.7854, and the product will be the area.

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 гROB. IX. Rule 1. Or, without plotting the field, calculate the area by PROB. IX. RULe 3.

Fig. 50.

9600

7200

16800

Acres 17)61600

4

Roods 2)46400

40

Rods 18)56000

Acres Roods Rods
Area 17 2 18.56

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

Observe, That in measuring with a chain, slant or inclined surfaces, as the sides of hills, should be 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, ac cording 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 opposite 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.

Fig. 51.

BOOK. See Fig. 51

.

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 ares as at D, from A and C as 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 from which the others may be seen, measure diagonals to