case, the work may be very readily done by measuring the three sides of a triangle, taking insets as before. After which construct the triangle, and from the opposite angle let fall a perpendicular upon the base, and proceed as before. 3. By this problem you may measure fields into which you are not permitted to enter, or which contain obstructions. B Examples. 1. Let the following figure represent a mere; its area is required. D Having fixed four marks, A, B, C, and D, forming a rectangle ; begin at A, and measure the line AB, taking the necessary insets, and entering them in your field-book. In the same manner proceed with the other three sides; and you will find noted the following dimensions. double areas of the insets taken on the different lines 291650 area of the whole insets 13-03350 ditto of the mere 4 •13400 40 5.36000 Area 13a. Or. 5p. 2. Let the following figure represent a wood; its area is required. B D Set up your cross at A, and let your assistant fix the marks B and D, so that the angle at A may be a right angle; and measure the line AB, taking insets as before. Then fix the mark C, as most convenient; measure the other three lines, and you will find in your field-book the following notes. Having constructed the figure, you will find the diagonal BD to measure 1930, the perpendicular A a 923, and c a 605 links. To find the Area of a Segment of a Circle, or any other Curvilineal Figure, by means of Equidistant Ordinates. H K M O F D A с E G I L N RULE-If a right line AN be divided into any even number of equal parts AC, CE, EG, &c.; and at the points of division be erected perpendicular ordinates AB, CD, EF, &c., terminated by any curve BDF, &c.; B and if A be put for the sum of the extreme or first and last ordinates, AB, NO; B for the sum of the even ordinates, CD, GH, LM, &c., viz., the second, fourth, sixth, &c.; and c for the sum of all the rest, EF, IK, &c., viz., the third, fifth, &c., or the odd ordinates, wanting the first and last; then the common distance AC, or CE, &c., of the ordinates, being multiplied by the sum arising from the addition of A, four times B, and two times c; one-third of the product will be the area A+4 B+2C ABON, very nearly; that is, XD the area, putting D = 3 AC, the common distance of the ordinates. = NOTE.-The foregoing rule being expressed in an algebraic form, is seldom properly understood by learners; but the following one may be easily comprehended and committed to memory. RULE. To the sum of the first and last ordinates, add four times the sum of all the even ordinates, and twice the sum of all the odd ordinates, not including the first and last; multiply this sum by the common distance of the ordinates, divide the product by 3, and the quotient will be the area required. NOTE. The length of the base must be ascertained before you begin to take the ordinates, in order that you may divide it into an even number of equal parts; or you may take the dimensions without doing this, and find the areas of the pieces at the end, by the rules for triangles and trapezoids, which being added to that part of the figure computed by equidistant ordinates, will give the whole area. See the following Examples. Examples. 1. Required the plan and area of a piece of land, measured by equidistant ordinates, from the following notes. |