MENSURATION 21 If the polygon be concave some of the triangles will have to be subtracted. Fig. 28. Fig. 29. 60. Area of a Polygon in terms of the Coordinates of its Angular Points. Let the coordi nates of P, Q, R (fig. 30) be ( x i, 2/i)i (#2, 2/a)> an 3 (3> 2/s) respectively, and let the axes be inclined at an angle a. Draw PL, QM, and RN parallel to OY, then II and O/ L M , _, NL=ON-OL = x 3 -x 1 . Fig. 30. Now PQR = PLMQ + QMNR-PLNR. But PLMQ = PLM + QMP = PLM + QLM jM sin (180 -a) (9, 0) _ J sin o. Similarly QMNR = ^(x 3 - x 2 )(?/ 2 + j/ 3 ) sin a , an< * PLNR =^(x 3 #1X2/3 + 2/1) sin o; hence area of PQE-isinafofas, -x 3 ) + yfa-xj + yfa-xj}; or in the notation of determinants 1 1 1 X 2/1 2/2 2/3 When the axes are rectangular sin a = sin 90 for the area becomes 1, and the formula 4 { 2/1(^2 - 1 1 1 2/1 2/2 2/3 61. The area of any rectilineal figure of n sides can be found by taking any point within the figure and joining it to the n vertices of the figure, thus dividing it into n triangles the area of each of which can be obtained as in the preceding case. We may, however, find the area of the figure directly. For example, in fig. 31 PQRST - PP T T + TT S S + SS R R - RR Q Q - QQ P P, and in fig. 32 PQRST U = PP U U + RR Q Q + TT S S - PP Q Q - RR S S - TT UU R X Fig. 31. Fig. 32. B. Irregular Curvilineal Figures. 62. Length of any Curve. If we divide the given arc into an even number of intervals and re- - gard these as approximately circu lar, we can find an approximation to the length of the arc by means of Huygens s formula, 32. For example, if we divide ABC (fig. 33) into four parts in D, B, and E, and draw the chords AD, AB, DB, BE, BC, and EC, then arc AC = AD approximately. Fig. 33. For other methods of approximation, see Rankine s Rules and Tables. 63. Area of an Irregular Curvilineal Figure. For rough ap proximations the following, called the trapezoidal method, may be used : Divide AjA,, (fig. 34) into n equal parts, and through the points A,. A ; An-z ATM An Fig. 34. of division draw the ordinates, called by surveyors offsets A,P, A 2 P 2 , &c. Let A 1 P 1 = s 1 , A 2 P 2 = s 2 , &c., A n P n = s,,, and A 1 A 2 = A 2 A 3 = . . . =A n _iA n = rt. Join P 1 P 2 , P 2 P 3 , & c ., then the area of the polygon A^P,,?,,?! = A 1 A 2 P 2 P 1 + A 2 A 3 P 3 P 2 + .... +A n -iA n P n P n -i 13, o) If we take n sufficiently great the difference between the area of the polygon and the Curvilineal figure can be made as small as we please, in other words, the smaller we make a the more accurately will the above formula represent the area of the curvilineal figure. The curve may either be wholly convex or wholly concave to the line AjA,,, or partly con vex and partly concave. 64. Simjyson s Rule. Let AjA,, (fig. 34) be divided into an even number of equal parts, and as before through the points of division draw the ordinates AjPj, A P 2 , &c. Let A" 1 A 3 P 3 P 1 (figs. 35, 36) be a part of the figure thus di vided; join P^s, and through P 2 draw BC parallel to PjP,, to meet AP in B and AP in C. A, Fi. 35. Fig. 36. meet Ajfj in n ana A 3 r 3 in u. Conceive a parabola to be drawn through PjP 2 P 3 having its axis parallel to the ordinates, then A 1 P 1 P 2 P 3 A 3 = trapezium A^DPjAsi parabolic segment PjP-jPs Now when the points PpP.jjP.j are near each other the parabolic curve will coincide very nearly with the given curve; hence ^iPiPaPa ^3 = k a ( s i + 4s 2 + s 3 ) very nearly. Similarly A 3 P 3 P 5 A 5 = ^a(s 3 + 4s 4 + s 5 ) , &c. ; hence whole area of figure whence the rule : add together the two extreme ordinates, twice the sum of the intermediate odd ordinates, and four times the sum of the even ones, and multiply this result by one-third of the common distance between the ordinates; the result is the area, accurately if the curved boundary be the arc of a parabola, in other cases ap proximately. The curve may either be wholly convex or wholly concave to the line AjA n , or partly convex and partly concave, provided in the latter case the points of contrary flexure occur only at the odd ordinates, for otherwise the intermediate arcs could not be even approximately parabolic. When points of contrary flexure occur ordinates may be drawn at these points, and the intermediate arcs being found separately may be added to obtain the whole area. 65. In the two preceding sections we investigated two formulas for approximating to the areas of curvilinear figures. We now proceed to consider the subject more generally. Ai Ap Ap*i A/i+i Fig. 37.
Let the equation to the curve PjPpPn+i (fig. 37) agree with the