by a line drawn from the angle A to the side BC, so that the triangle AEC may contain the proposed quantity. From the three sides, by Note 4, Part IV., the area of the given square triangle is found 488076 The difference = 290576, the area of the triangle ABE 2 1300)581152(447 links, the perpendicular DE. 5200 .6115 5200 .9152 9100 52 By the mode described in the last example, determine the point E, which you will find at the distance of 658 links from the angle B; measure this distance in the field from B to E, and proceed as before. 3. From a triangular field whose sides are 1500, 1200, and 1000 links respectively, part off 3A. 2R. 16P. by a fence made from the greater angle at the base to the opposite side. Ans. The perpendicular of the triangle required is found to be 480 links; and it rises upon the base at the distance of 537 links from the less angle. PROBLEM XIV. To part from a triangle any proposed quantity of land by a line RULE. The areas of similar triangles are to one another in the duplicate ratio of their homologous sides; hence, as the area of the triangle ABC is to the square of the side AC, or BC, the triangle DEC to the square of the side DC, or EC. Part I.) so is the area of (See Theo. XIII. Δ Examples. = E B 1. Suppose the base AB = 1200, the side AC 1000, and the side BC= 800 links; part off 1A. 2R. 16P. by the line DE parallel to AB. From the three sides, by Note 4, Part IV., we find the area of the triangle 396863 square links The difference = 236863, the area of the triangle DEC. DC; Then, as 396863: 1000 x 1000 :: 236863: 596838-20; and √596838.20 = 772.5 links hence 1000 - 772.5 = 227.5 links = AD. Again, as 396863: 800 × 800 :: 236863: 381976.45; and 381976.45 618 links EC; then 800 — 618 = 182 links = BE. = = Measure therefore in the field 227-5 links from A to D; and from B to E measure 182 links; stake out the line DE, and the work will be completed. 2. From a triangular field, whose sides are 1800, 1500, and 1200 links respectively, part off 3A. 2R. 32P. by a line parallel to the shortest side. Ans. The area of the given triangle is 892941 square links; the area of the triangle made by the line of division is 522941 square links; and one of its sides from the angle opposite the line of division to the commencement of that line, is 1147-9, and the other 1377-4 links. PROBLEM XV. To part from a rectangle or triangle any proposed quantity of land upon a line on which there are offsets, when the area of those offsets is to be considered as part of the portion to be parted off. RULE. Find the area of the offsets, which subtract from the portion to be parted off, and then proceed with the remainder as directed in the preceding problems. But, in a rectangle, when there are offsets on one or both of the lines adjoining that upon which the given quantity is to be parted off, reject these offsets and proceed as before directed. Then, having found the distance at which the line of division must be from that upon which the given quantity is to be parted off, find the area of the offsets contained between those lines, which area divide by the latter line, and the quotient will be the distance by which the former line must be approximated to the latter. Examples. 1. From a rectangular field whose dimensions are contained in the following notes, part off 2A. 3R. 32P, upon the chain-line AB, so that the offsets taken upon that line be included. may Hence the irregular figure AGBFE contains 2A. 3R. 32P. 2. From a rectangular field whose dimensions are contained in the following notes, part off 2A. 2R. 8p. by a line parallel to the chainline AB, so that the offsets taken upon this line, and also those upon the two adjoining lines contained between the chain-line AB and the line of division may be included. 2A. 2R. 8P. = 255000 40250 the area of the offsets taken on AB 1,000) 214,750 the difference 214.750 links = Ba or Am, which we may call 215 65 = ra = cm; and, by the scale, ae links. Now, 215 150 = is found to measure 58, and mn 53 links; hence the area of the offset Bae +the area of the offset Amn=13282, which, divided by 1000, gives 13 links, the distance by which the line en must be approximated to AB. Consequently, EF is the true line of division; and the regular figure AGBEF contains 2A. 2R. SP. minus the two shaded offsets. PROBLEM XVI. To part from a trapezium, or any irregular polygon whatever, any proposed quantity of land by a line drawn parallel to any of the sides, or by a line drawn from any of the angles, or from any assigned point in one of the sides to any of the opposite sides. RULE 1. Having laid down the whole figure, draw a guess-line in the direction required, parting off, as nearly as can be judged, the proposed quantity; after which, by the scale, measure, with the greatest accuracy, the guess-line, and also the quantity thus parted off. Then, if the guess-line or line of division be drawn from an angle, or from any assigned point in a side, divide the difference between the proposed quantity and the quantity parted off, by half the guessline, and the quotient will be the perpendicular to be set off, on one side, or the other, of the guess-line, accordingly as the quantity parted off is more or less than the quantity proposed. To the end of this perpendicular, from the point assigned, draw a new line of division, and it will part off the quantity required. 2. But if the guess-line be drawn parallel to any of the sides, divide the difference before mentioned by the whole guess-line, and the quotient will be the perpendicular to be set off from each end of the guess-line, on one side, or the other, as above. NOTE 1. When from a trapezium, approaching very nearly to a rectangle, it is required to part off any number of acres, &c., by a line parallel to one of its sides, it may be done as directed in Problem XI.; and if there be offsets upon any of the lines, they must be treated as in the last problem. 2 In using guess-lines, it is not necessary that the learner should draw them so as to coincide in measure with those of the examples which he is performing. It will be sufficient for him to proceed in a similar manner. Examples. 1. From a trapezium whose dimensions are contained in the following notes, part off 2A. 2R. 24P. by a line parallel to the side AB. |