The Figure may alfo be easily constructed by Prob. 1. Sect. 1. PROB. IX. To find the Content of a Trapezium Piece of Ground. Multiply the Diagonal, or Base, by half the Sum of the two Perpendiculars, or the two Perpendiculars by half the Bafe; or take half a Product of the Bafe into the Sum of the two Perpendiculars. A Line contained between the two remoteft Angles, is the Diagonal. EXAMPLE. Plate VII. Fig. 3. Let ABCD be à Piece of Ground, in Form of a Trapezium, whofe Diagonal, or Bafe, AC, is 32C, 10 L; its Perpendicular bB on one Side of that Line is 6 C. 41 L. and Perpendicular dD on the other Side is 13 C. 31L. Required the Content. Perp, The other two Methods are obvious, from what has been laid down (in the Triangle, Prob. 6.) and will produce a like Content. If both the Bafe and the Sum of the two Perpendiculars are odd, ufe the laft Method of the three. The Map may be drawn, by knowing in what Parts of the Diagonal the Perpendiculars were taken, and thence fetting them off, and joining their Extremity to thofe of the Diagonal. The Area of the Square, Oblong, Rhombus, or Rhomboides, may be obtained in the like Manner, by a Diagonal and Perpendiculars. Cc 2 If If a Map of any of the foregoing Figures be given, with the Scale by which it was laid down, take the Dimenfions from that Scale of Perches, and therby adjust their Contents, without reducing them to Chains. From this, and the fixth Problem, this following Method will be plain. First Method to find find the Area of Ground. How the Area of a Piece of Ground, be it ever fo irregular, may be obtained, by dividing it into Triangles, and Trapezia. Plate VII. Fig. 4. We here admit the Survey to be taken, and protracted; by having therefore the Map, and knowing the Scale by which it was laid down, the Cortent may be thus obtained. Difpofe the given Map into Triangles, by fine penciled Lines, fuch as are here reprefented by popped Lines in the Scheme, and number the Triangles with 1, 2, 3, 4, &c. Your Map being thus prepared, rule a Table with four Columns, the firft of which is for the Number of the Triangle, the fecond for the Base of it, the third for the Perpendicular, and the fourth for the Content in Perches. Then proceed to measure the Bafe of Number 1, from the Scale of Perches the Map was laid down, and and place that in the fecond Column of the Table, under the Word Bafe; and from the Angle oppofite to the Base, open your Compaffes fo, as when one Foot is in the Angular Point, the other being moved backwards and forwards, may juft touch the Bafe Line, and neither go the leaft above or beneath it; that Distance in the Compaffes, measured from the fame Scale, is the Length of that Perpendicular, which place is the third Column, under the Word Perpendicular. If the Perpendiculars of two Triangles fall on one and the fame Bafe, it is unneceffary to put down the Bafe twice, but infert the fecond Perpendicular oppofite the Number of the Triangle in the Table, and join it with the other Perpendicular by a Brace, as No. 1 & 2, 4 & 5, 6 & 7, 9 & 10, &c. Proceed after this Manner, till you have meafured all the Triangles; and then by Prob. 6, find the Content in Perches of each refpective Triangle, which feverally place in the Table opposite to the Number of the Triangle, in the fourth Column, under the Word Content. But where two Perpendiculars are joined together in the Table, by a Brace, having both one and the fame Pafe; find the Content of each, (being a Trapezium) in Perches, by Prob. 9, which place oppofite the Middle of those Perpendiculars, in the fourth Column, under the Word Content. Having thus obtained the Content of each re spective Triangle and Trapezium, which the Map contains, add them all together, and their Sum will be |