Chap. VIII.

GEOMETRY.

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correctly taken; but if the sum of the addition is more or less than 180°, measure the anglesagain, to ascertain whence the error has occurred, and by successive operations correct it.

To find the salient and re-entering Angle of a Plan, if correctly taken, it is requisiteto ascertain if the right angles of the irregular plan are equally so, viz. the salientangle, 1 BI), 76°, the re-entering angle 13 DC, 121° theangle 1)CE, 103°, CEF, 113°, EFG, 89°, the re-entering B

angle FG1I, 110°, GUI, 117°, and the salient angleIII B, 93°. Trace outside the two re-entering angles thelines B C and IIF, which form an artificial boun-dary, BCE Fill; measure all the angles of thisas if it were the true one, except the two CEF, 113°,and II IB, 103°, which are already known, and havingfound, by taking these angles, that I B C is 117°, BCE121°, EFH 126°, and FHI 150°, which, together withthe two angles CEF, 113°, and II IB, 93°, make 720°;then this shows that the angles are properly taken, be-cause the polygonal angles of a hexagon, whether regular or irregular, together make720°. But since the artificial boundary B C E FIII has the angle I B C, 117°, greater thanIBD , 76°, BCE, 121°, greater than DCE, 103°, EF1I, 126°, greater than EFG, 89°, andFill, 150°, greater than GUI, 117°, therefore we must ascertain if the four angles IBI),7G°, DCE, 103°, EFG, 89°, and GUI, 117°, are correctly taken. For the first two angles1131) and DCE, measure in the artificial triangle BCD, the angles 1)BC, 41°, andDCB, 18° ; then subtract DBC, 41°, from IBC, 117°; the remainder, 76°, being equal to theangle I B D , as measured, shows it has been accurately taken. If from the angle B C E,121°, the angle DCB 18° is subtracted, there will remain 103° for the angle DCE.Pursue the same course for the angle EFG and G II I, and the first will be found 89°, andthe second 117°.

To ascertain if the re-entering angles are properly taken, beginning with DBC, addto the artificial triangle BCD, the two angles DBC, 41°, and DCB, 18°, subtract theirsum 59° from 180°, the remainder, 121°, will be the value of the re-entering angle B 1) C,which was before found the value of that angle: the same rule must be followed for theartificial triangle G FII, to ascertain if the re-entering angle F G 11, 110°, has also been cor-rectly taken. Lastly, to verify the two angles CE F and f 1113, have recourse to the methodbefore given, by using the complement of the angle

Taking the Plan of Places wholly or the part ofa circular figure : such a plan must be enclosedwithin a cord, or in some other manner, so thatit may touch the enclosure as much as possible,as in the figure BCDEFGHIKL: if the en-closure or boundary is very irregular, and full ofretreats or sinuosities, as the lower part of the plan,their exact plan must be taken independently ;then remark where the two cords of the artificialboundary form an angle, as at G, and stretchanother cord B V, through the middle of theangle, to the undulations of which measure thelength ; then stretch cords from the sides of theangles to the undulations, at right angles to thesides, the length of which must be written down,as well as their distances from each other. If theartificial boundary whose plan is required hastowers, or other projections, they must be sur-rounded by cords, and the length of their sides and angles figured down, that their circum-ference may be defined : to lay such a plan on paper, draw a circle of any length or divisions,as 180 feet: place on one of the sides the length BC, 180 feet; draw with a protractor fromthe point C, the angle BCD, 101° : continue to draw in succession the length of the sidesand angles, until you have the measurement of the whole of the artificial boundary.

Fig.1109.

To have the Circular Boundary , observe at what distance from the angles the rea.boundary touches the lines shown by the cords, as at the points M, N, O, P; then take fromthe scale their relative distances, and mark them on the sides of the artificial boundary,through which the real outline must be traced.

To have the Circular Boundary correct, observe that the angles of the artificial boun-dary are divided into two by a right line drawn from the angle to the boundary, thelength of which is figured; this must also be set out on the paper at the respective angles :remark also, that to the right and left of each angle of the artificial boundary, where cordsare stretched, their distance and length must be set down; and when these lines are properlylaid down, the natural boundary may be traced through them.

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