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14.70 Diagonal AC
28. To protract this Trapezium. Draw the Side AB the given length; with the Diag. onal AC 28 and the Side BC 11.70 describe cross Ar. ches as at C, from A and B as Centres ; and the Point of intersection will represent that Corner of the Field : Then with the Side CD 21.50 and the Side AD 14.70 describe cross Arches as at D, from A and C as Centres ; and the Point of intersection will represent that Corner of the Field.
Acres Rood Rods Area 31-1 26.56 Note. The Perpendiculars need not be actually drawn ; their
length may be obtained as follows: From the Angle opposite the Diagonal open the Dividers so as when one Foot is in the angular Point, as at B, the other, being moved backwards and forwards, may just touch the Diagonal at a, and neither go the least above or below it; that distance in the Dividers being measured on the Scale will give the length of the Per
CASE III. To survey a Field which has more than four Sides, by the Chain only.
Measure the several Sides, and from some one of the Angles, from which the others may be seen, meas. ure Diagonals to them ; draw a Plot of the Field, and find the Area by PROBLEM XII. FIELD BOOK. See PLATE II. Fig. 52.
To protract this Field. Draw the Side AB, making it the given length 30. 60; with the Diagonal AC 45 and the Side BC 20.40 describe cross Arches as at C, from the Points A and B as Centres; and the Point of intersection will represent that Corner of the Field ; draw the Side BC and the dotted Diagonal AC: With the Diagonal AD 35 and the Side CD 22.40 describe cross Arches as at D, from the Points A and C; and draw the Side CD and the dotted Diagonal AD. Proceed in this manner till all the Sides and Diagonals are drawn.
To find the Area. The Field being plotted may be divided into one Trapezium and two Triangles ; the Area of which is calculated as follows.
The Trapezium ABCD. The Triangle ADE. Perpend. B a 11.68 Half Perp. Em
REMARKS. As each of the Sides of the several Triangles into which the preceding Plot of a Field is divided, is known from the Field Book, the Area of the Field may be calculated Arithmetically, by finding the Area of each Triangle, according to PROB. IX. Rule 3 ; and then adding the whole together. This method, though it may require more time, is preferable to the other, because more accurate. Indeed it is always better to calculate the Area of a Field Arithmetically than Geomet. rically ; for in the former no two persons can differ in their calculations; while according to the latter, which is the common method of casting the Contents of a Field, it is hardly to be expected that any two persons will perfectly agree. The inaccuracy of Scales, and the dif. ficulty of determining with precision the length of Sides and Perpendiculars, with a Scale and Dividers, render it almost if not quite impossible to obtain the exact Area of a Field, in the method commonly practised ; even if the Surveyor has measured it accurately in the first place.
Other methods of taking the Survey of a Field, by the Chain anly are mentioned in some Treatises on this subject, but they are rather curious than useful ; and it is much better to ascertain the Angles by an accurate Compass, or some Instrument designed purposely for
CASE IV. To survey a Field with a Chain and Compass. Measure the length of the Sides with a Chain, and take their Bearing or Course with a Compass ;* enter these in a Field Book ; plot the Field on Paper, and calculate the Area by the directions already given,
To protract or draw a Map of a Field. Draw a Line to represent a Meridian or North and South Line, from which lay off the Bearing or Course of the first Side of the Field, with a Protractor or from a Line of Chords; and from a Scale of equal Parts measure the length of the Side and draw a Line to represent it. At the end of this Line draw a Line parallel to the Meridian Line, and then lay off the second Side of the Field as before taught : Proceed in the same manner to draw parallel Lines and lay off the several Sides till the whole is protracted.
In protracting a field, let the Top of the Paper be considered as North ; the Bottom, South ; the Right hand, East; and the Left hand, West : Lay the Course to the Right or Left of the Meridian Line, according as it is East or West; and from the upper or lower part of the Line, according as it is North or South.
In all protractions, if the end of the last distance falls exactly on the Point from which you began, the Course also being right, the Field work and protraction are truly taken and performed ; if not, an error must have been committed in one of them : In such cases make a second protraction ; if this agrees with the former, it is to be presumed the fault is in the Field work; a re-survey must then be taken.
* A Compass may be so constructed with two Indexes, one moveable and the other fixed, as to ascertain the Angle made by two Sides, without reference to the Bearing of those Sides. Such a Compass would be particularly useful in surveying Land where there are mineral substances which have an influence upon the Compass Needle, attracting it one way or the other; and thus
AB. N. 7° 0 W.
Acres Rood Rods
REMARKS. The Sides of the several Triangles into which the Plot of a Field is divided may be found by Trigonometry; and then the Area of each Triangle may be calculated according to Prob. IX. Rule 3. The Sum of the Areas of the several Triangles will be the Area of the whole Field. This method may require more time but it is perfectly accurate, since no dependance is placed on the uncertain measurement of Scale and Di. viders.
In the preceding ExamPLE, suppose the Field divided into three Triangles. See Fig. 53. In the Triangle EAB, the Sides EA and AB are known from the FIELD Book, and their contained Angle is known from the Bearing of the Sides. The other Angles and the Side EB may be found by OBLIQUE TRIGONOMETRY, CASE III ; and then there will be the three Sides to find the Area. In the Triangle EBC, the Side BC is known from the FIELD Book, and the Side EB is found as above mentioned; the Angle EBA is also found as above ; this subtracted from the Angle ABC, which may be found from the Bearing of the Sides AB and BC, will leave the Angle EBC; there will then be two sides and their contained Angle to find the third Side ; and this being found there will be the three Sides to find the Area. In the Triangle EDC, thie Sides DE and DC are known from the FIELD Book, and their contained Angle is known from the Bearing of the Sides. The Side EC and the Arca may be found as above.
It is recommended to the Learner to make these calculations, as it will improve him in the knowledge of