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To find the area of a piece of ground, be it ever so
irregular, by dividing it into triangles and trapezia.
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 content may be thus obtained.
Dispose the given map into triangles, by fine pencilled lines, such as are here represented by poppd lines in the scheme, and number the triangles with 1, 2, 3, 4, &c. Your map being thus prepared, rule & table with four columns; the first of which is for the number of the triangle, the second for the base of it
, the third for the perpendicular, and the fourth for the content in perches.
Then proceed to measure the base of number 1,
place that in the second column of the table under the word base ; and from the angle opposite to the base, open your compasses so, as when one foot is in the angular point, the other being moved backwards, and forwards may just touch the base line, and neither go the least above or beneath it; that distance in the compasses, measured from the same scale, is the length of that perpendicular, which placed in the third column, under the word perpendicular.
If the perpendiculars of two triangles fall on one and the same base, it is unnecessary to put down the base twice, but insert the second perpendicular opposite to the number of the triangles 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 measured all the triangles ; and then by prob. 6. find the content in perches of each respective triangle which severally 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 same base; find the content of each (being a trapezium) in perches, by prob. 11. which place opposite the middle of those perpendiculars, in the fourth column, under the word content.
Having thus obtained the content of each respective triangle and trapezium, which the map contains, add them all together, and their sum will be the content of
the map in perches ; which being divided by 160, gives the content in acres. Thus, for
No Base Peipnd
15.0 S 8 38.7 6.7
259.20 15 10.0 %
Content in perches 4142.57
This being divided by 160, will give 25A. 3R. 22P. the content of the map.
Let your map be laid down by the largest scale your paper will admit, for then the bases and perpendiculars can be measured with greater accuracy thun when laid down by a smaller scale ; and if possible measure from scales divided diagonally.
If the bases and perpendiculars were measured by pezium, may be had as before, in problems 6 and 11. and consequently the whole content of the map.
If any part of your map has short or crooked bounds, as those represented in fig. 102. then by the straight edge of a transparent horn, draw a fine pencilled line as AB, to balance the parts taken in and left out, as also another, BC; these parts when small, may be balanced very nearly by the eye, or they may be more accurately balanced by method the third. Join the points A and C by a line, so will the content of the triangle ABC, be equal to that contained between the line AC, and the crooked boundary from A to B, and to C: by this method the number of triangles will be greatly lessened, and the content become more certain ; for the fewer operations you have, the less subject will you be to err : and if an errour be committed, the sooner it may be discovered.
The lines of the map should be drawn smail, and neat, as well as the bases; the compasses neatly pointed, and scale accurately divided; without all which you may err greatly. The multiplications should be run over twice at least, as also the addition of the column content.
From what has been said, it will be easy to survey a field, by reducing it into triangles, and measuring the bases and perpendiculars by the chain. certain the content only, it is not material to know at : what part of the base the perpendicular was taken: since it has been shewn (in cor. to theo. 13. sect. 1.) that triangles on the same base, and between the same parallels are equal : but if you would draw a
map from the bases and perpendiculars, it is evident that you must know at what part of the base the perpendicular was taken, in order to set it off in its due position; and hence the map is easily constructed.
To determine the area of a piece of ground, having
the map given, by reducing it to one triangle equal thereto, and thence finding its content.
Let A B C D E F G H be a map of ground, which you would reduce to one triangle equal thereto.
Produce any line of the map as AH, both ways; lay the edge of a parallel ruler from A to C, having B above it ; hold the other side of the ruler, or that next you, fast ; open till the same edge touches B, and by it, with a protracting pin, mark the point b on the produced line ; lay the edge of the ruler from b to D, having C above it ; hold the other side fast, open