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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 small, 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. To ascertain 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.

PROB. XVI.

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.

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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

till

on the produced line. A line draw from c to D will take in as much as it leaves out of the map.

Again, lay the edge of the ruler from H to F, having G above it, keep the other side fast, open till the same edge touches G, and by it mark the point g, on the produced line; lay the edge of the ruler from g to E, having F above it, keep the other side fast, open till the same edge touches F, and by it mark the point f, on the produced line. Lay the edge of the ruler from f to D, having E above it, keep the other side fast, open till the same edge touches E, and by it mark the point e, on the produced line. A line drawn from D to e, will take in as much as it leaves out. Thus have you the triangle c D e, equal to the irregular polygon A B C D E F G H.

If when the ruler's edge be applied to the points A and C, the point B falls under the ruler, hold that side next the said points fast, and draw back the other to any convenient distance; then hold this last side fast, and draw back the former edge to B, and by it mark b, on the produced line: and thus a parallel may be drawn to any point under the ruler, as well as if it were above it. It is best to keep the point of your protracting pin in the last point in the extended line, till you lay the edge of the ruler from it to the next station, or you may mistake one point for another.

This may also be performed with a scale, or ruler, which has a thin sloped edge, called a fiducial, or sure edge; and a fine pointed pair of compasses. Thus,

Lay that edge on the points A and C, take the dis tance from the point B to the edge of the scale, so that it may only touch it, in the same manner as you take the perpendicular of a triangle; carry that distance down by the edge of the scale parallel to it, to b; and there describe an arc on the, point b, and if it just touches the ruler's edge, the point b is in the true place of the extended line. Lay then the fiducial edge of the scale from b to D, and take a distance from C, that will just touch the edge of the scale; carry that distance along the edge, till the point which was in C, cuts the produced line in c; keep that foot in c, and describe an arc, and if it just touches the ruler's edge, the point c is in the true place of the extended line. Draw a line from c to D, and it will take in and leave out equally; in like manner the other side of the figure may be balanced by the line e D.

Let the point of your compasses be kept to the last point of the extended line, till you lay your scale from it to the next station, to prevent mistakes from the number of points.

That the triangle e D c, is equal to the right-lined figure ABCDEFGH, will be evident from prob lems 18. 19. sect. 1. for thereby if a line were drawn from b to C, it will give and take equally, and then the figure b CDEFGH, will be equal to the map. Thus the figure is lessened by one side, and by the next balance line will lessen it by two, and, so on, and will give and take equally. In the same manner an equa

The area of the triangle is easily obtained, as before, and thus you have the area of the map.

It is best to extend one of the shortest lines of the polygon, because if a very long line be produced, the triangle will have one angle very obtuse, and consequently the other two very acute in which case it will not be easy to determine exactly the length of the longest side, or the points where the balancing lines cut the extended one.

This method will be found very useful and ready in small enclosures, as well as very exact; it may be also used in large ones, but great care must be taken of the points on the extended line, which will be crowded, as well as of not missing a station.

PROB. XVII.

A map, with its area, being given; to find the scale, to which it was laid down.

CAST up the map by any scale whatsoever, and it

will be,

As the area found:

Is to the square of the scale by which you cast up, :: The given area of the map :

To the square of the scale by which it was laid down.

The square root of which will give the scale.

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