gle 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

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 than when laid down by a smaller scale, and if possible measure from scales divided diagonally.

If the bases and perpendiculars were measured by four-pole chains, the content of every triangle and trapezium, 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 plate 7. fig. 5. then by the straight edge of a transparent horn, draw a fine pencilled line, as AB, to balance the parts taken 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 error 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 shown (in cor. to theo. 13. geom.) 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.

PL. 8. fig. 5.

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 con

tent.

grou

Let ABCDEFGH be a map of which you would reduce to one triangle equal thereto.

ways,

Produce any line of the map, as AH, both lay the edge of a parallel ruler from A to C, having Babove 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 the same edge touches C, and by it mark the point c, on the produced line. A line drawn 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 Fabove 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 De, equal to the irregular polygon A B C D E F G H.*

The demonstration of this is evident from Prob, 19. Geo. Page 79, of twe Book.

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 edge; and a fine pointed pair of compasses.

Thus,

Lay that edge on the points A and C, take the distance 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.