Lay the ruler from 4 to E, move it parallel to F, and mark the temporary line at 5.

Draw a line from 5 to E, and produce it, at pleasure, for a temporary line.

Lay the ruler from E to C, move it parallel to D, and mark the temporary line at 6.

Lay the ruler from 6 to B, move it parallel to C, and mark the temporary line at 7.

Draw a line from 7 to B, and B 7 5 2 will be the trapezium required; hence the area of the figure ABCDEFGHKLM may be found by multiplying the diagonal B 5 by half the sum of the two perpendiculars 7 m and 2 n.

PROBLEM X.

It is required to reduce the figure ABCDEFGHKLMNPR to a trapezium by the parallel ruler.

Lay the ruler from A to C, move it parallel to B, and mark the temporary line at 1.

Lay the ruler from 1 to D, move it parallel to C, and mark the temporary line at 2.

Draw a line from D to 2, and produce it, at pleasure, for a temporary line.

Lay the ruler from 2 to P, move it parallel to R, and mark the temporary line at 3.

Lay the ruler from 3 to N, move it parallel to P, and mark the temporary line at 4.

Draw a line from 4 to N, and produce it, at pleasure, for a temporary line.

Lay the ruler from N to L, move it parallel to M, and mark the temporary line at 5.

Lay the ruler from 5 to K, move it parallel to L, and mark the temporary line at 6.

Lay the ruler from 6 to H, move it parallel to K, and mark the temporary line at 7.

Draw a line from 7 to H, and produce it, at pleasure, for a temporary line.

Lay the ruler from H to F, move it parallel to G, and mark the temporary line at 8.

Lay the ruler from 8 to E, move it parallel to F, and mark the temporary line at 9.

Lay the ruler from 9 to D, move it parallel to E, and mark the temporary line at T.

Draw the line DT, and D T 7 4 will be the trapezium required; hence the area of the irregular figure may be found by multiplying the diagonal D 7 by half the sum of the two perpendiculars Tm and 4 n.

PROBLEM XI.

It is required to draw an equalising line, by the parallel ruler, through the crooked fence ABCDE, so that the two fields which it separates may be reduced to trapeziums.

Lay the ruler from A to C, move it parallel to B, and mark the temporary line in KG at 1.

Lay the ruler from 1 to D, move it parallel to c, and mark the temporary line at 2.

Lay the ruler from 2 to E, move it parallel to D, and mark the temporary line at 3.

Draw a line from E to 3 (L), and the irregular figure ABCDEFG will be reduced to the trapezium LEFG, and the irregular figure ABCDEHK to the trapezium LEHK; hence their respective areas may be obtained by measuring diagonals and perpendiculars.

NOTE 1. Sometimes the proprietors of adjoining estates agree to straighten crooked fences or brooks by giving and taking equal quantities of land. When this is the case, you must first measure and plan the ground, then draw the equalising line as directed in the last problem, and take the distance from a to L very correctly by the scale. Measure this distance in the field from the angle A; range the division line E L, and stake it out; and the work will be completed. 2. It will be advisable to measure, both on the plan and in the field, the parts cut off on each side by the division-line, in order to prove the work; for an error committed in dividing land is of serious consequence, if it be not discovered and rectified before the new fence is made.

PROBLEM XII.

It is required to draw an equalising line by the parallel ruler, so that the curved fence which separates the two fields in the following figure may be reduced to a straight fence.

Lay the ruler from 1 to 3, move it parallel to 2, and mark the temporary line in AB at 1.

Lay the ruler from 1 to 4, move it parallel to 3, and mark the temporary line at 2.

Lay the ruler from 2 to 5, move it parallel to 4, and mark the temporary line at 3.

Lay the ruler from 3 to 6, move it parallel to 5, and mark the temporary line at 4.

Lay the ruler from 4 to 7, move it parallel to 6, and mark the temporary line at 5.

Draw a line from 7 to 5, and it will reduce the figure ABCD to two trapeziums; hence their respective areas may be found by measuring diagonals and perpendiculars.

The following general rule for the parallel ruler will be found of considerable service to learners, and may be easily committed to memory.

GENERAL RULE.

1. Lay the ruler from the first to the third angle, move it parallel to the second angle, and you will have the first mark on the temporary line.

2. Lay the ruler from the first mark on the temporary line to the fourth angle, move it parallel to the third angle, and you will have the second mark on the temporary line.

3. Lay the ruler from the second mark on the temporary line to the fifth angle, move it parallel to the fourth angle, and you will have the third mark on the temporary line.

4. Lay the ruler from the third mark on the temporary line to the sixth angle, move it parallel to the fifth angle, and you will have the fourth mark on the temporary line.

5. Lay the ruler from the fourth mark on the temporary line to the seventh angle, move it parallel to the sixth angle, and you will have the fifth mark on the temporary line.

6. Lay the ruler from the fifth mark on the temporary line to the eighth angle, move it parallel to the seventh angle, and you will have the sixth mark on the temporary line.

7. Lay the ruler from the sixth mark on the temporary line to the ninth angle, move it parallel to the eighth angle, and you will have the seventh mark on the temporary line.

8. Lay the ruler from the seventh mark on the temporary line to the tenth angle, move it parallel to the ninth angle, and you will have the eighth mark on the temporary line, &c.

NOTE. As the operations of the parallel ruler, in straightening crooked fences, are founded upon a mathematical truth, it is preferable to a lantern horn; but

the latter may be applied with much more expedition than the former; and if it be used by a skilful hand, its results will be sufficiently correct for general practice. (See Problems I. and II. Part IV.)

NOTE 1. In the first edition of this work, the content of the estate in Plate VIII. was found from a plan of 2 chains to an inch. The bases, diagonals, and perpendiculars were measured by the scale used in planning; the offsets taken in the field were used where convenient, and when those were insufficient, more were measured by the scale. In this edition, the crooked fences have been straightened by the parallel ruler, the bases and diagonals measured by a scale of 2 chains to an inch, and the perpendiculars by a scale of 1 chain to an inch; hence the area of each triangle was found by multiplying the base by the perpendicular, and the area of each trapezium by multiplying the diagonal by the sum of the two perpendiculars.

The diagonals, perpendiculars, and areas are entered in the foregoing Book of Castings; and it may also be observed that the bases of triangles are put down in the column of diagonals, and their perpendiculars in the first column of perpendiculars.

2 In straightening crooked fences by the parallel ruler, it frequently happens that one equalising line will serve for two adjoining fields; and almost every irregular figure may be reduced either to a triangle or a trapezium. (See Problems XVI. and XVII. Part 1.; and also the Use of the Parallel Ruler, Part V.)