Having laid down the figure, erect the perpendicular AP for a temporary line.

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

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

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

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

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

Draw a line from N to 5 (P), and NAP will be the right-angled triangle required.

Computation of the Area by equidistant Ordinates.

See Prob. IX. Part III.

Here the sum of the first and last ordinates is nothing; (180 +270 +190) × 4 640 × 4 the even ordinates; and (250 +260) × 2

=

=

2560, four times the sum of

twice the sum of the odd ordinates; then

510 x 2

=

7020

2560+1020 +200

3

Computation of the Area by Casting.

Measure the perpendicular AP, which you will find to be 398 links; 398 × 1200 477600

then

= 238800 square links 2A. 1R. 22P,

the area required; which differs only two-tenths of a perch from that found by equidistant ordinates.

PROBLEM VI.

It is required to reduce the following curve-line offset area to a rightangled triangle by the parallel ruler.

Erect a perpendicular at one end of the base for a temporary line, and assume a competent number of points in the curve to denote angles.

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

Lay the ruler from 5 to 8, move it parallel to 7, and mark the temporary line at 6.

Draw a line from 8 to 6, and 8 1 6 is the triangle required; hence the area of the irregular offset may be found by multiplying the base 1 8 by half the perpendicular 1 6.

NOTE. When a curve-line offset area has to be reduced to a triangle by the parallel ruler, a competent number of points must be assumed in the curve, to denote angles. These points must be taken at such distances from each other that a right line drawn between any two adjacent points would nearly coincide with the curve.

PROBLEM VII.

It is required to reduce the irregular figure ABCDEFGHK to a triangle by the parallel ruler.

Produce the base AB both ways, at pleasure, for a temporary line.

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

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

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

Draw a line from F to 3, and it will be a side of the required triangle.

Again, lay the ruler from в to D, move it parallel to c, and mark the temporary line at 1.

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

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

Draw a line from F to 3, and 3 F 3 will be the triangle required; hence the area of the irregular figure ABCDEFGHK may be found by multiplying the base 3 3 by half the perpendicular Fm.

NOTE. The method of reducing fields of four or five sides to triangles of equal areas may be seen in Problems XVI. and XVII. Part I.

E

PROBLEM VIII.

It is required to reduce the figure ABCDEFGHKLMN to a triangle by the parallel ruler.

Draw the temporary line 1 2 to touch the angle a.

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.

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, and produce it, at pleasure, for a temporary line.

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

Lay the ruler from a to L, move it parallel to м, and mark the temporary line at n.

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

Draw a line from m to K, and produce it, at pleasure, for a temporary line.

Lay the ruler from K to G, move it parallel to H, and mark the temporary line at 1.

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

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

Draw a line from E to 3, and E 3 m will be the triangle required; hence the area of the figure ABCDEFGHKLMN may be found by multiplying the base Em by half the perpendicular 3 x.

PROBLEM IX.

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

Produce the line AB, at pleasure, for a temporary line. Lay the ruler from A to L, move it parallel to M, and mark the temporary line at 1.

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

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

Lay the ruler from K to G, move it parallel to H, and mark the temporary line at 3.

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