has again stretched the chain and stuck the second arrow in the ground. Then the follower takes up the first arrow and walks towards the second. The process is continued until the proposed length is measured.

Whenever the follower has the ten arrows in his hands he records in the field-book that a length of ten chains has been passed over; then he gives the ten arrows again to the leader, and the work is continued. Thus on arriving at the second picket the field-book shews the number of tens of chains passed over, the arrows in the follower's hands correspond to the number of additional chains, and the number of links between the last arrow and the second picket can be counted. Thus the required length is found.

395. In measuring with the chain, great care must be taken to preserve the proper direction; and there is in general a double test of the accuracy with which this is effected. When the leader fixes an arrow he should take care that the straight line between this arrow and the first picket will pass through the follower's arrow; and at the same time the follower should take care that the straight line between his arrow and the second picket will pass through the leader's arrow.

396. If a field be in the shape of any rectilineal figure, and we measure the lengths of the appropriate straight lines, we can find the area of the field by the corresponding rule in the Third Section.

397. Examples,

(1) A rectangular field is 8 chains 95 links long, and 3 chains 26 links broad.

8 chains 95 links=8'95 chains; 3 chains 26 links = 3.26 chains. We use the rule of Art. 134,

The area of the field is 29.177 square chains, that is, 29177 acres; we may reduce the decimal of an acre to roods and poles: thus we obtain 2 acres 3 roods 27 poles very nearly. See Art. 126.

(2) The sides of a triangular field are 52 chains, 56 chains, and 6 chains respectively.

8.4 x 32 x 2.8 × 24 = 180.6336. The square root of 180 6336 is 13'44.

Thus the area is 13'44 square chains, that is, 1.344 acres, that is, 1 acre 1 rood 15'04 poles.

(3) The radius of a circular grass plot is 2 chains 50 links.

We use the rule of Art. 168,

2.5 x 2.5 × 3·1416=19.635.

Thus the area is 19.635 square chains, that is, 1.9635 acres, that is, 1 acre 3 roods 34 16 poles.

398. Some of the rules for finding the areas of rectilineal figures require us to know the length of the perpendicular from some given point to some given straight line. When the situation of such a perpendicular is known, the length of it can be measured in the manner explained in Art. 394; we shall now shew how the situation is determined.

399. To determine the situation of the perpendicular to a given straight line from a given point without it.

Let AB be the given

straight line, and the given point without it.

Fold a string into two equal parts. Let one person hold the middle at C; let two other persons hold

A D

the ends, and stretch the two parts, so that the ends may lie on the straight line AB, say at D and E. Take F the middle point of DE. Then CF is the perpendicular required.

400. The straight line AB in the preceding Article is supposed to be clearly marked on the ground in some manner. This may be effected by stretching a string or chain tightly between A and B, or by placing pickets at short distances in the direction passing through A and B. If, however, the straight line AB has not been thus marked out on the ground, a person standing beyond A must take care that the end of the string is properly placed at D, and then standing beyond B he must take care that the end of the string is properly placed at E.

401. To determine the situation of the straight line at right angles to a given straight line from a given point in it.

Let AB be the given straight line, and F the given point in it.

Take D and E points in AB, so that FD and FE may be equal. Fold a string, longer than DE, into two

A Ꭰ

equal parts. Let the ends of the string be fixed at D and E, and let a person take the middle and stretch both the parts. Suppose C the point to which the middle of the string is thus brought. Then FC is at right angles to AB, and is therefore the straight line required.

402. We see then that the situation of any required perpendicular can be determined with the aid of a string; but an instrument called the Cross is often employed by land surveyors for this purpose.

403. The Cross. This is usually a round piece of wood, about six inches in diameter, having two fine grooves in it, which are at right angles to each other. A staff with a pointed end is stuck upright in the ground, and the cross is fixed on the top of the staff so as to form a small round table.

404. To determine with the aid of the cross the situation of the perpendicular to a given straight line from a given point without it.

Let AB be the given straight line, and C the given point without it.

Place pickets at A, B, and C. Select a point in AB which appears by inspection to be at or near the intersection of the required perpendicular with AB; let D denote this point. Fix the staff at D, and place the cross with one of its grooves

T. M.

A

D

16

parallel to AB; so that in looking along the groove in one direction, the picket at A is seen, and in looking along the same groove in the other direction the picket at B is seen. Look along the other groove; if the picket at C is seen in the line of this groove, then the point D is the intersection of the perpendicular from C with AB: but if the picket at C is not seen in the line of the groove, the staff must be taken up and moved along AB, to the right or to the left of the assumed position, according as the picket at C appeared to the right or to the left of the groove. By a little trial the proper position will be found for the staff, such that the pickets at A and B can be seen in looking along one groove, and the picket at C in looking along the other; and this position of the staff determines the situation of the perpendicular on AB from C.

405. To determine with the aid of the staff the situation of the straight line at right angles to a given straight line from a given point in it.

Let AB be the given straight line, and D the given point in it.

Fix the staff at D, and place the cross with one of the grooves parallel to AB. Then the other groove determines the direction of the straight line at right angles to AB.

406. Thus in the preceding Chapter and the present we have explained how the operations are to be performed which will furnish the lengths required for calculating the areas of fields. We will now give some examples of the calculations,

407. Examples.

(1) The base of a triangle is 132 chains, and the height is 83 chains.

The area of the field is 54'78 square chains, that is, 5.478 acres, that is, 5 acres 1 rood 36-48 poles.