crooked line from a map of the survey, or ascertain it from an actual survey, which may be done by drawing any line, QR, passing either through several points of the crooked boundary, or close by it, and measuring offsets from QR to the several corners or indentations in the crooked boundary. From these dimensions, lay down, upon any scale, the old mearing, draw the right line AH, equalizing the portions added to, and taken from, each property; join BA, and measure, upon the plan, the angle BAH, by the protractor or the scale of chords; then proceed to the ground, plant the theodolite at A, and a staff at B, and measure the same number of degrees, from B round to S, as had been measured on the plan; then prolong the line, in the direction of AS, to H.

PROP. 6.- To change a figure from one scale to another.

to EF; then shall the figure bcdef A be similar to the figure BCDEFA, and in the given proportion.

OF THE DIVISION OF LANDS.

When land, held in common by any number of proprietors, is to be divided, survey and compute the whole first, deduct the quantities to be allowed for roads; then divide the remaining quantity among the proprietors by the rule of double fellowship, in proportion to the real value of their estates. If all the land to be divided be of equal value, the rule of single fellowship will solve the problem. In this case, you add the numbers expressing the proportions of claimants' estates, into one sum; then say, as this sum is to the number of acres to be divided, so is the number expressing the proportion of each particular proprietor, to his share of the commons to be divided. Now, having each man's portion, it may be separated by the directions given in the last problem but one.

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When the entire of the land to be divided is not equally valuable, and when any particular part to be marked off for any one's share is better or worse than the general mean quality of the land, then you must diminish or increase the quantity of his share, in the same proportion ; or which comes to the same thing, divide the ground among the claimants, in the direct ratio of the value of their claims, and the inverse ratio of the quality of the ground allotted to each;" that is, in proportion to the quotients arising from the division of the value of each person's estate, by the number which expresses the quality of the ground in his share.

It is required to divide 400 acres of land among ABC, whose claims on it are respectively in proportion to the numbers 2, 3, 5.

It is required to divide 800 acres of land among A, B, C, whose estates are worth £1000, £2000, £3000 a year; the ground in their shares being worth 5, 10, and 20 shillings an acre, respectively.

Here the claims are as 1, 2, 3; and the qualities of their land are as 1, 2, 4: therefore their quantities must be as,, ; or as 1, 1, 2: then

1:29010 acres = A's share. 1+1+1=2:800::1:29010 acres=B's share. 3 : 218 acres = C's share.

4

2

These respective quantities may be cut off by the method described in one of the preceding problems.

PROP. 7.- To measure any inaccessible distance by the chain as the breadth of a river.

Let us suppose BC to measure 4 perches, BE 5, and ED 6; required the distance AB.

6

4

2:5::4:10 perches, the distance AB.

This problem is very useful in conducting a chain survey. Let us suppose, in the progress of the survey, that one of your lines runs in the direction of EB; you then place a mark, in a line with EB, at A; and proceed as above. But if you wish to shew, on your plan, the method employed in crossing the river, EC or BD, which, with the measurements already made, will enable you to plot the work to a scale; then applying the scale to BA, you at once see the distance.

PROP. 8. To reduce a Field, in the form of a trapezium, to a triangle of equal area.

Let it be required to construct a triangle CEB, which shall be equal in area to the trapezium ABCD.

D

C

A

B

E

Produce the line BA, towards E; lay a parallel ruler on AC, and slide it back to DE; then join EC, and

the triangle ECB will be equal to the trapezium ABCD.

This problem sometimes facilitates the finding of areas, as it easily transforms the trapezium into a triangle, the area of which is found from its base and perpendicular.

PROP. 9.- To reduce a five-sided figure to a triangle, having the same area.

Let ABCDE be the figure to be transformed into a triangle of equal area. Produce AB both ways,

towards F and G. Apply the parallel ruler to AD, move it back to EF, and join FD. Again, apply it to BD, move it back to GC, and join GD; then the triangle FDG is equal in area to the figure ABCDE.

PROP. 10. In a triangular Field, to draw from one of its angles, right lines to the opposite sides, dividing it into any number of parts, which shall be to each other in any assigned proportion.

Divide the base AB into the same number of parts,