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The following Field-Notes (from A. BURNS) are of a piece of land, which is proposed, as an exam, ple, to be divided into three equal parts by two right lines running from the sixth and seventh stations : and proved, by calculating the content of the middle part.
PL. 8. fig.5.
The plot ABCDEFGHA is proposed to be divided, geometrically, in the proportion of 2 to 3, by a right line from a given point in any boundary or angle thereof, suppose the point D.
Reduce the plot to the triangle c De, as already taught; divide the base ce in the point N, so that cN be to Ne in the ratio of two or three, by prob. 14. page 53; draw DN, and it is done.
Example 2d may likewise be performed geometrically.
Produce CD both ways for a base, and reduce the whole to a triangle, making 1 the vertical point, then bisect the base in N, and draw IN. But,
Notwithstanding this geometrical method is demonstrably true in theory, it is not as safe, on practical occasions requiring accuracy, as the calculation, even when performed with the greatest care ; for which reason we will not enlarge on it here.
Suppose 864 acres to be laid out in form of a right angled parallelo
gram, of which the sides shall be in proportion as 5 to 3 ; required their dimensions ?
For the greater side, multiply the area by the greater number of the given proportion, and divide
by the less, or, for the less side, multiply by the less number, and divide by the greater; the square root of the quotient will be the side required: thus,
If it be required to lay out any quantity of ground, suppose 47 A. 2R. 16P. in form of a parallelogram, of which the length is to exceed the breadth by a given difference, for instance 80 perches, then add the square of balf this difference to the area, and take the square root of the sum; to which add half the difference for the greater side, and subtract it therefrom for the less; thus, 2)80 47A. 2R. 16P.=7616 perches,
1600 half diff. add and subt.-40
Any proposed quantity of ground may be laid out or inclosed in the form.
by prob. 2d. Parallelogram,1 side giv. by pro. 4th. sec. of a
Triangle of a given base, by pro. 7th. 4.
by prob. 13th. It is sometimes most convenient, when land is to be laid out adjacent to a creek, river, or other crooked boundary, to measure off-sets to the angles or bending thereof, from a right line or lines taken near such boundary, and to deduct the area of these off-sets from the given quantity, and then to lay off the remainder from the right line or lines, in the desired form.
In laying out new lands, attention must be paid to the allowance for roads, as exemplified in prob, 14th,
OF SURVEYING HARBOURS, SHOALS, SANDS, &c.
Px. 13. fig. 1.
THERE are three methods, whereby this may be performed; for the observations may be made either on the water or on the land. Those made on the water are of two kinds, one by the log-line and compass (as in plane sailing measuring) the course and distance round the sand; and then to be plotted as a large wood, or any inclosure taken by the circumferentor.
This method I omit for two reasons; first, because it is to be deduced from the writers of navigation: and, secondly, because the distances thus measured are liable to the errors of currents, which generally attend shoals or sands near the shore.
The second method, when there are no distances to be measured on the water, though still there is one inconvenience, cominon also to the former, because the bearings or observations are to be taken on that unstable element (an error scarce mentioned by practical artists) I shall briefly hint at; and so rather choose a third, which is liable to neither of these imperfections.