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, 864A. = 138240P. 1.98240 5 8 Answ. V 230400 = 480. V 82944 = 288. EXAMPLE IX. If it be required to lay out any quantity of ground suppose 47A. 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 half 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. 15P.= 7616 perches. 1600 ✓ 9216 = 96 1600 half diff. add and subt. -40 (the length= 136 Answ. the breadth Any proposed quantity of ground may be laid out or inclosed in the form 56 of a Square by prob. 2d. Parallelogram, 1 side giv. by pro. 4. sec. IV. Triangle of a given base, by pro. 7. Circle by prob. 13. 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, page 203. EXAMPLE X. It is required to divide off 30 acres, to the southeast end of the tract, of which the field-notes are given in example 4th, by a right-line to run N. 20 E. See example 4th. SECTION VIII. Containing the surveying of Harbours, 8c. and Levelling OF SURVEYING HARBOURS, SHOALS, SANDS, &c. Plate XIII. fig. 1. THA HERE 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 logline 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 enclosure 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, where there are no distances to be measured on the water, though still there is one inconvenience, common also to the former, because the bearings or observations are to be taken on that unstable element (an error scarce mention ed' by practical artists) I shall briefly hint at; and so rather choose a third, which is liable to neither of these imperfections. Let a boat be manned out with a signal flag, a log and line, lead and line, and to observe the bearings of any land-mark, a compass with sights. Take two or more objects or places, as A, B, C, on the shore, from whence the boat may be seen on the several parts of this shoal, and determine their relative position by bearing and distances either before or after the other necessary observations are made. One of the boat's crew is to sound till he finds himself on the edge of the sand, by the depth of water, and then to come to an anchor ; which he is to signify to two persons on the shore, at B and C, by his signal. And then from those known land-marks, B, and C, the observers are to take the bearings of the boat, and to register their observations, which, when done, they are to signify to the crew by waving a flag, or by some other signal. And in the mean time, to prevent mistakes, let the crew take the bearings of each of these landmarks: Then weigh anchor, which suppose at D. Then by sounding, proceed to E, and make like observations. And so at E, F, G, &c. till you have surrounded your sand. And if in this process you are about to loose the sight of one of your land-marks suppose C, let your assistant at C, or B, who at that time will also be; about to loose the sight of the boat, by signals (before agreed on) remove to some other object before-hand agreed on, suppose to H, or K; and then to proceed as before. Lastly, if the sand runs so far out at sea, that the object cannot be seen by the boat, nor the boat by the observer on shore; there may be rockets fired by the boat's crew, and also by the observers on the shore in the night, whereby those bearings may be taken almost at as great a distance as the light can be seen. For supposing they rise but a quarter of a mile above the apparent horizon, its stay will be about 9 seconds, and its distance for this quarter of a mile will be visible about '44 miles.. But rockets rise much higher, and then the distances are much greater, whereby they are visible. Or two boats may lay at anchor instead of the land marks and then you may work as before. Now, since the land-marks B and C are fixed, their position may be laid down in the draught, as in common surveying, by plotting the distance between B, and C. And then, by plotting the line, BD, and the line DC, according to their position, their common intersection will give the point D. And in like manner E, F, G, &c. may be plotted ; and so the shoals completed. And this from the bearings taken at B and c. If this be a standing lake, environed by bogs, or other impediments, the observations at D, E, F, &c. by taking their opposites, may suffice to plot the same from the land-mark, A, B, C, &c. as well as those taken on the land; or, indeed, by |