It was already shown, that the diameter of a circle containing 640 acres is 90,27 chains; if then we multiply this sum into itself, and divide the product by the transverse axis=100 chains, the quotient 81,456799 chains will be the conjugate diameter of an ellipse containing 640 acres : and with these dimensions, agreeably to the directions already given, the ellipse may be traced out on the ground.

In this manner may any regular figure be laid down on the ground, containing a given quantity of land: but where the boundaries of the figure are irregular curves, the surveyor must exercise his judgment, in arriving at the truth by gradual approximations, which will give results suffieiently accurate for the ordinary purposes of land-measuring.

Another part of the surveyor's business is the making exchanges of ground, where he is required to lay out such a quantity of land at a given value as may be equivalent to another quantity of a different value. In this case he has only to mu tiply the given quantity by its value, and to divide the product by the value of the land to be given in exchange for it, when the quotient will be the quantity required: thus, if 50 acres of land, valued at 45 shillings per acre, of yearly rent, are to be exchanged for land worth 37. per acre yearly, the product of 50 by 45 2,250 shillings, divided by 60 shillings, the value of the other land, will give for a quotient 35,5 acres, or 35 acres 2 roods, the quantity to be given in exchange for the 50 acres at 45 shillings.

In surveying a county or other extensive tract of land, the usual way is to select a number of commanding positions for stations, such as the tops of hills, steeples, towers, &c. which can be easily observed the one from the other, and from which a number of intermediate towns, villages, &c. may also be observed. The greater the distance between the stations for observation, the more convenient will it be to make the survey. Then with the theodolite, or other proper instrument, measuring the angles formed by the different

different objects observed at the first station, the distance. from it to the second station must be carefully and accurately measured and reduced to the true level or horizontal line, by making the proper allowances for the different elevation of the two stations, as well as for the effects produced by the state of the atmosphere. When this is performed, the surveyor observes the angles formed at the second station by those objects which he had observed before from the first station: and then removing the instrument to a third station, he again observes the objects already noticed; and if the points ascertained by the intersections of these three sets of observations shall coincide, it is to be presumed the angles have been accurately measured. In this way the angles, hearings, and intersections formed by the most remarkable objects, are to be determined all over the tract of country to be surveyed but the distances between the several stations need not be calculated, because the distance between the first and second stations, which ought always, if possible, to be four, five, or six miles, being ascertained, it becomes a sort of standard by which to compute all the other distances. for the sides of all plain triangles being proportionally as the angles respectively opposite to each, and having in the first triangle measured all the angles and one of the sides, the other sides are readily calculated to be employed in computing the sides of all the other triangles in sucession. however, it can seldom happen that the distance between any two stations, sufficiently elevated to give a proper com.. mand of the country round them, is so level as to allow it to be measured with due accuracy, unless with great attention and labour, the practice in surveys of this kind is to choose some open level tract from which the first and second stations can be observed, on which to measure a line, the longer the better, to serve as a base at which the calculation of the sides of the several triangles is to commence; and at the conclusion of the survey, as also in some intermediate situations

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situations, if requisite, to select another level tract on which another base for verification is to be measured, to check the distances obtained by the trigonometrical calculations. In performing the operations for ascertaining the true distance between the meridians of Greenwich and Paris, already noticed, the first base was measured on Hounslow Heath in a direction from north-west to south-east, and extending 27,404,0137 feet, or nearly 5 miles; and the base for verification afterwards measured on Romney Marsh, on the coast of Kent, lying in a direction nearly parallel to that of the first base, extended to 28,535,6773 feet, or a little more than 5 miles.