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structions given in the 7th Prop. and in the 4th Case of Oblique-angled Trigonometry Let this be performed with respect to the triangle DBC, where stating this proportion, as the base BD to the sum of the opposite sides BC + CD, so is the difference of these sides BC-CD to the differ ence of the segments of the base formed by a perpendicular let fall on it from the angle at C. or B b-b D, which difference will be 2,114, and, consequently, by adding half this quantity to half the base, we obtain the greater segment Bb3,697, and the less segment b D 1,583: then, by means of either of these segments and the adjoining side of the triangle we discover the perpendicular, or altitude, of the triangle, to be 3,542.

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In the 5th Example of Mensuration of Surfaces, the methods were shown for calculating the area of a triangle, according to which the area of the triangle DBC will be 9,3509 square chains, or 9 square chains and 3,509 square links, to be brought into their proper values in landmeasure. It was already observed, that the measuring chain both in England and Scotland is so adapted to the acre of each country, that 10 square chains are equal to 1 acre but the contents of the triangle DBC not amounting to 10 chains, its area will not be an acre; the contents 9,3509 square chains, therefore, multiplied by the next inferior denomination (Reduction, page 202) or 4, the number of roods in an acre, and the product, 37,4036, divided by 10, the square chains in an acre, we obtain 3 roods and a decimal fraction, ,74036, which again multiplied by 40, the poles in 1 rood, we obtain 29 poles with another fraction, 6144, to be multiplied by the number of square yards in 1 pole, or 30,25, producing 18,5856 square yards: and the whole contents of the triangle DBC will be 3 roods, 29 poles, 18,5856 yards of English measure.

Proceeding in the same way with the triangle DFE,

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whose sides are known, its area will be found = 1 acre, roods, 0 poles, and 4,098576 yards: the triangle DFA will be found to contain 3 acres, 0 roods, 13 poles, and 26,609 yards; and the triangle DAB will contain 2 acres, oroods, O poles, and 21,6 yards.

In this manner we discover the quantity of land contained within the right-lined figure ABCDEF: but there remains still to be calculated the space inclosed by the measured line FA, and the crooked boundary of the field between the points F and A, consisting of 4 right lines, Fc, ce, ef, and ƒ A: let perpendiculars be drawn from the three angular points to the line FA, of which c d measures 56 links, that from e 36 links, and that from ƒ 34 links. By inspection of the plate it will appear that the two figures at the extremities F cd, and hf A, are right-angled triangles, and that the two intermediate figures are trapezoids; consequently if the spaces F d, dg, gh, and h A, be measured, we may calculate the areas of the several figures: let, therefore, Fd be measured 92 links, d g = 1,44 links, gh = 1,84 links, and h A = 2,12 links. Having now the bases and the altitudes of these triangles and trapezoids, we discover the area of the triangle Fed to be 2,576 square links, that of the trapezoid ce gd to be 6,532 square links, that of the trapezoid e f h g to be 9,108 square links, and that of the triangle fA h to be 6,784 square links. (See Examples 5th and 7th of Mensuration of Surfaces.) Then, adding these four quantities together, we have, for the area of the whole figure, cut off by the measured line FA, 3 square chains and 3392 links, which being brought into its proper value, will give 1 rood, 13 poles, and 12,9228 yards, to be added to the contents of the 4 triangles already computed. . It was already observed, that the measured line AB falls partly within and partly without the boundary of the field on that side, cutting it in the point a; this boundary con

sisting

sisting not of right lines, but of a sweeping curve, the area comprehended between it and the line AB can only be calculated by an approximation to the truth, as is done with circles and every other sort of curve: but this approximation may be carried so near to the true area, as always to differ from it by a quantity less than any assigned quantity: at the same time that in land-measuring such nicety of computation can never be required.

The line AB falls within the boundary of the field all the way from A to a, a distance of 4 chains: beginning at A set off on A a, a space equal to a chain or 100 links, and there erecting a perpendicular to the boundary line, let the distance be measured = 40 links; again, setting off another chain on A a, erect another perpendicular to the boundary measuring 52 links; lastly, setting off another chain more on A a, let the perpendicular there erected to the boundary, measure 36 links. As the boundary line from the angular points to the extremity of the adjoining perpendiculars, and between the extremity of two. adjoining perpendiculars, may be considered as consisting of right lines, the first and last divisions of the figure to be measured may be regarded as right-angled triangles, and the intermediate divisions as trapezoids. In such a case the area of the figure cut off by A a will be found = 12800 square links = 20 poles, 14,52 yards, to be added to the contents of the field already found.

On the other hand, as the measured line AB passes on the outside of the boundary of the field from a to B, the space inclosed between it and the boundary must be measured and calculated in the same way as the foregoing space; and its area, which will be found to be 17900 square links, or 28 poles, 19,36 yards, is to be deducted from the total contents of the field. In this last position the perpendicuJars are set off from a B at the distance of 1, 2, and 3 chains from az and their lengths are 44, 58, and 76 links.

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holes, by rivetting the extremities of the pins they became perfectly fast, and, as it were, united to the short plates, while the embraced ends of the long plates turned freely round on the middle part of the pins.

At every tenth link the joint now described had a position at right angles to the former, that is to say, the short plates lay horizontally, and the pins passing through them stood vertically; thus the whole chain containing 200 cast-steel pins, 180 lay horizontally, and 20, including the two by which the handles were attached to the chain, stood vertically. These cross joints, which were chiefly intended for the purpose of folding up the chain in small compass, by returning upon itself at every tenth link, were useful in another way, by presenting a horizontal surface to which small circular pieces of brass were screwed, with the figures 1, 2, 3, &c to 9, engraved on them, denoting the decimal parts of the length, as 10, 20, 30, &c. to 90 links.

The chain, on its first construction, was in length one hundred feet, including the two brass handles, in the extremity of which was a semicircular hole of the same diameter with the steel arrows employed to keep the account of the chains when applied to common measurements; but although the instrument performed its duty to entire satisfaction, yet, as it was afterwards supposed that still greater accuracy might be obtained by making a few slight alterations, the two end links were changed, each being made equal to one foot in length, exclusive of the handles.

The whole chain weighed about 18 pounds, and when folded up is easily contained in a deal box about 14 inches long, 8 inches bread, and the same in depth.

The perambulator is an instrument for measuring distances on roads, streets, &c.; it is also called the surveying-wheel, and a particular sort is named the pedometer, or way-wiser. This wheel is fitted to measure out a pole, or 16 feet, in making two revolutions, consequently its cir

cumference

çumference is 8 feet, and its diameter 2,626 feet. The instrument is commonly pushed forward by a person walking and holding the handle; but is also frequently connected to a carriage, and so made to indicate the distances travelled on a road. Within the machine are various movements acting on indexes on the face, to point out the space passed over in miles, furlongs, poles, yards, &c, The advantages of the perambulator are its read ness and expedition in measuring distances; but proper allowances must be made to bring the distance shown to the level line, because the instrument in its revolutions falling into hollows, ruts, or other inconsiderable depressions, must always give the distance somewhat greater than the truth.

II. The second branch of surveying, called plotting, protracting, or mapping, is performed by means of the scale and the protractor.

In the introduction to Trigonometry were given directions for constructing scales of chords, sines, tangents, and secants (p. 392), and also scales of equal parts (p. 385). ' That principally used in laying down a survey is called the plotting scale, being made of box, brass, or ivory, six, nine, or twelve inches long, by an inch and a half broad. This instrument contains several scales laid out on both sides: on the one are a number of scales of equal parts, the inch being divided into various numbers of such parts; also scales of chords, &c. for laying down angles; and sometimes the degrees of a circle are marked on one edge, answering to a centre on the opposite edge, by which means the instrument may be employed as a protractor. On the other side are several diagonal scales of different sizes, or numbers of divisions in the inch, serving to take off lines, expressed by numbers, to three dimensions, as units, tens, and hundreds, and also a scale where the foot is divided into 100 parts. Some of these scales have also a line of equal parts laid down on both edges, made thin for the purpose,

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