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TO FIND THE CONTENT OF A FIELD WITHOUT PLOTTING. 259

should be entered in a column marked E., and those west in a column marked w., the column of double meridian distances being made double for the purpose. By means of this double column of double meridian distances, and the double column of differences of latitude, the content of the field may be computed by the following rule.

The difference between the northings multiplied by the eastings plus the southings multiplied by the westings, and the northings multiplied by the westings, plus the southings multiplied by the eastings, will be equal to double the area of the land.

The proof of this is left as an exercise for the student.

In the following example the bearings were taken with a compass resembling the mariner's in principle. A metallic graduated circle, one diameter of which, that joining the zero and 180° points, being an attached needle, the graduated circle was held stationary in space by the magnetic force of the earth. The numbering was from zero to 360 in the direction shown in the annexed diagram. The compass sights were plain, and the number on the line of sights towards the extremity next the eye was the one read and recorded in the 1st column p. 261. The equivalents of these readings in bearings of the compass courses or sides of the field from the meridian, are recorded in the 2d column. These are ascertained by considering in what part of the circumference in the diagram above the No. in the 1st column would fall; the course would be in the direction from this point to the centre of the circle.

90

180

270

The third column contains the lengths of the courses or sides of the field, measured with a chain. Then follow the columns of difference of latitude and departure,* and the columns of corrected diff. of lat. and departure. The assumed meridian from which to estimate the double meridian distances is taken through the point at which the survey commenced. The double meridian distance of the first coure then, according to the rule, will be equal to its departure 234-1, and is w. because to departure is w. Double the meridian distance of the second course is equal to that of the preceding course 234·1+ the departure of the preceding course 234•1 + its own departure 50.3. (See rule.) All these numbers being w., their sum in the arithmetical sense, 518.5, is taken as the double meridian distance of the 2d course. For the next course the departure 17.6 is E., and on the general analytic principle that quantities estimated in a contrary sense must have contrary signs, this may be

* The sum of the column N. 264.5 exceeds that of the column s. 263.0 by 1·5; half this, or 8 is subtracted from the numbers of the first column N., 6 from the 1st No. in the column, and 1 from each of the other two, to obtain the Nos. in the 2d column N., and ⚫7, in the same manner, is added by distribution among the five numbers of the 1st column s., to produce those of the 2d column s., &c.

+ It would be most simple to assume it through the westernmost point of the land, in the present example at the commencement of the 3d course, where the reading was 1630. Here the courses, which were previously all w., begin to turn E. The advantage of this is that the double meridian distances would be all E.

tances.

considered positive, if we regard the previous numbers employed in computing the double meridian distances, which are all w. as negative. To add this 17.6 then, in the algebraic sense, to the sum of 518.5, and 50-3, according to the rule, will be in reality to subtract it, which gives 551.2 for the 3d D. M. D. For a similar reason the sum of 17.6 and 73.2 being both E., must be subtracted from 551.2, which is w., to produce the 4th D. M. D.; and so on till we arrive at the 7th course, marked 80°, in the 1st column. Here the sum of the two departures, 96.7 and 59-2, both E., viz., 155.9 exceeds the last D. M. D., 135.9, which is still w., and as they have contrary signs, their algebraic sum will be their difference with the sign of the greater, which is E., and this difference 20.0 must be entered in the column E. of double meridian disThe D. M. D. of the last course is obtained by adding 59.2 and 20.0 both E., and subtracting 40.1, which is w., from their sum. The double meridian distances E., which have corresponding difference of latitude, N., are now multiplied by them according to the rule; and the double meridian distances w., which have corresponding differences of latitude s., and the products all entered in a column entitled N. X E. + s. X w., and their sum taken. Of the N. X w. + $. X E., which the remaining part of the rule requires to be formed, there is but one product in this example an N X w. 196°2 by 234.1, or 45930-42, for which an additional column, which would ordinarily be employed, is not worth while. This product is subtracted from the sum of the former, and the remainder, 68103.90, is by the rule equal to double the area of the land in square links, 10,000* of which make a square chain. Half this will be the area, which is converted into square chains by removing the decimal point 4 places to the left; and this again into acres, by removing the decimal point one place further to the left still, since there are 10 sq. chains in an acre. The decimals of an acre are converted into roods and perches by multiplying by 4 and by 40.

The plotting of the above example will be an exercise. A circle should be described on the paper, and points marked on it, according to the compass readings in the first column, the N. and s. line corresponding to the 0° and 1800 points, as in the last diagram. Lines drawn from the points thus marked to the centre of the circle will be parallel to the boundary lines of the survey. For further directions see p. 237 at bottom.

* Which is the square of 100, the No. of links in a chain.

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HYDROGRAPHIC SURVEYING.

In the survey of harbors, after having surveyed and plotted the outline of the shore, it becomes necessary to set down upon the map the depths of the water in feet or fathoms at a sufficient number of points to serve as a guide to navigators. The depth is ascertained by sounding, and the problem is to fix upon the map the points at which the soundings were made. One method consists in rowing a boat uniformly in a straight line from one point on the shore to another opposite, casting the lead at regular intervals by a watch; this line being drawn on the map and divided into as many equal parts as there were casts, the points of division will be the points required; upon these the numbers obtained by the soundings are to be put down. Another method is to place three signals upon the shore, not in the same straight line, and with a sextant* in the boat to measure the angles subtended by the lines joining these signals; then having these lines plotted upon the map, construct upon each of them a segment capable of containing the observed angle subtended by it (see Plane Geom., Prob. 21), and the intersection of the arcs of these segments

will determine the points on the map

at which the boat was situated at the time of observation. The sounding of course should be taken at the same point, and recorded at its position (6) thus determined on the map.

A third method consists in having two theodolites, and taking the angles with them from the extremities of a

base line on the shore, by which means the position of the boat is determined. In this method a system of signals is requisite, by which the observers on shore may know the instant at which the sounding is made.

A very perfect one was invented by MR. THOMAS H. NORRIS, of New York, and practised in the survey of the mouths of the Mississippi. This consisted in having at one of the two stations on the shore (which in the low lands of the Mississippi were elevated platforms of wood) a flag which could be run up and a chronometer. The boat also carried a chronometer. The intervals of time at which the soundings should be made having been previously agreed upon, about 10 seconds before one of these intervals expired, the flag was run up, and both theodolites brought to bear upon the boat, or rather upon a staff at its bow, from which the

* See the instrument of this name described at p. 290, note.

sounding was made. The tangent screws served to keep the instruments steadily upon this point, and the instant the 10 seconds were up, the flag was lowered, the lead was cast, and the readings taken from the horizontal limbs of the instruments and recorded. This mode was found to be very rapid and accurate. By way of experiment the boat was frequently made to cross its track, and the agreement was exact. By rowing the boat along at oar's length from the shore, and determining its position at frequent intervals, as above described, the line of shore could be traced upon the map. This was found particularly convenient in the survey of the bayous or inlets of the low muddy banks.

Horizontal sections of the bottom of a harbor may be determined, making the plane of the water a plane of reference, in an obvious manner, and the bottom represented in the same manner as a hill. This, however, is not often practised.

Points at great distances out at sea are obtained by triangulating outward with three vessels successively moored at points more and more remote from the shore.

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