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GEODESY is a higher kind of surveying, which takes into account the curvature of the earth's surface. It has for its object to determine, with the utmost possible accuracy, the geographical positions of points on the earth's surface by the process of triangulation already repeatedly described in this work, but requiring for the present purpose certain modifications, which it will be the object of the following pages to unfold.

The triangles composing the chain best fulfil their destination when the largest possible, and nearly equilateral. The sides are ordinarily from ten to fifty miles, and limited only by the want of distinct vision with the instruments, or interruptions from the nature of the ground. The primary chain being finished, a secondary chain of smaller triangles, having their vertices within the larger, is surveyed, and a still smaller chain of tertiary triangles. A great number of small bases and points of reference are thus determined, from which the surveys with the plane table or compass may originate, to complete the map in all its details. One of the chief problems, after the triangulation is finished, is the determination of the difference of latitude and longitude between the vertices of the triangles, so that when the absolute latitude and longitude of some of the vertices shall have been determined by astronomical observations, that of all the others may be known by differentiation. Previous to the explanation of the method employed for this purpose, it will be necessary to give some account of the mode of conducting the triangulation and, first, of the


The measured base, from which the triangulation commences, should be selected upon ground which will admit of its extending several miles. Its length should be ascertained with the greatest care, and for this purpose it has been customary to use metallic rods, allowing for their expansion and contraction from changes of temperature, the amount of which

had been previously determined by experiment, with a thermometer which was always observed when making the measurements.* For greater accuracy in placing the ends of the rods together, an optical contact has been employed. This is produced by placing horizontally upon tressels the rods, one a little above, and its end projecting over that of the next, a notch being cut out of the ends of the rods, across which a thread is horizontally stretched; a microscope supported on a stand, from which an arm holding the microscope projects over the ends of the rods, so that the microscope can be placed vertically over the threads, looking down upon them; the rods are then moved by screws in the tressels, till the threads are seen to coincide in the microscope, or till each coincides with a mark on the stand below. The optical contact is then complete.

The best base apparatus now in existence, probably, is that employed upon the coast survey of the United States, and of this we shall attempt a more particular description. It is made self-adjusting under changes of temperature. The measuring rods consist each of two bars, the one above the other; the lower of brass and the upper of iron, which is less expansible. The two bars are connected firmly by a cross-piece at one end, but allowed each to expand freely, being unconnected at the other, at which, however, there is a short lever of the second order placed vertically, having its fulcrum or extremity at the end of the lower bar, on which it works as a hinge, and its point of resistance at the end of the upper bar, the lever continuing above this, and having the power applied horizontally to its upper end, through a steel rod, which projects horizontally from the next measuring rod, above which it is sustained by a short support. The contact between this steel rod and the upper end of the lever is that of a blunt knife edge against a plane of agate. The other end of the steel rod works freely against the lower end of a very short vertical lever of the first order, the upper end of which supports a spirit level. The end of the spirit level farthest from the steel rod has a tendency to fall, occasioned by a counterpoise weight, which projects from it, but this is prevented by pressure of the steel rod against the lower end of the short lever, on the top of which the spirit level rests. This delicate mode of contact, first suggested by Bessel, depends on the sensibility of the spirit level. The tressels on which the rods rest, at two

* For each degree of the centigrade thermometer, platina expands 0.000008565 of its dimensions in every direction, iron 0.000010666, and brass 0.000017843. For 10 Fah. the expansion is for brass, 0.00001050903, iron 0.000006963535.

+ The effect of this arrangement is evident; the whole rod, composed of the two bars, lengthens by the effect of heat, but the lower more than the upper bar, so that

points only, are strongly trussed, and their upper parts at least metallic, and admit, by means of screws, of the various motions required in placing the apparatus. In measuring the base line, the measuring rods need not be placed exactly in a horizontal position, but the measure they give can easily be reduced to a horizontal one, if only the inclination of the rod to the horizon be known*.

The contrivance to indicate this consists of a sector, the plane of which is vertical, attached to the rods and graduated; a brass radius of this sector, moving about its centre, and supporting a spirit level, stands at the zero of the sector when the rod and spirit level are both horizontal. When the rod is inclined, the number of degrees passed over on the sector, in making the spirit level horizontal, shows the inclination of the rod. the upper end of the vertical lever, which makes the contact, is thrown back. The ratio of the length of the arms of the lever will depend on the ratio of expansibility of brass and iron.

*The formula would be evidently

B =l cos a

B being the horizontal measure required, I the oblique measure given by the apparatus, and a its inclination shown by the sector. But a being very small, it is better to compute the correction l -B to be subtracted from 1 to obtain B. It is

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a being very small. In practice it is customary to tabulate this formula.

The base must next be reduced to the level of the neighboring sea. Leto be the radius of the earth (or better the normal) for the level of the sea; p + h the radius for the level of the base, h being the mean height of the ground on which the base is measured above the level of the sea, and b the reduced base.

Then, since similar arcs are as their radii, we have


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Which is the correction, always subtractive.

But p being very great in comparison of h, it is sufficient for all ordinary values of B to take

B- b=

The horizontal alignment is made by means of a small transit, altitude, and azimuth instrument, or theodolite, and pickets with cards of pasteboard in their tops, placed along the line. The telescope of the instrument is directed to the distant extremity of the base, and then brought down on its horizontal or supporting axis, to indicate the proper position of each picket. The correctness of the ratio of the length of the arms of the vertical lever, to prevent changes in the distances between the contacts from changes of temperature, is tested by a pyrometer, invented by Mr. Saxton, of the Coast Survey. The rod is placed level upon two marble piers, sunk in the ground, which support its ends. The rod is then subjected to changes of temperature, artificially created, and being prevented from expanding in one direction by an upright stancheon, against which it abuts, its expansion, if any, in the opposite direction, acts against the vertical axis of a small plane mirror, giving the mirror an angular rotation about its vertical axis, the plane of the mirror always continuing vertical. At the opposite side of the room is a telescope placed horizontally on a pier, and directed to this mirror, and directly under the object glass of the telescope, is a horizontal scale of three feet in length, the divisions of which are of an inch. The movement of one of these divisions over the vertical wire of the telescope, occasioned by a motion of the small mirror in which it is reflected, corresponds to a change in length of the rod of 5 of an 25000 inch. The distance of the piers, which is liable to change, by hygrometric changes, is tested by a standard bar at the temperature of freezing. The standard bar is compared with a bar from France, the only one in the country. Its length is an exact French metre, which is the ten millionth part of a quarter of the meridian, the value of which in units already in use has been found by means of two measured degrees in distant latitudes. (See Appendix VI. p. 368.)

A bar of brass and iron exposed to the same temperature will not heat equally in equal times; this is well known to depend upon the different conducting powers of the two metals, their different specific heats, and the different powers of their surfaces to absorb heat. The bars, then, if of equal sections, when the temperature is rising or falling, will not have the same temperature, and the system is not compensating. By adapting the sections according to rules deduced, partly by theory and partly by experiment, a small residual quantity remains to be corrected, which is detected. only by the delicate tests of the Saxon pyrometer, or the lever of contact and level of Bessel. This correction is made by applying a covering more absorbent of heat to one bar than to the other.

The rods are surrounded by a tin covering, and of about twenty

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