facilitated the comprehension of the action of the instrument had the dial been graduated to show the difference of the atmospheric pressure in absolute weight or pounds;" but seeing that "the density of the atmosphere would decrease in geometrical progression, for altitudes in arithmetical progression, and since this density also varies directly as the pressure to which it is subjected, and which is measured by the height of the barometric column, it follows that, if at different altitudes these columns decrease in geometric progression, the altitudes will increase in arithmetical progression, and will therefore be proportional to the logarithms of the barometric columns. Hence, if the temperature remained constant, the difference of two altitudes would vary as the difference of the

logarithms of the barometric columns at those altitudes; so that if h be taken to represent the height of a higher station above a lower one, and if в be the height of the barometer at the lower station, and the height at the higher station, we should have

where k is a constant quantity, to be determined by experiment.' I cannot do better than refer the student to the very exhaustive consideration of the theory of the aneroid in the work by Mr. Heather, from which the foregoing is extracted.

It should be stated that the variation of temperature greatly affects the results of observation with the aneroid. To guard against any error a thermometer is attached so that the difference of temperature may be noted at the various stations.

Fig. 139 is a plan of the aneroid with the dial removed, and Fig. 140 is an isometrical view of the same. The instrument is from 4 to 5 in. in diameter (some even larger) and about 1 in. thick. "The pressure of the atmosphere is indicated by the hand h (Fig. 139) pointing to a scale, which is graduated to correspond with

the common barometer." There is also a scale compensated to agree with the altitudes in feet, which is attached by a movable rim so that its zero may be regulated as necessary. Referring to Fig. 140, A is the screw adjusting the hand, B B the fulcrum, c c the principal lever, DD the vacuum vase, 1 vertical rod connecting lever co with levers 2 and 3, e b adjusting screws for leverage, s spiral spring, м socket in vacuum vase, K pin attached to socket. The cost of an aneroid barometer varies from £3 3s. to £8 8s. The Stadiometer. This is one of those instruments for expeditiously measuring distances (illustrated in Fig. 141). It consists

of a telescope м B, fitted to a vertical arc o, which works in the frame PG which is attached to the body-piece of the instrument, and above which a round disc or table A rotates. "There is a scale D D' fastened to the frame, the centre of which corresponds with the centre of the instrument, and which is graduated to the scale to which the surveyor wishes to plot his work. The teleScope MB is fitted with a diaphragm, with two horizontal hairs, distant from one another a hundredth part of a foot of the focal length of the object glass. From this proportion it follows that, when any ordinary levelling-staff is held on any distant point and the telescope brought to bear upon it, if the readings, in feet and decimals of a foot, of the intersections of the hairs on the staff be

G

observed, their difference multiplied by a hundred gives the true distance in feet of the staff from the instrument."

The Omnimeter." This instrument (Fig. 142), like the stadiometer, is intended to measure base lines and distances without chaining, and also the differences of altitudes and angles.'

It consists of a graduated limb (1) reading by means of a vernier to ten seconds for the measurement of horizontal angles; a good telescope revolving in a plane perpendicular to the limb; a powerful microscope (3) closely united to the telescope; a highly sensitive

Fig. 142.

level (4) lying upon the rule (5) which has a fixed length (of twenty centimetres for example); a scale (6) fixed vertically at the extremity of the said rule, in the plane of the optical axis of the microscope, and divided into half-millimetres, the millimetres only being indicated by figures from 1 to 100; a micrometrical screw (7) attached to the base of the scale, and giving a correct reading of the scale to the Too of a millimetre, denoted on the graduated circle of the screw; an extremely sensitive level (8) capable of being applied to the telescope, and for determining in case of necessity its horizontally.

OMNIMETER AND TACHEOMETER.

83

Accompanying the instrument is a staff of a fixed length-say three metres-having a white mark upon a black ground at either end. This staff is held at the point of which the distance from the instrument is required, and the telescope having been directed to the staff, which must be held perfectly plumb, the inclination is read off by means of the microscope from the scale. This being done the distance may be calculated. There is also another instrument for determining distances other than by measurement, called the tacheometer; but, whilst for military or approximate purposes these appliances may be very useful and expeditious, I do not hesitate to confess a predilection for ascertaining the lengths of my base and other lines by actual measurement.

CHAPTER IV.

TRIGONOMETRY AS APPLIED TO SURVEYING.

It is not intended in this chapter to do more than explain the general principles of trigonometry as applied to surveying.* It is a science of great scope and interest, involving a vast amount of patient study if its higher branches are required; but for "Practical Surveying" it is quite possible, in such a chapter as the present, to give a sufficiently general outline of trigonometry to enable the student to apply it thereto.

Trigonometry has for its object the solution of triangles, and its application to surveying is the " art of measuring and computing the sides of plane triangles, ‡ or of such whose sides are right lines."

Triangles consist of six parts, viz. three sides and three angles; and in every case in trigonometry three parts must be given in order to find the other three; and of those three given parts one must be a side, because with the same angles the sides may be greater or less in proportion.

It way be well, in order to render the consideration of this subject complete, to give here a few of the principal definitions of Euclid's geometry which bear upon trigonometry.

1. Plane Surface. A plane surface, or plane, is a surface in which if any two points be taken, the straight line between them lies wholly in that surface.

2. Plane Angle. A plane angle is the inclination of two lines to each other in a plane, which meet together, but are not in the same direction.

Note. This definition includes angles formed by two curved *The word trigonometry is derived from two Greek words rρiуwvov trigo-non), a triangle, and μerpew (met-re-o), measure.

+ Trigonometry within the limits of its earlier definition is geometrical; its advance beyond those limits is due to the introduction of purely algebraical methods. The quantities with which geometrical trigonometry deals are certain lines definitely placed with respect to an angle, and consequently varying with it.-Chambers's Encyclopædia.

"Plane Trigonometry" is the science which deals with straight lines, as compared with "Spherical Trigonometry," which involves the consideration of curved figures.