lengths X Y and X, Y, measured off on the scale as before, XY + X1 Y1

or the area

=

2

When the area is large, the instrument will move through a large angle, and consequently, if approximately square with A B at starting, it will be a long way out at the finish. In such a case all that is necessary is to see that the mean position of the instrument is square with A B.

The following precautions are to be observed:

Do not allow the hatchet to work on a rough surface

smooth writing or drawing papers are suitable; do not work on wood, as the hatchet tends to travel along the grain.

Do not allow the hatchet to go off the edge of the paper or over ridges. Hold the instrument freely so that the motion of the hatchet shall not be interfered with. It is always advisable to use the weight on the hatchet to prevent side slip.

Use the instrument on a flat table, not on a sloping desk. On no account attempt to sharpen the hatchet, either with an oilstone or otherwise.

See that the instrument is held with its legs fairly vertical.

In measuring the area off always work from the zero of the curved scale.

This instrument is an improvement on the Stang planimeter made by Knudsen, of Copenhagen. In a pamphlet published by him he gives the complete mathe

matical theory of the instrument and arrives at the follow

ing result:

2

Where I the area traced out by the pointer in sq. inches, C1 = the distance between the dents X and Y in inches,

X1 and Y

p the length of the instrument in inches,

R2 the mean square of the radii of the figure

(Le milieu des carrés des distances de T (a) jusqu'à la circonférence)

The making of such a calculation for every area measured is far too tedious an operation. By using Goodman's instrument as described above, first in one direction and then

in the other, the quantity C+C is obtained by a direct

2

reading. If the instrument were of very great length as compared with the dimensions of the area, the quantity in the brackets would vanish, then putting = c, the area would simply be equal to the product pc, and a scale for measuring the distance in which I square inch was

2

I
P

FIG. 4.

made equal to inches would at once give the area in square inches. But as the areas generally dealt with are of somewhat large dimensions as compared with the length of the instrument, the quantity in brackets must not be neglected. Assume for a moment that we are measuring the area of a circle with the instrument-its area is proportional to R2, hence as (2p)2 is a constant for any given instrument, the whole quantity () is proportional to the area traced out by the pointer. Thus by making a scale

2p

with gradually increasing divisions this quantity may be entirely eliminated. This is what is accomplished in Professor Goodman's planimeter, and instead of having to solve the equation as given above in order to find the area, it is read off direct from the scale without any calculation.

If the area dealt with is not a circle, the error involved in assuming that its R2 is equal to the R2 of a circle of equal area is so small that it is quite inappreciable on a scale which only reads to tenths of a square inch, it would indeed seldom be appreciable on a scale reading to hundredths of a square inch.

Planimeters are now so well understood, and their results are found to be so reliable, that no Surveyor's office is complete without them. A level table is essential to ensure accuracy. The instruments shown upon pages 155, 156 are known as Amsler's planimeters. An example of their employment is furnished in the chapter on Contours (pages 197-199).

CHAPTER XV.

TAKING LEVELS.

THE chief use of instruments for taking levels is to compare the heights of different stations with reference to a fixed datum or horizontal line. The datum may be the sea level or any arbitrary horizontal line that can be localised. The method of procedure is virtually the same in all cases. An accurate plan is absolutely essential, and the surveyor has first to obtain a detailed map of the country upon which the line he is to follow is marked, or else he must make a plan for himself in order to keep a record of any line that may be marked upon the ground, or otherwise indicated. Hence surveying operations precede those of levelling. Hutton, an eminent mathematician who was created a Fellow of The Royal Society in 1774, and wrote at the end of the eighteenth century upon mathematical subjects, states “that two or more places are on the same level when they are equally distant from the centre of the earth. Also one place is higher than another, or above the level of it, when it is further from the centre of the earth, and a line, equally distant from that centre in all its parts, is called a line of true level."

Levelling-The field work connected with the process of taking levels may be illustrated by the diagram headed "flying levels," that is, levels in which a description is not required of all the intermediate points upon which the staff is to be held in order to connect the work. (See pages 166168). Suppose the level of the point A is known by previous levelling to be 60 ft. above a given datum, and that it is simply required to ascertain the level of the point H. In the first place, attention should be given to the adjustments of the instrument employed. The verticality of its axis, and consequently the horizontal position of the azimuth, must be proved.

It