The integrator shown in P1. I., fig. 6, was designed by Professor Hele-Shaw for determining areas, moments of stability and inertia by a single tracing of the figure. The principle of the instrument is similar to that of Amsler's integrator, but the instrument is designed specially to avoid slipping of the measuring wheel upon the moving surface, which in this case is a sphere.
In 1876 F. Hohmann invented a "precision" planimeter, which since 188o has been made with various modifications by Amsler, Coradi, Ott and others. In this type the recording wheel rests lightly on the specially prepared fine surface of a disc, so as to reduce friction due to slipping.
Important improvements due to Coradi are embodied in the instrument shown in Pl. I., fig. 7, which was made in 1915. The pivot end of the tracer arm is constrained to move in a circle whose centre is the same as that of the base plate, or pole disc. The edge of the pole disc is milled and engages with a small wheel on the axis of the revolving disc. This latter is made of aluminium and its upper surface is covered with smooth paper. As its axis, which is attached to the pole arm, revolves with the pole arm, the small wheel gears with the edge of the pole disc and the aluminium disc rotates through an angle proportional to the angle described by the pole arm. Upon this rotating disc rests the recording wheel, which is in turn rotated by its contact with the disc through an angle proportional to the area swept over by the tracer. The length of the tracer arm can be varied to suit the scale required, and the tracer point is provided with a support and a spring contact.
Ott (of Kempten, Bavaria) manufactures an "universal" plani meter which is made in three sections, and can be set up in differ ent ways so as to form a compensation polar planimeter, a roll ing planimeter, or a radial averaging instrument.
In 1887, Captain Prytz invented the simple knife-edge or "hatchet" planimeter, introduced and popularized in England by Professor Goodman. In its original form it consisted of a metal bar, bent at right angles at both ends, one of which (the tracer) was pointed, and the other in the form of a curved knife-edge. In using this instrument, a point is chosen at or near the centre of the area to be measured, and a radial line is drawn to the boundary. The point of the instrument is placed at the centre, and the hatchet pressed into the paper to form a dent. The point
of the instrument is then made to follow the radial line and the boundary line, ultimately returning to the central point of the area, along the same radial line. The hatchet is again pressed into the paper to form a dent. If AB, AB' be the initial and final positions of the arm, the area described is equal to the length of the arm multiplied by the length of the arc BB'. Within certain limits, the length of the chord the linear distance be tween the initial and final marks made by the knife edge—may be taken instead of the length of the arc. In 1890 Prof. Goodman added a scale on the back of the instrument, which when applied to the distance between the two dents gave a direct reading of area in square inches.
For any value of x the steepness of the curve drawn by the instrument is proportional to the ordinate of the given curve for the same value of x. The ascent then made by the new curve in passing from one ordinate to the other is a measure of the area between the given curve, the axis of x and the two ordinates.
The frame work is a kind of T-square (which can slide along a horizontal straight edge) carrying a fixed centre B, which moves along the axis of x of the given curve. A rod, passing always through B, carries a pointer A, which is constrained to move in the vertical line e e of the T-square; A can then be made to follow any given curve. The distance from B to e e is constant (k), therefore the inclination of the rod AB is such that its tan gent is equal to the ordinate of the given curve+ k; so that AB has always the inclination of the required curve.
The curve is drawn by means of a three-wheeled cart of lead whose first wheel C is mounted like the steering wheel of a bicycle. By means of epicyclic gearing this wheel is kept parallel to AB, and can move only in the direction of its own plane. As C is always in e e produced, the wheel draws the required curve if allowed to pass over a sheet of carbon paper.