Patented in the German Empire from 25 November 1910.
The present invention relates to an inexpensive, simple planimeter, which, after a single circuit round the contour, gives the area of curvilinear, flat figures with the accuracy and speed of precision polar planimeters, but is also suitable for solving some other tasks discussed below.
The drawings in Figures 1 to 7 show different versions of the instrument, Figures 8 and 9 serve to explain the measuring method.
The instrument consists of a rod a g (Fig. 1 to 7) with tracing pin b and index i, as well as a wheel- or ball-shaped roller r, over the periphery of which the rod is guided in a groove f during use, so that, in contrast to Prytz's rod planimeter, it moves along the moving rod without being firmly connected to it. As is already known with other planimeters, the tracing pin can be replaced by a transparent plate with a tracing point.
At b, the rod has a support c which also serves as a handle (Fig. 5 and 6) and, due to its weight, the edge g presses laterally against the roller r, which is prevented from falling over by a counterweight k (Fig. 5); or the rod rests with its groove on a ball as a roller (Fig. 7); or the wheels r h, which sit on a common axle, support the rod. Some marks on the edge g indicate the points where the bar is to be placed over the roller for the purpose of calculating the area. If the tracing pin falls within the circumference of an area to be calculated, you read a scale e placed on the paper at index i before and after tracing the contour and the area is obtained from the difference, which is multiplied by 5, 10, etc., depending on the starting position of the roller on the rod. For a more precise calculation of larger areas, this difference, which increases with the size of the content, requires a correction from a table.
The area calculation is explained from the theory of the "optical planimeter"described in patent specification 238499 (see also Zeitschrift für Instrumentenkunde 1911, 3rd issue, page 65 ff.). When tracing around a surface (Fig. 8), the center of the roller moves relatively half as fast as the tracing pin b in the direction of the rod, because the rod rests on the periphery of the roller. The tracing pin will therefore always be half as far away from the roller r than from the intersection point F of the extension g with a fixed line l (Fig. 8). Accordingly, the tracing pin b corresponds to the so-called "optical bypass point" in relation to the intersection F of the "thread line" g with the circular line l, if one imagines a reflective plane above the roller, which follows the relative movement of the roller under the rod.
The radius of curvature of l increases with the distance of the roller from the tracing pin and with the area k. According to the theory of the optical planimeter, the area is equal to the sector b F F1. Since Fb = F1b is constant, the area is proportional to F F1, i i1, or r r1, provided that a small correction is added to the difference i i1, which is taken from a refinement table assigned to the planimeter and increases with i i1. The achievable accuracy of the area calculations is relatively high, and the error is, for example, on average only ±0.2 per cent of the area for areas of medium size after a single circuit of the figure and only slightly more for small figures. With old, cracked maps, the roller can be run over a disc to increase accuracy. According to the rules of the "optical Planimeter", the device can be used to determine the mean ordinate of a registered curve and to measure it without having to travel along the abscissa axis (zero line).
To calculate the static and moment of inertia, for example, a roller as shown in Fig. 4 is used. Here, the rod a rests on the edge of a smaller disc positioned centrally to the roller, whereby the smaller the disc, the more the relative movement of the roller against the moving rod is slowed down. The same is achieved in Fig. 7 with a smaller diameter ball. If, for example, the ratio of the diameter of the roller and disc is 2:1 or 3:1, then in Fig. 8 Fr : rb = 2:1 or 3:1, and F F1 or i i1 is proportional to the static or moment of inertia of k, according to the theory of the optical planimeter.
If a silver or metal mirror with a small, circular opening is attached to the planimeter, and placed vertically on a stand at a height of 10 to 20 cm above the table level, it can be used very effectively as a surface measuring machine for measuring hides, leather, etc. The stand with the mirror perforated at S (Fig. 9) is placed on a piece of paper at the edge of a table. From B, the observer sees on the paper, apparently through the opening S, the image k' of a piece of leather k lying on the floor. If he allows this mirror image to coincide optically with the tracing pin of a planimeter and traces round it, the difference in reading before and after the circuit of the planimeter is proportional to the area of the leather, provided the mirror has not moved during the circuit. The height of S above the table or the position of the rod can be selected so that the area of the leather is read off directly.
1. compensating rod planimeter, characterised in that the rod is mounted on a shaftless roller which performs a relative movement against the moving rod when the surface is traversed.
2. compensating rod planimeter according to claim 1, characterised in that the moving rod rests on a smaller disc of 1/2 or 1/3 of the diameter of the roller which is concentrically mounted on the roller r.
3. compensating rod planimeter according to claim 1 for use as a surface measuring machine, characterised in that the planimeter is associated with a mirror with a small opening which is mounted on a frame and stands vertically on the plane of the table, so that the observer, with one eye looking into the opening, traces the mirror image of a piece of leather lying on the floor, which is produced on the table, with the moving pin of the planimeter, and that the leather surface can be read off after the tracing.