Les instruments du calcul savant > Instruments d'intégration conservés au musée des arts et métiers |
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Orthogonal Planimeter/Integrimeter Orthogonal planimeter/integrimeter with wheel-and-cylinder mechanism, according to Abdank-Abakanowicz (type II)
Orthogonal planimeter/integrimeter according to Abdank-Abakanowicz, type I This instrument makes use of the continuous screw (vis à pas variable); the screw is realised like in the demonstration model for the continuous screw by a wheel, pressed on the mantle of a cylinder. One may discern up to three versions which differ in technical details and will be called type I, II and III here. This numbering scheme will most probably also indicate the chronological order of the devices. In the type I instrument (shown in the drawing above), the point of contact between wheel r and cylinder c can be moved along a generator of the cone by turning the screw p at its milled head. At the same time the toothed bar YY is moved parallel to itself in the direction of the x-axis XX. The direction of the wheel r can be changed by the arm t; this is achieved by turning the other screw p at its milled head. Turning both screws has to be done simultaneously in order to keep the tracer pin P on the boundary line of the given figure; this will obviously need some exercise. In the type II instrument, the point of contact between wheel r and cylinder c can again be changed by turning a screw at its milled head. But now the bar named YY in type I is no longer a toothed bar; instead, the direction of the wheel r is changed by the arm t which is now coupled in a gliding motion to YY. In other words, the second screw p of the type I instrument has been eliminated and replaced by a gliding joint. To keep the tracer pin P on the curve, it is still necessary to synchronise the turning of the remaining screw and the movement of the arm t. This is probably almost as difficult as it was with the type I instrument. A type II instrument is preserved in the collections of the CNAM (13300-0002-):
Orthogonal planimeter/integrimeter according to Abdank-Abakanowicz, type II Finally, in the type III instrument both screws have now replaced by gliding joints. These, of course, are technically much more elaborate than screws, but their use allows controlling both the point of contact between wheel and cylinder and the direction of the wheel by the position of the tracer pin alone. Now the instrument is easy to use. Type III can be seen in Abdank-Abakanowicz's publications, but there is also a real instrument of this kind preserved in the CNAM collections (see CNAM 13300-0003-). Generally the plane of the wheel r will be oblique to the axis of the cylinder. Changing the point of contact and the direction of the wheel will thus result in a turning of the cylinder around its axis (like in the continuous screw model CNAM 13300-0004-; the only difference is that now the wheel moves along a generating line, while the cylinder is not moveable along its axis. But as the decisive movement is relative, this does not make a principal difference). The total turning of the cylinder can be read off on the drum to its left, with the help of a vernier. The theory of the continuous screw has shown that at any time, the position of the cylinder, counted from its initial position, has accumulated the area surrounded until that time. This is the characteristic property that makes this planimeter an integrimeter. |
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