Les instruments du calcul savant > Instruments d'intégration conservés au musée des arts et métiers

Spirograph

Instrument for drawing the logarithmic spiral ("spirograph"), according to Abdank-Abakanowicz
Maker: unknown; instrument no.: none (probably one-of-a-kind); c1885 (estimated)
Inventory: CNAM, inventory no. 13300-0005-
Details: Entry CNAM: 1900
References: Abdank-Abakanowicz 1886, 135-137 and Abdank-Abakanowicz 1889, 8-9 (both with illustrations, slightly different)

Spirograph by Abdank-Abakanowicz, c1885, CNAM 13300-0005

Abdank-Abakanowicz's instrument for drawing the logarithmic spiral is an immediate application of the principle of the continuous screw. The cylinder here has infinite radius and is represented by the plane. The wheel is knife-edged and movable in a carriage along a straight line. The angle of inclination of the wheel can be adjusted with the help of scales from -90° to +90°. The instrument can be turned around the centre of the plate K, the pole of the instrument; let design the angle of rotation of the whole instrument, counted clockwise from an initial position.

Fig. 90, Abdank-Abakanowicz 1886, 137

The carriage is moved completely to the left and the inclination of the wheel is fixed. If now the instrument is turned clockwise by its handle L (i.e. with permanently increasing from its initial position ), the knife-edge of the wheel sinks into the paper and forces the wheel to keep its direction. As the direction of the wheel is oblique to the circle which it would describe if allowed to glide, there results an evasive movement - like the displacement of the cylinder in the continuous screw mechanism -, made possible by the mobility of the carriage along the straight line. The carriage will move towards L, its distance from K being proportional to the sine of the angle of inclination , i.e. proportional to , and of course also proportional to the arc s. (Note that compared with the continuous screw the roles of sine and cosine have been reversed.)

To determine the trace of the knife-edged wheel one has to consider small ("infinitesimal") changes dr and . If the wheel is a distance r away from K and if the instrument is turned by a small angle , the length s of the arc mentioned above will be . Consequently, the displacement of the carriage will be proportional to both s and , that is, proportional to . As is fixed (and by that also ), it is constant, and one has , k a constant. This is a differential equation, which is written (after separating the variables, i.e. separate grouping of terms containing r and , respectively, which is possible in this case) . Integrating both sides now gives

, or

(C is a constant of integration, and a = e^C). But this is the equation of the so-called logarithmic spiral in polar coordinates, where in addition r(0) = a. If one desires to obtain this curve not only by the trace of the knife-edged wheel, but drawn with a pen, the pen ought to have exactly the position of the wheel - which is impossible. Therefore, one has only two alternatives: Either one finds an additional mechanism, governed by the carriage, and resulting in a position of the pen which describes another logarithmic spiral, or one considers the curve described by any point on the carriage, not identical with the position of the wheel. As things are, the first alternative is possible, but would lead to a rather complicated mechanism. So there remains the second alternative. If the pen is placed at E, for example, lying on the straight line and having distance b from the point of contact of the wheel, it will describe a curve given by

But this curve can be used in almost exactly the same way as the logarithmic spiral. This spiral is important in graphical methods of computing and is used when - by graphical means only - the length ln r of a line segment is sought, while r is given.

The following drawing taken from the enlarged and revised German edition (1889) of Abdank-Abakanowicz's original (French) publication of 1886 shows more clearly how the "shifted" logarithmic spiral looks like.

Fig. 7, Abdank-Abakanowicz 1889, 8

This instrument also illustrates the fact that the integrating mechanism knife-edged wheel-and-plane really solves a differential equation. In the case of the logarithmic spiral this is a rather simple differential equation, which is determined by the "form" of the device. It is to be expected that the knife-edged wheel-and-plane mechanism might be able to solve other differential equations, too; it is "only" necessary to find new mechanisms whose forms "represent" these differential equations. Besides the integraphs invented by Abdank-Abakanowicz, which graphically solve the (trivial) differential equation y' = f(x), there were invented many other so-called "integrators" on the basis outlined above. Especially the Italian mathematician Ernesto Pascal (1865-1940) figured prominently among the scientists dealing with these instruments, which were mostly built as one-of-a-kind, if ever.