Amsler's polar planimeter, model 4 (standard model, German silver)
Amsler's polar planimeter, model 4 (German silver), no. 3691 (CNAM 08004-0000)
Amsler's polar planimeter, invented in 1854, patented in 1855 and published for the first time in 1856, gave mechanical integration its final breakthrough. Contrasting the relatively complex, complicated and thus also expensive orthogonal planimeters, Amsler's instrument was of extreme simplicity in construction and use (but this does not hold for its theory!). It suffices to place the instrument with its fixed point E (called the "pole" of the instrument, hence its name) on the drawing, such that the tracer pin may be led once around the boundary of the area to be measured. The difference between the readouts of the measuring wheel at the end and at the beginning of the tracing is proportional to the area of the figure circumscribed.
The wheel, gliding and rolling on the surface, establishes the integrating mechanism; it is absolutely necessary that its axis be parallel to the direction of the tracer arm. Perhaps one can see best what happens if only the tracer arm is considered. If its length is f, and if the tracer arm is transported a distance w perpendicularly to itself while remaining parallel to its initial direction, then it has swept out a rectangle of area f · w; at the same time the length w has been registered by the "roll" of the wheel. Thus the roll of the wheel, multiplied by f, is equal to the area swept out. If the tracer arm is moved in a straight line, but obliquely with angle to its perpendicular direction, while still remaining parallel to its initial position, it traces a parallelogram of height ; the wheel simultaneously rolls and glides. When rolling, it registers , when gliding, it registers nothing. Thus, the wheel again registers the distance between the initial and the final position of the tracer arm, this distance being the height of the parallelogram. If again multiplied by f, the wheel has registered the parallelogram's area. (The previous case is covered by this description, too: one has , which gives thus , as required).
If the tracer arm is turned around its, say, left endpoint, the wheel will register this movement, too. But it is not necessary to interpret this turning of the wheel, because after a complete tracing of a closed boundary, the tracer arm has turned around his left endpoint as much to the left as to the right, so the registering of these movements has cancelled in the end. Consequently, only the displacements of the tracer arm perpendicular to itself have been registered, i.e. e. accumulated. After multiplication by f they will give the total of the area covered during the movement of the tracer arm.
By making the length of the tracer arm adjustable, one has the possibility to adapt the instrument to different units of measurement (e. g. cm2 or square inches). Amsler's planimeter models 1 (brass) and 2 (German silver) do not have an adjustable tracer arm length and thus are only measuring in a single, invariable unit; this unit, however, could be chosen when ordering the instrument.
Amsler's polar planimeter, model 1 (brass) / 2 (German silver). Amsler prospect, 1874