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

Integrator (Moment Planimeter)

Amsler Integrator 1
Maker: Amsler, Schaffhausen/Switzerland; instrument no. 194; c1880
Inventory: CNAM, inventory no. 11417-0000-
Details: Signed »J. Amsler No. 194«; entry CNAM: 1888 (see file).
References: Amsler 1856; Amsler, Instruction […], 1886. Instrument mentioned in Cat. CNAM 1905, 156, Cat. CNAM 1942, 129

Amsler Integrator, model 1, instrument no. 194, c1880, CNAM 11417-0000

"Integrator" is the name given by Jakob Amsler(-Laffon) (1823-1912) to instruments, which determine from a graphically given f(x) in rectangular coordinates the integral of f(x)², f(x)³ - without drawing these functions - as the result of a measurement along f(x). The integral of f(x) is the area between the curve described by f(x) and the x-axis; the integrals of f(x)² and f(x)³ give the static moment and the moment of inertia of this area relative to the x-axis, respectively. This is why later the probably more suitable name moment planimeter was introduced for instruments of this kind.

The principle underlying these instruments is the use of appropriate formulas between powers of the sine function and the sines of certain multiples of the arguments, i.e. angles. This is most conveniently shown by considering the power 2: If one end of the tracer arm remains on the x-axis, while the other follows a given function f(x), then at any time , where is the angle (changing with x) between the x-axis and the tracer arm. From trigonometry we learn that , whence

When integrating along a closed curve, the first of these two integrals vanishes; it remains . Since a "regular" (linear) planimeter already measures , it is "only" necessary to double the angle to , and to measure the cosine instead of the sine; the latter is easily done by turning the measuring wheel by 90° degrees, as shows. For f(x)³ one has similar but more complicated formulas.

A precise date for the beginning of Amsler's production of moment planimeters is not known. They were already described in his 1856 breakthrough publication. It is known, on the other hand, that they were for the first time shown in the Paris 1867 Exhibition and that they had been commercially manufactured some time earlier. A most probable guess would be 1865, suggesting that the definitively wrong year 1856, given by Dyck in his landmark catalogue, is just a printer's fault for 1865.