# Integraph

Integraph per Abakanowicz design, 1915 catalog drawing
Coradi of Zurich's integraph, 1911 drawing

An Integraph is a mechanical analog computing device for plotting the integral of a graphically defined function.

## History

It was invented independently about 1880 by the British physicist Sir Charles Vernon Boys and by Bruno Abdank-Abakanowicz, a Polish-Lithuanian mathematician/electrical engineer from the Russian Empire. Abakanowicz's design was constructed by Coradi of Zurich.

## Description

The input to the integraph is a tracing point that is moved to trace the input curve. The output is defined by the path a disk that rolls along the paper without slipping. The mechanism sets the angle of the output disk based on the position of the input curve: if the input is zero, the disk is angled to roll straight, parallel to the x axis on the Cartesian plane. If the input is above zero the disk is angled slightly toward the positive y direction, such that the y value of its position increases as it rolls in that direction. If the input is below zero, the disk is angled the other way such that its y position decreases as it rolls.

The hardware consists of a rectangular carriage which moves left to right on rollers. Two sides of the carriage run parallel to the x axis. The other two sides are parallel to the y axis. Along the trailing vertical (y axis) rail slides a smaller carriage holding a tracing point. Along the leading vertical rail slides a second smaller carriage to which is affixed a small, sharp disc, which rests and rolls (but does not slide) on the graphing paper. The trailing carriage is connected both with a point in the center of the carriage and the disc on the leading rail by a system of sliding crossheads and wires, such that the tracing point must follow the disc's tangential path.

## Mechanism

The integraph plots (traces) the integral curve

${\displaystyle Y=F(x)=\int f(x)dx,}$

when we are given the differential curve,

${\displaystyle y=f(x).}$

The mathematical basis of the mechanism depends on the following considerations:[1] For any point (x, y) of the differential curve, construct the auxiliary triangle with vertices (x, y), (x, 0) and (x − 1, 0). The hypotenuse of this right triangle intersects the X-axis making an angle the value of whose tangent is y. This hypotenuse is parallel to the tangent line of the integral curve at (X, Y) that corresponds to (x, y).

The integraph may be used to obtain a quadrature of the circle. If the differential curve is the unit circle, the integral curve intersects the lines X = ± 1 at points that are equally spaced at a distance of π/2.[1]

## References

1. ^ a b Klein, Felix (1956) [1930], Famous Problems of Elementary Geometry, Dover, pp. 78–80
• Granville, William. Elements of Differential and Integral Calculus, New Edition. Ginn and Company. NY, NY, 1934
• Bruno Abdank-Abakanowicz, Les intégraphes, la courbe intégrale et ses applications: étude sur un nouveau système d'intégrateurs mécaniques.

Gauthier-Villars, 1886 available at Google Books