Graphical and Mechanical Methods for Solving Polynomial Equations


Before the digital computer era, some ingenious minds came up with ideas of mechanical and electro-mechanical devices that can solve various types of equations. Most of these devices are probably not very practical due to their accuracy. Also most of these devices could only calculate the real roots of a polynomial equation. Nonetheless, I find these devices intriguing. Maybe some of these devices can be used for educational purposes since they make an abstract concept more tangible or visible. Some of these devices can be especially useful to the more mechanical oriented students. I think that these devices ilustrate the beauty of mathematics, physics, mechanics and engineering in general. I am pretty sure that some of these devices can be simulated by various computer programs or software.

Machines for Solving Algebraic Equations
This paper is 17 pages long if we include the references. People interested in the history of mechanical and electro-mechanical equation solvers should read this paper. It cites about 58 sources that can be very useful for additional research.

A mechanism for solving: equations of the nth degree
This is a paper by Dr. R.F. Muirhead. The mechanical devices discussed in this paper are very simple devices made of bars. He discusses how to make a device to solve polynomial equations and devices that can solve systems of equations (even systems of non-linear equations).

Machine for solving numerical equations
This is a link to an article that describes a machine proposed by George B. Grant. The device is a scale with multiple horizontal beams, and can be used to calculate the real roots of a polynomial equation. The coefficients are represented by the mass of the weights, with the negative or positive sign being determined by the position of the weights to the left side or the right side of the scale.

Two Hydraulic Methods to Extract the nth Root of Any Number
The link contains 2 articles by Dr. Arnold Emch. The second article extends the first method discussed in the first article in order to make the method applicable to a general polynomial of degree n.