Indexing the archive…
Your Universe of Digital Possibilities
Watt fitted a spinning pair of balls to his engines in 1788: as the engine sped up the balls flew out, lifted a sleeve and closed the steam valve. It held the speed — until better machining took away the friction it had secretly relied on, and engines began to hunt. James Clerk Maxwell explained why in 1868; Ivan Vyshnegradsky drew the map in 1876. Run the engine, kick it, take its friction away, and let the auditor read the instability straight off the tachometer.
The flywheel speeds up when the steam torque k cos φ beats the load F; the balls swing out when the spindle’s centrifugal pull beats gravity, and the sleeve’s friction b resists them. The throttle closes as they rise. Linearised about the working speed, the pair becomes one cubic.
Every root of the cubic has a negative real part — every small disturbance dies away — exactly when the coefficients are positive and the middle pair’s product beats the outer pair’s. No root need ever be found.
Scale the governed engine’s cubic until its outer coefficients are 1, and stability is the region above a hyperbola in the (X, Y) plane. In this model it reads b J ω₀ > 2 m l F: friction and inertia on one side, ball mass and load on the other.
The fifth Heron machine computes nothing on purpose: it regulates. The balls measure the speed and the linkage subtracts it from the set point — feedback, in iron. Whether that feedback settles or hunts is decided by a cubic the machine never sees, and Maxwell, Vyshnegradsky, Routh and Hurwitz between them learned to read it without solving it. The bench integrates the full nonlinear engine; the auditor reads the trace and checks the cubic. The integrators that solve equations by feedback are in The Integrator; a pendulum clock’s escapement, feedback of another kind, is The Metronome; and many governors coupled together, each pulling the others into step, are The Chorus.