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Chapter 08 · Not written yet

Gears in a chain

How a nudge travels through a machine.

Down a train of gears the ratios multiply, and sensitivities behave the same way. One jammed gear stops everything, which is a gradient dying, seen before it is named.

  • chain rule
  • composition
  • sensitivity

How this chapter will go

Multiply sensitivities along a chain, by hand, on three links.

The image it is built onA train of gears. If A turns three times faster than B, and B twice as fast as C, then A turns six times faster than C. The factors multiply.

  1. Two gears, one ratio

    Turn the first and the second follows, faster or slower by a fixed factor. Change the factor and the output speed changes with it.

  2. Three gears, one product

    Add a third and the total factor is the two multiplied. The rule is stated in gears before it is stated in anything else.

    × 3.00× 2.000.00 turns0.00 turnsyou turn here× 6.00 comes out
    • the crank you turn
    • the far end
    • the links between
    first ×
    3.00
    second ×
    2.00
    multiplied
    6.00
    if added
    5.00
    turns in
    0.00
    turns out
    0.00

    Press Turn it and watch the two ends, or drag the crank yourself.

    The factor from end to end is the product of the factors of each link. Set the two ratios and read the total.

  3. The machine is a train of gears

    Each step of a calculation is a gear: a small change on its input becomes a change on its output, multiplied by some factor. Sensitivity travels the same way speed does.

  4. One jammed gear stops everything

    If any link has a factor near zero, the whole product collapses and nothing reaches the far end. That is a gradient dying, seen before it is given a name.

Written on three links, with words rather than letters, so the product can be read aloud. Each bracket is one gear: how much its output moves when its input moves.

Show it in symbols

total factor = (third gear) × (second gear) × (first gear)

With symbols instead of words, this is the chain rule. It says one thing: multiply along the chain.

The misreading to head off

Sensitivities multiply, they do not add. Two links that each double give four, not four by addition, and the figure should make that mistake visible rather than describe it.

There is no date, no list to join and nothing to subscribe to. The chapter simply appears here once the drawing works.