Marks that look alike and do different jobs

What do Δ, d, a prime and a small raised number each ask of the reader?

Δ is a change by a definite amount. In the Brownian paper Δ is how far a particle moves in one short interval: an actual distance, such as 1 micrometre.

d, as in dx/dt, is a rate at one instant: how fast x changes as t changes, the value that a change in x divided by a change in t settles on as both shrink. It is not d times x divided by d times t. The curly ∂ is the same idea when other quantities are held fixed.

A prime marks another observer's value. In §7 of the relativity paper ν is the frequency of a light wave for one observer and ν′ the frequency of the same wave for an observer moving relative to the first; φ and φ′ are its direction for each.

A small raised number is repeated multiplication. In the light-quanta paper's §5, W = (v/v₀)ⁿ multiplies v/v₀ by itself n times: with v/v₀ = ½ and n = 4 that is 1/16.

The papers of 1905 print lg for the natural logarithm, which a modern text writes ln. Today lg often means the logarithm to base 10; in these papers it never does.

Worked example: Five marks, five jobs

1. A particle at 3 μm moves to 4 μm: Δx = 1 μm, a definite change.
2. A ball that has gone 3t² metres after t seconds has average speeds of 9, 6.3 and 6.03 m/s over shorter and shorter intervals from t = 1 s; they settle on dx/dt = 6 m/s.
3. An observer at rest finds a frequency ν; one moving away from the source finds ν′, a smaller number for the same wave.
4. (½)⁴ = ½ × ½ × ½ × ½ = 1/16.
5. lg 2 in the 1905 papers is ln 2 = 0.693, not the base-10 value 0.301.