This page assumes nothing. Before you can read the derivation you must be fluent in a small dictionary of symbols and pictures. We build them one at a time, each resting on the one before.
Look at the figure: the black incoming ray splits at the surface into a reflected arrow and a transmitted arrow. Why do we care? Because Newton's rings are made by two rays that split apart, travel slightly different distances, then recombine. To track them we need a language for "how far each went."
The red curve is one wave; the red bracket marks one λ. For visible light λ is tiny — about λ=589nm=589×10−9m, roughly a five-hundredth the width of a human hair.
In the figure, the top pair of waves are shifted by exactly one full λ — crest on crest, they reinforce (red = bright result). The bottom pair are shifted by half a λ — crest meets trough, they flatten to nothing (dark). This single picture is the engine of the whole topic. See Interference — path difference & coherence for the deeper story of why recombining waves add this way.
The figure shows the plano-convex lens (curved on top, flat below) resting on the flat plate. Why three different lengths? Because the payoff formula links a thing you can measure (r, sideways with a microscope) to a thing you cannot directly see (t, the microscopic gap) using the fixed lens shape (R). Their relation is the near-parabola t≈r2/2R, which is why rings crowd outward. Compare with Wedge fringes / air wedge, where the gap grows linearly instead.
Read it top-down: the ruler and the difference feed the optical path; the flip corrects it; the geometry supplies t; together they produce the ring conditions that the parent derivation turns into radius formulas. The same machinery also powers the Michelson interferometer.