A companion to two-lines-one (two curves that look like one, then diverge). This is the sharper case: a 接線 (sessen, a tangent line) and a curve that agree at exactly one point — same value, same slope — and are identical to first order there. Zoom into that point and you cannot tell them apart. The tangent is the best straight lie the curve permits. And it is still a lie: the curve carries a curvature the line has no term for, so the moment you step off the point, a gap opens — and it opens as the square of how far you stepped. Agreement at a point is not agreement.
The tangent at a is built from two facts about the curve there: its height f(a) and its slope f′(a) — the 一次近似 (ichiji-kinji, first-order approximation). That is all a line can carry; a line has no second-order term. But the curve has one — its curvature f″(a) — and that is precisely the number the tangent drops. Taylor makes the loss exact: the vertical gap is f(a+Δ) − [f(a)+f′(a)Δ] ≈ ½·f″(a)·Δ². So the gap is zero at the point, and grows as Δ² — quadratically, curvature-scaled. This is why the zoom trick works: halve your distance to the point and the gap quarters, faster than the eye, until the two look like one. It is also why it is a trick. The curve was never the line; it only agreed to first order, and first order is a choice of how much of the truth to keep. Where f″(a)=0 — an inflection — the tangent agrees to second order and the lie gets much smaller, but a third-order gap still waits. There is no order at which a line becomes a curve.