The curve table says where the curve is. It does not put a single stake in the ground. The classic way to do that needs one instrument and one idea: set up over the PC, zero the circle along the back tangent, and turn a small angle to each stake in turn.

That angle is the deflection angle: $\delta = \dfrac{\ell}{2R}$. Read aloud: delta equals ell over two R. $\delta$, a small delta, is the angle turned off the back tangent at the PC to sight the stake. $\ell$, a script ell, is the arc distance from the PC to that stake, measured along the curve in metres. $R$ is the radius in metres. You solve for $\delta$ when the stations are chosen, or for $\ell = 2R\,\delta$ when a stake has been set and its station must be booked. Keep $\delta$ apart from the capital $\Delta$ of the last three lessons: $\Delta$ is the whole turn between the straights, and $\delta$ is one sight to one stake.

Where the 2 comes from is the whole lesson. An arc of length $\ell$ subtends $\ell/R$ at the centre of the circle. The instrument is not at the centre. It stands on the curve, and the angle between a tangent and a chord is half the central angle the chord subtends. Same arc, seen from the rim, looks half as wide.

The ratio $\ell/2R$ is two lengths divided, so it comes out in radians. The horizontal circle reads degrees. Multiply by $\dfrac{180}{\pi} = 57.2958$: with $R = 200$ m and the first stake 20 m round, $\delta = 20/400 = 0.05$ rad, which is 2.86°. Going the other way, degrees become radians before they meet the radius, or $2R\delta$ is not an arc at all.

Two facts make the method trustworthy. Every deflection is turned from the same back tangent, so the angles grow in step with the arc: twice as far round, twice the angle. And the table checks itself. Run the arithmetic out to the PT, where $\ell$ is the whole curve length $L$, and the deflection must come to exactly $\Delta/2$. If it does not, the curve data is wrong before anyone has driven a stake.

The named mistake is dropping the half: turning $\ell/R$, the central angle. Every stake then lands inside the curve, and the table ends on the full $\Delta$, which looks reassuring and is the signature of the error. Its quieter cousin is booking 0.05 and forgetting that it is radians.

Two notes for the field. What the tape measures from the PC to a stake is the chord, $c = 2R\sin\delta$, a little shorter than the arc, and on a sharp curve the difference is worth carrying. And on a North American sheet in 100 ft stations the arithmetic collapses to a rule: each full station turns $D/2$, half the degree of curve. That shortcut is imperial by construction, like $D$ itself, so this lesson stays in metres and works from the radius.