The following procedure provides an approximate value (segment) for the side of a regular 19-sided polygon inscribed in a circle. The midpoint of the ray inclined at $60^{\circ}$ to the radius $O A$ is $B$, its projection on $O A$ is $C$, the projection of $C$ on $A B$ is $D$, and the projection of $D$ on $O A$ is $E$. Then the approximate value of the side is $D E$. - If the obtained segment is measured 19 times consecutively as a chord in the circle, what angle does the ray drawn to the starting and ending points enclose? Give an informed opinion on how accurate the value obtained for this angle is.
a) In the triangle $OAB$, there is a right angle at $B$, so each of the triangles $OBC$, $BCD$, and $CDE$ is half of an equilateral triangle. Thus,

$$
DE = \frac{\sqrt{3}}{2} CD = \left(\frac{\sqrt{3}}{2}\right)^2 BC = \left(\frac{\sqrt{3}}{2}\right)^3 OB = \frac{3 \sqrt{3}}{16} \cdot OA
$$

Accordingly, denoting the intersection of the arc with center $A$ and radius $DE$ with $G$, for the angle $\angle AOG = \alpha$,

$$
\sin \frac{\alpha}{2} = \frac{AG / 2}{OA} = \frac{DE}{2 \cdot OA} = \frac{3 \sqrt{3}}{32} = 0.16238
$$

(to five decimal places, since $\sqrt{3}$ can be approximated to any desired precision), and thus $\alpha / 2 = 9^\circ 20.6'$. After measuring the chord 19 times, the initial radius turns by 38 times this amount, which is $355^\circ 3'$, so the acute angle between them is $\beta = 4^\circ 57'$.
b) The part of the angle $\alpha / 2$ above $9^\circ 20'$ is calculated from the ratio

$$
\delta = \frac{0.16238 - \sin 9^\circ 20'}{\sin 9^\circ 30' - \sin 9^\circ 20'}
$$

The table gives $\sin 9^\circ 20' = 0.1622$ and $\sin 9^\circ 30' = 0.1650$ to four decimal places, which means to five decimal places, $0.16215 \leq \sin 9^\circ 20' \leq 0.16225$ and $0.16495 \leq \sin 9^\circ 30' \leq 0.16505$.

If we only consider these, the numerator of $\delta$ would be maximized if $\sin 9^\circ 20' = 0.16215$, and the denominator would be minimized if $\sin 9^\circ 30' = 0.16495$, thus

$$
\delta = \frac{0.00013}{0.00280} > 0.46'
$$

The accepted fractional part of $\alpha / 2$ is $0.6'$, and its maximum error according to this would not exceed $0.83' - 0.6' = 0.23'$. The maximum error of the above $\beta$ would not exceed 38 times this, and certainly not $9'$.

(Another question would be how precise $\alpha = 18^\circ 41.2'$ is compared to the central angle of a regular 19-sided polygon.)

Remarks. 1. When estimating the error, the data adjacent to the used data in the table could be considered: $\sin 9^\circ 10' = 0.1593$, $\sin 9^\circ 40' = 0.1679$, both differ from the used data by $29 \cdot 10^{-4}$, while between them, the difference is $28 \cdot 10^{-4}$. From this, it can be seen that the rounding of 0.1622 and 0.1650 reduced the difference between them, and the difference between the two adjacent values increased due to the rounding—i.e., the rounding of 0.1622 and 0.1650 was successful in the direction of each other. However, this cannot make the table data more precise. (To six decimal places, $\sin 9^\circ 20' = 0.162178$ and $\sin 9^\circ 30' = 0.165048$, with the latter almost the maximum possible rounding.)

2. The fact that $\cos \alpha$ is rational tempted many to use it. Denoting the projection of $G$ onto $OA$ with $H$ and the point opposite $A$ on $k$ with $A^*$, from the right triangle $AA^*G$,

$$
\begin{gathered}
AH = \frac{AG^2}{AA^*} = \frac{DE^2}{2 \cdot OA} = \frac{27}{512}, \quad \text{and thus} \\
\cos \alpha = \frac{OH}{OG} = \frac{485}{512} = 0.94727
\end{gathered}
$$

This value falls between the table values $\cos 18^\circ 40' = 0.9474$ and $\cos 18^\circ 30' = 0.9465$. Since the difference between these is only $9 \cdot 10^{-4}$—the cosine function changes much more slowly in this region compared to the sine function around $9^\circ 20'$—calculating $\beta$ and its error limit from this would yield a larger value for the latter.

Every table has parts that allow for a smaller error limit in reverse calculation, where the absolute difference between adjacent data is larger, intuitively: where the corresponding curve is steeper.