J. HartMANN, an Argentine mathematician, in 1928, published the following procedure for constructing a segment approximately equal to a given semicircular arc [f̄]. Let the center of the semicircular arc be $A$, the diameter $B C$, and the radius perpendicular to it $A D$, with the midpoint of $A D$ being $E$. Let $F$ be the point on the extension of $B C$ beyond $C$ such that $B F = 5 B A$, and let $k$ be the circle with diameter $B F$. Let one of the intersection points of the circle centered at $F$ with radius $B C$ and circle $k$ be $G$. Draw the line $B G$. Let the midpoint of $C E$ be $H$, and let $J$ be the point on the segment $B C$ such that $B K = B J$. Then the segment $K G$ is approximately equal to the length of the semicircular arc $B D C$. What is the accuracy of this procedure? ( $\pi=3.141592653589 \ldots$ ).[^0]

[^0]:    ${ }^{1}$ Boletin Matematico, I. (1928) 89 o.
Let's choose the length unit to be the radius $AB$, so the length of the semicircle arc $BDC$ is $\pi$. On the other hand, in the right-angled triangle $BFG$, $BF=5; FG=2$, so $BG=\sqrt{21}$, and $AE=1/2, EC=\sqrt{5}/2$, so $HC=JC=\sqrt{5}/4$, therefore $BJ=2-\sqrt{5}/4$, and finally $KG=\sqrt{21}-2+\sqrt{5}/4$.

![](https://cdn.mathpix.com/cropped/2024_05_02_51d17659deb53946f974g-1.jpg?height=479&width=786&top_left_y=281&top_left_x=664)

For the approximation with decimal fractions, let's write the third term of $KG$ as $\sqrt{5/16}=\sqrt{0.3125}$. Thus, $\sqrt{0.3125}=0.559016994 \ldots$, and $\sqrt{21}=4.582575694 \ldots$, so $KG=3.141592689 \ldots$. According to this, the approximate value's 8th significant digit (7th decimal place) is still accurate, the approximate value is greater than the true value of $\pi$, and the difference is less than $3.7/10^8$, which is $1.2/10^8$ of the true value.

Dóra Gyula (Budapest, Árpád Gymnasium II. o. t.)

Note. When looking for the measure of accuracy and the magnitude of the difference, we cannot know in advance how many decimal places of the two square roots we need to determine. Therefore, it is advisable to form the sum and compare after each new digit of the roots is known. After finding the digit that definitely differs, it is useful to estimate one more digit to determine the percentage error. - In this example, this was possible because the two terms that represent "slow calculation" are independent of each other.