343. Given two lines, find a third such that its product with the sum of itself and the first given line equals the square of the second given line.

## Problems of Christopher Clavius.
343. If the sought value is $x$, and the given values are $a$ and $b$, then according to the condition $x(x+a)=b^{2}$, or $x^{2}+a x=b^{2}$, i.e., a quadratic equation.

![](https://cdn.mathpix.com/cropped/2024_05_21_926f0bb262c8f9569516g-154.jpg?height=468&width=420&top_left_y=1274&top_left_x=361)

Fig. 78.
From this,

$$
x=-\frac{a}{2}+\sqrt{\frac{a^{2}}{4}+b^{2}}
$$

Benedetti provides a construction for the root (Fig. 78): construct a right angle and on its sides, mark $A B=a$ and $B C=b$. Dividing the line $A B$ at point $O$, connect points $O$ and $C$. From $O$, with radius $A O$, draw an arc intersecting $O C$ at point $E$. The segment $C E=x$.

Christopher Clavius (1537-1612), a mathematics teacher in Rome (in the Society of Jesus), author of a treatise on algebra (1609) and publisher of Euclid's "Elements" with commentaries.