379. To construct a mean proportional between two given lines without describing circles and drawing perpendiculars.

## Pascal's Problem.
379. Thomas Strutt (late 17th century) mentions this structure without, however, specifying to whom it belongs.

On an arbitrary straight line, from an arbitrary point \( A \), we lay off the given lines \( A B = a \) and \( A C = b \) (Fig. 84). On the same straight line,

![](https://cdn.mathpix.com/cropped/2024_05_21_926f0bb262c8f9569516g-165.jpg?height=506&width=560&top_left_y=1249&top_left_x=357)
in the opposite direction from point \( B \), we lay off the length \( B D = b \) and with radius \( b \), we draw arcs from points \( D \) and \( C \). The line \( A E \), connecting the point of intersection of these arcs with point \( A \), will be the mean proportional between \( a \) and \( b \). Indeed: the isosceles triangles \( A E B \) and \( A E C \) are similar, therefore

\[
a : A E = A E : b
\]

Blaise Pascal (1623-1662), a famous French mathematician, known for his research on the properties of the cycloid and conic sections. In his small treatise "Essai pour les coniques," there is a remarkable lemma known as "Pascal's hexagon." Together with Fermat, he shares the glory of creating the theory of probability.