Mechanica Vol. II (1736)

the body will press against the channel at M along the direction Mn by an amount =
$\frac{2v}{r}$ Euler concludes the proof from the previous page: the pressure is equal to twice the "height due to velocity" divided by the radius of curvature.

Corollary 2.

22. If the body were to move with a greater or lesser speed along the curve AM, then the pressure at M would be greater or lesser in duplicate ratio The "duplicate ratio" of the speed refers to the square of the speed ($v^2$). to the speed, because the height v is proportional to the square of the speed.

Corollary 3.

23. The direction of this pressure is normal Perpendicular to the tangent of the curve at that point. to the curve, and is directly opposite to the position of the radius of osculation MO. Therefore, the radius of osculation extended to the other side of the curve will give the direction of this pressure.

Corollary 4.

24. If the body moves in a straight line, this pressure will be zero, because the radius of osculation is infinite. This is also clear from the very nature of motion itself. For a body moving uniformly in a straight line proceeds forward by its own accord, and for this reason, it does not press against a straight channel Euler uses the term "canalis" (channel) to describe any physical constraint, like a tube or a rail, that guides the body's path..

Corollary 5.

25. If the curve AM were a circle, the pressure will be the same everywhere. It will be greater, however, the smaller the radius of the circle is. For while the speed remains the same, the pressure will be inversely proportional to the radius of the circle.

Scholion 1.

26. In order for a body to be able to move freely along the curve AM