Editing Brachistochrone curve (section)

==== Analytic solution ====
[[File:Brachistochrone Bernoulli Direct Method.png|Brachistochrone Bernoulli Direct Method]]

A body is regarded as sliding along any small circular arc Ce between the radii KC and Ke, with centre K fixed. The first stage of the proof involves finding the particular circular arc, Mm, which the body traverses in the minimum time.

The line KNC intersects AL at N, and line Kne intersects it at n, and they make a small angle CKe at K. Let NK = a, and define a variable point, C on KN extended. Of all the possible circular arcs Ce, it is required to find the arc Mm, which requires the minimum time to slide between the 2 radii, KM and Km. To find Mm Bernoulli argues as follows.

Let MN = x. He defines m so that MD = mx, and n so that Mm = nx + na and notes that x is the only variable and that m is finite and n is infinitely small. The small time to travel along arc Mm is <math> \frac{Mm}{MD^{\frac{1}{2}}} = \frac{n(x + a)}{(mx)^{\frac{1}{2}}} </math>, which has to be a minimum (‘un plus petit’). He does not explain that because Mm is so small the speed along it can be assumed to be the speed at M, which is as the square root of MD, the vertical distance of M below the horizontal line AL. Plus MD=mx via Pythagoras theorem.

It follows that, when differentiated this must give 
:<math>
\frac{(x - a)dx}{2x^{\frac{3}{2}}} = 0
</math> so that x = a.

This condition defines the curve that the body slides along in the shortest time possible. For each point, M on the curve, the radius of curvature, MK is cut in 2 equal parts by its axis AL. This property, which Bernoulli says had been known for a long time, is unique to the cycloid.

Finally, he considers the more general case where the speed is an arbitrary function X(x), so the time to be minimised is <math> \frac{(x + a)}{X} </math>.
The minimum condition then becomes 
<math>
X = \frac{(x + a)dX}{dx}
</math>
which he writes as :<math>
X = (x + a)\Delta x
</math>
and which gives MN (=x) as a function of NK (= a). From this the equation of the curve could be obtained from the integral calculus, though he does not demonstrate this.