Editing Brachistochrone curve (section)

=== Galileo's problem ===
Earlier, in 1638, [[Galileo Galilei]] had tried to solve a similar problem for the path of the fastest descent from a point to a wall in his ''[[Two New Sciences]]''. He draws the conclusion that the arc of a circle is faster than any number of its chords,<ref>{{citation |author=Galileo Galilei |title=Discourses regarding two new sciences |page=239 |year=1638 |chapter=Third Day, Theorem 22, Prop. 36 |chapter-url=http://galileoandeinstein.physics.virginia.edu/tns_draft/tns_160to243.html |author-link=Galileo Galilei}} This conclusion had appeared six years earlier in Galileo's ''Dialogue Concerning the Two Chief World Systems'' (Day 4).</ref><blockquote>From the preceding it is possible to infer that the quickest path of all [lationem omnium velocissimam], from one point to another, is not the shortest path, namely, a straight line, but the arc of a circle.
 
...

Consequently the nearer the inscribed polygon approaches a circle the shorter the time required for descent from A to C. What has been proven for the quadrant holds true also for smaller arcs; the reasoning is the same.</blockquote>
[[File:Galileo's_Shortest_Time_Curve_Conjecture.jpg|alt=Diagrams for Wikipedia entry regarding Galileo's Conjecture|center]]
Just after Theorem 6 of ''Two New Sciences'', Galileo warns of possible fallacies and the need for a "higher science". In this dialogue Galileo reviews his own work. Galileo studied the cycloid and gave it its name, but the connection between it and his problem had to wait for advances in mathematics.

Galileo’s conjecture is that “The shortest time of all [for a movable body] will be that of its fall along the arc ADB [of a quarter circle] and similar properties are to be understood as holding for all lesser arcs taken upward from the lowest limit B.”

In Fig.1, from the “Dialogue Concerning the Two Chief World Systems”, Galileo claims that the body sliding along the circular arc of a quarter circle, from A to B will reach B in less time than if it took any other path from A to B. Similarly, in Fig. 2, from any point D on the arc AB, he claims that the time along the lesser arc DB will be less than for any other path from D to B. In fact, the quickest path from A to B or from D to B, the brachistochrone, is a cycloidal arc, which is shown in Fig. 3 for the path from A to B, and Fig.4 for the path from D to B, superposed on the respective circular arc.
<ref>{{cite book |last1=Galilei |first1=Galileo |title="Dialogue Concerning the Two Chief World Systems – Ptolemaic and Copernican translated by Stillman Drake, foreword by Albert Einstein " |date=1967 |publisher=University of California Press Berkeley and Los Angeles |isbn=0520004493 |edition=Hardback |page=451}}</ref>