"a body
made to oscillate along a cycloid will always take the same time to
cover it"
http://www.scitechantiques.com/cycloidhtml/
The cycloid posses interesting physical properties. It is brachistochronous and tautochronous: brachistochronous, because it represents the path completed in the shortest time between two points for a given type of motion (such
as a fall under the effect of gravity); tautochronous, because a body
made to oscillate along a cycloid will always take the same time to
cover it, whatever the amplitude of the oscillation..........
更新1:
我看書看過的這個theorem的 我學過少少coordinate geometry但證極都證不到 我問過老師他說某國（忘）的科學院以此為公開題 一共有５人交答案 其中一人沒有在答案寫上名字，有人認出了牛頓獨特的符號就說 「雖然魔鬼沒有留下名字，但我們看到魔鬼的爪子」 有無高手可以教路？
更新2:
感謝煩惱即是菩提大師 這個問題我在兩年前已看過 最近偶爾再看到便拿來 我以前嘗試做時都能找到x,y方向的加速度 但要證"最短時間"便完全不知如何入手了
