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XIX. Jfotes on Times of Descent under Gravity, suggested 

 by a proposition of Galileo's. By W. B. Morton, M.A., 

 andf. C.Tobin, M.A* 



IN the series of propositions which form the subject- 

 matter of the third day of his " Dialogues " Galileo 

 established, by Geometrical methods, all the well-known 

 properties of uniformly accelerated rectilinear motion, and, 

 in addition, a number of theorems, many of them oE great 

 interest and beauty, which are not included in modern text- 

 books. One of these is Prop. 36, which states that a 

 particle which slides to the lowest point of a vertical 

 circle (fig. 1) starting from rest at any point B of the cir- 

 cumference below the level of the centre, will make the 



Fig-. 1. 



journey in shorter time if it moves along two successive 

 chords BA, AO of the circle than if it goes directly along 

 the one chord BO. This proposition is noteworthy for two 

 reasons. In the first place its geometrical proof, as given 

 by Galileo, is quite simple, but if one attempts it by the 

 straightforward algebraic methods of the present day 

 the work is unexpectedly complicated. And, secondly, the 

 theorem has an historical interest, for it was the first step 

 towards the solution of the problem of the brachistochrone. 

 * Communicated by the Authors. 

 Phil. Mag. S. 6. Vol. 41. No/'2d2. Feb. 1921. Q 



