226 Prof. Morton and Mr. Tobin on Times of Descent 



It brings out clearly the fact that a path longer in distance 

 may be shorter in time. 



Accordingly Galileo, in a Scholium to the proposition, 

 states that it appears to follow (" colligi posse videtur") that 

 the most rapid passage from one fixed point to another is 

 not by the shortest geometrical path, the straight line, but 

 by an arc of a circle. The argument which follows is 

 limited to the proof that the time of descent is continually 

 shortened as more and more chords are taken between the 

 starting point and the bottom of the circle, approaching the 

 arc as a limit. In order to establish this Galileo has to 

 make an unproved assumption, which he introduces by the 

 words " verisimile est' 5 to the effect that the superiority of 

 the two-chord path continues to hold when the particle starts 

 with an initial velocity due to a fall from a level which is 

 not above the centre of the circle. 



The "Dialogues" were printed in 1638. In 1696 John 

 Bernoulli discovered the cycloid to be the true curve of 

 quickest descent, and published the problem as a challenge 

 to the learned world. He doss not appear to have been 

 aware of Galileo's earlier attempt until his attention was 

 called to it by Leibnitz. In a paper which the latter 

 contributed to the 'Acta Eruditorum ' in 1697*. it was 

 pointed out that Galileo had, through lack of the methods of 

 the differential calculus, gone wrong on two questions, 

 viz., the form of the catenary which he identified with the 

 parabola, and the curve of quickest descent which he 

 supposed to be a circle. To this at a later date Bernoulli 

 added a rather curious comment f. The catenary, as he 

 had himself shown, can be constructed by the rectification 

 of a parabola, and the cycloid by the rectification of a circle, 

 therefore, in both instances, " Galilee a devine quelque chose 

 d'approchant." 



To return to the proposition, Galileo's method of proof, 

 slightly simplified, may be stated as follows. B is the point 

 of departure on the circle OABC whose lowest point is 0. 

 The circle BDA is described with B as its highest point. 

 Then the times BD and BA from rest at B are equal, and 

 the velocity at A is the same whether the particle slides along 

 BA or GA. Therefore we have to show that the time along 

 DO from rest at B is greater than that along AO from rest 

 at G. The proof turns on the fact that DO is longer than AO. 

 This is proved by Galileo, as a lemma, in a rather clumsy 

 way. It is most readily seen by expressing the angles ODA 



* Leibnitz, Math. Schrifteh (Gerhardt's edition), y. p. 333. 

 f John Bernoulli's works, i. p, 199. 



