under Gravity, suggested by a proposition of Galileo 1 s. 227 



and OAD in terms of the inclinations, a f$ say, of the chords 

 OA OB to the horizontal. It is easy to show that 



ODA=« + /3, 

 OAD = 180°- 2/3. 

 Since a</3 and /3<45°, 



.-. a + /3<2£<180°-2/3, 

 ODA<OAD. 



The inclined planes OB, OG having the same height, the 

 times of fall along them are proportional to their lengths and 

 maybe represented by these lengths. Take GX the mean 

 proportional between GA, GO and BY that between BD, 

 BO. Then BY, GX represent on the same scale the times 

 of sliding from rest down BD, GA, and OY, OX the times 

 for the remaining parts DO, AO. It remains to show 

 that OY>OX. 



We have OB<OG and OD>OA, 



.'. OB:OD<OG:OA, 

 .'. OB:DB>OG:AG. 

 But OB : YB = YB : DB = Y : YD, 



.-. OB :DB = OY 2 : YD 2 



OG:AG = OX 2 :XA 2 , 

 .'. OY:YD>OX:XA. 



In other words, OY is a larger fractional part of the 

 longer line OD: so OY>OX, and the proposition is proved. 



In connexion with the proposition and scholium the 

 following questions suggest themselves, and in spite of the 

 well-worn character and the comparative unimportance of 

 the subject, some of the results obtained are perhaps not- 

 devoid ot interest. 



(1) The range of validity of the proposition as the upper 

 point rises out of the lower quadrant of the circle and tho 

 truth or falsehood of the assumption made in the scholium. 

 This suggests a comparison of the times down one chord and 

 down two chords for the general case when the particle 

 starts with an initial velocity and leads, incidentally, to an 

 analytical proof of the proposition. 



(2) Extension to the case where the intermediate point A 

 is not on the circle. This includes an examination of the 

 locus of A for a given time of descent down the two lines 



Q2 



