186 Mr Sang on the manner in which 



produce SA to meet the arc described from S with the radius 

 SG, and draw the horizontal line HKI ; HI will be the ten- 

 dency down the arc, and SI will be the pressure of the knife 



edge against the plate : wherefore, since the friction is propor- 

 tional to the pressure, IK will represent the friction in the posi- 

 tion SA, leaving the surplus HK to accelerate the descent of 

 the pendulum. From S with the radius SF describe an arc, 

 produce SH to meet it in L, and draw HN, LO, perpendicular 

 to SF. The trigon KHN is clearly similar to FSG, whence 



KH : HN : : FS : SG : : LS : SH : : LO : HN 



to that HK is equal to LO. 



Now, if gravity, instead of acting in the direction of and with 

 the intensity SG, were to act in the direction of and with the 

 intensity SF, LO would just be the tendency of the pendulum 

 to descend from the point A ; and, therefore, the first oscilla- 

 tion will be performed equally on each side of the line SF, and 

 will be accelerated just as if gravitation were increased from SG 

 to SF. The next oscillation would be performed equally on 

 each side of a line, making with SG an angle equal to FSG, 

 and thus the arcs of oscillation would decrease in arithmetical 

 progression ; the extreme distances from the vertical line being 

 diminished at each oscillation by twice the angle GSF. 



