OF AHTS AND SCIENCES. 297 



scendental, but became algebraical when the pericxls of the component 

 motions were in some simple ratio, either exactly or approximately. 

 He developed in detail: 1. The unison; 2. The octave; 3. The 

 twelfth; 4. The double octave: and. all of them whether perfect or 

 out of tune. His investigation had no reference to acoustics ; and he 

 describes these cases V>y their ratios, and not by the musical intervals 

 which these ratios may represent. He adds that similar results would 

 be oh)tained if the longer period, divided by the shorter, were expressed 

 by any whole number, either exactly or approximately. Such ratios 

 as 2 : 3 or 3 : 4, etc., did not come within the scope of his paper. Dr. 

 Bowditch obtains the equations for the few cases which he specially 

 examines, and finds that for the unison the path is expressed by an equa- 

 tion of the first or the second order ; for the octave, by an equation of 

 the second or fourth order ; for the ratio 1 : 3, by an equation of the 

 third or the sixth order ; and for the double octave, by an equation 

 of the fourth or eighth order : in each case the equation being of a 

 lower order when the difference of phase in the two movements is 

 zero or 180°. 



Dr. Bowditch adds : " 1 made a few experiments in order to com- 

 pare the preceding theory with observation." In his first experiment, 

 the two points of suspension were 4.75 inches apart, the single branch 

 of the thread was 46.5 inches, and the vertical height of the double 

 branch only .65 of an inch. Therefore the two movements were more 

 nearly in unison than in Professor Dean's experiment. A ball of 

 lead about half an inch in diameter hung at the bottom. By calcula- 

 tion one movement would gain a whole vibration upon the other in 

 286 vibrations of the more rapid movement: by experiment the num- 

 ber came out 282. In another experiment, the two points of suspen- 

 sion fvere 69 inches asunder: the single branch of the pendulum was 

 21.9 inches, and the vertical height of the double branch 64.4 inches. 

 A ball of lead If inches in diameter was attached. In this case the 

 two movements differed in period by the interval of an octave, im- 

 perfect to the extent of one vibration in sixty-seven of the slower move- 

 ment ; and experiment confirmed this conclusion, the whole cycle of 

 orbits being completed in that time. In regard to the two other ratios, 

 Dr. Bowditch says : " A few rough experiments were, however, made 

 in these cases, and the results appeared to be suflaciently conformable 

 to the theory." This kind of experiment is not suited to such ratios 

 as 1 : 3 and 1 : 4. For in these cases the single branch of the cord 

 is only one ninth and one sixteenth of the total length, and the 

 amplitude of its vibration comparatively small. 



