MATHEMATICAL SCIENCES-SEVENTEENTH CENT. 165 



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the drum of the ear. He says that the vibrations due to the. sound 

 may be made visible by rubbing the finger round a gla^ss set in a large 

 vessel of water, and if by pressure the note is, made suddenly to rise* 

 to the octave above, every one of the undulations, which will be seen 

 regularly spreading round the glass, will suddenly split into two, proving 

 that the vibrations that occasion the octave are double those .belonging 

 to the single note. He then describes how he accidentally discovered 

 a method of more accurately measuring the relative lengths of sound- 

 waves. He was scraping a brass plate with an iron chisel, and, moving 

 the tool rapidly upon the plate, he- occasionally heard a hissing and 

 whistling sound, very shrill and loud, and he observed that whenever 

 this occurred, and then only, the light dust on the plate arranged itself 

 in a long row of small parallel streaks equidistant from each other. In 

 repeated experiments he produced different tones by scraping with 

 greater or less velocity, and he remarked that the streaks produced by 

 the acute sounds stood closer together than those produced by the low 

 notes. Among the sounds were two which, by comparison with the 

 notes of a violin, were found to differ by an exact fifth ; and counting 

 the streaks in each case, he found thirty of the one to occupy the same 

 space as forty-five of the other, which is exactly the known propor- 

 tion of the lengths of identical strings that will yield notes having 

 the interval of a fifth. We shall have occasion on a subsequent page 

 to revert to the subject of vibrating plates in mentioning Chladni's 

 acoustic researches. Galileo goes on to give the general outline of 

 the theory of concords and discords. "The immediate cause of the 

 form of musical intervals is neither the length, the tension, nor the 

 thickness, but the proportion of the number of the undulations of the 

 air which strike upon the drum of the ear, and make it vibrate in the 

 same intervals. Hence we may gather a plausible reason of the dif- 

 ferent sensations occasioned to us by different couples of sounds, of 

 which we hear some with great pleasure, some with less, and call them 

 accordingly concords, more or less perfect, whilst some excite in us 

 great dissatisfaction, and are called discords. The disagreeable sen- 

 sation belonging to the latter probably arises from the disorderly 

 manner in which the vibrations strike upon the drum of the ear ; so 

 that, for instance, a most cruel discord would be produced by sounding 

 together two strings of which the lengths are to each other as the side 

 and diagonal of a square, which is the discord of the false fifth. On 

 the contrary, agreeable consonances will result from strings of which 

 the number of vibrations made in the same time are commensurable, 

 so that the cartilage of the drum may not undergo the incessant torture 

 of a double inflexion from disagreeing percussions." 



Galileo, in his Dialogues, uses an expression invented by the older 

 school of philosophers, namely, Nature's abhorrence of a vacuum; 

 but he must not be understood as putting this phrase which is con- 

 venient enough as representing the difficulty of obtaining a vacuous 



