68 ON THE AUGMENTATION OF SOUNDS. 



to ~. Eighth, Though time is infinitely divifible in a ma- 

 thematical fenfe, M. Euler has (hewn *, that the ear con- 

 ceives it to confift of indivifible or elementary particles ; con- 

 sequently all the forces which arrive at P during one of thefe 

 elementary intervals, make a tingle indivifible impreffion on 

 an car placed at that point; becaufe this organ cannot take 

 cognizance of a fmaller particle of time ; therefore the force 

 of a found, produced at the center of a hollow fphere by the 

 undulations of its furface, is truly denoted by the expref- 



fion — . Ninth, In general, the force of a found, at any in- 



ftant of its duration, is equal to the fum of the forces exerted 

 by thofe pulfes, which ftrike the ear in the elementary inter- 

 val : Confequently, if the dittance between the hearer and 

 the founding body be fitch, that lines drawn from his ear to 

 every point of the vibrating furface, may be confidered as 



equal, the force of the found will be exprefled by -~-. Tenth,, 



To (hew the nature of this theorem by an example ; fuppofe 

 a bell A, the diameter of whofe mouth = 1-i, to be an octave 

 below a bell B, the diameter of which = 1 : aifo let A be 

 heard twice as far as B, when each receives an equal firoke ; 

 the ratio of the breadths of the (wells in A and B is required. 

 Let V, v, be the velocities of the fvvells in A and B, L, I, the 

 breadths of thefe (wells ; alio let 2 and 1 be their refpe&ive 



ranges: then fince/ is given, — - = -—-, by the theorem ; 



hence, as V : 4o : : L : /. Now the femi-circumfererices of 



A and B are in the ratio of 1-i. to 1 ; but while (he vibrations 

 pafs over half the circumference of A, (hey pafs and repafs 

 over the fame part of B : therefore, as V: v : :Ij : 2: hence 

 aifo, as L ; I : : 1 : 6. Q. E. I. 

 The principle Should the principle explained above be admitted, the fol- 

 extended to lowing conclufion mud, aifo be received as a neceflary confe- 



voices and other r . , . » . , ,, , 



founds. quence or it : the voices o( animals, as well as the notes of 



mufical inftruments, and the reports produced by blows given 

 to lefs elaftic ("abftances, derive no inconhderable portion of 

 their refpeciive forces from the vibrations of parts which are 



* Tcntamen de Mus. Cap. I. Sec. 13. 



not 



