VARIATION IN GENERAL 19 



whereas it only shows that the variations of the peach are often 

 discontinuous, with wide gaps representing spaces not filled by 

 variation. 



To get a full understanding of this ground the student must 

 form a clear conception of the distinction between continuity and 

 discontinuity as used in this connection. 



A man grows from childhood to maturity. In doing so he 

 passes through all possible weights and heights between those of 

 infancy and maturity. We cannot represent all these values by 

 any of our units of weight or measurements because all numbers 

 are by nature discontinuous. The only measure of continuity is 

 a line, because a line, curved or straight, represents all values 

 sensible and insensible between its two extremes. We can thus 

 plot continuity, but we cannot measure it except by cutting it 

 into sections and measuring it at stated points as if it were dis- 

 continuous, ignoring the intervals. 



Changes of temperature are continuous, as are those of humidity, 

 illumination, and all growth in the sense of extension in size, 

 whether plant or animal. All motion, whether regular or irregular, 

 is continuous because all intervening spaces are included. 



Discontinuity, on the other hand, implies vacant spaces not 

 represented by values. The good singer goes abruptly from one 

 note to the next, giving a discontinuous series of tones, while the 

 unskilled vocalist slides up or down the scale, giving rise to a 

 continuous series of tones in his effort to find the proper note. 

 The latter is not music because the ear is not pleasantly affected 

 by this confused jumble of sound waves arising from the inter- 

 mediate tones. Good music consists of a series of tones not 

 flowing into one another but cut sharply off and cast into a 

 discontinuous series, striking the ear at intervals with sound 

 waves that fit with mathematical precision. 



All number is by nature discontinuous. By fractions we attempt 

 to bridge the space between contiguous units, as between I and 

 2 ; but however small the fraction, there is yet a space, and a 

 sensible and measurable one, between the fraction and the next 

 anit. i T 9_9^_9_ i s no j- 2) nor will it ever become 2 this side of 

 infinity by the addition of any number of nines to the numerator 

 and ciphers to the denominator. It will constantly approach 2, 



