150 THE QUANTITATIVE METHOD IN BIOLOGY 



watery solutions, containing respectively an acid and an alkaline 

 substance The solutions are prepared in such a way that lo c c 

 ot the acid neutrahze exactly lo c.c. of the alkali. By means of 

 the (exact) pipette A of the preceding example (§ no) lo c c 

 ot the acid are taken ; by means of the (imperfect) pipette B 

 we take rather inexactly lo c.c. of the alkaline solutionf Both 

 volumes being mixed into one portion, we add one drop of turn- 

 sol, by which the reaction (acid red, alkaline blue) is indicated 

 A number of portions being obtained, we see that the red 

 and the blue portions are (about) equaUy numerous. How 

 can this result be explained ? 



All the operations have been carried out exactly as in the 

 preceding example (§ no). Since the acid has been measured 

 by means of A, the volume acid is exactly lo c.c. in all the 

 portions. The alkaline solution has been measured by means 

 of B, which is imperfect, and therefore the alkaline volume is 

 variable from one portion to another. In all the portions in 

 which the alkal. vol. <io c.c, the colour is red (acid pre- 

 dominant), whereas the portions in which alkal. vol.> lo c.c. 

 are blue.^ 



We know from the example in § io8 (density) that the errors 

 depend on chance. A curve of errors (Fig. i8, p. 146) is sym- 

 rtietrical, positive and negative errors being equally numerous. 

 Since each positive error [alkal. vol. >io) produces blue and each 

 negative error (alkal. vol. <io) red, both colours are observed an 

 equal number of times. 



In this example the variation produced by chance (the 

 colours only being taken into account) is not expressed by 

 means of figures. Chance may produce only two events (states 

 of equilibrium) : red or blue. When chance is alone at play, 

 red and blue are equally frequent. (If, however, a simple cause 

 interfered, augmenting, for instance, slightly the acid volume 

 in each portion, the frequency of the event red would increase, 

 and vice versa.) 



§ 112.— REMARKS ON THE PRECEDING EXAMPLES : 

 CONTINUOUS AND DISCONTINUOUS VARIATION.— In 



the examples mentioned in §§ 108-110 (density, sand, two 

 volumes of water) the variation of the observed values is con- 

 tinuous : it is expressed by a series of figures in which no gaps 

 occur, the transition from one value to the next one being 

 gradual (see the figures, p. 145) ^ and the frequency of the 



1 It is thinkable that the reactions of certain portions were neutral, both 

 constituent volumes being strictly equal. The probability (frequency) of 

 this event is exceedingly small, and therefore it may be practically excluded. 

 Coinpare the possible states of equilibrium of a coin, § 92, p. 117. 



" fii the mentioned series of observed values (p. 145) the difference between 

 two successive figures is not an existing gap : its value depends on the used 



