152 THE RATE OF GROWTH [ch. 



progression so long as nutriment is enough and to spare; that is to 

 say, the rate of growth is proportional to the number present: 



dt ^ 



But in a test-tube colony the supply of nourishment is hmited, 

 and the rate of multiphcation is bound to fall off. If a be the 

 original concentration of food-stuff, it will have dwindled by time t 

 to (a — y). The rate of growth will now be 



% = by{<^-y), 



which means that the rate of increase iS proportional to the number 

 of organisms present, and to the concentration of the food-supply. 

 It is Verhulst's case in a nutshell; the differential equation so 

 indicated leads to an S-shaped curve which further experiment 

 confirms; and Sach's "grand period of growth" is seen to accom- 

 phsh itself*. 



The growth of yeast is studied in the everyday routine of a 

 brewery. But the brewer is concerned only with the phase of 

 unrestricted growth, and the rules of compound interest are all he 

 needs, to find its rate or test its constancy. A population of 1360 

 yeast-cells grew to 3,550,000 in 35 hours: it had multiplied -2610 

 times. Accordingly, 



1^8^51^^^^ = 0-098 = log 1-254. 



That is to say, the population had increased at the rate of 25-4 per 

 cent, per hour, during the 35 hours. 



The time (^2) required to double the population is easily found : 



log 2 0-301 ^ ^^ , 



* The sigmoid curve illustrates a theorem which, obvious as it may seem, is of no 

 small philosophical importance, to wit, that a body starting from rest must, in order 

 to attain a certain velocity, pass through all intermediate velocities on its way. 

 Galileo discusses this theorem, and attributes it to Plato: "Platone avendo per 

 avventura avuto concetto non potere alcun mobil passare daUa quiete ad alcun 



determinate grado di velocita se non col passare per tutti gli altri gradi di 



velocita minori, etc."; Discorsi e dimostrazioni, ed. 1638, p. 254. 



