5i6 THE GRAMMAR OF SCIENCE 



gradual variation or change. Among the sciences which 

 deal especially (if not entirely) with discrete quantity, the 

 best known are probably Arithmetic and Algebra; but 

 there are a number of others we ought to briefly note. 

 We want to know how to measure quantity and what 

 errors are likely to arise in its measurement. Closely 

 allied to this is the discussion of probable and average 

 quantities, dealing with cases where we cannot measure 

 individual quantity, but only approximate and average 

 results. Hence arise the Theory of Measurement^ Theory 

 of Errors, Theoiy of Probability, Theory of Statistics, 

 etc. 



Passing to change in quantity, we remark that if one 

 quantity varies with another it is said to be ^ f unction of 

 the second. Thus temperature is a function of time and 

 of position, brightness of distance, and speed of time. To 

 understand the mutual relationship of quantities which are 

 functions of each other is the scope of sciences like the 

 Theory of Functiojts, which teaches us how functions can 

 be represented and handled. Examples of this representa- 

 tion will be found in our chapter on the Geometry of 

 Motion, Figs. lO and 13. Special branches are the 

 Differential Calculus or Calculus of Fhixions, which deals 

 with the rates of change, and of which we have had 

 examples in determining speed and curvature (pp. 2 1 5 

 and 226) ; and the Integral Calculus, or Calculus of Sums, 

 which passes from the relation between the rates back to 

 the relation between the changing quantities, and of which 

 we have had an example in the process of summation by 

 which we passed from acceleration as a function of position 

 to the map of the path of a moving body (p. 232). 



We next turn to the special relations of Space, and 

 we note that conceptual space may be considered from 

 two standpoints. We may deal solely with the relative 

 position of points and lines and surfaces without taking 

 any quantitative measurements of distances, areas, or 

 volumes. This forms a very important and valuable sub- 

 division of Geometry, which has been much developed of 

 recent years and has been largely used by theoretical 



