3& ELEMENTS OF SCIENCE 



Even the beginner may see how, in some instances, a 

 geometrical proposition may be more conveniently treated 

 algebraically. Thus, e.g., there is one * which declares 

 that if a right line be divided into any two parts, the 

 square of the whole line is equal to the square of the two 

 parts together with twice the product of those parts. 



Now evidently this is equivalent to saying that if we 

 take a to represent one of the two parts into which the 

 right line is divided and b to represent its other part, 

 then the square of the whole line is equal to the squares 

 of a and b together with twice the product of a and b, 

 and this must be 2 + 2ab + fr 2 , which, as we saw before,! 

 is the result of multiplying a + b by itself, or in other 

 words is equivalent to (a + 6) 2 . 



Of late years a converse process has taken place, and 

 various algebraic processes have been converted into geo- 

 metrical demonstrations, which, as less highly abstract, 

 are more readily apprehensible. 



By a number of elaborate processes (which, however 

 elaborate, are essentially similar to and wholly based upon 

 the elementary matters herein pointed out) the most 

 varied properties of objects may be investigated, includ- 

 ing complex reciprocal relations of increase, decrease, and 

 variation. When two quantities vary, they may do so 

 equally or in different proportions or ratios. When one 

 quantity varies with another, it is said to be a. function of 

 the latter. There are many other divisions of the science, 

 whereof one is known as the Differential Calculus ar.d 

 deals with computations concerning the rates of change 

 between quantities, while another, called the Integral 

 Calculus, passes from the relation between such rates 

 back to the relations existing between the changing 



* Euclid, Book II., Proposition IV. t See ante, p. 30. 



