34. ELEMENTS OF SCIENCE 



braic truths with arithmetical ones, let us, for example, 

 represent a by 2 and b by 3. 



We now know that a + b multiplied by a + b equals 



- Similarly (2 + 3) x (2 + 3) = 2* + 2 (3x2) + 3 2 . 

 j For 2 + 3 = 5 an d 5x5 = 25. 

 Also 2 2 = 4, while 2 (3 x 2)= 12 and 3 2 = 9. 

 And 4+12 + 9 = 25. 



( The sciences of numbers and quantity apply, as before 

 said, to all things without exception. A less universal 

 branch of mathematics relates to all things with length, 

 breadth, and thickness. This is geometry. A brief 

 account of its simplest truths will serve to conclude 

 our introduction to the elements of mathematical 

 sciences. 



, The simplest way of introducing the reader to the ele- 

 ments of geometry will be to explain a proposition of 

 Euclid. The first of his propositions solves the problem 

 how to draw an equilateral triangle (i.e., one all the 

 three sides of which are equal) upon a given straight 

 line of a certain definite length. 



To do this we must take the following premisses for 

 granted : 



1. That a straight line may be drawn from one point 

 to another ; 



2. That a circle may be drawn from any given centre 

 at any practicable distance from it ; 



3. That a circle is such a figure that all straight lines 

 drawn from its centre to its circumference (i.e., to the 

 single line which bounds it) are equal to one another ; 

 and, 



4. That things which are equal to the same thing are 

 equal to each other. 



These truths (which are some of the definitions and 



