11 



BEGINNINGS IN GREECE 51 



The series of squares is formed by adding the odd numbers 

 successively ; 1+3 = 4, 1+3+5 = 9, etc. The series 2, 6, 

 12, 20, 30, etc. is formed by adding the even 

 numbers, or again by multiplying adjacent 

 natural numbers. If we construct a series of 

 squares or parallelograms with a common angle 

 and sides of length 1, 2, 3, 4, 5, etc. the figure 

 which must be added to any one to produce 

 the next larger was called by the Greeks a 

 gnomon, the area of which would be repre- 

 sented by one of the series of odd numbers, - - an interesting and 

 typical example of the Greek habit of combining geometry with 

 number-theory. As products of two numbers were associated with 

 areas-- "square" or "oblong" -so products of three factors 

 were interpreted as volumes. A later Pythagorean calls the cube 

 the " geometrical harmony ' - an expression embodying the as- 

 sociation of mathematics with music. The cube has indeed 6 

 faces, 8 vertices, 12 edges ; 6, 8, and 12 are in harmonic progres- 

 sion, that is, 8 is the harmonic mean between 6 and 12. 



PYTHAGOREAN GEOMETRY. - - In geometry the Pythagoreans 

 formulated definitions of the fundamental elements, line, sur- 

 face, angle, etc. They are credited with a number of theorems 

 depending on the application of one surface to another, 1 and im- 

 plying a knowledge of methods of determining area and of the 

 properties of parallel lines. They developed a fairly complete 

 theory of the triangle, including the fundamental proof that the 

 sum of the angles of a triangle is two right angles, by a method 

 not very different from our own. The theory of the "cosmical 

 bodies" mentioned in the register is of special in- 

 terest. Any solid angle must have at least three 

 faces. If three equal equilateral triangles have a 

 common vertex they will when cut or folded so 

 that their edges are brought together, form a solid 

 angle, and a fourth equal triangle will complete a regular tet- 

 rahedron. Similarly, if we start with four triangles, we may 



1 Some of these are equivalent to the solution of the quadratic equation. 



