MATHEMATICS FOR CHILDREN. 



807 



This can be demonstrated, but it strikes the eyes without that. We 

 see, too, that the interior figure is a square, and that it is constructed 

 on the hypotenuse of the triangles in question. 



It is easy to see in the other figure, which is formed after the 

 same measures as its alternate, that the triangles 1, 2, 3, 4 can be 



B 



D. 



FIG. 5. 



FIG. 6. 



arranged so as to occupy the positions 1', 2', 3', 4' in such way as to 

 leave in the main square two smaller squares constructed on the 

 sides of one of the right-angled triangles. It follows that the square 

 A is equivalent to the sum of the squares B and G. The theorem 

 thus becomes a kind of intuition, a thing evidently indisputable. 



It is a curious fact that the origin of this demonstration is lost 

 in the obscurity of the past; it probably goes back to thirty or 

 forty centuries, at least, before the Christian era, and apparently to 

 India. Bhascara, in his Bija Ganita, after tracing a figure, a sim- 

 ple combination of these two, says, " There you see it." I remark 

 that such a demonstration, even if dressed with geometrical terms, 

 assuming a character that conforms to existing ways of teaching, 

 would be vastly superior, even in secondary schools, to the demon- 

 strations of Legendre and others, which are much harder. The 

 return to what was done very long ago in this case constitutes a 

 great advance upon what we are doing now. 



Having given our little one an initiation into the mysteries of 

 arithmetic and geometry, we introduce him to algebra, a branch 

 which passes in the majority of families as the hardest, most com- 

 plicated, and most abstruse that can be imagined. I do not pre- 

 tend that algebraic theories enter easily into the child's delicate 

 brain; rather the contrary; but I declare that some ideas in algebra 

 can be made comprehensible to children without fatigue. We can, 

 for instance, make them understand, in the way of amusement 

 and without great difficulty, the formula that gives the sum of the 



