1^6 



THB PROPOSITIONS 03? PYTHAGORAS AND PAPPtlS. 



E C. This affords an easy proof of the well known theorem of Pappus, 

 ■of which the Pythagorean is a particular case. For, referring to the 

 triangle H E F, (E being not now a right-angle,) the parallelograms 

 ■are those constructed on the sides H E, E F, and we can change them 

 into any others on the same bases and between the same parallels with- 

 out altering their areas, the point C remaining also fixed. Also, if the 

 parallelograms be constructed on the outside, the paint corresponding 

 to C will be on E C produced backwards to an equal length, ^ence 

 we have the following proposition which is that of Pappus slightly 

 -extended, " If on the two sides of a triangle as bases any two paral- 

 lelograms be constructed, (both on the inside, or both on the outside,) 

 they will together be equal to a parallelogram constructed on the base 

 of the triangle and having its other side equal and parallel to the line 

 joining the vertical angle of the triangle with the point which is the 

 intersection of the sides which, in each of the two parallelograms, are 

 ^opposite to the respective bases." 



The follomng mode of dissecting the square on the hypothenuse so 



as to fill up the squares on the sides of a right-angled triangle is pi'D= 

 bably not new, though I have nowhere met with it. It is at 



