RESPECTING THE THEORY OF PARALLELS. 449 



it some special definition of the particular parallels which form the parallelogram. 

 For example, we may define a parallelogram as a four-sided figure, of which the 

 opposite sides are parallel, hy making the alternate angles with one of the diagonals 

 equal. It will thus be a symmetric figure. 



Euclid's definition of a square is at any rate vicious. To alter it consistently 

 with Euclid's construction (I. 46), reading it simply as an equisided quadrilateral 

 that has one angle a right angle, becomes an impossibility on our hypothesis 

 We may adopt the following 



Definition. — A square is a four-sided figure, which has all its sides equal and 

 all its angles equal. 



Prop. XXII. The opposite sides and angles of a parallelogram are equal to one 



another. 



Let ABCD be a parallelogram ; AC the defining diagonal. 

 The triangles ABC, DCA have the angles CAB, ACB respec- 

 tively equal to ACD, DAC and the diagonal common ; there- 

 fore they are equal in every respect ; whence the truth of the 

 proposition. 



Prop. XXIII. The alternate angles which the sides of a parallelogram make mth 



both diagonals are equal. 



The triangles ABD, CDB are equal in every respect (Euc. I. 8) ; whence the 

 truth of the proposition. 



Cor. Either diagonal bisects the parallelogram. 



Prop. XXIV. Parallelograms upon the same base cannot be between the same 



parallels, 



For if they could, the diagonals of the two parallelograms drawn from the 

 same point in one of the parallels would make the alternate angles equal ; which 

 is impossible by Prop. VII. 



Prop. XXV. The point of intersection of the two diagonals of a parallelogram is the 

 . middle point of the common perpendicular to each pair of parallels, which con- 

 stitute its sides. 



For if from this point a perpendicular be drawn to each of the opposite 

 parallels, there will be formed two triangles equal in every respect (Euc. I. 26) ; 

 and consequently the perpendiculars will be in a straight line. 



VOL. XXIII. PART III. 6 F 



