Then, since AB is parallel to CD , and BC has fallen upon them, the alternate angles ABC , BCD are equal to one another. [[I. 29](elem.1.29)]

And, since AB is equal to CD , and BC is common, the two sides AB , BC are equal to the two sides DC , CB ; and the angle ABC is equal to the angle BCD ; therefore the base AC is equal to the base BD , and the griangle ABC is equal to the triangle DCB , and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend; [[I. 4](elem.1.4)] therefore the angle ACB is equal to the angle CBD .

And, since the straight line BC falling on the two straight lines AC , BD has made the alternate angles equal to one another, AC is parallel to BD . [[I. 27](elem.1.27)]

And it was also proved equal to it.

Therefore etc.

Q. E. D.

I have for clearness' sake inserted the words in brackets though they are not in the original Greek, which has joining...in the same directions

or on the same sides,

ἐπὶ τὰ αυτὰ μέρη ἐπιζευγνύουσαι . The expression tiwards the same parts,

though usage has sanctioned it, is perhaps not quite satisfactory.

and 18. DCB. The Greek has BC , CD

and BCD

in these places respectively. Euclid is not always careful to write in corresponding order the letters denoting corresponding points in congruent figures. On the contrary, he evidently prefers the alphabetical order, and seems to disdain to alter it for the sake of beginners or others who might be confused by it. In the case of angles alteration is perhaps unnecessary; but in the case of triangles and pairs of corresponding sides I have ventured to alter the order to that which the mathematician of to-day expects.