Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/290

266 divided into pairs of pieces admitting of superposition and coincidence; see;also his Preface, page x.

I. 38. An important case of I. 38 is that in which the triangles are on equal bases and have a common vertex.

I. 40. We may demonstrate I. 40 without adopting the indirect method. Join BD, CD. The triangles DEC and DEF are equal, by I. 38; the triangles ABC and DEF are equal, by hypothesis; therefore the triangles DBC and ABC are equal, by the first Axiom. Therefore AD is parallel to BC, by I. 39. Philosophical Magazine, October 1850.

I. 44. In I. 44, Euclid does not shew that AH and FG will meet. "I cannot help being of opinion that the construction would have been more in Euclid's manner if he had made GH equal to BA and then joining HA had proved that HA was parallel to GB by the thirty-third proposition." Williamson.

I. 47. Tradition ascribed the discovery of I. 47 to Pythagoras. Many demonstrations have been given of this celebrated proposition; the following is one of the most interesting.

Let ABCD, AEFG be any two squares, placed so that their bases may join and form one straight line. Take GH and EK each equal to AB, and join HC, CK, KF, FH.

Then it may be shewn that the triangle HBC is equal in all respects to the triangle FEK, and the triangle KDG to the triangle FGH. Therefore the two squares are together equivalent to the figure CKFH. It may then be shewn, with the aid of I. 32, that the figure CKFH is a square. And the side CH is the hypotenuse of a right-angled triangle of which the sides CB, BH are equal to the sides of the two given squares. This demonstration requires no proposition of Euclid after I. 32, and it shews how two given squares may be cut into pieces which will fit together so as to form a third square. Quarterly Journal of Mathematics, Vol.

A large number of demonstrations of this proposition are collected in a dissertation by Joh. Jos. Ign. Hoffinann, entitled ''Der Pythagorische Lehrsatz...Zweyte...Ausgabe. Mainz.'' 1821.