NOTE ON EUCLID. 319 



Let A B C be a triangle having tlie side A B 

 equal to the side A C. Then the angle ABC 

 shall be equal to the angle A C B. For because 

 A B is equal to A C, the two sides B A, A C are 

 equal to the two, C A, A B, each to each, and 

 they contain a common angle ; therefore the angles 

 are equal to which the equal sides are opposite; 

 therefore A B C is equal to A C B. Q. E, D. ^ 



The only objection which I can imagine to be raised against this 

 proof is that we cannot compare a triangle with itself by superposition, 

 and consequently this method of demonstration is a departure from 

 Euclid's method. For of necessity the question could only be one of 

 method not of abstract truth. 



I would submit that Euclid himself by no means restricts the ap- 

 plication of the 4th proposition to triangles capable of superposition, 

 in fact the ordinary proof twice compares triangles having a common 

 part, and which could not possibly be superimposed, and propositions 

 6 and 7, &c. afford instances of the same — proposition VI a remarka- 

 ble one. 



The demonstration given above is a particular 

 case of one given by Proclus in his commentary, he 

 takes the points D E in A B, AC, and the proof 

 follows the same order as that given by Euclid. It 

 will be at once evident that if D and E coincide with 

 A, the angles A C D, A B E vanish, and it is no 

 more a departure from Euclid's method to prove 

 the equality of A B C to A C B than of E B C to 

 D C B since the triangles D B C, E C B could not be proved equal by 

 superposition ; or more simply still we may consider it to be what both 

 Euclid's and Proclus' constructions become when D, E are coincident 

 with B, C. Not that we are to suppose that they anticipated any 

 such pushing of their constructions to the limit, all we wish to infer 

 is, that thei/ considered their case proved whenever they had two 

 respectively equal sides containing equal or common angles, whether 

 these sides could be superimposed as in proposition IV or not, as in 

 most other cases. 



The following elegant proof of the proposition, which is logically 

 true, is liable to serious objections as a real departure from Euclid's 

 method. I know not to whom it was originally due, but it is pub- 

 lished in a slightly different form in the edition of Euclid in Messrs. 



