fHfi l*R6t*dSl'rtONS Ot^ tlffflAOOilAS A*fe i»A:eptTS. la 



Me inconsistent with pagus in the nominative case. His reference to 

 Sedce Pagus seems to confirm the conjecture, that Ricagmbedce was 

 composed of two words, of which the latter bedce was the name of the 

 goddess. Henee Beda vicus^ (now Bitburg), in the route a Treviris 

 Agrippinam, as giren in the Itinerary of Antoninus, derived its appeP 

 lation ; and from it came Pagus Bedensis, which is noticed in Wessek 

 ing'snote* Vide Vet. Rom. Itiner. Amstel. 1735, p. 373. 



KOTE Olt THE PEOPOSITIONS OF PYTHAG-OHAS 

 AND PAPPUS. 



Read before the Canadian Institute, Dee. \Qth, 1857= 



The following elegant construction is given by Prof. De Morgan iiS 

 the Quarterly Mathematical Journal, (VoL I. page 327,) as due to the 

 Astronomer Royal. 



Let ABC D, BE FG be ^^ 



two squares forming a gno** "'' ' 



mon ; take A H equal to 

 BE; joinHF, HD. Trans= 

 late without rotation each of Dp 

 the triangles A H D, H E F 

 along the hypothenuse of 

 the other (coming into the 

 positions indicated by the 

 dotted lines) : a square H K 

 is then formed equal to the 

 original two together. 



Professor De Morgan re» 

 marks that the proof thus 

 obtained of the Pythagorean 



Proposition (Euclid I. 47) is the siiilplest that has yet been devised^ 

 The original squares are plainly those on the aides of the right-angled 

 triangle H E F, and the new square that on the hypothenuse. 



Precisely the same method may be employed when, instead of 

 squares, we have two parallelograms forming a gnomon ; in this case 

 the resulting figure will also be a parallelogram^ and, adopting the 

 same letters as in the figure, its other side will be equal and parallel to 



